Topological features of magnetically-ordered and correlated crystals

Depa men o Physics
Topological ea u es o
magne ically-o de ed and co ela ed
c ys als
Mikel Ga cía Díez
Supe ised by
Juan Luis Mañes Palacios
Maia Ga cía Ve gnio y
Disse a ion submi ed o he Uni e si y o he Basque Coun y UPV/EHU as pa ial
ul illmen o he equi emen s o he Ph.D deg ee in Physics
May 16, 2025
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(cc) 2025 Mikel Ga cía Díez (cc by-nc-sa 4.0)
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Abs ac
The classi ica ion o condensed ma e sys ems in e ms o opological classes has
wi nessed signi ican ad ancemen s o e he pas decade. Al hough ansi ions be-
ween opological phases canno be igo ously es ablished h ough changes in he
symme y o a local o de pa ame e , symme y none heless plays a c ucial ole in
p o ec ing opologically non- i ial s a es. This is pa icula ly ele an in c ys alline
media, which, in addi ion o in e nal symme ies such as ime- e e sal and cha ge
conjuga ion, exhibi a high deg ee o spa ial symme y desc ibed in e ms o space
g oups. Recen p og ess in he comp ehensi e abula ion o magne ic space g oups
and hei ep esen a ions has acili a ed he ex ension o symme y-based opological
me hods, no ably Topological Quan um Chemis y, o sys ems exhibi ing magne ic
o de ing, such as e omagne s.
In his wo k, we employ hese me hods ex ensi ely o add ess a di e se ange o
p oblems conce ning he classi ica ion and cha ac e iza ion o opological phases in
c ys alline media. Following a b ie in oduc ion o g oup heo y, densi y unc ional
heo y, and opology in c ys alline sys ems, we p esen an upda e o I Rep, a compu a-
ional ool w i en in Py hon designed o symme y analysis o ab-ini io calcula ions.
In his i e a ion, I Rep has been enhanced o suppo magne ic space g oups and
expanded o pe o m a comp ehensi e symme y-based opological cha ac e iza ion
using elemen a y band ep esen a ions and symme y indica o s.
Wi h his imp o ed ool, we i s analyze wo hexagonal compounds, Co and
Fe
3
GeTe
2
, which a e g ouped oge he due o hei sha ed magne ic space g oup
symme y. In bo h ma e ials, nodal lines a e p o ec ed by mi o planes, and we
conduc an exhaus i e sea ch o hese ea u es nea he Fe mi le el. In he case
o Co, ou heo e ical p edic ions a e co obo a ed by expe imen al angle- esol ed
pho oemission spec oscopy measu emen s. Fo Fe
3
GeTe
2
, we u he in es iga e he
o igins o he anomalous Hall conduc i i y in his ma e ial, iden i ying h ee p incipal
con ibu ions: nodal lines li ed by magne ic o de , spin-o bi coupling gaps be ween
spin-up and spin-down s a es, and Weyl nodes.
Subsequen ly, we e isi he opology o he FeSe i on based supe conduc o . Fi s ,
we model he elec onic band s uc u e o he doped FeTe
0.55
Se
0.45
c ys al h ough DFT
and a symme y-based igh binding app oach and shed ligh in o he mechanism o
i
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ii
he ansi ion in o a non- i ial
Z2
s a e suppo ing a Di ac su ace s a e as e idenced
by he ARPES measu emen s o ou collabo a o s. Second, we p edic ha uniaxial
s ain in p is ine FeSe is capable o d i ing ansi ions in o s ong and weak opological
phases by modi ying he la ice cons an along he
a
and
c
di ec ions o he o iginal
e agonal cell.
Las ly, we explo e he applica ion o opological quan um chemis y o s ongly
co ela ed sys ems. In his con ex , we analyze he honeycomb Ki ae model and i s
analy ical solu ion in e ms o Majo ana e mions o cha ac e ize he symme y o
spin exci a ions in he lux- ee sec o and p edic he exis ence o opological edge
s a es. This app oach p esen s an al e na i e pa hway o he sys ema ic s udy o
in e ac ing phases, complemen ing exis ing me hods based on G een’s unc ions and
opological Hamil onians.
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Lis o publica ions
I
Topological semime als wi hou quasipa icles. Hu, H., Chen, L., Se y, C., Ga cia-
Diez, M., G e e, S. E., P oko ie , A., Ki chne , S., Ve gnio y, M. G., Paschen, S.,
Cano, J., & Si, Q.. a Xi :2110.06182 (2021).
II
Cubic 3D Che n pho onic insula o s wi h o ien able la ge Che n ec o s. De esco i,
C., Ga cía-Díez, M., Rob edo, I., De Paz, M. B., Lasa-Alonso, J., B adlyn, B.,
Mañes, J. L., Ve gnio y, M. G., & Ga cía-E xa i, A.. Na u e Communica ions,
12(1) (2021).
III
Vec o ial Bulk-Bounda y co espondence o 3D pho onic Che n insula o s. De-
esco i, C., Ga cía-Díez, M., B adlyn, B., Mañes, J. L., Ve gnio y, M. G., &
Ga cía-E xa i, A.. Ad anced Op ical Ma e ials, 10(20) (2022)
IV
T ans e sali y-En o ced Tigh -Binding Model o 3D Pho onic C ys als aided by
Topological Quan um Chemis y. Mo ales-Pé ez, A., De esco i, C., Hwang, Y.,
Ga cía-Díez, M., B adlyn, B., Mañes, J. L., Ve gnio y, M. G., & Ga cía-E xa i,
A.. a xi :2305.18257 (2023)
V
O bi al ing edien s and pe sis en Di ac su ace s a e o he opological band
s uc u e in FeTe
0.55
Se
0.45
. Li, Y., Chen, S., Ga cía-Díez, M., I aola, M. I., P au,
H., Zhu, Y., Mao, Z., Chen, T., Yi, M., Dai, P., Sobo a, J. A., Hashimo o, M.,
Ve gnio y, M. G., Lu, D., & Shen, Z.. Physical Re iew X, 14(2) (2024).
VI
Axion opology in pho onic c ys al domain walls. De esco i, C., Mo ales-Pé ez,
A., Hwang, Y., Ga cía-Díez, M., Rob edo, I., Mañes, J. L., B adlyn, B., Ga cía-
E xa i, A., & Ve gnio y, M. G.. Na u e Communica ions, 15(1) (2024).
VII
Band ep esen a ions in S ongly Co ela ed Se ings: The Ki ae Honeycomb
Model. Fün haus, A., Ga cía-Díez, M., Ve gnio y, M. G., Kopp, T., Win e , S.
M., & Valen í, R.. a xi :2501.11396 (2025).
VIII
O igins o he anomalous Hall conduc i i y in he symme y en o ced Fe3GeTe2
nodal-line e omagne .Ga cía-Díez, M., Beidenkop , H., Rob edo, I., &Ve gnio y,
M. G.. a xi :2502.07420 (2025)
iii
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i
IX
Fla band d i en i ine an magne ism in he Co-pnic ides (La,Ca)Co
2
(As,P)
2
.
Subi es, D., Ga cía-Díez, M., Ka , A., Lim, C.-., Li, V. M., Yannello, V., Ca -
bone, D., Ga giani, P., Yilmaz, T., Dai, J., Talla ida, M., Vesco o, E., Sha uk, M.,
Ve gnio y, M. G., & Blanco-Canosa, S.. a xi :2503.02728 (2025)
X
Concu en mul i ac ali y and anomalous hall esponse in he nodal line semime al
Fe
3
GeTe
2
nea localiza ion. Ma himala , S., Gup a, A., Roe , Y., Galeski, S.,
Waw zynczak, R., Ga cia-Diez, M., Rob edo, I., Vi , P., Kuma , N., Schnelle, W.,
Ka in, V. A., Küspe , J., Wang, Q., Chang, J., Sassa, Y., S e n, A., Felix, V. O.,
Ve gnio y, M. G., Felse , C., . . . Beidenkop , H.. a xi :2503.04367 (2025)
XI
Topological phase ansi ions o FeSe unde uniaxial s ain.Ga cía-Díez M., P o e,
J.B., Da ignon, A., Backes, S., Valen í, R. & Ve gnio y, M.G. In p epa a ion.
XII
Mani old o magne ic nodal lines in and elemen al e omagne . Cla k, O.J.,
Ga cía-Díez, M., Fink, J., Rade , O., Mi anda, R., Ve gnio y, M.G. & Sánchez-
Ba iga, J. In p epa a ion.
XIII
Magne ic I Rep: symme y and opological analysis o magne ic c ys als om
Densi y Func ional Theo y.Ga cía-Díez M., I aola, M., Rob edo, I., Mañes, J.L.
S. Tsi kin, S epan & Ve gnio y, M.G.. In p epa a ion.
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Lis o abb e ia ions
BCS Bilbao C ys allog aphic Se e
BZ ( i s ) B illouin Zone
DFT Densi y Func ional Theo y
EBR Elemen a y Band Rep esen a ion
FQHE F ac ional Quan um Hall E ec
IQHE In ege Quan um Hall E ec
(M)SG (Magne ic) Space G oup
(M)TQC (Magne ic) Topological Quan um Chemis y
QAHI Quan um Anomalous Hall Insula o
QSHI Quan um Spin Hall Insula o
SI Symme y Indica o
SOC Spin-O bi Coupling
SSG Si e-Symme y G oup
TB Tigh -Binding
TI Topological Insula o
TRIM Time-Re e sal-In a ian Momen um
TRS Time Re e sal Symme y
WL Wilson Loop
WP Wycko Posi ion
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Con en s
Con en s ii
Lis o Figu es ix
Lis o Tables xi
1 In oduc ion 1
2 G oup heo y o c ys alline sys ems and hei band s uc u es 11
2.1 The ansla ional symme y o la ices . . . . . . . . . . . . . . . . . . 12
2.2 The c ys allog aphic poin g oups . . . . . . . . . . . . . . . . . . . . 17
2.3 Spaceg oups................................ 18
2.4 In oducing ime e e sal: magne ic space g oups . . . . . . . . . . . 25
2.5 Elec ons in pe iodic sys ems . . . . . . . . . . . . . . . . . . . . . . . 32
3 Densi y unc ional heo y 37
3.1 S a emen o he p oblem . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Independen elec ons and Ha ee-Fock app oxima ion . . . . . . . . 39
3.3 The Hohenbe g-Kohn heo ems and he Kohn-Sham me hod . . . . . 40
3.4 Compu a ion o he DFT g ound s a e . . . . . . . . . . . . . . . . . . 44
4
Topology in condensed ma e sys ems and i s geome ical in e p e-
a ion 47
4.1 De i a ion o he geome ic phase . . . . . . . . . . . . . . . . . . . . 48
4.2 Gauge heo y o mula ion . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Wilson loops: gene aliza ion o he Be y phase . . . . . . . . . . . . 51
4.4 The Che n numbe : a opological in a ian . . . . . . . . . . . . . . . 55
4.5 Applica ion o c ys alline sys ems o elec ons . . . . . . . . . . . . . 62
5
Topology om eal-space symme y: Magne ic Topological Quan-
um Chemis y 75
5.1 Elemen a y band ep esen a ions . . . . . . . . . . . . . . . . . . . . . 76
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1. In oduc ion
As condensed ma e Physics esea che s, one o he i s hings we o en do when
discussing a gi en ma e ial is o classi y i . The highes -le el classi ica ion one usually
conside s is be ween solid, liquid and gas, ocusing mainly on he i s g oup. O
cou se, his sepa a ion in o h ee b oad ca ego ies is insu icien o i ually any
desc ip ion ha aims o accu a ely model he beha io o a physical sys em. Fo
example, one soon inds ha solids can be spli in o insula o s, which a e no able
o conduc elec ici y unde an ex e nal ol age di e ence, and me als, which do
ha e low-ene gy exci a ions ha can anspo cha ge. Ma e ials can change be ween
phases depending usually on he he modynamic condi ions o he en i onmen ,
ypically p essu e and empe a u e.
Up o 1980, he desc ip ion o all phase ansi ions was amed in he heo y
de eloped by Le Landau, which dis inguished each phase by i s symme ies [1]. In
his amewo k, he usual way o p oceed is o expand he ee ene gy o he sys em in
e ms o a local o de pa ame e , usually physical quan i ies like a local magne iza ion,
and model pa ame e s ha encode he e ec o p essu e and empe a u e. The ac ual
alue he o de pa ame e akes is he one ha minimizes he ee ene gy gi en
he ex e nal condi ions. A change o such condi ions incu s a change on he ee
ene gy landscape such ha i is ene ge ically a o able o he local o de pa ame e
o acqui e a non-ze o equilib ium alue which, in u n, b eaks a leas one symme y
o he sys em. The p o o ypical case is a o a ion-in a ian sys em whose ene gy we
expand in e ms o a local magne iza ion ha akes a non-ze o alue when he em-
pe a u e is lowe ed. Since he local magne ic ield chooses a di ec ion in an o he wise
in a ian sys em, he symme y is lowe ed o only ope a ions ha lea e his axial
ec o in a ian : o a ions abou i s axis, in e sion and hei combina ions. Landau
heo y has p o en o be a e y powe ul me hod no only o discuss e omagne ic
ansi ions like in his example bu supe luids and liquid c ys als o example.
Howe e , as success ul as Landau heo y migh ha e been, he expe imen s con-
duc ed by Klaus on Kli zing in ha yea on low- empe a u e wo-dimensional
elec on gasses unde a s ong ex e nal magne ic ield showed ha his heo y could
no be he comple e pic u e. In pa icula , on Kli zing obse ed ha he esis i i y
o he samples ans e se o he applied ol age displayed quan ized pla eaus o al-
ues
h/e2·1/n
, whe e
n
is in ege , which inc eased wi h he alue o he magne ic
ield [2]. This poin ed a a se ies o phase ansi ions, one pe pla eau, ha sha ed
he same symme ies and hence could no be desc ibed unde Landau’s pa adigm. As
we know now, his was he i s ealiza ion o a opological phase ansi ion and he
in ege quan um Hall e ec (IQHE). This phenomenon also displayed a hallma k o
opological e ec s: i s obus ness. The IQHE e ec was no a me e luke, i could be
consis en ly ep oduced o di e en samples and, mo eo e , he quan iza ion o he
2
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Figu e 1.1: Longi udinal
ρxx
and ans e se
ρxy
esis i i y as a unc ion o he mag-
ne ic ield in a low- empe a u e 2D elec on gas showing pla eaus o quan ized Hall
esis i i y. Rep oduced om Re . [2].
Hall conduc ance has now been measu ed o be ex emely p ecise [3].
The ac ha he IQHE is a opological e ec was la e sugges ed by Thouless e
al. in 1982, when hey de i ed an analy ical o mula o he Hall conduc i i y in a 2D
elec on gas [4]. The exp ession, which ul ima ely a ises om he single- aluedness
o he elec onic wa e unc ion unde closed loops in he pe iodic B illouin zone (BZ),
showed ha he ans e sal conduc i i y ollowed he o mula
σxy =e2
2πℏC,
whe e
C
is an in ege called he i s Che n numbe o , in he con ex o ha a i-
cle, he TKNN in a ian . The obus ness and p ecise quan iza ion o he ans e se
conduc ance in he IQHE could hen be explained by he ac ha
C
canno change
smoo hly unless he e is a se e e change in he magne ic ield s eng h ha causes
a phase ansi ion. The de i a ion also explained ha he Hall esis i i y changed
due a numbe o p o ec ed su ace s a es con ined o he bo de o he 2D sample
which a e he o igin o he ans e se conduc ance o he elec on gas. Wha makes
he phenomenon pe haps e en mo e s iking is ha he exis ence o hese chi al
edge s a es was p edic ed om bulk p ope ies only, as conside ing pe iodic bounda y
condi ions (which makes he BZ desc ip ion possible) neglec s he exis ence o su -
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1. In oduc ion
aces comple ely. This ea u e was la e ecognized o be a case o he bulk-bounda y
co espondence, ano he gene al cha ac e is ic o opological e ec s in condensed
ma e sys ems.
The obus ness o hese e ec s and exo ic cha ac e is ics like he bounda y modes
a ac ed he a en ion o he condensed ma e Physics communi y, bo h o he inhe -
en in e es in unde s anding opological phenomena and he possible applica ions in
spin onics, quan um compu ing o supe conduc i i y. In 1988, D. Haldane p oposed
a model o elec ons on a honeycomb la ice ha displayed IQHE in he absence o
an ex e nal magne ic ield, showing ha he e can be ma e ials which a e inhe en ly
opological [5]. This was modeled by in oducing a complex hopping e m whose
ci cula ion is ze o, he e o e showing no ne magne ic lux h ough each plaque e,
bu none heless b eaking ime e e sal symme y (TRS). O he wise, chi al edge s a es,
which p opaga e in only one di ec ion and a e incompa ible wi h a ime- e e sed
pic u e, could no exis in he model. This was he i s model o a Che n insula o , due
o he opological in a ian ha dis inguishes his phase. A ew yea s la e , in 2005,
C.L. Kane and E.J. Mele p oposed a model wi h bo h TRS and opologically p o ec ed
edge s a es, essen ially by coupling wo laye s o he Haldane model and conside ing
spin ul e mions wi h componen s coupled by spin-o bi coupling (SOC) e ec [6].
Na u ally, single bounda y modes we e o bidden by TRS bu , uning he pa ame e s
o he model, one can ob ain an insula o displaying pai s o bounda y modes. To
sa is y he TRS cons ain s, he edge modes p opaga e in opposi e di ec ions and
ca y opposi e spin. Due o his, hese kind o sys ems we e called quan um spin
Hall insula o s (QSHIs). Con a y o Che n insula o s, whe e he in a ian can be
inc eased wi h no limi , QSHIs a e desc ibed by a
Z2
numbe aking only alues
{0,1}
. This is because he hyb idiza ion o wo di e en pai s o helical edge s a es
is no o bidden by TRS and gene ically gaps any phase ha has an e en numbe o
bounda y modes, des oying he e ec . Topological insula o s we e no con ined o
wo dimensions, and expe imen al ealiza ions o 3D sys ems like Bi
2
Se
3
, Bi
2
Te
3
o
Sb2Te3we e achie ed in he ollowing yea s [7,8].
As he opological na u e o hese phenomena was e ealed, we ealized why
Landau’s heo y o phase ansi ions ailed o desc ibe hem. While he cen al objec
in he heo y is a local o de pa ame e , he in o ma ion abou opological e ec s is
smea ed globally h oughou he sys em. T acking how he elec onic wa e unc ion
changes along closed loops in he BZ, as in he TKNN o mula, equi es knowledge o
he wa e unc ion along he pa h, in pa icula how he phase changes om one poin
in ecip ocal space o he o he s. This does no mean ha symme y is i ele an in he
s udy o opological ma e ials. The in e play be ween in e nal symme ies and opol-
ogy c ys allized in wha we know as he en- old classi ica ion [9
–
11], which desc ibes
4
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he possible opological phases o di e en dimensions acco ding o hei ans o ma-
ion unde ime- e e sal, cha ge conjuga ion and hei combina ion. Addi ional wo k
showed ha c ys al symme ies could also be ela ed o he opological in a ian s.
Fo example, L.Fu and C.Kane showed ha he
Z2
in a ian in in e sion-symme ic
c ys als could be compu ed di ec ly wi h he in e sion eigen alues o he occupied
se o bands a especial TRS-in a ian wa e ec o s [12]. Fang e al. also disco e ed
a co espondence be ween o a ion eigen alues and he Che n numbe [13]. These
kind o ma e ials, whe e he opology is p o ec ed by he c ys al symme ies, we e
la e called opological c ys alline insula o s (TCIs) by L.Fu [14].
Up o hen, he diagnosis o opology in ealis ic ma e ials was a di icul compu-
a ional p oblem. Mos o he analysis elied on complex calcula ions in ol ing, o
example, line in eg als o e closed pa hs in he B illouin zone. Fo model Hamil onians
like Haldane’s he compu a ion was ai ly simple bu eal insula o s, o en desc ibed
ia Densi y Func ional Theo y (DFT) calcula ions, we e much ha de o analyze.
Howe e , The e o s o link symme y eigen alues o opological in a ian s showed a
way owa ds a simple me hod o diagnose opological p ope ies on sys ems ha , o
he mos pa , showed a high deg ee o symme y, as c ys als. In 2016, a amewo k
uni ying all hese me hods was de eloped, called Topological Quan um Chemis y
(TQC) [15
–
18]. TQC ansla ed he p oblem o symme y-p o ec ed opology om
ecip ocal o eal space by ela ing he eigen alues o symme y ope a ions a high-
symme y wa e ec o s wi h he localiza ion [19,20] o Wannie unc ions si ing a
speci ic Wycko posi ions in he uni cell o he c ys al. The disco e y o opological
ma e ials was educed o an algeb aic p oblem: compu e he symme y p ope ies
o he occupied s a es a all high-symme y poin s o he BZ and check i he bands
can be ob ained by placing o bi als o ce ain symme y a Wycko posi ions in he
eal-space cell. I his is possible, hen he ma e ial is i ial o he opology is no p o-
ec ed by c ys al symme ies, o he wise i is opological. The me hod also con i med
ha , unless he e is a gap closing and eopening in he elec onic band s uc u e,
opology was insensi i e o smoo h (adiaba ic) de o ma ions o he ene gy le els. The
key insigh is ha i ial insula o s a e adiaba ically equi alen o a ays o isola ed
a oms a Wycko posi ions o a omic limi s, so any o he ma e ial ha is smoo hly
unable in o ha limi mus also be i ial. Mo eo e , om ha same analysis, one
can ex ac he so-called symme y indica o s (SIs), which a e quan i ies compu ed
om symme y eigen alues ha a e ela ed o opological in a ian s [21,22]. Wi h
his addi ion, TQC can no only iden i y opological ma e ials bu also ell he speci ic
kind o opology, as long as i is p o ec ed by symme y.
Such an s eamlined me hod opened he doo o he sys ema ic sea ch o opologi-
cal ma e ials om all expe imen ally- ealized s oichiome ic c ys als. The au oma ion
o a wo k low including DFT calcula ions and he subsequen TQC analysis allowed
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1. In oduc ion
Symme y
eigen alues
Inpu
s uc u e DFT
Is
a omic limi ?
No
symme y
indica ed
Topological
T i ial Wilson
loops,
e c.
Symme y
indica o s Topological
in a ian s
No Yes
I symme y indica ed
Figu e 1.2: The TQC analysis wo k low. S a ing om an expe imen al inpu s uc-
u e, DFT calcula ions a e pe o med om which he symme y eigen alues a high-
symme y poin s in he BZ can be ex ac ed. Then, one checks i his is an a omic
limi . I i is no , hen he ma e ial is opological, o he wise he opology canno
be diagnosed only by symme y (2D elec on gas o example). In ha case, mo e
complex me hods such as Wilson loops a e equi ed. I he opological phase is
symme y-indica ed, hen he opological in a ian s can be in e ed om he SIs.
o inc ease he numbe o p edic ed opological ma e ials o he o de o ens o
housands [23,24], including old examples whose opology had been o e looked due
o a lack o a s aigh o wa d way o iden i y i . Fu he mo e, all he machine y
necessa y o a TQC analysis such as space g oup symme y ables, a omic limi s and
SI o mulas, was made a ailable o he public, accele a ing he ield o TCIs.
Howe e , he analysis o TRS-b oken c ys als such as hose e omagne s and an i-
e omagne s emained ou side he each o TQC, which only conside ed space g oup
symme y wi h TRS included. The e o s in o abula ing all he ep esen a ion o he
1651 Shubniko g oups, which desc ibe all possible magne ic a angemen s o a oms
in a la ice wi h SOC, led o he ex ension o he me hod o magne ic TQC (MQTC) in
2021 [25]. Wi h his comple e enume a ion o all space g oup symme ies, he analysis
o impo an cases, such as Che n insula o s which in insically b eak TRS due o he
non-ze o local magne ic momen dis ibu ion, was unlocked. The di ec ex ension
o TQC o magne ic ma e ials immedia ely allowed he mass sea ch o opological
magne ic ma e ials [26], wi h a comple e opological classi ica ion hanks o he a
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heo y o SIs ha included all Shubniko g oups.
All o he p e ious cases a e p ima ily unde s ood wi hing he amewo k o
non-in e ac ing o weakly in e ac ing elec on models, in which we can ely on
band heo y o compu e opological in a ian s, whe he ia symme y indica o s
o o he me hods such as Wilson loops. Howe e , eal ma e ials can show s ong
elec on-elec on co ela ions ha signi ican ly modi y hese opological ea u es.
The p ime example o hese co ela ed opological phases is he ac ional quan um
Hall e ec (FQHE), whe e in e ac ions be ween elec ons leads o cha ge ac ionaliza-
ion and quasipa icles wi h cha ge smalle han he elec on’s and exo ic exchange
s a is ics [27
–
32]. Elec on co ela ions usually a ise in c ys alline ma e ials due o
he Coulomb epulsion be ween elec ons in localized o bi als. This is especially
impo an in
4
and
3d
o hea y a oms whe e he s onge pull om he nucleus and
he smalle numbe o nodes in he wa e unc ion leads o an enhanced localiza ion.
F om he poin o iew o he band s uc u e, localized elec ons gi e ise o e y
li le dispe sing bands, which implies ha hei quasipa icle eloci y is small and
hence can in e ac mo e e icien ly. Among hese co ela ed ma e ials, we ind Mo
insula o s [33,34], hea y- e mion sys ems [35,36], Kondo insula o s [37,38], quan um
spin liquids [39] and wis ed and Moi é sys ems [40,41].
The e ha e been a numbe o p oposals o add ess he opology o in e ac ing
sys em in a sys ema ic way. Fo example, a numbe o exp essions o opological
in a ian s in e m o o G een’s unc ions [42,43], which cap u e he dynamics o
in e ac ing media, ha e p oposed wi h hei main d awback being he di icul y o
apply hem. O he esea ch shows ha i is possible o use g oup cohomology, cobo -
dism heo y and highe - o m symme y o s udy “symme y-p o ec ed” [44] and
“symme y-en iched” [45] opological phases, equi ing complex ma hema ical con-
cep s. An al e na i e pa h has also been pu o wa d based on mapping he in e ac ing
sys em in o a non-in e ac ing pic u e, cons uc ing a so-called “ opological Hamil-
onian” [46,47]. Recen ly, he use o TQC applied o his al e na i e single-pa icle
pic u e has had i s i s esul s [48
–
50], opening he doo o ex ending his sys ema ic
analysis o in e ac ing sys ems. Resea ch in his di ec ion is s ill ongoing and a he
p omising, since o he wise he diagnosis o opology in co ela ed sys ems s ill elies
on ad-hoc me hods, ailo ed o speci ic cases.
In his hesis, we will add ess he ollowing p oblems and objec i es:
Ex ension o he open-sou ce Py hon package I Rep o magne ic ma e ials and
upg ade o enable he ull TQC analysis ia elemen a y band ep esen a ions
and SIs. P e ious e sions o I Rep [51] could no be applied o ma e ials showing
o de o he local magne ic momen s o he a oms, o example wi h
3d
and
4
o bi als.
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1. In oduc ion
Thanks o he abula ion o all he symme y in o ma ion o he Shubniko g oups,
we can now ex end he analysis o magne ic ma e ials s ill main aining a simple,
easy- o-use in e ace wi h he use wi h new quali y-o -li e imp o emen s. Mo eo e ,
he package is now able o pe o m a ull symme y-indica ed opology analysis,
pe o ming he decomposi ion o he selec ed se o bands in e ms o elemen a y
band ep esen a ions and compu ing all he co esponding symme y indica o s o
bo h single and double space g oups. This enables a ully local and au oma ized
wo k low o he s udy o opological ma e ials a he single-pa icle le el.
Using magne ic space g oups o diagnose sys ems wi h opological nodal lines
and hei anomalous Hall conduc i i y. We pe o m an exhaus i e symme y
analysis o Fe
3
GeTe
2
and hexagonal close-packed (hcp) Co ha e eals how he e -
omagne ic o de can gi e ise and also des oy opological nodal lines, which we
cha ac e ize ia Wilson loop calcula ions. Addi ionally, on he one hand, Fe
3
GeTe
2
has been expe imen ally shown o display la ge anomalous Hall conduc i i y and
ou p ecise symme y unde s anding combined wi h ma e ials modeling ia Wannie
unc ion models un eils he in insic sou ces o his e ec . On he o he hand, we
p esen expe imen al esul s on he opological nodal lines in hcp-Co, con i ming he
p edic ed ea u es ia densi y unc ional heo y and symme y analysis.
S udy o a p essu e-induced opological ansi ion in he FeSe supe conduc o .
FeSe is one o he mos s udied membe s o he amily o i on-based supe conduc-
o s. Despi e i s s uc u al simplici y, a numbe o compe ing phenomena such as
nema ici y, magne ic o de and o bi al-selec i e co ela ions in e ac wi h supe con-
duc i i y. The sensi i i y o he elec onic p ope ies wi h espec o he Fe-Se and
Fe-Fe dis ances has been used o une i s p ope ies (such as he c i ical empe a u e)
h ough ex e nal p essu e. Addi ionally, chemical p essu e h ough Te doping has
been shown o achie e a opological ansi ion in o a s ong opological insula o
(TI). In his wo k, we pe o m densi y unc ional heo y calcula ions o FeSe unde
uniaxial s ain and a subsequen analysis in e ms o TQC and SIs. Con a y o wha
p e ious esea ch sugges s, FeSe is a weak TI a ambien p essu e. Mo eo e , we also
ind ha uniaxial s ain d i es he sys em in o an o ho hombic phase and p omp s a
opological phase ansi ion in o a s ong TI, sugges ing ano he ou e o une he
opological p ope ies o FeSe.
Topological ansi ions in he FeSe supe conduc o ia doping and p essu e.
FeSe is one o he mos s udied membe s o he amily o i on-based supe conduc o s.
Despi e i s s uc u al simplici y, a numbe o compe ing phenomena such as nema ic-
i y, magne ic o de and o bi al-selec i e co ela ions in e ac wi h supe conduc i i y.
The sensi i i y o he elec onic p ope ies wi h espec o he Fe-Se and Fe-Fe dis-
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ances has been used o une i s p ope ies (such as he c i ical empe a u e) h ough
ex e nal p essu e. Addi ionally, chemical p essu e h ough Te doping has been shown
o achie e a opological ansi ion in o a s ong opological insula o (TI). In his wo k,
we pe o m densi y unc ional heo y calcula ions o he FeTe
0.55
Se
0.45
, a e y well
s udied doped sibling compound, and de elop a symme y-based igh -binding model
ha help explain he elec onic ea u es obse ed in angle- esol ed-pho oemission
spec oscopy measu emen s. Ou wo k sheds ligh in o he he mechanism o he
ansi ion in o a s ong TI phase upon doping ha hos s su ace Di ac s a es, as
e idenced by he om ou collabo a o s. We also p opose uniaxial s ain in p is ine
FeSe as a way o d i e simila opological phase ansi ions. Based on an analysis
in e ms o TQC and SIs, we ind ha modi ying he
a
and
c
la ice cons an s o he
o iginal e agonal cell gi es ise o s ong and weak TI phases, depending on he
di ec ion and he s eng h o he ex e nal pe u ba ion.
Resul s on he applica ion o TQC o highly-co ela ed sys ems in he honey-
comb Ki ae model. We conside he ex ension o TQC o sys ems wi h opological
o de and s ong co ela ions. The pa icula case o he honeycomb Ki ae model
allows us o di ec ly apply he me hod using he non-in e ac ing pic u e ob ained by
ans o ming he spin deg ees o eedom in o Majo ana e mions, wi hou elying
on he de ini ion o a opological Hamil onian. In his se ing, we p opose a concep
o “spin o bi als” composed o mo e han one spin ope a o which gi e ise o a band
s uc u e o exci a ion ene gies. The analysis o his band s uc u e wi h TQC and
SIs e eals ha , upon in oducing a ime- e e sal-b eaking e m co esponding o
an ex e nal magne ic ield, he sys em can be uned in o a Che n insula o phase
wi h chi al edge modes and we p o e i by compu ing he spec um o a ini e s ipe
geome y. Thus, his wo k sheds ligh on he ques ion o how TQC can be gene alized
o s ongly co ela ed sys ems, adding o he p e ious esul s we desc ibed abo e.
The hesis is s uc u ed as ollows. In Chap e s 2 and 3 we gi e an in oduc ion o he
g oup heo y o Shubniko space g oups and densi y unc ional heo y. Chap e s 4
and 5 concen a e on he opology o condensed ma e sys ems and i s diagnosis
h ough TQC and SIs. The esul s s a wi h Chap e 6, whe e we desc ibe he new
unc ionali ies o I Rep o he opological analysis o magne ic ma e ials. In Chap e
7, his is used o un eil he nodal lines in Fe
3
GeTe
2
and hcp-Co. Chap e 8 s udies he
e ec o doping and uniaxial s ain in he FeSe supe conduc o . Finally, Chap e 10
explo es he applica ion o TQC in he honeycomb Ki ae model o s ongly co ela ed
spins.
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CHAPTER 2
G oup heo y o c ys alline sys ems
and hei band s uc u es
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
P imi i e Body-cen e ed
Face-cen e ed
Figu e 2.2: The h ee cubic B a ais la ices. G ay nodes co espond o new poin s due
o he cen e ing wi h espec o he p imi i e la ice.
2.3 Space g oups
Space g oups a e he combina ion o he ansla ional an o a ional symme ies o
eal c ys als. Thei ope a ions a e usually deno ed wi h Sei z symbols
{R| }
, whe e
R
is an elemen o a poin g oup and
is a ansla ion. F om he composi ion o
ope a ions, we can de i e he in e se o any o hem
{R2| 2}{R1| 1}={R2R1|R2 1+ 2}=⇒ {R| }−1={R−1|−R−1 }.(2.12)
The simples way o ob ain a space g oup om a B a ais la ice and a poin g oup is
by di ec ly combining he wo o gi e a symmo phic space g oup. In hese, bo h
g oups a e essen ially sepa a ed: he g oup o o a ions is by i sel a poin g oup
and he ansla ions
a e always ansla ions o he B a ais la ice. Only 73 o hem
can be cons uc ed in his way. The emaining 157, o a o al o 230, a e called
non-symmo phic. In hese, he ope a ions may ha e a ansla ional pa
which is
no pa o he B a ais la ice. This ansla ional pa mus be a ac ion o a la ice
ansla ion acco ding o he o de o he ope a ion. Fo example, we may ind elemen s
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2.3. Space g oups
Cubic Te agonal O ho hombic
T igonal Hexagonal Monoclinic
T iclinic
120º
Figu e 2.3: The se en c ys al sys ems. Edges wi h di e en labels a e assumed o be
o di e en leng h in gene al. Unlabeled angles a e also assumed o be 90◦.
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
o he o m {2001|0,0,1/2}, whe e 2001 is a o a ion abou he hi d axis o he basis
and he ansla ion is gi en also in his basis. In his example, i is easy o see ha we
canno sepa a e o a ions and ansla ions since
{2001|0,0,1/2}2
is a ansla ion o
he la ice wi h no o a ional pa and hus he e is no a closed poin g oup wi hin
he space g oup.
Rep esen a ions o space g oups
The p ocess o ob aining he ep esen a ions o a gi en space g oup s a s by picking
a basis in which he ansla ions a e diagonal, e.g he Bloch basis. Fi s , no ice ha a
s a e ψka kis ans o med by an ope a ion {R| }in o a s a e a Rk
{E| }{R| }ψk={R| }{E|R−1 }ψk=e−ik·R−1 {R| }ψk=e−iRk· {R| }ψk,
(2.13)
whe e we ha e used ha
R
is uni a y so
k·R−1 =Rk·
. Le
G
be he space g oup
o he c ys al, hen we de ine
De ini ion 2.3. The subg oup
Gk⊆G
o ope a ions ha lea e he wa e ec o
k
in a ian up o a ecip ocal la ice ansla ion is called he li le g oup o k
Gk={g∈G:|gk=k′+K},(2.14)
whe e Kis a ecip ocal la ice ec o .
No ice ha , by de ini ion,
Gk
is also a space g oup, since ansla ions ha e no
e ec on ecip ocal la ice ec o s. The li le co-g oup
¯
Gk
o
k
is ob ained by aking
he di e en o a ional pa s o he elemen s o
Gk
and is isomo phic o a poin g oup.
Gkis ei he Go a s ic subg oup o G, which mo i a es he ollowing de ini ion
De ini ion 2.4. Gi en a wa e ec o
k
, he se o non-equi alen ec o s ob ained by
ac ing wi h all he ope a ions o Gon kis called he s a o k
S a (k) = {ki∈1BZ|∃g∈G:gk=ki∧k=ki+K}.(2.15)
The wa e ec o s o he s a a e usually called a ms o he s a .
The li le g oups o he ec o s o he s a a e ac ually isomo phic and conjuga e
in he sense ha
gk=q=⇒Gq=gGkg−1.(2.16)
The usual way o p oceed is o induce a ep esen a ion o
G
om he smalle
subg oup
Gk
and hose o he a ms o he s a . All he i eps o
G
a e i s labeled
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2.3. Space g oups
by a wa e ec o
k
which indexes a s a , whe e
k
mus be in he i educible pa o
he BZ. Once
k
is ixed, he space o s a es o he s a o
k
is in a ian unde any
ope a ion o
G
. The ull space g oup i eps a e o med by blocks co esponding o an
i ep o he li le g oup Gk. In mo e de ail, we can decompose Gin o cose s o Gk
G=[
gα/∈Gk
gαGk(2.17)
such ha gi en a ep esen a ion
σ
o
Gk
, we can induce a ep esen a ion
Σ
o
G
,
which is deno ed
Σ=σ↑G
. Le
(α, β)
index he blocks o
Σ
and
(i, j)
he elemen s
o Σ, hen o any g∈G
[Σ(g)]iα,jβ = [˜σ(g−1
αggβ]]ij,(2.18)
whe e
[˜σ(g)]ij =([σ(g)]ij i g∈Gk
0 o he wise .(2.19)
I
σ
is
d
-dimensional i ep o
Gk
and assuming he e a e
m
a ms, he ull i ep
Σ
is
(m×d)
-dimensional. The block s uc u e hen a ises om he ha gi en a gene al
{R| }∈G, we can always ind a ela ion wi h an elemen {h| }∈Gko he o m
{R| }{Rβ| β}={Rα| α}{h| },(2.20)
whe e
Rαk=kα
and
Rβk=kβ
a e a ms o he s a . The only non-ze o
(α, β)
blocks
o
Σ
induced om
σ
a e p ecisely hose o which he abo e ela ion holds. In ui i ely,
ocusing on he o a ional pa
R=RαhR−1
β
, he e ec o
{R| }
on he s a es a
kβ
is ela ed o b inging hem o
k
, applying he o a ion
h
acco ding o he i ep o
Gk
and hen mo ing hem o kα.
To ind all he i eps o
G
one mus also ind all he i eps o
Gk
in he i s place.
This poses a p oblem since
Gk
is in ini e and he e o e has an in ini e numbe o
i eps. One can ac ually ind a ini e numbe o hem by ealizing ha
Gk
is a space
g oup ha can be decomposed in o le cose s
Gk=T∪[
gi/∈T
giT, (2.21)
whe e
gi∈Gk
. We i s poin ou ha he e is a ini e numbe o cose s. When
Gk
is
symmo phic, he poin g oup ope a ions o i s li le co-g oup and he ansla ions a e
essen ially sepa a ed, i.e.,
Gk=¯
GkT
. In his case, a ini e se o i eps can be ound
om he i eps o
¯
Gk
, which is ini e. This is no he case o non-symmo phic space
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
g oups. In ha si ua ion, wo addi ional cons ain s can be imposed. Fi s , we will
only accep i eps σin which pu e la ice ansla ions ha e he o m
∆σ( ) = e−ik·
1
d,(2.22)
whe e
∈T
,
d
is he dimension o
σ
and
∆σ( )
is he ma ix ep esen a ion o
in
he i ep
σ
. These a e called small i eps o
Gk
and ha e he p ope y ha elemen s
which di e by a la ice ansla ion ha e he same ma ix ep esen a ion up o a phase
∆σ( gi) = e−ik· ∆σ(gi),(2.23)
wi h
gi∈Gk
and
gi∈giT
. Second, we also impose ha wo small i eps
σ
and
σ′
a e equi alen i he e exis s a uni a y
N
( he same o all elemen s o
Gk
) ha ela es
hem
∆σ(g) = N∆σ′(g)N−1,∀g∈Gk⇐⇒ σ′≡σ. (2.24)
Along wi h he es ic ion on he ep esen a ion o ansla ions, his is enough o ind
only a ini e se o non-equi alen i educible small ep esen a ions o
Gk
and hus a
ini e se o ull i educible ep esen a ions o G.
Al hough being amilia wi h he undamen al p ope ies, especially he block
s uc u e o he ep esen a ions o space g oups, is ce ainly bene icial, i is no
necessa y o know he de ails in p ac ice. All he i eps o all he space g oups o
high symme y poin s and lines o he BZ a e a ailable a he Bilbao C ys allog aphic
Se e websi e [52], which i is assumed ha is used om his poin onwa d.
Induc ion o space g oup ep esen a ions om eal space
o bi als.
Conside a c ys al wi h he symme y o some space g oup
G
. The s uc u e is com-
posed o a oms in a uni cell which a e coupled when b ough close o one ano he due
o elec omagne ic in e ac ions o he elec ons and ions. The disc e e single-a om
le els ge dis o ed in such a way ha , upon aking he Fou ie ans o m and o
well-de ined quasi-pa icle ene gies (such as in he case o he single-elec on app oxi-
ma ion), a desc ip ion in e ms o ene gy bands a ises. Each s a e (o se o degene a e
s a es), which ca ies a label o a ecip ocal
k
ec o , ans o ms acco ding o a space
g oup ep esen a ion in a space o med by he s a o
k
, as explained p e iously.
The e o e, we see ha a omic o bi als loca ed in he c ys alline s uc u e gi e ise
o a se o bands ans o ming as a space g oup ep esen a ion. Each a omic o bi al
displays a pa icula symme y due o he i s en i onmen desc ibed by a symme y
g oup. I u ns ou ha he ep esen a ions o his g oup can be p omo ed o a ull
ep esen a ion o he space g oup, as we will see nex .
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2.3. Space g oups
To begin he discussion, gi en a posi ion
q
in he uni cell, we de ine he Wycko
posi ion (WP) as he se o loca ions ob ained by ac ing wi h all he symme y
elemen s o
G
on
q
ha s ill lie in he same uni cell. This is some imes called he
o bi o
q
, wi h he es ic ion ha only posi ions in he same uni cell a e e ained.
WPs a e labeled by a numbe , which is he numbe o unique si es o mul iplici y, and
a le e which a ies o each space g oup (e.g. 2a).
WPs can be iewed as he eal-space analogue o
S a (k)
: each si e in he WP is
in a ian unde a subse o
G
called he si e-symme y g oup (SSG), which we will
deno e Gq:
Gq={g∈G|gq=q}.(2.25)
The di e ence is ha
Gq
is ini e, since i canno con ain la ice ansla ions by
de ini ion, and isomo phic o a poin g oup, whe eas he li le-g oup o
k
is in ini e
(and a space g oup in ac ). The SSGs a di e en si es a e isomo phic and conjuga e,
ha is, i qα=gαq hen Gqα=gαGqg−1
α.
In gene al, WPs can ha e ixed posi ions o can be pa ame e ized by some coo di-
na e, o example
(1/2,0, z)
whe e
z∈(0,1/2)
, such ha i connec s wo di e en
WPs a
(1/2,0,0)
and
(1/2,0,1/2)
. A Wycko posi ion is called maximal when he
SSG o any o i s si es is no a p ope subg oup o any o he si es o ano he WP ha
i is connec ed o. In ou example,
(1/2,0,0
) may display mo e symme y han he
mo e gene al (1/2,0, z)such ha he SSG o he la e is a subg oup o he o me .
Once in a c ys alline en i onmen , he o bi als o an a om a
q
o a gi en WP a e
dis o ed. The single-a om o bi als ans o m unde he
O(3)
o a ion g oup wi h
in e sion, whose ep esen a ions can be subduced in o ep esen a ions o
Gq
, such
ha we can o m linea combina ions o sphe ical ha monics ha o m a basis o
each i ep o he SSG. This basis is mo e na u al because i is adap ed o he c ys al
symme y and i is he one we will use om his poin .
Le
ρ
label one i ep o he
Gq
which belongs o a WP wi h si es
{q,q1,q2...qn}
such ha
qα=gαq
o some
gα∈G Gq
. The p ocess o inducing an space g oup
ep esen a ion
Σρ
om
ρ
is simila o he one used o induce i om li le-g oup
ep esen a ions, elying again in a cose decomposi ion o he o m:
G=
n
[
α=1
gα(Gq⋉T),(2.26)
whe e
T
is again he g oup o la ice ansla ions and
⋉
deno es he semi-di ec
p oduc 5. The ull space g oup ep esen a ion Σρis he induced om ρsuch ha
[Σρ(g)]iα ,jβ ′= [˜ρ(g−1
α{E| }g{E| ′}gβ]]ij,(2.27)
5The semi-di ec p oduc applies he e because Tis a no mal subg oup o Gand Gq∩T=∅
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
whe e
[˜ρ(g)]ij =([ρ(g)]ij i g∈Gq
0 o he wise .(2.28)
The e o e, we see ha he same in ui ion applies as o he induc ion om he li le
g oup o
k
: he eal-space ep esen a ion
Σρ
mo es o bi als be ween si es o he WP
and o a es hem acco ding o
ρ
o
Gq
. Fo s a es a
qα
, hey a e i s shi ed o
q
,
whe e he o a ion acco ding o
ρ
is applied, and hen mo ed o he inal posi ion
qβ
.
I
g∈Gqα
hen
qα=qβ
. The ole o he la ice ansla ions
{E| }
and
{E| ′}
is o
adjus o he ansla ions misma ches due o
gα
and
gβ
, which may con ain ans-
la ions by hemsel es (e en non-in ege ones). The choice o cose ep esen a i es
gα
amoun s o he selec ion o a speci ic basis. Fo example, i
ρ
is spanned by
pz
o bi als only and
q1
is ela ed o
q
by he in e sion
{I|0}
such ha i is chosen as
ep esen a i e, hen he basis o Σρwill ha e a −pza q1.
Once his p ocess is es ablished,
Σρ
can be b ough o ecip ocal space by a Fou ie
ans o m. Le
ϕi
α
be he se o s a es ans o ming as
ρ
o
Gq
(o i s analogues in he
conjuga e Gqα). A change o basis o ecip ocal space gi es
ϕi
α(k, ) = 1
√NX
T
eik·Tϕi
α( −T).(2.29)
Following he exp ession o
Σρ
, he ac ion o
g∈G
on
ϕi
α( −T)
a some uni cell
Tcan be shown o be
Σρ(g)ϕi
α( −T) = 1
√N
dim(ρ)
X
j=1
[ρ(h)]jiϕi
β( −RT− αβ),(2.30)
whe e αβ =gqα−qβi gqα=qβand h∈Gqis he unique elemen ha sa is ies
g={E| αβ}gβh. (2.31)
I ollows ha he ac ion in he ecip ocal-space basis is
Σρ(g)ϕi
α(k, ) = e−iRk· αβ
√N
dim(ρ)
X
j=1
[ρ(h)]ijϕi
β(Rk, ),(2.32)
which we see changes a s a e a
k
o one a
Rk
, as i should. The space-g oup ep e-
sen a ion so induced is called a band ep esen a ion, usually deno ed as
ρ@WP ↑G
indica ing ha i was induced o
G
om he si e-symme y g oup ep esen a ion
ρ
a he Wycko posi ion "WP”. Since he e a e
n
si es in he uni cell in he WP,
dim(ρ)×nbands a e ob ained in ecip ocal space.
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2.4. In oducing ime e e sal: magne ic space g oups
2.4 In oducing ime e e sal: magne ic space
g oups
Con en ional space g oups only desc ibe he spa ial symme ies o he a omic a -
angemen o c ys als. Howe e , he e may be addi ional physical p ope ies ha
mus be aken in o accoun in he symme y analysis o a condensed ma e sys em.
Such is he case o he magne iza ion o a oms in e omagne ic, an i- e omagne ic
and e imagne ic c ys als. In gene al, in he absence o spin o magne ic ields, he
mos common si ua ion is ha physical sys ems show ime- e e sal symme y
(TRS). When magne iza ion is included in he pic u e, he symme y g oup may no
include TRS by i sel , bu a combina ion o TRS and some space g oup ope a ion. The
esul an space g oup is called a magne ic space g oup (MSG), which a e pa o he
se o all space g oups, called Shubniko g oups, which include hose ha we ha e
discussed in he p e ious sec ions.
K ame s’ degene acy
Time e e sal ac s i ially on eal space bu may ha e an e ec on o he physical
quan i ies, which o he pu pose o his wo k a e:
•
I e e ses magne ic ields and spin componen s (and o he angula momen a).
•Changes he wa e ec o k o −k
I is implemen ed by an an i-uni a y ope a o , which we will label
Θ
, and ac s on a
Bloch s a e ψk( )as
Θaψk( ) = a∗Θψk( ) = a∗˜
ψk( ),(2.33)
whe e we no e ha
¯
ψk( )
is he ime- e e sed pa ne o
ψk
and
Θ
is an i-linea . I
can be p o en ha
Θ
is always ep esen ed by
Θ = UK
, whe e
U
is a uni a y ma ix
o he dimension o he space i ac s on and
K
deno es complex conjuga ion o e e y
e m si ing on i s igh . Fo example, o a spin-1/2
Θ = ηeiπSy/ℏK=−iησzK, (2.34)
whe e
Sy
is he
y
componen o he spin ope a o and
σy
is he associa ed second
Pauli ma ix. In p ac ice,
U
can always be ound using he ac ha
Θ
commu es wi h
all he space g oup ope a ions and ha i is uni a y.
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
Objec In e sion Time- e e sal
−
k−k−k
S S −S
Table 2.1: The ans o ma ion p ope ies unde in e sion and ime e e sal o eal-
space ec o s , ecip ocal-space kand spin S.
Ano he impo an p ope y is ha i beha es di e en ly o e mions and bosons,
in pa icula
Θ2=(−1 e mions
+1 bosons .(2.35)
This has a - eaching consequences in he elec onic sys ems ha conce n his hesis.
Conside , in he mos gene al case, a s a e
|ϕ⟩
and i s ime- e e sed pa ne
Θ|ϕ⟩
. We
may ask whe he hese wo a e eally he same s a e, up o a phase:
Θ|ϕ⟩?
=eiφ|ϕ⟩.(2.36)
C ucially, in he case o elec ons Θ2=−1so we ha e
Θ2|ϕ⟩=e−iφeiφ|ϕ⟩=|ϕ⟩=−|ϕ⟩.(2.37)
We see ha we a i e a a con adic ion because ou assump ion o he wo s a es
being p opo ional is alse. The e o e, o elec onic s a es, he ime- e e sed pa ne
is in ac a di e en s a e and, i Θis indeed a symme y, hey ha e he same ene gy,
which is doubly degene a e. This simple de i a ion o a non-degene a e ene gy
can be ex ended o he degene a e case, so all he ene gies in a e mionic, ime-
e e sal-in a ian sys em a e a leas doubly-degene a e. This is called K ame s’
degene acy.
In sys em wi h ansla ional symme y, he ac ion o Θon Hamil onian a kis
ΘHkΘ−1=H−k(2.38)
and he s a e o ene gy
En
a
k
,
|ψnk⟩
is ela ed o
Θ|˜
ψnk⟩=|ψn−k⟩
, which has he
same ene gy. Addi ionally, i
k≡ −k
, i ollows ha all ene gies a
k
a e a leas
doubly degene a e. This special se o poin s a e he Time-Re e sal-In a ian
Momen a (TRIMs), which a e loca ed a he co ne s o he BZ cube (in educed
coo dina es and h ee dimensions) and
Γ = (0,0,0)
. Mo eo e , i in e sion
I
is also
a symme y,
Iθ
is an an i-uni a y symme y as well, which means ha all ene gies a
all kpoin s a e a leas doubly-degene a e.
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2.4. In oducing ime e e sal: magne ic space g oups
Classi ica ion o magne ic space g oups
Once he TR ope a o is in oduced, we may di ide he esul ing 1,651 Shubniko
space g oups in o ou ca ego ies. The mos basic one is g oups ha simply do no
con ain he TR ope a o , which is wha we ha e been wo king wi h implici ly, and
a e named Fedo o o Type-I g oups. No e ha some con igu a ions o magne ic
momen s may no p esen TRS in any way, no e en combined wi h space g oup
ope a ions.
Ano he possibili y is ha
Θ
by i sel is a symme y o he sys em, which is he
case o Type-II o g ay space g oups. These ha e he s uc u e
MII =G∪ΘG, (2.39)
whe e
G
is a uni a y (Fedo o ) space g oup. Consequen ly,
MII
has double he ope -
a ions o
G
: hal o hem co espond o he o iginal elemen s and he o he hal o
he same ones combined wi h
Θ
. In pa icula , he iden i y can be combined wi h TR
so
Θ
is an elemen in i s own igh . Consequen ly,
MII
canno desc ibe c ys als wi h
a non-ze o magne ic momen s, since he g ound-s a e magne iza ion a e e y si e
would be e e sed by Θ, leading o a dis inc ly di e en s a e.
The hi d case occu s when
Θ
is no a symme y bu some o i s combina ions
wi h uni a y space-g oup ope a ions a e, gi ing as a esul a Type-III MSG wi h he
gene al s uc u e
MIII =H∪Θ(G H),(2.40)
whe e
H⊂G
is he so-called hal ing subg oup because i has index wo in
G
, i.e.,
i has hal he ope a ions. As a consequence, and as i happens o all magne ic g oups
o he han Type-I, hal he elemen s a e uni a y and he o he hal a e an i-uni a y.
The o me a e ob ained by combining he emaining elemen s in he se di e ence
G H
wi h
Θ
. Type-III MSGs can desc ibe bo h e omagne ic and an i e omagne ic
con igu a ions.
Finally, Type-IV MSGs di e om Type-III in ha he e a e pu e ansla ions
combined wi h TR. Consequen ly, he gene al s uc u e is
MIV =H∪Θ 0H, G =H∪ 0H, (2.41)
whe e
H
is isomo phic o a Type-I MSG and
0
is a cen e ing ansla ion whose leng h
is hal o ei he o he ollowing
a+b+c,a+b,a+c,b+c,a,b,c,(2.42)
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2. G oup heo y o c ys alline sys ems and hei band s uc u es
whe e Hij couples he subspaces o i eps ρiand ρj. F om Eq.(2.67), we ha e
Γ(g)HΓ†(g) = ρmδmiHijρ†
jδjn =ρmHmnρ†
n=Hmn.(2.70)
The e o e, conside ing
H
as a
G
-linea map, only he blocks be ween equi alen i eps
a e non-ze o acco ding o he i s lemma. Fu he mo e, he second lemma s a es ha
e e y block be ween equi alen i eps is p opo ional o he iden i y ma ix. Fo he
sake o cla i y, conside a case whe e
Γ=ρ1⊕ρ2⊕ρ2
. Then Wigne ’s heo em is
he s a emen ha he ma ix Hhas he o m
H=


H(1)
11 0 0
0H(2)
11 H(2)
12
0H(2)†
12 H(2)
22


,(2.71)
whe e we ha e al eady used ha
ˆ
H
is He mi ian and
H(i)
is a block ha ac s on
he subspace o an i ep
ρi
. Mo eo e , he blocks o i eps o mul iplici y g ea e
han one, such as
ρ2
, can be block-diagonalized by eo de ing he bases. Assume ha
each
ρ2
has a basis
{ 1, 2, . . . , d}
and
{w1, w2, . . . , wd}
, whe e
d
is he dimension
o
ρ2
. Then, by aking a basis
{ 1, w1, 2, w2, . . . , d, wd}
o
ρ2⊕ρ2
we achie e a
ully block-diagonal o m o he Hamil onian ma ix
H=


H(1)
11 0 0
0˜
H(2)
11 0
0 0 ˜
H(2)
22


.(2.72)
We a e in e es ed in using his esul o a sys em obeying he symme y o a space
g oup. Fi s , we conside he Bloch basis, which diagonalizes he ansla ion ope a o s
such ha each ansla ion
T
by a ec o
o he la ice is assigned a diagonal ma ix
p opo ional o
e−ik·
. This implies ha
H
can be decomposed in o a diagonal o m
wi h blocks
H(k)
which a e decoupled by symme y. Fu he mo e, each
H(k)
is
in a ian unde he li le g oup o
k
, so Wigne ’s heo em can be u he applied o
in e he o m o each block. Conside ing a symme y-adap ed basis o he subspace
co esponding o
k
, we know ha we can diagonalize
H(k)
analogously o Eq.(2.72).
Finally, gi en his s uc u e, we can see ha each
d
-dimensional i ep o he
li le g oup o
k
gi es ise o a
d
-dimensional subspace o s a es ha ha e he same
degene a e ene gy and span a basis o he i ep. This exac degene acy is no
acciden al and is p o ec ed as long as he symme ies o he li le g oup a e p ese ed.
Howe e , g oup heo y does no p edic he ac ual alues o hese ene gies and hey
depend on he de ails o he c ys al. Ope a ions ha a e no in he li le g oup o
k
ela e he s a es a one wa e ec o wi h he es o he s a , implying ha e e y
ene gy is epea ed on e e y a m wi h he same degene acy.
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2.5. Elec ons in pe iodic sys ems
Sys em wi h spin-1/2 pa icles
Be o e concluding his chap e , a inal no e on he ea men o spin-1/2 pa icles is
due. When spin-o bi coupling (SOC) is neglec ed, he spin up and down sec o s can
be e ec i ely decoupled, ea ing elec ons in each subsys em as scala . Howe e ,
when SOC is conside ed, we ha e o gi e he exci a ions a ull spino ial ea men .
The e a e wo app oaches o he same p oblem. The i s one consis s in aking in o
accoun ha he elec on wa e unc ion has o bi al and spin pa s. An elemen
g
in
he space g oup Gac s on bo h deg ees o eedom as
g|ψjσ⟩=D(g)ijS(g)σ′σ|ψiσ′⟩,(2.73)
whe e
|ψiσ⟩
deno es a s a e o o bi al pa
i
and spin
σ
. The e o e, we a e e ec i ely
conside ing ha he spin ul exci a ions ans o m unde a double- alued i ep
Γ=D×S
, which is he p oduc o he o bi al and spin ep esen a ions. The ca ea
is ha he ull o a ion g oup ac ing on spin is
SU(2)
a he han
SO(3)
. Because he
o me is a double co e o he la e , e e y eal-space o a ion by an angle
θ
ac ually
co esponds o wo possible spin o a ions ha di e by a minus sign: one h ough
θ
and ano he one h ough
θ+ 2π
. To a o a ion h ough an angle
θ
along an axis
ˆn
,
we can assign he SU(2) ma ix o spin
S=e−iθ
2(ˆn·σ)= cos(θ
2)−isin(θ
2)(ˆn ·σ),(2.74)
whe e
σ
is he ec o o Pauli ma ices. The mapping makes explici ha he e is one-
o- wo co espondence due o he
θ/2
ac o . Ins ead o bo he ing wi h he p oduc
ep esen a ion, he second and equi alen app oach is o conside ha he space g oup
has double he elemen s, gi ing ise o double g oups, whe e each new elemen is
dis inguished by a
2π
addi ional o a ion om he o iginal one. The ma ices o he
spin o a ion and double i eps di e by a minus sign be ween he wo la o s o he
same eal-space ope a ion and he in e sion is assigned he iden i y.
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CHAPTER 3
Densi y unc ional heo y
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3. Densi y unc ional heo y
E en in small samples, he shee numbe o pa icles in ol ed in he physical
beha io o a ma e ial ende s he p oblem comple ely in ac able i exac accu acy
is o be p ese ed. Some me hods, such as exac diagonaliza ion, can indeed sol e
he exac physics o a model Hamil onian wi hou any simpli ica ion. Howe e , hey
a e usually limi ed by he dimension o he Hilbe space o he sys em, which g ows
exponen ially wi h i s size. Fo his eason, as usual, we equi e ce ain deg ee o
app oxima ion ha allows o e ain he mos impo an aspec s o he physics in ol ed
in mac oscopic ma e ials.
In pa icula , since we a e in e es ed in he opology o he elec onic eigens a es
o c ys alline sys ems, we wish o be able o compu e he ene gy le els and eigen-
unc ions o a wide ange o ma e ials. This chap e is dedica ed o explain one o
he mos ex ended me hods, called Densi y Func ional Theo y o , by i s ac onym,
DFT.
3.1 S a emen o he p oblem
The physics o he ionic and elec onic deg ees o eedom in condensed ma e
sys ems such as c ys als, lea ing aside spin o simplici y, a e go e ned by a uni e sal
Hamil onian
ˆ
H=−ℏ2
2mX
i∇2
i−X
i,I
ZIe2
| i−RI|+1
2X
i=j
e2
| i− j|
−X
I
ℏ2
2MI∇2
I+1
2X
I=J
ZIZJe2
|RI−RJ|.
(3.1)
The co esponding e ms a e, in o de o appea ance:
1. Kine ic ene gy o he elec ons o mass m.
2.
Coulomb a ac ion o each elec on o cha ge
e
a posi ion
i
o e e y o he
ion o cha ge ZIea posi ion RI.
3. Coulomb epulsion o each elec on a iwi h e e y o he elec on a j.
4. Kine ic ene gy o he ions wi h mass MI.
5. Coulomb epulsion o each ion a RIwi h e e y o he ion a RJ.
We will neglec he kine ic ene gy o he a oms, as i lies in a much lowe ene gy scale
due o he highe masses
MI
compa ed o he mass o he elec ons. The p oblem
is hus simpli ied o a pic u e whe e he elec ons mo e in a ixed, pe iodic a ay
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3.2. Independen elec ons and Ha ee-Fock app oxima ion
o ions o posi i e cha ge. This is called he Bo n-Oppenheime app oxima ion.
F om he e onwa d, we will use Ha ee a omic uni s, which amoun s o se ing
ℏ=m=4π
ϵ0= 1
, whe e
ϵ0
is he acuum pe mi i i y. Thus, in his app oxima ion,
he Hamil onian eads
ˆ
H=−1
2X
i∇2
i−X
i.I
ZI
| i−RI|+1
2X
i=j
1
| i− j|+
+1
2X
I=J
ZIZJ
|RI−RJ|=ˆ
T+ˆ
VIe +ˆ
Vee +VII,
(3.2)
aking in o accoun ha he ion-ion ene gy
VII
is a cons an since he in e -a omic
dis ance emains ixed (apa om small displacemen s due o la ice ib a ions). The
complexi y esides p ecisely in he
ˆ
Vee
e m, since i in ol es he in e ac ion o he
elec ons by pai s and, as such, canno be sepa a ed in one-pa icle ope a o s. As a
consequence, he ue g ound s a e wa e unc ion o he sys em depends a he same
ime on all he elec on posi ions
Ψ( 1,..., N)
, wi h a o al o
3N
deg ees o eedom
( h ee pe pa icle in h ee dimensions wi h
N
elec ons), which is huge al eady e en
neglec ing spin. The p oblem is now how o ind me hods ha can app oxima e he
elec on-elec on in e ac ion e m.
3.2 Independen elec ons and Ha ee-Fock
app oxima ion
The mos basic app oxima ion o he elec onic p oblem is o assume ha he elec onic
deg ees o eedom can indeed be sepa a ed. The wa e unc ion
Ψ
hus is a p oduc
o single-elec on unc ions, which only depend on one posi ion. Since elec ons a e
iden ical e mions,
Ψ
should also be an i-symme ic wi h espec o he exchange
o wo pa icles. The esul ing o m o he s a e o he sys em is known as a Sla e
de e minan
Ψ = 1
√N!

ϕ1( 1, σ1)ϕ1( 2, σ2)··· ϕ1( N, σN)
ϕ2( 1, σ1)··· ··· ···
.
.
..
.
..
.
..
.
.
ϕN( 1, σ1)ϕN( 2, σ2)··· ϕN( N, σN)

,(3.3)
whe e each elec on has he posi ion
i
and spin
σi
deg ees o eedom. The an i-
symme y o he de e minan when exchanging wo ows o columns ensu es ha
Ψ
is an i-symme ic when exchanging
( i, σi)↔( j, σj)
. The p e- ac o is he e o
no malize he s a e o uni y.
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3. Densi y unc ional heo y
This app oxima ion is known as he Ha ee-Fock me hod. Al hough i uses a
unc ion o independen elec ons, some co ela ions a e s ill included. In pa icula ,
o cing he an i-symme y o
Ψ
includes he co ela ion due o he Pauli exclusion
p inciple. Tha is, he only wo-body co ela ion included is he one ha p e en s
wo elec ons wi h he same spin being a he same posi ion in space. Using his o m
o he solu ion in he Hamil onian in Eq.(3.2) yields he expec a ion alue
⟨Ψ|H|Ψ⟩=X
i,σ Zϕσ∗
i( )−1
2∇2+VIe( )ϕσ
i( )d3 +
+1
2X
i,j,σi,σjZϕσi∗
i( )ϕσj∗
j( ′)1
| − ′|ϕσi
i( )ϕσj
j( ′)d3 d3 ′−
−1
2X
i,j,σ Zϕσ∗
i( )ϕσ∗
j( ′)1
| − ′|ϕσ
j( )ϕσ
i( ′)d3 d3 ′.
(3.4)
The second e m is known as he di ec in e ac ion while he hi d is he exchange
in e ac ion. The e ec o he la e is always o educe he ene gy. The in e p e a ion
in his app oxima ion is ha each elec on c ea es a "hole” due o o he pa icles
ending o be away om i because o he exclusion p inciple. The in e ac ion o he
nega i ely cha ged elec on wi h he hole gi es an o e all nega i e con ibu ion o
he o al ene gy.
In he nex sec ions, we will see ha his kind o independen -pa icle app oxima-
ion can be used in he DFT amewo k.
3.3 The Hohenbe g-Kohn heo ems and he
Kohn-Sham me hod
To begin wi h he o mula ion o DFT, we mus i s in oduce he heo e ical g ounds
p o ided by wo impo an heo ems, which we will now o mula e wi hou p oo .
Theo em 3.1. The g ound-s a e densi y
n0( )
o he sys em de e mines uniquely,
up o a cons an , he ex e nal po en ial, such as
VIe
, o any sys em o in e ac ing
pa icles.
This means ha he elec onic Hamil onian in Eq.(3.2) has a uni e sal pa com-
p ised o he kine ic and in e ac ing ene gy o he elec ons, and a speci ic e m o
he in e ac ion o he ions wi h he elec ons. Rema kably, knowing
n0( )
implies
ull knowledge o he Hamil onian and, he e o e, all he physical p ope ies o he
sys em.
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3.3. The Hohenbe g-Kohn heo ems and he Kohn-Sham me hod
Theo em 3.2. Fo any ex e nal po en ial, we can de ine a uni e sal unc ional
E[n]
,
which depends on he densi y, whose global minimum is achie ed exac ly o he
g ound-s a e densi y E[n0].The unc ional has he o m
E[n] = T[n]+Eee[n] + ZVIe( )n( )d3 +EII,(3.5)
whe e
T
and
Eee
a e he kine ic and in e ac ion ene gy o elec ons and
EII
is he
ion-ion epulsion ene gy.
The second heo em p o ides a way o de e mine he g ound-s a e densi y. P o-
ided we ound he unc ional
E[n]
, hen
n0( )
is compu ed simply by minimiza ion
o he ene gy. Mo eo e ,
E[n]
depends only on he densi y, which ce ainly has less
in o ma ion ha he ue wa e unc ion o he sys em, as di e en s a es may lead o
he same elec onic densi y.
Un o una ely, gi en he elec onic densi y o a ma e ial, in p ac ice he e is no
known way o ex ac all he physical p ope ies, al hough he i s heo em p o es
ha i is indeed possible. Fo example, e en e alua ion o he kine ic ene gy
T[n]
is no possible using only he densi y, wi hou gi ing an explici o m o he wa e
unc ion om whe e i o igina es (e.g. a Sla e de e minan ). The p ac ical solu ion
gi en by Kohn-Sham app oach is p ecisely o s a e he p oblem in e ms o a wa e
unc ion ha is a p oduc o one-pa icle s a es, which allows us o e alua e all he
e ms in he unc ional. This wa e unc ion is no he ue many-body s a e o he
sys em, bu ha o an auxilia y, independen -pa icle sys em. The key insigh is ha ,
i we can design he unc ional o he auxilia y sys em such ha i has he same
g ound s a e densi y as he ue sys em, hen he compu ed elec onic densi y is
also he exac densi y o he o iginal Hamil onian.
Kohn-Sham equa ions
The s a e o he auxilia y independen -pa icle sys em can be desc ibed by a Sla e
de e minan as in Eq.(3.3), whose densi y is gi en by
n( ) = X
σ
Nσ
X
i|ϕσ
i( )|2,(3.6)
whe e
ϕσ
i
a e one-pa icle s a es o spin
σ
, wi h a o al o
Nσ
pa icles pe spin. The
kine ic e m can be explici ly e alua ed as
TKS =−1
2X
σ,i ⟨ϕσ
i|∇2|ϕσ
i⟩=1
2X
σ,i Z|∇ϕσ
i( )|2d3 , (3.7)
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3. Densi y unc ional heo y
whe e i is unde s ood ha he e is an
i
-index summa ion o each
σ
. Al hough
TKS
is
e alua ed in e ms o he wa e unc ion, i is in ac a unc ional o he densi y, as
s a ed by he second Hohenbe g-Kohn heo em. F om he Ha ee-Fock heo y, we
know ha he e is also a di ec in e ac ion e m o he densi y wi h i sel
EHa ee =1
2Zn( )n( ′)
| − ′|d3 d3 ′(3.8)
and and exchange-co ela ion e m
Exc
. I is p ecisely his las , unspeci ied e m
ha is used o design he sys em such ha i has he eal g ound-s a e ene gy, de ining
he Kohn-Sham unc ional as
EKS[n] = TKS[n] + ZVIe( )n( )d3 +EHa ee[n]+EII +Exc[n].(3.9)
The o mal exp ession o he exchange-co ela ion pa is
Exc[n] = ⟨ˆ
T⟩−TKS[n]+⟨ˆ
Vee⟩−EHa ee[n],(3.10)
accoun ing exac ly o he di e ence be ween he eal elec onic ene gy
⟨ˆ
T+ˆ
Vee⟩
and
he independen -pa icle ene gy. This gua an ees ha he ene gy and g ound-s a e
densi y o he Kohn-Sham sys em a e exac ly he eal quan i ies. The exp ession o
TKS
is only known in e ms o he wa e unc ion. Howe e , using he chain ule and
Eq.(3.6), we can minimize EKS[n]subjec o he o hono mali y cons ain s
⟨ϕσ
i|ϕσ′
j⟩=δσσ′δij.(3.11)
As a esul , we ob ain he Kohn-Sham equa ions
Hσ
KSϕσ
i=−1
2∇2+Vσ
KSϕσ
i=ϵσ
iϕσ
i,(3.12)
whe e he Kohn-Sham po en ial is de ined as
Vσ
KS( )=VIe( ) + δEHa ee
δnσ( )+δExc
δnσ( ).(3.13)
The exchange-co ela ion e m
Al hough he Kohn-Sham me hod is exac , he p oblem o ea ing he in e ac ion
e ms has been ansla ed o he compu a ion o he exchange-co ela ion unc ional,
which is essen ial o ep oduce he ue elec onic densi y. As a consequence, much
o he e o in de eloping DFT-based me hods is pu in o modelling o he exchange-
co ela ion e m accu a ely and e icien ly. One impo an aspec is ha , gi en ha
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3.3. The Hohenbe g-Kohn heo ems and he Kohn-Sham me hod
he long- ange in e ac ion e ms a e al eady accoun ed o in he Ha ee e m,
Exc
mus ha e a local exp ession o he o m
Exc[n] = Zεxc([n], )n( ),(3.14)
such ha
Vσ
xc ≡δExc
δnσ( )=εxc([n], )+n( )δεxc([n], )
δnσ( ).(3.15)
Thus, he modeling o
Exc
is ul ima ely a p oblem o modeling
εxc
. Some o he mos
common app oxima ions a e:
•
Local-spin-densi y app oxima ion (LSDA) [53]: hey assume he exchange-
co ela ion ene gy o an homogeneous elec on gas
ELSDA
xc [n↑, n↓] = Zn( )ϵhom
x(n↑( ), n↓( ))+εhom
c(n↑( ), n↓( )),(3.16)
sepa a ing linea ly he exchange and co ela ion pa s. The ad an age is ha
he exchange ene gy is known analy ically
εhom
x(nσ) = −3
46
πnσ1/3
,(3.17)
while he co ela ion pa has been i ed o Mon e Ca lo simula ions. Su p is-
ingly, despi e i s simplici y, LSDA can be a he accu a e, especially o sys ems
close o homogeneous elec on gases such as nea ly- ee-elec ons me als.
•
Gene alized-g adien app oxima ion (GGA): he exchange-co ela ion en-
e gy is expanded also in e ms o de i a i es o he densi y, which helps accoun
o i s non-homogenei y away om he cons an LSDA app oxima ion. This
co ec s i s endency o unde es ima e he exchange and o e es ima e he co -
ela ion. One common implemen a ion is he Pe dew-Bu ke-Enze ho [54]
(PBE) o m. Beyond, he basic GGA, he e exis s he me a-GGA unc ionals,
which u he expand he ene gy in e ms o second de i a es o he densi y
and o he ele an quan i ies. One example is he modi ied Becke-Johnson
po en ial [55,56] (mBJ), which akes he LSDA co ela ion po en ial and adds
an exchange e m which also depends on he local kine ic ene gy densi y.
•
Hyb id unc ionals: hey combine he exac Ha ee-Fock exchange wi h
empe ically i ed exchange-co ela ion e ms and p o ide he mos accu a e
app oxima ion. Howe e , he compu a ional cos is also g ea ly inc eased. Some
examples a e he B3LYP [57,58], PBE0 [59,60] and M06 [61] unc ionals.
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
phase o e e y pa h
λ( )
. Only i
⟨ψ′(λ)|∂µ|ψ′(λ)⟩
is exp essed as he g adien o a
scala unc ion can he phase be adjus ed o anish o any pa h, since he in eg al
does no depend on he speci ic cu e ollowed in pa ame e space. I
λ
aces a closed
loop in pa ame e space, such ha
λ(0) = λ( )
, he single- aluedness o he wa e
unc ion equi es ha
∆α=α( )−α(0) = 2πm
, whe e
m∈Z
. This can only be
made ze o o any loop i we can cancel he phase o any open pa h, since a loop
can be decomposed o a se ies o open pa hs. This ac was popula ized by Michael
Be y in a pape published in 1984 [62] and i means ha he geome ic phase canno
be emo ed in closed loops, hence i was named he Be y phase in his con ex .
He ealized ha he ue dependence o he he eigens a es is on he pa ame e s
λ
a he han jus
. Whe eas o he la e one can always exp ess he in eg and as a
de i a i e o a unc ion since i is a one-dimensional pa ame e space, o he o me
i is no always possible.
4.2 Gauge heo y o mula ion
The eade amilia wi h he opic may ealize ha his is in ac a gauge heo y, whe e
in his case he gauge g oup is
U(1)
, i.e., a phase. Howe e , his is simpli ied wi h
espec o o he cases such as elec omagne ism since he ield deg ees o eedom
a e s a ic. The p e ious discussion can be e o mula ed in his language by no icing
ha he Be y phase can be e-w i en
γ( ) = IAµdλµ(4.11)
wi h
Aµ≡ −i⟨ψ(λ)|∂µψ(λ)⟩
being he gauge po en ial, which is usually called Be y
po en ial o Be y connec ion. Unde a gauge ans o ma ion such as in Eq.(4.9),
he gauge po en ial ans o ms wi h he amilia exp ession
Aµ→Aµ+∂µα. (4.12)
We can also de ine a gauge-co a ian de i a i e
Dµ≡∂µ−iAµ(4.13)
such ha unde a gauge ans o ma ion
D′
µ|ψ′⟩= (∂µ−iA′
µ)eiα|ψ⟩=eiαDµ|ψ⟩.(4.14)
The ield s eng h o Be y cu a u e associa ed o he connec ion is de ined as
Ωµν =∂µAν−∂νAµ−i[Aµ, Aν] = i[Dµ, Dν],(4.15)
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4.3. Wilson loops: gene aliza ion o he Be y phase
whe e
[·,·]
deno es he commu a o , which anishes in gene al when he gauge heo y
is Abelian, as in he
U(1)
Be y phase desc ibed abo e. Howe e , his is no necessa y.
I ins ead o a single s a e
|ψ(λ)⟩
we had acked a subspace o degene a e eigens a es,
he Be y phase would in ac be a ma ix ans o ming hese s a es in o each o he
ins ead o a simple phase. Then he ull exp ession o he cu a u e is impo an
when dealing wi h non-Abelian gauge heo ies. Fo such cases, he ans o ma ion
in Eq.(4.12) is no qui e co ec and mus be eplaced by he mo e gene al exp ession
Aµ→U†AµU+iU†∂µU, (4.16)
we e
U
, in his con ex , is a uni a y ma ix ha exp esses he change o he subspace
s a es a e a e sing he loop. I is easy o e i y ha i
U=e−iα
, e e y hing in he
exp ession commu es and Eq.(4.12) is eco e ed.
Remaining now in he simpli ied
U(1)
e sion o he sake o cla i y, suppose we
λ( )
is es ic ed o a mani old o
d
dimensions and ha i aces a loop
C
which
encloses an a ea Σ. Then he Be y phase is he line in eg al
γ( ) = IC
Aµdλµ=ZΣ
Ωµν dsµ∧dsν(4.17)
whe e S okes’ heo em was used o ans o m he line o a su ace in eg al, assuming
ha i holds, i.e., ha
A
is de ined on
λ( )
and
Ω
in
Σ
(i could be wo ked ou in
pa ches o he wise).
dsµ∧dsν
is a di e en ial o he su ace
Σ
in some coo dina es.
This is p ecisely he Be y phase we ound in Sec ion 4.1.
To e lec on his, we ha e ound ha he e is a eedom o choose any linea
combina ion o degene a e s a es a each poin along he pa h
λ( )
o ep esen he
physical s a e o he sys em. This eedom comes a he cos o ha ing o de ine
addi ional s uc u e ha exp esses how o ans o m he s a es om one poin o
an in ini esimally close one, which is he connec ion
A
. We ha e also ound ha ,
a e a e sing a loop, he inal s a e may di e om he ini ial one by a uni a y
ans o ma ion. This di e ence is p ecisely he lux o he Be y cu a u e h ough
he a ea enclosed by he cu e. In ac , Ωgi es he in ini esimal con ibu ion o his
di e ence o e a plaque e o di e en ial a ea such ha he o al change is he in eg al
o e he su ace.
4.3
Wilson loops: gene aliza ion o he Be y phase
The de i a ion in Sec ion 4.1 is a simpli ied e sion o a mo e gene al concep called
Wilson loop. In ha p ocedu e, we acked he change o
|ψ(λ)⟩
such ha , a e e y
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
poin in pa ame e space, i was bo h no malized and pe pendicula o s a es wi h
o he ene gies. This can be iewed as a o m o pa allel anspo in he sense we
iden i y he o hono mali y condi ion wi h he ac ha he “angle” o he ec o wi h
i sel along a line angen o he cu e
λ( )
emains cons an . The pa allel anspo
condi ion can be exp essed as he anishing o he co a ian de i a i e along he
cu e λ( )
λµDµ|ψ(λ)⟩=λµ(∂µ|ψ(λ)⟩−iAµ|ψ(λ)⟩) = 0 (4.18)
and aking he inne p oduc wi h ⟨ψ(λ)|we jus ind again ha
⟨ψ(λ)|∂µψ(λ)⟩=iAµ.(4.19)
The e o e, wha we ound in he de i a ion o he geome ic phase is ha , despi e
ansla ing
|ψ(λ)⟩
in a pa allel ashion along he cu e, he inal and ini ial s a es
may di e by a phase. This idea is gene alized by conside ing a closed subspace o
eigens a es o
H( ){|ψn(λ)⟩}
such ha we expand he ins an aneous s a e o he
sys em as
|χ(λ)⟩=cn(λ)|ψn(λ)⟩,(4.20)
whe e we assume summa ion o e epea ed indices. The pa allel anspo condi ion
eads
Dµcm=∂µcm−i(Aµ)mncn= 0,(4.21)
no ing ha
Aµ
is ma ix- alued. The ans o ma ion om he ini ial o inal coe i-
cien s is gi en by he o mal exp ession
cm(λ( )) = hPeiRλ( )
λ(0) Aµdλµimn cn(λ(0)),(4.22)
whe e we iden i y he Wilson line ope a o
W(λ)mn =hPeiRλ( )
λ(0) Aµdλµimn .(4.23)
The symbol
P
means ha he in eg a ion is pa h-o de ed in he di ec ion o he cu e.
We emphasize ha he spec um o
W(λ)
depends bo h on he pa h
λ( )
and he
connec ion
A
, which means ha i is no gauge in a ian . I can be shown ha , unde
a uni a y ans o ma ion (a gauge) o he basis s a es
|˜
ψn(λ)⟩=Umn|ψm(λ)⟩
,
W(λ)
ans o ms as
˜
Wmn(λ)=U†(λ( ))miWij(λ)U(λ(0))jn.(4.24)
Thus, we see ha in he special case ha he line is closed such ha
λ(0) = λ( )
,
Eq.(4.24) is ac ually a simila i y ans o ma ion, which means ha bo h he eigen al-
ues and he ace a e gauge in a ian . This speci ic case o he Wilson line ope a o
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4.3. Wilson loops: gene aliza ion o he Be y phase
is called a Wilson loop
1
and is a undamen al objec o s udy he opology ha will
be used h oughou his wo k. Wilson loops ha e some in e es ing p ope ies ha we
enume a e he e:
•
Conside ing all he loops ha s a om a poin
λ0
, Wilson loops along all
hese pa hs o m a g oup called he holonomy g oup
Φλ0(M, A)
. The ans-
o ma ion co esponding o a composi ions o loops is he p oduc o ma ices
o each pa h. The in e se ans o ma ion can be ound by a e sing he loop
in he opposi e di ec ion, while he iden i y ma ix co esponds o he cons an
loop (no loop a all).
•
The choice o base poin along he pa h is i ele an o he spec um. To see
his, conside loops
α,β
wi h base poin s
λα,λβ
espec i ely as in Fig.4.1.
Assuming he pa h-connec edness o he pa ame e space, he loop
α
can be
composed by going along a pa h om
λα
o
λβ
, pe o ming he
β
loop and he
going back om λβ o λα. The co esponding ope a o is
Wα=Wλα→λβWβWλβ→λα=W†
λβ→λαWβWλβ→λα(4.25)
and since he ma ices a e uni a y, his is a simila i y ans o ma ion.
•
The Wilson loops o e pa hs ha a e smoo hly de o mable in o each o he
(homo opic) a e in gene al di e en . Howe e , hey a e equal i he connec ion is
la , so he cu a u e anishes e e ywhe e. In ac , he in eg al o he cu a u e
o e he a ea enclosed by wo homo opic loops measu es he di e ence in
pa allel anspo along bo h pa hs.
All in all, gauge eedom allows us o de ine a basis a each poin in pa ame e space
and he connec ion is he objec ha exp esses how he basis changes om one poin
o ano he . When pa allel- anspo ed, he p ecise way in which he di e en spaces
a each poin in pa ame e space a e glued oge he may esul in s a es e u ning o
a ans o med e sion o hemsel es when acing a loop. This change is p ecisely
he holonomy and i is compu ed by he Wilson loop ope a o . Addi ionally, he se
o such ans o ma ions ound by conside ing all loops h ough a base poin o ms
he holonomy g oup.
Why Wilson loops p obe he opology
So a , all he concep s we ha e in oduced such as pa allel anspo , connec ion and
cu a u e, a e pu ely geome ical. Howe e , he holonomy in oduces a way o ela e
1
The e m “Wilson loop” may some imes be used o e e o he ace o he Wilson line ope a o
o e a closed pa h.
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
Figu e 4.1: Ske ch o he Wilson loop cons uc ion ha shows i s independence om
he base poin . The loop s a ing and ending a
λα
is equi alen o going om
λα
o
λβ
, pe o ming he loop a
λβ
and going back o
λα
. This is ma hema ically equi alen
o a simila i y ans o ma ion and, he e o e, he spec um does no change.
hese ideas o he opology o he unde lying mani old. As we will see in ollowing
sec ions, he inal link is gi en by ela ing he holonomy o he cu a u e. In his
sec ion, we will i s see how Wilson loops cap u e he opology o a sys em.
Fi s , we need o in oduce he concep o homo opy. Conside a mani old
2M
.
A pa h on
M
s a ing a
x0
and ending a
x1
is jus a map
λ
om he uni in e al
I≡[0,1]
o poin s in
M
such ha
λ(0) = x0
and
λ(1) = x1
. A loop is pa h whose
s a and end poin s a e he same. A homo opy is a con inuous map
F:I×I→M
ha maps loops α(s), β(s)wi h he same base poin x0 o each o he such ha
F(s, 0) = α(s), F(s, 1) = β(s)and F(0, )=F(1, )=x0.(4.26)
Two loops a e he e o e homo opic i he e exis such
F
ela ing hem. Wi h his
no ion in mind, he undamen al g oup
π1(M, x0)
o loops wi h base poin
x0
is
o med by all loops ha s a and end a
x0
whe e pa hs ha a e homo opic a e
conside ed he same.
π1(M, x0)
knows abou he opology o
M
because he e may
be non-con ac ible loops on
M
ha canno be de o med o nei he he iden i y o
cons an loop (because hey a e non con ac ible) no o o he loops, which may
happen when he e a e holes in
M
. As a ypical example, he undamen al g oup o
he sphe e
S2
is i ial, since all closed pa hs can be de o med in o each o he wi hou
lea ing
S2
. Howe e ,
π1
o he o us
S1×S1
is no i ial and in ac
π1(T2) = Z×Z
,
whe e we gi e one in ege o he winding along one di ec ion enclosing he “handle”
o he o us and ano he one o he o he di ec ion ha goes a ound he hole, which
canno be de o med in o each o he (see Fig.(4.2). No e ha in hese examples, he
2In ac , i need no be a di e en iable mani old, bu only a opological space
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4.4. The Che n numbe : a opological in a ian
Figu e 4.2: The wo kinds o non-con ac ible loops in a 2- o us, wi h wo examples
o he homo opic amilies in ed and blue. The undamen al g oup
π1(T2) = Z×Z
,
co esponding o he numbe o imes a loop goes a ound each handle. Because
T2
is
pa h-connec ed, π1(T2)is independen o he base poin .
base poin o
π1
is i ele an , which happens o e e y mani old whe e any wo poin s
can be connec ed by a con inuous pa h. We assume ha his is he case om he e
onwa d.
The connec ion o
π1
o he holonomy g oup, comes om he ac ha Wilson
loops a e compu ed p ecisely o e closed pa hs such as hose in he undamen al
g oup. In ac , he e is a map be ween he loops in
π1(M)
and he Wilson loops o
he holonomy g oup
Φ(M, A)
such ha o each loop in
π1(M)
we assign he Wilson
loop ope a o along i . This map is in gene al no su jec i e because
Φ(M, A)
is
la ge han
π1
because Wilson loops along homo opic pa hs a e no equal unless
he e is no cu a u e. F om he homo opies ha iden i y equi alen loops, we also
ob ain a con inuous map be ween Wilson loops whose pa hs a e homo opic. I can be
shown ha Wilson loops along pa hs ha can be de o med o he cons an pa h a e
connec ed ia his con inuous de o ma ion o he iden i y, while he ones ha canno
be de o med in his way a e comple ely disconnec ed.
Apa om he ela ionship wi h he undamen al g oup, he ollowing sec ion
shows ha he Wilson loops can also be used o compu e opological in a ian s, adding
o he connec ion be ween geome y and opology.
4.4 The Che n numbe : a opological in a ian
The opology o mani olds can be dis inguished by compu a ion o opological in-
dices
3
. In his sec ion, we in oduce pe haps he mos widely known o hem in he
con ex o opological ma e ials called he Che n numbe and show how i can be
compu ed om he Be y cu a u e enso . Mo eo e , we will also see ha he e
3
Mani olds wi h di e en indices a e opologically inequi alen . Howe e , wo mani olds wi h
he same se opological indices canno be said o be equi alen , since he e could be o he unknown
indices ha dis inguish hem.
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
α
λ
A1
A2
Figu e 4.3: Ske ch o he wo le el sys em. Since
|λ|
is ixed, he pa ame e mani old
is a sphe e
S2
. Since eigens a es ha e o ex singula i ies ei he a he no h o he
sou h pole, wo gauges mus be chosen co esponding o wo di e en connec ions
A1
and
A2
. These a e ela ed by a gauge ans o ma ion gi en by
α
, whose winding
numbe is he Che n numbe .
is a link be ween Wilson loops and he cu a u e which comple es he ela ionship
be ween opology and geome y.
Fi s , suppose ha he pa ame e space is a sphe e
M=S2
. This exe cise will
co e he mos common example o opology, which is he wo le el sys em, whose
Hamil onian in some basis can be desc ibed by a wo-dimensional ma ix
H(λ)=aσ0+λ·σ,(4.27)
wi h ene gies
E±=a±|λ|,(4.28)
whe e
λ
is a se o pa ame e s which may e ol e adiaba ically wi h ime and
σ
a e
he Pauli ma ices assembled in o a ec o ,
σ0
being he iden i y. Fo he sake o ou
discussion, he o e all shi in he ene gy le els gi en by
aσ0
,
a
being a eal numbe ,
can be neglec ed. Fu he mo e, he modulus o λis ixed so he only eal deg ees o
eedom i we ake s anda d sphe ical coo dina es a e wo angles. This means ha
λ
is es ic ed o a sphe e
S2
as we ha e in oduced. This gene al Hamil onian desc ibes
he ene gy le els o a spin-
1/2
pa icle in a magne ic ield
B
i we ake
λ≡B
, o
example.
Conside now he in eg al o e he sphe e o he Be y cu a u e
ZS2
Ωµν dsµ∧dsν(4.29)
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4.4. The Che n numbe : a opological in a ian
whe e
dsµ∧dsν
is an in ini esimal a ea in some coo dina es. We could conclude nai ely
ha his in eg al anishes since, by applica ion o S okes’ heo em, he exp ession
can be ans o med in o he in eg al o
A
o e he bounda y
∂S2
, bu he sphe e has
no bounda y (we should no be con used by he pic u e o embedding o a sphe e
in h ee-dimensional space). Howe e , we will see ha S okes’ heo em canno be
blindly applied he e. No ice i s ha i
|λ| = 0
he e a e wo eigens a es sepa a ed
by a spec al gap. This means ha bo h eigens a es a e no mixed as long as he pa h
aced by
λ
is es ic ed o
S2
, so we can ack, o example, he e olu ion o he
lowe ene gy s a e
|ψ0(λ)⟩
. The connec ion
A
is compu ed om
|ψ0(λ)⟩
and i is
a well known ac ha a smoo h ec o ield canno be de ined o e all
S2
, which
implies ha we ha e o wo k by pa ches, o example di iding i in o hemisphe es.
This o ces us o use a leas o di e en connec ions
A1
and
A2
, which a e ela ed in
by a gauge ans o ma ion implemen ed on he s a es by
eiα(λ)
on he a ea whe e he
wo pa ches o e lap
A2
µ=A1
µ−∂µα(4.30)
whe e i is mos con enien in his case o wo k in sphe ical coo dina es (i.e.
µ
e e s
o
θ
and
ϕ
angles). The ac ual o m o he eigens a es o he Hamil onian in Eq.(4.27)
is
|ψ0(λ)⟩=cos(θ/2)
sin(θ/2)eiϕ,|ψ1(λ)⟩=sin(θ/2)
−cos(θ/2)eiϕ,(4.31)
whe e
λ=|λ|(sin θcos ϕ, sin θsin ϕ, cos θ)
and he eigen alues a e
E0=−|λ|
and
E1=|λ|
. We see ha
|ψ0(λ)⟩
has a phase singula i y a he sou h pole whe e
θ=π
.
Howe e , his can be undone by a gauge
e−iϕ
such ha he esul ing ec o is ill-
de ined a he no h pole. P oceeding now o compu e he in eg al in Eq.(4.29), we
ind
ZS2
Ωµν dsµ∧dsν=ZM1
Ωµν dsµ
1∧dsν
1+ZM2
Ωµν dsµ
2∧dsν
2(4.32)
=Z∂M1
A1
µdλµ−Z∂M2
A2
µdλµ(4.33)
=IL
∂µα dλµ.(4.34)
whe e
L
deno es he cu e ha delimi s he wo hemisphe es. In going om he i s
o he second line, we ha e o ake in o accoun ha he wo hemisphe es
M1
and
M2
ha e he opposi e o ien a ion so he cu es mus be a e sed in opposi e di ec ions,
gi ing ise o he minus sign. Remembe ing ha he single- aluedness o he wa e
unc ion
|ψ0(λ)⟩
equi es ha he gauge ans o ma ion be pe iodic, we inally a i e
a ZL
∂µα dλµ= 2πC (4.35)
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
whe e
C
is known as he Che n numbe
4
. I is an in a ian o he s a e
|ψ0(λ)⟩
ha
has he e y in e es ing p ope y o being an in ege . This means ha , as i s alue
canno con inuously change, ex eme changes in he s uc u e o he eigens a e as
a unc ion o
λ
mus ake place in o de o i o a y, a ea u e which is gene al
o opological in a ian s. In pa icula , unless he gap be ween
|ψ0(λ⟩)
and he
highe -ene gy s a e is closed, he Che n numbe will emain unchanged, no ma e
he ac ual alues o he ene gy le els o e
S2
. In passing, we ha e also seen ha
C
is in ac ela ed o he winding numbe o he gauge ans o ma ion o e he
sphe e. When gene alizing om he wo-le el sys em, we could ack mo e han one
eigens a e such ha Ais non-Abelian. Then Eq.(4.29) is ew i en o a mani old M
ZM
T (Ωµν)dsµ∧dsν,(4.36)
he gauge ans o ma ion be ween connec ions in wo pa ches is
A1
µ=U†A2
µU+iU†∂µU(4.37)
and he Che n numbe is IL
∂µχ dλµ,(4.38)
whe e we de ine
χ≡Im log de U
. Taking in o accoun ha
U
is uni a y so ha i s
eigen alues a e complex numbe s o uni modulus
i(U†)ij(∂µU)jk =ie−iϕiδij(i)(∂µϕj)eiϕjδjk =−∂µϕiδik (4.39)
so
χ
is jus he sum o he angles o he phase eigen alues o
U5
. No e also ha aking
he ace o Ω emo es he commu a o e m in Eq.(4.15).
Going back o he wo le el sys em, we can compu e he Be y connec ion om
he eigens a e |ψ0(λ)⟩in Eq.(4.27)
Aθ=−i⟨ψ0(λ)|∂θ|ψ0(λ)⟩= 0,(4.40)
Aϕ==−i⟨ψ0(λ)|∂ϕ|ψ0(λ)⟩= sin2θ
2,(4.41)
which is alid o any pa ch on he sphe e ha does no include
θ=π
. The only
non-ze o independen componen o he Be y cu a u e is hen
Ωθϕ =−Ωϕθ =∂θAϕ−∂ϕAθ=∂θAϕ= sin θ
2cos θ
2.(4.42)
4
This is he i s Che n numbe . The e could be o he s ela ed o in eg als o powe s o
Ω
. Howe e ,
as a di e en ial o m,
Ω
is al eady o o de wo, which means ha highe powe s a e al eady o
dimension highe ha S2and hence anish.
5We ha e used a basis whe e Uis diagonal wi hou loss o gene ali y.
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4.4. The Che n numbe : a opological in a ian
Pe o ming he in eg al o e he whole sphe e
Z2π
0
dϕ Zπ
0
dθ sin θ
2cos θ
2= 2π−1
2cos θϕ
0
= 2π, (4.43)
which means ha he Che n numbe is
C= 1
. One can also show ha , in Ca esian
coo dina es, he cu a u e has only a adial componen o he o m
Ω ∝1
,(4.44)
which is simila o he exp ession o he magne ic ield o a hypo he ical magne ic
monopole.
Rela ing he Che n numbe o he holonomy
In Sec ion4.3 we saw how he holonomy exp esses he change o he eigens a es when
hey a e pa allel- anspo ed along closed pa hs. We ha e also de ined he Che n
numbe as he in eg al o he Be y cu a u e o e he pa ame e space and we ha e
seen how i p obes he non- i iali y o he Be y connec ion de ined in ha same
space. We now show ha he ela ion be ween he holonomy o a connec ion and he
cu a u e ela es he Che n numbe wi h he Wilson loop ope a o s.
The missing link is p o ided by he Amb ose-Singe heo em. In a b oad sense,
i ells us ha he in o ma ion o he holonomy g oup a a poin
λ0
can be ound in
he cu a u e a ha same poin . In o he wo ds,
Ω
gi es he di e en ial con ibu ion
o he holonomy o an in ini esimally small loop
C
a
λ0
. Gi en he o m o he
Wilson line (o loop) in Eq.(4.23), by expanding he pa h-o de ed exponen ial o
second o de [63], one can ind ha
W(C) = Pexp iIC
Aµdλµ≈1+iPIC
Aµdλµ+
+i2
2PIC
Aµdλµ2
+···
(4.45)
The i s o de e m can be exp essed by S okes’ heo em as
PIC
Aµdλµ=ZZΣ
1
2(∂µAν−∂νAµ)dλµ∧dλν,(4.46)
whe e Σis he a ea delimi ed by C, and he second o de e m is
1
2PIC
Aµdλµ2
=−1
2ZZΣ
[Aµ, Aν]dλµ∧dλν+··· (4.47)
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
The e o e, he spec um o he Wilson loop ope a o is ela ed o ha o he posi ion
ope a o
8
. Fu he mo e, he cen e o he
n
- h Wannie unc ion
⟨wn0|x|wn0⟩=xn
is he e-scaled loga i hm o he
n
- h diagonal elemen o
W
, p o ided ha
{|˜
ψnk⟩}
we e chosen o be maximally localized. Na u ally, he spec um o he Wilson loop
does change unde gauge ans o ma ions and so do he Wannie cen e s. Howe e ,
he ace is gauge-in a ian so he sum o cen e s also emains unchanged, a leas up
o a la ice ec o ansla ion. This inde e minacy comes om he in a iance o he
Wilson loop ope a o when an addi ional
2π
phase is included. Thus, he e a e some
gauge ans o ma ions ha shi he la ice o Wannie cen e s by some ec o , bu he
la ice i sel emains he same since i epea s inde ini ely. The e o e, we can ega d
his sum o cen e s as he ac ual cen e s a ound which he cha ge o he elec ons in
he selec ed bands is cen e ed.
The abo e in e p e a ion can be gene alized o 3D cases by conside ing hyb id
Wannie unc ions. These a e a se o s a es which a e localized in only one di ec ion
⊥
ob ained by Fou ie ans o ming only in he associa ed ecip ocal space di ec ion
k⊥:
|wnR⊥,k∥⟩=a
2πZ2π/a
0
e−ik⊥R⊥|˜
ψnk⊥,k∥⟩dk⊥,(4.78)
whe e he cell has leng h
a
in he
⊥
di ec ion and
k∥
a e he wo emaining momen a
in he pe iodic di ec ions. These s a es can be isualized as "shee s” ha ex end o e
∥
bu ha e a de ini e cen e in
⊥
. Jus as in he 1D case, he Wilson loop compu ed
by in eg a ing o e
k⊥
p o ides he posi ion o hese posi ions. The se o hyb id
Wannie unc ions is in ac a se o eigens a es o he posi ion ope a o
⊥
p ojec ed
on o he se o bands i hey a e maximally localized in
⊥
. Howe e , in gene al i
is no possible o ind a se o unc ions ha diagonalizes wo posi ion ope a o s a he
same ime, since hey do no commu e. I is only possible when he Be y cu a u e
anishes a all poin s o he BZ.
Hyb id Wannie unc ions and he Che n numbe
We can inally p o ide a physical in e p e a ion o he Che n numbe in his pic u e.
Assume o simplici y a 2D sys em o which we ha e cons uc ed hyb id Wannie
unc ions
{|wnR⊥,k∥⟩}
which a e smoo h in
k⊥
. We can compu e Wilson loops
W(k∥)
by in eg a ing o e
k⊥
which a e dependen on he emaining
k∥
and, by he p e-
ious explana ions, hese gi e he posi ions o he hyb id Wannie cha ge cen e s.
8
Mo e p ecisely, i is ela ed o he spec um o he posi ion ope a o p ojec ed on o he se o bands
o in e es .
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4.5. Applica ion o c ys alline sys ems o elec ons
Fu he mo e, we ha e seen ha acking he e olu ion o
W(k∥)
o e closed loops is
equi alen o compu ing he Che n numbe .
Fo he Che n numbe o be non-ze o, he necessa y bu no su icien condi ion
is ha ime- e e sal is b oken. This is because upon ime- e e sal, he cu a u e
ans o ms as
Ω(k)TR
−→ −Ω(−k)(4.79)
which means ha he in eg al o e a closed mani old mus anish by cancella ion o
opposi e wa e ec o s. In passing, we men ion ha in e sion ans o ms
Ω(k)in e sion
−−−−→ Ω(−k),(4.80)
which implies ha when bo h TR and in e sion a e p esen , he cu a u e mus
anish e e ywhe e. A non-ze o Che n numbe
C⊥
compu ed by in eg a ing along
k⊥
implies ha he la ice o hyb id cen e s is shi ed by
C⊥
uni cells in he
R⊥
di ec ion upon e u ning o he s a ing poin . In his sense, i is in e p e ed as cha ge
being pumped in ha di ec ion. This also implies ha he e is an obs uc ion o
compu ing exponen ially-localized Wannie unc ions in all di ec ions, since
hey a e no pe iodic in he
k∥
. We also saw in he wo-le el sys em ha he Che n
numbe is ela ed o he winding o a gauge ans o ma ion, which is necessa y when
a global gauge canno be ound. The e o e,
C⊥= 0
p e en s us om inding a smoo h
gauge in all di ec ions, so he Fou ie ans o m gi es badly localized unc ions as a
esul .
Quan um Anomalous Hall insula o s
Insula o s wi h non-ze o Che n numbe a e known as quan um anomalous Hall
insula o s (QAHI). They a e he hallma k o non i ial opology in condensed ma e
sys ems, p o iding a mani es a ion o hese ea u es in he o m a Hall cu en e en
in he absence o an ex e nal magne ic ield. This can happen o example when
he c ys al p esen s an a angemen o magne ic momen s obeying he symme y
o a Shubniko space g oup wi hou TRS, as he b eaking o his symme y is equi ed.
Fo simplici y, conside i s he case o a 2D ma e ial. Suppose ha we ha e
cons uc ed hyb id Wannie unc ions in he di ec ion
1
and ha he Wilson loop
winds o e a loop along
k2
by
2π
. This implies ha he Che n numbe
C= 1
and
ha he Wannie cen e s a e shi ed by a la ice ec o
R1
, which means ha he e is
cha ge lowing in ha di ec ion. A consequence o his is ha , when he sys em is cu
pe pendicula o
R1
, he e should be cha ge accumula ing in he su ace. Howe e ,
we know ha his is no possible since he cha ge in he su ace does no change.
A e all, he Hamil onian is he same a he base poin o he loop so he con ibu ion
o he bands o he su ace cha ge canno a y.
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
The pa adox is esol ed by he p esence o a pai o s a es, one a each su ace, in
he gap o he insula o . While one o he s a es injec s one cha ge in he su ace om
he bulk, he o he one mus ex ac he same cha ge om he su ace o he bulk.
Mo eo e , hese s a es a e chi al: hey mo e in opposi e di ec ion. This is equi ed by
he b eaking o ime- e e sal, as he TR-pa ne s a e mo ing in he opposi e di ec ion
canno exis on he same su ace. These s a es a e esponsible o he quan ized Hall
conduc i i y
σQHAI =Ce2
h,(4.81)
whe e
h
is he Planck cons an . This is an example o he bulk-bounda y co e-
spondence: he p ope ies o he bulk ma e ial ema kably equi e ea u es on he
su ace.
Sys ems in 3D a e ac ually cha ac e ized by a iple o Che n numbe s
(C1, C2, C3)
,
co esponding o he e olu ion o hyb id Wannie cha ge cen e s in each o he di-
ec ions in eal space. They can be ega ded as weakly coupled laye s o 2D QAHIs
s acked in he di ec ion o he Che n iple . Fo example, a Che n insula o wi h
(0,0, C)
displays a Hall conduc i i y as in Eq.(4.81) when made ini e in
(R1, R2)
. In
his pic u e, he edge s a es o e e y 2D laye combine o o m s a es ha a e ex ended
o e he su aces o he ull c ys al.
To conclude, we p o ide a heu is ic explana ion o he bulk-bounda y heo em.
Rega ding acuum (o ai , o all in en s and pu poses) as a i ial insula o wi h
anishing Che n numbe , he in e ace be ween a Che n insula o in such en i onmen
mus p esen some gap closing. This is because opological in a ian s can only
a y upon closu e and e-opening o an ene gy gap. Since when mo ing om he
opological insula o o he acuum he Che n numbe a ies, he e mus be gapless
edge s a es ha a e loca ed p ecisely in he gap.
Quan um Spin Hall insula o s
While Che n insula o s canno exis when TRS pe sis s, i u ns ou ha he e ano he
classi ica ion o TR-in a ian insula o s wi h a di e en opological index. Conside
a Hamil onian
H
o a 2D sys em o spin ul elec ons.
H
can be decomposed in o spin-
up and down pa s, wi h coupling e ms be ween he wo spin alues co esponding
o SOC
H=H↑∆†
∆H↓.(4.82)
Suppose now ha he sys em is a combina ions o wo subsys ems, one o each spin
o ien a ion, such ha each subsys em o spinless e mions has a Che n numbe
C↑,↓
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4.5. Applica ion o c ys alline sys ems o elec ons
E
R R R
a) b) c)
Figu e 4.5: Ske ch o bands in a TR-in a ian , one-dimensional insula o be ween
wo TRIMs
Γ
and R.. a) Quan um spin Hall insula o b) T i ial insula o wi hou
SOC c) Same i ial insula o wi h SOC. The blue and g een egions ep esen he
alence and conduc ion bands, espec i ely. The a ows o e he bands indica e he
spin componen .
when hey a e decoupled. As a whole,
H
, wi h coupled spins, mus be TR-in a ian
since TRS in e changes
H↓
and
H↑
while also ans o ming spin-up elec ons o spin-
down. The e o e, we mus ha e ha
CT=C↑+C↓= 0
, such ha
C↑=−C↓
. This
means ha no ne cha ge can low on he bounda y o he 2D sys em, since spin-up
and down elec ons mo e in opposi e di ec ions due o he opposi e Che n numbe .
Howe e , we say ha he e is a spin cu en along he edge, co esponding o wha
we call he quan um spin Hall e ec (QSHE), which is quan ized and p o ec ed
unless he e is a gap closing because C↑,↓canno change con inuously.
To de ine he opological in a ian iden i ying he quan um spin Hall insula o s
(QSHIs), we need o ake in o accoun he ac ion o TRS, deno ed
Θ
, on Bloch s a es.
Recall om Chap e 2 ha he Bloch Hamil onian sa is ies
ΘHkΘ−1=H−k(4.83)
and ha he ac ion o TRS on s a es is
Θ|ψ↑
k⟩=|ψ↓
−k⟩,
Θ|ψ↓
k⟩=−|ψ↑
−k⟩.(4.84)
F om his, we see ha TR-pa ne s a es a e degene a e a TRIMs, whe e
k
is equi -
alen o
−k
, e en wi h SOC. This en o ces wo ypes o connec i i y o he bands
be ween wo TRIMs, as ske ched in Figu e 4.5 o a 1D insula o . In any case, TRS
equi es ha e e y s a e a any TRIM is doubly degene a e, o cing a connec ion o
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
wo bands a hose poin s. Figu e 4.5a ep esen s a QSHI, whe e a TR-p ese ing pe -
u ba ion can mo e he pai s up and down in o he conduc ion and alence bands (in
g een and blue), bu a single in-gap s a e always emains. The emaining TR-pa ne
edge s a e is along he pa h ela ed o
Γ−R
by TRS. In Figu e 4.5b, an insula o
wi hou SOC and
C↓=−1, C↑= 1
is depic ed. No ice ha spin-up ( ed a ows) and
down (blue) edge s a es c oss wi hou coupling because he e is no SOC and
Sz
is
a good quan um numbe . Finally, in Figu e 4.5c, we ac i a e SOC and he wo spin
o ien a ions a e coupled. Howe e , his is a i ial insula o because pe u ba ions
ha espec TRS, which keep s a es pai ed up a TRIMs, can push he bands o he
conduc ion and alence egions such ha he e is no in-gap s a e le .
QSHIs a e he e o e desc ibed by a
Z2
index, since adding a second pai o edge
s a es i ializes he opology due o hyb idiza ion o he spins ia SOC, as opposed o
Che n insula o s ha can display an unbounded numbe o modes and a e desc ibed
by a
Z
in a ian . A isual way o compu e he
Z2
numbe is o ace an ho izon al line
om
Γ
o
R
and coun he numbe o edge s a es i c osses. I he e is any line ha
cu s an e en numbe o bands, hen he insula o is i ial. Fo in e sion-symme ic
c ys als, Fu and Kane [12] ound ha he opological index can be compu ed by
knowing he pa i y
ξm
o he Bloch s a es
|um(k)⟩
o e e y band
m
a all he TRIMs
Λaas
(−1)ν=Y
aY
m
ξm(Λa).(4.85)
The same discussion we ha e used o he Hamil onian can be done o he Wilson
loop ope a o s. In pa icula , he
ν
index can be compu ed by calcula ing he low o
Hyb id Wannie unc ions o he occupied bands o he 2D sys em o e hal he BZ
and coun ing he numbe o WL lines c ossed by an imagina y ho izon al line.
In 3D c ys als, Fu, Kane and Mele [65] demons a ed ha he e a e ou in a ian s
desc ibing he opological p ope ies o he QSHIs, which we a ange in he o m
o a ou - uple
(ν0,;ν1, ν2, ν3)
. The six
ki= 0,1/2
planes in he BZ a e e ec i ely
TR-in a ian sys ems ha can be assigned a
Z2
in a ian . I u ns ou ha only ou
o hem a e independen due o K ame s’ degene acy. The co esponding o mulas o
in e sion eigen alues a e
(−1)ν0=Y
aY
m
ξm(Λm),
(−1)νi=Y
bY
m
ξm(Λb(ki= 1/2)),(4.86)
whe e
Λb(ki= 1/2)
a e he TRIM poin s on he plane
ki= 1/2
. Equi alen ly, one can
compu e he low o Wannie cen e s on he planes
ki= 0,1/2
and do he ho izon al
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4.5. Applica ion o c ys alline sys ems o elec ons
line cons uc ion o ob ain he indices
ν′
i, νi
, espec i ely. This esul s in six indices
ha a e ela ed as
(−1)ν0= (−1)ν1+ν′
1= (−1)ν2+ν′
2= (−1)ν3+ν′
3,(4.87)
which o ces
νi, ν′
i
o be di e en when
ν0= 1
and equal i
ν0= 0
. Any insula o
wi h a leas one non-ze o index is conside ed non- i ial. When
ν0= 1
, ega dless
o he alues o
νi
, he QSHI is called a s ong opological insula o , while c ys als
wi h a leas one
νi= 1
and
ν0= 0
a e called weak. Weak opological insula o s
can be conside ed as a se ies o 2D quan um spin Hall laye s s acked in he di ec ion
o he
(ν1, ν2, ν3)
iple . Co espondingly, hese show su ace s a es only on selec
ace s. Fo example, a c ys al wi h iple
(0,0,1)
has bounda y s a es on he
x
and
y
su aces. The e is no such laye cons uc ion o s ong QSHIs, which show su ace
s a es on all he aces.
Finding Wannie unc ions om DFT calcula ions
The compu a ion o a maximally localized Wannie basis om DFT can be ca ied
ou in a s aigh o wa d way hanks o he Wannie 90 [66] so wa e. He e, we will
b ie ly desc ibe he p ocedu e ollowed by Wannie 90, wi h a ocus on he p ac ical
poin s when pe o ming ou own calcula ions.
The ole o he ab-ini io is o compu e he Bloch eigens a es
|ψmk⟩
. Howe e ,
hese eigens a es will no be op imally smoo h in
k
, o which we ha e o ind a
k
dependen uni a y ans o ma ion ha will i on ou he gauge h oughou he BZ,
ob aining a new se
|˜
ψnk⟩=Uk
mn|ψmk⟩.(4.88)
This op imally smoo h se is he Fou ie ans o med o ob ain he maximally local-
ized Wannie basis. The pu pose o Wannie 90 is o ind he
Uk
ans o ma ion by
minimizing he sp ead σo he WFs, de ined as
σ2=X
n⟨wn0( )| 2|wn0( )⟩−|⟨wn0( )| |wn0( )⟩|2.(4.89)
The sp ead has a gauge-in a ian componen
σ2
I
and ano he componen
˜σ2
which
can be educed by a sui able choice o Uk
σ2=σ2
I+ ˜σ2=σ2
I+σ2
D+σ2
OD,(4.90)
whe e
σ2
D
and
σ2
OD
a e he “diagonal” and “o -diagonal” componen s. In a ealis ic
DFT calcula ion, we may no ind an isola ed se o bands o , i i exis s, his se may be
o med by a la ge numbe o s a es. Wannie 90 implemen s o his cases he so called
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4. Topology in condensed ma e sys ems and i s geome ical
in e p e a ion
“disen anglemen p ocedu e”. The me hod equi es ha we se an ene gy window o
selec he s a es we wan o include in he calcula ion o a o al o
Nk
s a es a each
k
poin . I we wan o ob ain
N
WFs, he p og am will ind a ec angula
Nk×N
ma ix Uk
dis ha minimizes σ2
Iinside he window, ob aining a se o op imal s a es
|˜unk⟩=
Nk
X
m=1
Uk
dis|umk⟩.(4.91)
Fu he mo e, we can also selec an inne “ ozen window” whe e he s a es a e in-
cluded as-is, which helps accu a ely ep oduce he band s uc u e inside ha ene gy
ange and is usually se o be na ow and cen e ed a ound he Fe mi le el.
In p ac ice, wha we need o p o ide is a se o ial localized unc ions
|ϕn⟩
. When
no be e chemical in ui ion is a ailable (e.g. hyb id o bi als), he way o selec hese
o bi als is o include he a omic o bi als wi h he la ges con ibu ion a ound he
Fe mi le el. F om his, Wannie 90 will compu e
•P ojec ion o he Bloch eigens a es on o he localized unc ions
Ak
mn =⟨ψmk|ϕn⟩.(4.92)
•An o e lap ma ix o he cell-pe iodic pa o he Bloch s a es
Mk,b
mn =⟨umk|unk+G⟩.(4.93)
Apa om his, we will selec he ou e and ozen ene gy windows and a su icien
numbe o s eps bo h in he disen anglemen and gauge-smoo hing p ocedu es. I is
usually bene icial o con e ge an addi ional se o bands abo e and below he selec ed
se and comple ely dis ega d s a es which a e e y a isola ed and e y a in ene gy.
Compu ing he Be y phase nume ically
The compu a ion o he Wannie basis allows us o exp ess he Hamil onian in e ms
o ansi ion ampli udes be ween localized o bi als, ha is, a igh -binding (TB) model.
In p ac ical calcula ions o Be y phases and cu a u es, we will use his kind o
app oxima ion since i is compu a ionally mo e ac able. This is because we need
a ine
k
-g id o ob ain a e sion o he Be y phase using a disc e ized e sion in
ecip ocal space.
Fo simplici y, conside i s he case o an isola ed band wi h cell-pe iodic eigen-
s a es
|unki⟩
along a closed pa h
C
disc e ized in
N
wa e ec o s
ki
. Fo wo consec-
u i e poin s which a e su icien ly close
Im ⟨umki|unki+1 ⟩(4.94)
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4.5. Applica ion o c ys alline sys ems o elec ons
gi es a no ion o he phase di e ence be ween he s a es a
ki
and
ki+1
. The e o e,
we can de ine he Be y phase along Cbu he ollowing p oduc o o e laps
ϕ=−Im ln ⟨unk0|unk1⟩⟨unk1|unk2⟩···⟨unkN−1|unk0⟩,(4.95)
since
Im ln  eiϕ=ϕ
. In he mul i-band case, we ha e o calcula e he o e lap ma ix
Mki,ki+1
mn =⟨umki|unki+1 (4.96)
and we ob ain he ollowing o mula o he aced holonomy
ϕ=−Im ln Y
i
de Mki,ki+1 .(4.97)
Finally, o app oxima e he Be y cu a u e, we compu e he eigens a es in a su i-
cien ly inely-g ained
k
-g id. Gi en na u e o he cu a u e as he local con ibu ion
o he holonomy, we can calcula e i by compu ing he Be y phase along small loops
in he g id and di iding i by he a ea o he loop.
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CHAPTER 5
Topology om eal-space symme y:
Magne ic Topological Quan um
Chemis y
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
In Chap e 2 we in oduced he s uc u e and ep esen a ion heo y o he Shub-
niko space g oups desc ibing he symme y o c ys alline solids. Conside ing i s
applica ion o he elec onic band s uc u e, one o he key akeaways was ha eigen-
s a es o he Hamil onian a a gi en
k
poin ans o m as an i educible ep esen a ion
o he li le g oup o
k
. Mo eo e , a
d
-dimensional i ep co esponds o a
d
- old se o
s a es wi h he same degene a e ene gy. In Chap e 5, we showed ha an isola ed
se o bands can be ep esen ed by a ec o con aining he mul iplici y o each i ep
a e e y high-symme y
k
poin in he BZ om he poin o iew o an analysis o
symme y-indica ed opology. This ema kable ac implies ha , in many cases, we
can cha ac e ize he opology o sys em by using algeb aic me hods ins ead o mo e
demanding calcula ions such as Wilson loops (as in Chap e 4).
Gi en he abo e and conside ing ha nume ical me hods, in pa icula DFT, a e
necessa y o compu e ealis ic ma e ials, any way o au oma ically iden i y he sym-
me y con en and he subsequen opological analysis o elec onic band s uc u es
ob ained om ab-ini io me hods is ex emely use ul. Mo eo e , he p oblem is na u-
ally well sui ed o a compu e implemen a ion since, as we will see, we only ha e
o compu e he ace o symme y ope a ions o e subspaces co esponding o he
same ene gy, which is in essence a enso ope a ion. Au oma ion o he ask, howe e ,
equi es a big e o by i sel . In pa icula , an applica ion which is easy o use and
gene al o any c ys al has o ake in o accoun ha he DFT calcula ion can be done
in an a bi a y uni cell, while he symme y ables a e usually exp essed in he
con en ional se ing.
In his chap e , we p esen a ool ha s eamlines his p ocess in a use - iendly
way and which we ha e enhanced o wo k wi h MSGs and analyze he opology in
e ms o EBR decomposi ion and symme y indica o s.
6.1 Compu ing he symme y o eigens a es om
DFT calcula ions
i ep
[51] is a Py hon package whose main pu pose is o calcula e he symme y
con en o he elec onic bands compu ed by a DFT plane-wa e so wa e in a simple
and easy- o-use way. One o i s s eng hs in compa ison o simila p og ams is ha
he co ec symme y SG is au oma ically compu ed gi en he inpu s uc u e o he
calcula ion. O he solu ions equi e he DFT calcula ion be pe o med in a pa icu-
la se ing which ag ees wi h he s anda d uni cell in he some SG i ep da abase.
Howe e ,
i ep
can be used in any se ing and all he symme y ope a ion ans o -
ma ions a e compu ed au oma ically o ma ch hem wi h he symme y ables o he
Bilbao C ys allog aphic Se e (BCS) [52]. The consis en use o he BCS no a ion
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6.1. Compu ing he symme y o eigens a es om DFT calcula ions
also p e en s any con usion ega ding he labeling o he i eps, denomina ion o he
space g oups and uni -cell se ing, o example. Ano he key ea u e is ha
i ep
can be in e aced wi h some o he mos popula plane-wa e DFT sui es, such as VASP,
Abini and Quan umEPRESSO, as well as Wannie 90 o he analysis o in e pola ed
igh -binding models in a Wannie basis calcula ed om ab-ini io me hods.
Howe e ,
i ep
was p e iously unable o accoun o he symme y p ope ies o
c ys als displaying magne ic o de ing. This has limi ed i s applica ion o he analysis
o e omagne ic, an i- e omagne ic and e imagne ic sys ems, which ha e been
shown o be a p omising pla o m o he s udy o opological phases [67]. Mo eo e ,
many compounds wi h elemen s con aining alence
d
and
o bi als display some
kind o g ound-s a e, spon aneous magne iza ion [68
–
70]. This leads o an mo e com-
plex classi ica ion o he c ys al symme y in 1621 MSGs, compa ed o he 230 egula ,
TR-in a ian SGs. One can also a gue ha he co ec analysis o pa amagne ic phases
wi h SOC is bes done using MSG heo y, since TRS can be be e unde s ood in he
con ex o g ay o Type-I MSGs.
i ep
is packed wi h all he symme y ables o
high-symme y poin s equi ed o a MTQC analysis o all MSGs ob ained om he
BCS [25,26], enabling he local analysis wi hou any ex e nal dependencies.
Fu he mo e, p e ious i e a ions o he p og am whe e limi ed o only he compu-
a ion o he li le-g oup i eps a high-symme y poin s and only limi ed opological
analysis was a ailable. This upda e in oduces he capabili y o compu e all he
symme y indica o s o all MSGs, gi en in he so-called physical basis and hus wi h
well known in e p e a ion. Addi ionally, one can check he opological classi ica ion
o a gi en se o elec onic bands, compu ing i s decomposi ion in o EBRs [15] when
he sys em displays i ial o agile opology.
Wo king wi h a plane-wa e basis se
Many DFT sui es such as VASP, Abini and Quan umESPRESSO use a plane wa e
basis o ep esen he elec onic wa e unc ions. In pa icula , he eigens a es
1|ψnk⟩
o
H(k)
a a gi en ecip ocal space poin a e expanded in a se ies o plane wa es
|k+G⟩|ψnk⟩=X
G
Cnk(G)|k+G⟩(6.1)
whe e we impose a
G
cu o o a maximum plane-wa e ene gy o
Ecu
such ha
ℏ2(k+G)2/2m < Ecu
. The ac ual cu o used by ou p og am need no be he
1
When using PAW pseudo-po en ials, we ob ain he expansion o he pseudo wa e unc ions
ep esen ing he s a es ou side he co e. Howe e , since hey a e ela ed o he ue wa e unc ion by
a linea ans o ma ion ha commu es wi h he symme y ope a ions, hei ans o ma ion p ope ies
a e he same.
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
calcula ion cu o . In gene al,
Ecu ∼50
eV is enough o calcula e he aces o
symme y ope a ions wi hou e o s. To exp ess he ans o ma ion o he eigens a es
using he plane-wa e coe icien s, we no e ha an ope a ion
g={R| }
ac s on
|k+G⟩as
g|k+G⟩=e−iR(k+G)· |R(k+G)⟩.(6.2)
I ollows ha he ace can be compu ed as
⟨ψnk|g|ψnk⟩=X
G,G′
C∗
nk(G′)Cnk(G)⟨k+G′|g|k+G⟩
=X
G,G′
C∗
nk(G′)Cnk(G)e−iR(k+G)· ⟨k+G′|g|R(k+G)⟩
=X
G,G′
C∗
nk(G′)Cnk(G)e−iR(k+G)· δG′,Rk−k+G
=X
G,G′
C∗
nk(Rk−k+G)Cnk(G)e−iR(k+G)· ,
(6.3)
whe e we ha e used he o hono mali y condi ion
⟨k′+G′|k+G⟩=δG′,k−k′+G.(6.4)
When using spino wa e unc ions
|ψσ
nk⟩
whe e
σ
is he spin componen , an addi ional
ans o ma ion o he spin pa o he eigens a e mus be aken in o accoun
⟨ψnk|g|ψnk⟩=X
σ,σ′⟨ψσ
nk|g|ψσ′
nk⟩.(6.5)
The
SU(2)
ma ix
S(g)
can be ob ained by he mapping om he eal-space ope a ion
S({R| }) =
1
2cos(θ
2)−i(ˆn ·σ) sin(θ
2),(6.6)
whe e
1
2
is he
2×2
iden i y ma ix,
σ
is he ec o o Pauli ma ices,
ˆn
is he axis
o
R
and
θ
i s angle. The e o e, o e e y
k
poin , he p og am g oups eigens a es
by iden ical (up o a h eshold) ene gy and compu es he ace o all uni a y cose
ep esen a i es in he li le g oup o
k
and he co esponding i eps can be compu ed
by using Equa ion 2.66.
Handling changes o basis be ween di e en uni cells
One o he s eng hs o
i ep
is ha i allows he use o choose any uni cell o
he calcula ion o elec onic eigens a es. This is especially use ul since in o ma ion
abou he symme y ope a ions and ep esen a ions o space g oups is con en ionally
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6.1. Compu ing he symme y o eigens a es om DFT calcula ions
a1
a2
a3
p
x' x
c1
c2
c3
Figu e 6.1: Ske ch o wo axes se ings ela ed by a ma ix
P
and shi o o igin
p
.
The same poin is exp essed in di e en coo dina es x′in he cise ing and xin ai.
de ined wi h axes o ming a non-p imi i e cell, while he calcula ions a e usually done
in he p imi i e uni cell, which shows he ue pe iodici y o he la ice and equi es
a smalle basis due o he smalle numbe o a oms. This means ha a use - iendly
p og am mus ma ch he in o ma ion (ma ix ep esen a ions, aces,
k
poin s, e c.)
om he DFT calcula ion, which depends on how he use de ined he calcula ion,
wi h he symme y ables, which is ixed o he con en ional se ing.
Conside he calcula ion cell wi h ec o s
(a1,a2,a3)
de ining he uni cell and
he basis ec o s o he con en ional se ing o he symme y ables
(c1,c2,c3)
. The
wo cells a e ela ed by a linea ans o ma ion Pas
(c1,c2,c3) = (a1,a2,a3)·P, (6.7)
whe e
P
is in e ible. The e can also be a shi o o igin be ween he cells gi en by
he ec o
p=Oc−Oa,(6.8)
whe e
Oc
and
Oa
a e he o igins o he con en ional and calcula ion cells, espec i ely.
The exp ession o a eal-space poin in bo h se ings is
=x′
ici=x′
iPjiaj+pjaj=xjaj,(6.9)
We can exp ess i in ma ix no a ion as


x1
x2
x3
=P·

x′
1
x′
2
x′
3
+

p1
p2
p3
.(6.10)
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
The in e se ans o ma ion is (P, p)−1= (P−1,−P−1p)and, consequen ly


x′
1
x′
2
x′
3
=P−1·

x1
x2
x3
−P−1

p1
p2
p3
.(6.11)
Upon a cell ans o ma ion, he ecip ocal la ice ec o s a e only a ec ed by he
ans o ma ion
P
. The ec o s
qi
co esponding o he s anda d cell mus be ela ed
o he kio he calcula ion cell by some ye - o-be-de e mined ans o ma ion
qi=Ajikj.(6.12)
The key o iden i y Ain e ms o Pis ha bo h bases mus sa is y
ki·aj=qi·cj= 2πδij.(6.13)
Using he eal-space basis ans o ma ion
qi·cj=X
n,m
Amikm·anPnj = 2πX
m
AmiPmj = 2πX
m
[A ]imPmj = 2πδij.(6.14)
This implies ha
A= [P ]−1
. I is s aigh o wa d o see ha , i he bases a e ela ed
by a pu e o a ion (wi h a possible ansla ion) hen
A=P
. This is no he case o
ans o ma ions be ween uni cells o di e en olume, which a e indeed possible.
The ans o ma ion o ecip ocal space coo dina es o a ec o
l=y′
iqi
is he e o e
gi en by


y1
y2
y3
=A·

y′
1
y′
2
y′
3
= [P ]−1·

y′
1
y′
2
y′
3
.(6.15)
The ep esen a ion o a symme y ope a ion
{R| }
also changes om he s anda d
o he calcula ion cell. The ac ion on a eal-space posi ion exp ess in he s anda d
se ing is
˜
={R′| ′} = (R′
ijx′
j+ ′
i)ci= ˜x′
ici.(6.16)
F om he poin o iew o he calcula ion cell, he same ans o ma ion is exp essed as
˜
= (Rijxj+ i)ai= ˜xiai.(6.17)
Deno ing he ec o s o coe icien s in bo h cells as ˜x′and ˜x, we ha e
˜x′=R′·x′+ ′,˜x =R·˜x + .(6.18)
Since ˜x′and ˜x mus ag ee upon cell ans o ma ion, we ha e
˜x =P˜x′+p=P[R′·x′+ ′]+p=P·R′·x′+P· ′+p
=P·R′(P−1·x−P−1·p) + P· ′+p=P·R′·P−1·x
−P·R′·P−1·p+P· ′+p=R·x+ ,
(6.19)
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6.1. Compu ing he symme y o eigens a es om DFT calcula ions
which means ha
{R| }={PR′P−1|P· ′+p−PR′P−1·p}.(6.20)
This exp ession implies ha a symme y wi h only ac ional ansla ion in one se ing
may be a combina ion o a non-symmo phic ope a ion and a la ice ansla ion in he
o he , which has meaning ul e ec s o poin s o he han Γ.
Spin
SU(2)
o a ions equi e special a en ion due o he one- o- wo mapping om
eal-space ope a ions o spino space. By inspec ing Equa ion 6.6, we see ha he e
is eedom o choose he axis
ˆn
and angle
θ
o o a ion. The p oblem is exace ba ed
by he change o sign by a shi in he angle
θ→θ+ 2π
, which also means ha
wo old o a ions h ough
π
and
−π
a e di e en . All in all, his means ha he e can
be a minus sign be ween he ma ix in he symme y ables and he ones gene a ed
by inding
ˆn
and
θ
p og ama ically and applying Equa ion 6.6. To ci cum en his
p oblem, we implemen ed he ollowing me hod:
1.
Gi en he known eal-space o a ion ma ices o all Shubniko g oups, we
compu e a
SU(2)
ma ix using Equa ion 6.6 axis
ˆn
compu ed om he eal-
space ma ix and angle
θ
. In he BCS, he
SU(2)
ma ix is exp essed in he spin
axes se ing, which is di e en om he con en ional cell se ing. We ound
ha all SGs in he igonal and hexagonal c ys al sys ems sha e he same spin
axes, while he i e emaining sys ems sha e ano he di e en se o axes (see
Table 6.1). This means ha we can gene a e he
SU(2)
ma ix o any ope a ion
in he ables up o a sign, due o he a bi a y choice o ±ˆn and θ.
2. T ans o m he axis om he con en ional cell o he calcula ion cell using he
change o basis
Mca =P−1
which is au oma ically ound o gi en by he use
ˆna=Mca ·ˆn.(6.21)
I
|Mca|>1
, we ha e o no malize he axis. We also impose he cons ain
|Mca|>0so he handedness o he axes does no change.
3.
Change he axis
ˆna
om he calcula ion cell axes o Ca esian coo dina es, since
implemen ed DFT sui es exp ess spin in his con en ion. The ans o ma ion
ma ix
Mae
is simply
(a1,a2,a3)
and ano he no maliza ion o he axis
ˆne=
Maeˆnamay be equi ed.
4.
Find he
SU(2)
ma ix in Ca esian coo dina es using
ˆne
and
θ
in Equa ion 6.6
and mul iply by he sign ound in s ep 1. This ensu es ha he exac same
o a ion as in he symme y ables is pe o med in he calcula ion se ing.
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
igonal and hexagonal es
s1(0,−1,0) (1,0,0)
s2(−√3
2,1
2,0) (0,1,0)
s3(0,0,−1) (0,0,1)
Table 6.1: Spin axes used in he BCS o he
SU(2)
ma ix depending on he c ys al
sys em.
6.2 New unc ionali ies o i ep
Analysis o magne ic-g oup symme y
Wi h hese new e o s,
i ep
now suppo s c ys al displaying magne ic o de ing,
allowing he use o au oma ically de ec he co ec Shubniko SG and analyze he
symme y p ope ies o he ene gy eigens a es.
The use is equi ed o c ea e a plain- ex ile wi h he magne ic momen s o all
he a oms in he uni cell, a ow wi h h ee numbe s (in Ca esian coo dina es) o
each one. i ep can be p omp ed o use MSG symme y as ollows
>> i ep -magmom pa h/ o/magne ic/momen a/ ile [o he op ions]
Wi h his new op ion,
i ep
will de ec he MSG o he inpu s uc u e (e.g. om a
POSCAR
ile) and magne ic momen a by using he
spglib
[71,72] lib a y, e ie ing
he symbol and numbe in he BNS con en ion. Al hough i ep iden i ica ion only
equi es he uni a y symme ies,
i ep
will also lis al he an i-uni a y ope a ions
in bo h he DFT calcula ion and s anda d se ings, acco ding o he BCS.
This new capabili y equi es he ull lis o li le-g oup co eps a all high-symme y
poin s o all 1,651 Shubniko space g oups. This in o ma ion is a ailable a he BCS
in he COREPRESENTATIONS
2
applica ion. The new i ep ables a e made a ailable
locally and packed in a companion package, which is au oma ically ins alled when
i ep is downloaded.
---------- INFORMATION ABOUT THE SPACE GROUP ----------
Space g oup Cm'cm'(# 63.464) has 4 uni a y symme y ope a ions
### 1
o a ion : | 1 0 0 | o a ion : | 1 0 0 |
| 0 1 0 | ( e UC) | 0 1 0 |
2h ps://www.c ys .ehu.es/cgi-bin/c ys /p og ams/
co ep esen a ions.pl
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6.2. New unc ionali ies o i ep
| 0 0 1 | | 0 0 1 |
gk = [kx, ky, kz] | e UC: gk = [kx, ky, kz]
spino o . : | 1.000+0.000j 0.000+0.000j |
| 0.000+0.000j 1.000+0.000j |
spino o . ( e UC) : | 1.000+0.000j 0.000+0.000j |
| 0.000+0.000j 1.000+0.000j |
ansla ion : [ 0.0000 0.0000 0.0000 ]
ansla ion ( e UC) : [ 0.0000 0.0000 0.0000 ]
axis: [0. 0. 1.] ; angle = 0 , in e sion : False
Since DFT p og ams may no ake in o accoun he ull MSG, we ecommend
adjus ing he
-degenTh esh
pa ame e , which con ols he minimum di e ence
be ween wo ene gies o conside hem degene a e.
MTQC analysis and decomposi ion o DFT bands in o EBRs
i ep
now includes all he symme y ec o s o all he EBRs a maximal
k
poin s
o he 1,651 Shubniko SGs, as a ailable in MBANDREP
3
applica ion o he BCS.
Addi ionally, he Smi h no mal o m o e e y EBR ma ix is s o ed as pa o he
da ase .
The EBR analysis can by ac i a ed using he
–eb -decomposi ion
command
line lag, equi ing he iden i ica ion o he i eps a all maximal
k
poin s in he same
i ep
call. The se o bands o analyze is es ic ed ia he
-IBs a
and
-IBend
lags. The p og am will do he ollowing:
1.
Find he symme y ec o o he inpu bands and check i he e a e in ege
solu ions o he EBR decomposi ion p oblem using he Smi h no mal o m. This
allows he iden i ica ion o s ong opological ma e ials.
2.
I he e exis s an in ege solu ion, some possible linea combina ions o EBRs
will be compu ed. This co e s bo h he i ial and agile opology cases.
The compu a ion EBR decomposi ion is ound using he openly-a ailable OR-Tools
[73] package, cas ing he ask in o a cons ained p og amming Boolean sa is iabili y
(CP-SAT) p oblem. I i s looks o linea combina ions wi h all-posi i e coe icien s.
I none is ound, he sub ou ine will elax he cons ain s o allow nega i e alues.
Among he many iable solu ions, we chose o show hose wi h smalle coe icien s.
3h ps://www.c ys .ehu.es/cgi-bin/c ys /p og ams/mband ep.pl
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
Sea ch o ans o ma ion o con en ional cell
P e ious e sions o
i ep
equi ed ha he DFT calcula ion (in he
(a1,a2,a3)
basis) be done di ec ly in he con en ional cell (wi h ec o s
(c1,c2,c3)
) o ha use s
p o ide he co ec cell ans o ma ion ia he
- e UC
and
-shi UC
command-line
a gumen s such ha


c1
c2
c3
= e UC ·

a1
a2
a3
.(6.22)
The se ings may also be ela ed by a change o o igin
shi UC = Ocon −ODF T
.
This new e sion o
i ep
can ind bo h he o a ion and ansla ion ela ing he
calcula ion and con en ional cell, which is equi ed o he i ep iden i ica ion since
he ables a e w i en in he la e se ing. This can be done wi h he a gumen
-sea chcell .
I p omp ed o do so,
i ep
will use he o a ion p o ided by
spglib
and y
o ma ch he symme y ope a ions agains he i ep ables. I i does no wo k, he
p og am can s ill ind he co ec ans o ma ion by conside ing he ollowing wo
cases:
•
I he space g oup is cen osymme ic, he o igin o he p imi i e cell is placed
in he in e sion cen e .
•
I i does no ha e in e sion symme y, di e en cen e ing ansla ions a e ied
o he co esponding c ys al sys em.
High-symme y poin s in con en ional cell
Since he p og am allows he use o any compu a ion cell, his also has an impac
in he ecip ocal space coo dina es o he high-symme y poin s. Mo e speci ically,
he coo dina es o a ecip ocal space ec o
G
w i en in he con en ional cell ecip-
ocal basis
(q1,q2,q3)
a e usually no he same in he calcula ion ecip ocal basis
(k1,k2,k3)
. This is impo an since he high-symme y poin coo dina es o he
calcula ion mus ma ch wi h hose o he ables upon ans o ma ion, indica ing ha
he co ec kpoin s whe e compu ed in he calcula ion cell.
To accoun o he ela ion be ween ecip ocal bases in Equa ion 6.14,
i ep
now includes he p e-p ocessing a gumen
–p in -hs-kpoin s
o display he high-
symme y poin s in bo h he DFT and con en ional cells:
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6.2. New unc ionali ies o i ep
########## High-symme y k poin s ##########
Reading symme ies om ables o SG 63.464
Change o coo dina es om con en ional o DFT cell:
| 0.50 -0.50 0.00 |
| 0.50 0.50 0.00 |
| 0.00 0.00 1.00 |
Change o coo dina es om DFT o con en ional cell:
| 1.00 1.00 0.00 |
| -1.00 1.00 0.00 |
| 0.00 0.00 1.00 |
Coo dina es in symme y ables:
GM : 0.000000 0.000000 0.000000
R : 0.500000 0.500000 0.500000
S : 0.500000 0.500000 0.000000
T : 1.000000 0.000000 0.500000
Y : 1.000000 0.000000 0.000000
Z : 0.000000 0.000000 0.500000
Coo dina es o DFT calcula ion:
GM : 0.000000 0.000000 0.000000
R : 0.000000 0.500000 0.500000
S : 0.000000 0.500000 0.000000
T : 0.500000 0.500000 0.500000
Y : 0.500000 0.500000 0.000000
Z : 0.000000 0.000000 0.500000
The use can hen compu e he eigens a es a he co ec high-symme y poin s in
o de o iden i y he i eps la e . The example clea ly shows ha he coo dina es in
bo h ecip ocal bases can be di e en .
Calcula ion o symme y indica o s
Al hough calcula ion o he SIs h ough he EBR ma ix is possible as hey a e all
a ailable in
i ep
, we choose o implemen he SI o mulas in he physical basis
o all single and double SG s and MSGs. To enable his unc ionali y, one mus
add he lag
–symme y-indica o s
o he op ions used o iden i y he i eps a all
maximal
k
poin s. We ha e s o ed all he SI o mulas in e ms o i ep mul iplici-
ies as ob ained om he BCS, which can be used once he iden i ica ion o i eps
is done. We also epo he cases whe e he (M)SG does no ha e a non- i ial SI g oup.
One no e is due ega ding he se o SIs e u ned o a gi en (M)SG. In acco dance
o he in o ma ion gi en in he BCS, we also chose o epo e e y SI o he (M)SG
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6. Diagnosing band opology om ab-ini io calcula ions using he I Rep
package
Solu ion 1
1 x [ -E->G(4) @ 2a(4'3m',23) ] + 1 x [ -1Eu->G(4) @ 4b(3m',3) ] +
1 x [ -2Eu->G(4) @ 4b(3m',3) ] + 1 x [ -Eu->G(4) @ 4b(3m',3) ] +
1 x [ -1Eu->G(4) @ 4c(3m',3) ] + 1 x [ -2Eu->G(4) @ 4c(3m',3) ] +
1 x [ -Eu->G(4) @ 4c(3m',3) ] + 1 x [ -1E->G(12) @ 12 (2'2'2,2) ] +
2 x [ -2E->G(12) @ 12 (2'2'2,2) ]
...
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CHAPTER 7
Magne ic-symme y-en o ced
opological nodal lines: he cases o
Fe3GeTe2and Co
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
In Chap e 2, we desc ibed he ools o analyze he symme y o a physical sys em,
wi h an emphasis on c ys alline s uc u es, which show ansla ional and o a ional
symme y. We also showed how o ea he p esence o a local magne ic o de ing by
conside ing he ype-I, III and IV Shubniko space g oups. Chap e 4 also explained
how opology can gi e ise o cha ac e is ic esponses in he ma e ial, such as he
anomalous Hall conduc ance and how i is ela ed o ea u es in he elec onic band
s uc u e such as Weyl nodes. In his chap e , we will make ex ensi e use o hese
ools o in es iga e wo speci ic ma e ials: Fe3GeTe2and hexagonal Co.
Coinciden ally, bo h Fe
3
GeTe
2
and his phase o Co sha e he same pa en space
g oup when magne iza ion is no conside ed, which is why he wo cases o s udy a e
p esen ed in his same chap e . Al hough his acili a ed he analysis o bo h sys ems
in he i s place, each one shows unique ea u es ha should be s udied in de ail.
7.1 Mi o -p o ec ed nodal lines
Nodal lines a e closed, con inuous lines in ecip ocal space whe e wo o mo e bands
in e sec . Al hough c ossings be ween wo bands can gene ically appea in a 3D c ys-
al wi h no addi ional equi emen s (i.e. Weyl o Di ac nodes), he e is no gua an ee
ha hey will assemble in o a con inuous se o nodal poin s, which is why nodal
lines equi e some kind o symme y p o ec ion.
To see why symme y may gi e ise o p o ec ed c ossings, conside wo Bloch
s a es
|ψ(k)⟩
and
|ϕ(k)⟩
ans o ming as di e en co eps
D
and
˜
D
o he li le g oup
Gk
o
k
, which we conside o be one-dimensional o simplici y. A gi en
g∈Gk
ac s on he s a es wi h
D(g)
and
˜
D(g)
espec i ely. Because
g
is in he li le g oup,
he Bloch Hamil onian H(k)mus be in a ian
g−1H(k)g=H(gk) = H(k),(7.1)
which ollows because, by de ini ion,
gk=k+G≡k
, whe e
G
is a ec o o he
ecip ocal la ice. I ollows ha he ollowing expec a ion alue is in a ian
⟨ψ(k)|g−1H(k)g|ϕ(k)⟩=⟨ψ(k)|H(k)|ϕ(k)⟩.(7.2)
Equi alen ly, we could apply he ope a o s o he s a es ins ead o ans o ming
H(k)
,
which would yield
D(g)−1˜
D(g)⟨ψ(k)|H(k)|ϕ(k)⟩=⟨ψ(k)|H(k)|ϕ(k)⟩.(7.3)
This shows ha his ma ix elemen can only be non-ze o i
D(g)−1˜
D(g)
is he iden i y,
ha is,
|ψ(k)⟩
and
|ϕ(k)⟩
ans o m as equi alen ep esen a ions. I ollows hen
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7.1. Mi o -p o ec ed nodal lines
ha hyb idiza ion and gapping o bands be ween s a es o he di e en symme y
is o bidden. Fo Fe
3
GeTe
2
and Co, we a e in e es ed in mi o symme ies due o
he pa icula space g oup hey belong o. Fo example, conside a mi o plane
m001
pe pendicula o he
c
axis. Any poin on a plane o he o m
kP= (kx, ky, kz =
0,1/2)
is in a ian wi h espec o
m001
, which means ha
m001 ∈GkP
. In mo e
de ail, i
GkP
has wo co eps whose ace o he
m001
symme y elemen di e , we
say ha he c ossing be ween he s a es o di e en i eps is p o ec ed by he mi o
symme y. The hyb idiza ion is o bidden a all o he poin s
kP
on he plane, which
means ha wo bands can in e sec and gi e ise o a con inuous line o degene acies,
a nodal line. This is p ecisely he con igu a ion we ha e in bo h Fe3GeTe2and Co.
How o de ec mi o -p o ec ed nodal lines
In his wo k, we cha ac e ized he symme y-p o ec ed nodal lines by a me hod based
on Wilson loops (WLs). Following ou example o he
m001
mi o symme y, we
compu ed he e olu ion along pa hs on he
kP
plane o he ace o WLs compu ed
along he pe pendicula di ec ion
kz
. Due o he mi o symme y, loops o his
kind a e quan ized o ei he 0 o
π
, modulo
2π
. A discon inui y o
π
iden i ies as
symme y-p o ec ed nodal line ha canno be emo ed by a small, local pe u ba ion
ha p ese es he
m001
symme y [79]. The eason o his is ha he ace o he
WLs o he o m
W(kx, ky)
is ela ed o he numbe o s a es o posi i e o nega i e
eigen alue which a e occupied o a gi en poin on he plane. The e o e, as ske ched
in Figu e 7.1, when acking he e olu ion o he bands jus below a nodal c ossing,
he numbe o s a es wi h posi i e o nega i e eigen alue changes by one as one
c osses a nodal line.
In p ac ice, we model eal ma e ials om ab-ini io calcula ions in he amewo k
o DFT in his case. Al hough compu ing WLs and o he quan i ies ela ed o he
Be y connec ion is possible om DFT, compu ing he eigens a es in a su icien ly
dense g id o
k
poin s is ou o he ques ion, since i would make he calcula ions
ex emely cos ly. Fo his eason, we i s ind a basis o Wannie unc ions ha
accu a ely ep oduce he elec onic band s uc u e in a window a ound he Fe mi
le el using Wannie 90. This p ocedu e also p o ides an in e pola ed igh -binding
Hamil onian w i en in he Wannie basis, which o a educed numbe o bands o
in e es is mo e manageable. Ob aining he eigens a es and ene gies o a
k
poin
is he e o e educed o diagonaliza ion o he Bloch Hamil onian, which is as and
can be e y easily pa allelized. To compu e he Wannie unc ions, one speci ies a
se o s a ing o bi als o p ojec he Bloch s a es on o, usually hose wi h he la ge
con ibu ion a ound he Fe mi le el. In essence, Wannie 90 inds hen a se o Bloch
s a es om he ini ial p ojec ions ha is op imally smoo h in
k
and cons uc s he
basis se .
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
k
a) b) kx
ky
Ene gy
Figu e 7.1: Ske ch o nodal line and iden i ica ion ia WLs. a) Ene gy dispe sion whe e
he colo indica es he mi o eigen alue o he s a es. The b oken line sepa a es he
s a es acked by he WLs (below) om he es (abo e). b) Cu o BZ o a cons an
kz= 0,1
2
in ac ional coo dina es. The ci cle depic s he nodal line in (a) and he
b oken line is a di ec ed pa h along which we compu e he WLs in eg a ing in he
pe pendicula di ec ion ou o he plane. The ace o he WL unde goes a
π
jump
when i in e sec s he ci cle and, a he same ime, he numbe o s a es wi h a gi en
mi o eigen alue ( ed band in (a)) changes by one.
Upon compu ing he Wannie Hamil onian, we use a pos -p ocessing ool called
Wannie Tools [80], which enables wo calcula ions in a simple way. Fi s , we can
compu e all he c ossing poin s be ween one band and nex one h oughou he BZ,
allowing us o loca e he nodal line candida es. Secondly, we can also pe o m WL
calcula ions along any pa h in he BZ o con i m he nodal c ossings. This is necessa y
in p ac ice because he algo i hm o sea ching c ossing poin s may ail o ind all o
hem, depending on he
k
g id and speci ic ea u es o he band s uc u e. The e o e,
lines ha may seem b oken in he calcula ion may be symme y-p o ec ed nodal lines,
which a e unequi ocally de ec ed wi h his me hod.
D um-head s a es
The Wilson loop me hod has he ad an age o being able o iden i y p o ec ed su ace
s a es ha li e inside he p ojec ion o he nodal line on o he 2D su ace BZ, called
d um-head s a es [81]. The name e e s o hese s a es ex ending all o e he p ojec-
ion o he nodal line, esembling he ba e head o a d um whose hoop is p ecisely he
p ojec ion o he c ossings (i.e. inside he ci cle in Figu e 7.1b). D um-head s a es a e
pa icula ly in e es ing because hei dispe sion is usually ai ly la , which p omo es
he de elopmen o s ong co ela ions.
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7.1. Mi o -p o ec ed nodal lines
a)
b)
Figu e 7.2: Ske ch o he wo possible posi ions o he hyb id Wannie unc ion cen e s
o he nodal lines o a slab o h ee cells in he
c
-axis di ec ion. The colo ed sphe es
ep esen Fe (o ange-b own) Te (b ass) and Ge (pu ple) a oms. The ed squa e is
he posi ion o he hyb id Wannie unc ion o he occupied bands o a Wilson loop
calcula ion along
kz
.a) The ace o he Wilson loop is
ϕ= 0
and he HWF cen e
is a
z= 0
.b)
ϕ=π
and he cen e is a
z= 1/2
, which gi es ise o su ace s a es
since he su ace e mina ion is exac ly a he HWF cen e .
We ecall om Chap e 4 ha he ace o he WL ope a o along
kz
(ou o he
in a ian plane) is p opo ional o he a e age
z
posi ion o he cen e s o he hyb id
Wannie unc ions, ex ended in
x, y
and localized in
z
. The ace is quan ized o 0
o
π
, meaning ha he cha ge cen e o he bands jus below de nodal c ossing is
ei he a he o igin o a he uni cell bounda y. I , when making he c ys al ini e in
his di ec ion, we cu he s uc u e a he poin whe e he cha ge cen e is, he e will
be a dangling cha ge a he su ace, which is p ecisely he d um-head s a es. This is
shown in Figu e 7.2, whe e we ha e ske ched he wo si ua ions o Fe3GeTe2.
Ve y impo an ly, he ac ual alue o he WL ace depends on he posi ion o he
mi o plane inside he uni cell, which is con en ion dependen . Fo example, a shi
o hal he
c
axis o all a omic si es would exchange he 0 and
π
(modulo
2π
) alues
o his kind o WLs. This is p ecisely because he posi ion o he hyb id cha ge cen e
inside he uni cell depends o he e e ence o igin. Wi h his in mind, Figu e 7.2b
is he case ha shows su ace d um-head s a es in ou speci ic choice o o igin o
Fe
3
GeTe
2
, which co esponds o a WL ace inside he nodal-line p ojec ion o
ϕ=π
.
This dependence on he choice o o igin aises an impo an ques ion, since he
ace o he WL ope a o mus be gauge in a ian , as shown in Eq.4.24. A displacemen
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
o he a oms by hal he
c
axis co esponds o a gauge ans o ma ion o he Bloch s a es
by a phase, which is a uni a y ans o ma ion, leading o an appa en con adic ion.
Howe e , gauge in a iance holds o an isola ed se o bands, which is no his case
since he bands jus below he nodal c ossing a e indeed no isola ed om a leas he
i s abo e, as ske ched in Figu e 7.1a.
7.2 S udy o Fe3GeTe2
Fe
3
GeTe
2
is an i ine an e omagne wi h a Cu ie empe a u e
Tc≈220K
and a
hexagonal laye ed s uc u e [82]. The bulk ma e ial is o med by slabs o Fe
3
Ge lying
be ween laye s o Te bound by weak an de Waals o ces o o m a 3D s uc u e,
which makes i easily ex oliable [83] o g own by deposi ion [84]. The wo non-
equi alen Fe posi ions in he uni cell display a local magne ic momen o a ound
1.4µB
pe a om along he s acking
c
axis [85,86]. P e ious s udies ha e shown ha
Fe
3
GeTe
2
displays a la ge anomalous Hall conduc i i y [87], anomalous Ne ns e -
ec [88] and non-linea quan um Hall e ec [89], which has d awn he a en ion
o hei applicabili y in he de elopmen o no el spin onic de ices. I s magne ic
p ope ies ha e been s udied in-dep h [90
–
94], disco e ing he possibili y o une i s
magne ic con igu a ion ia s ain [95] o applied ol age [96]. Mo eo e , o he s ud-
ies claim ha Fe
3
GeTe
2
shows Kondo la ice beha io [97] and sky mionic phases [98].
Full knowledge o he symme y o Fe
3
GeTe
2
is necessa y o unde s and i s opo-
logical p ope ies. I o e s ano he pla o m o in es iga e he in e play be ween
opology and symme y in magne ically o de ed ma e ials, since i s g ound s a e is
desc ibed by a MSG o a spin space g oup when SOC is neglec ed. The non-ze o local
magne ic momen s b eak TRS in insically, which is necessa y o some ea u es o
appea , such as Weyl nodes [99].
P e ious esea ch also shows ha Fe
3
GeTe
2
hos s one nodal line loca ed e y
nea he Fe mi le el, when SOC is neglec ed [100]. In pa icula , he band s uc u e
shows a wo old-degene a e c ossing a he
K= (1/3,1/3,0)
poin . This degene acy
ex ends along he high-symme y
P
line joining he
K
and
H= (1/3,1/3,1/2)
wa e ec o s. When SOC is aken in o accoun , he nodal line was shown o be
gapped, leading o an a oided c ossing wi h a e y small ene gy gap along which he
Be y cu a u e is concen a ed, which esul s in a la ge lux. This, in u n, induces a
la ge anomalous Hall conduc i i y (AHC), which was o bidden by he TRS in he
non-magne ic s uc u e. This mechanism was pu o wa d o explain he high AHC
epo ed in expe imen s.
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7.2. S udy o Fe3GeTe2
Symme y analysis, DFT and wannie iza ion me hods
In o de o s udy he elec onic band s uc u e o Fe
3
GeTe
2
, we pe o med DFT
calcula ions as implemen ed in he Vienna Ab-ini io Simula ion Package (VASP)
[74,75]. We used P ojec o Augmen ed Wa e [101] (PAW) pseudo-po en ials wi h he
Pe dew-Bu ke-E nzhenho [54] (PBE) implemen a ion o he Gene alized G adien
App oxima ion (GGA) o he exchange-co ela ion unc ional. Addi ionally, an de
Waals o ces we e included using he DFT-D3 me hod wi h Becke-Johnson damping
unc ion [102]. The ene gy cu -o o he plane wa e basis was se o 600eV and he
k
poin g id was a Gamma-cen e ed, egula Monkho s -Pack poin se o dimensions
11×11×7
o a uni o m densi y o poin s gi en he uni cell aken om he e e ence
li e a u e [82].
To compu e he maximally localized Wannie unc ions om DFT, we s a ed wi h
a basis o
d
o bi als o Fe and
p
o Te and Ge, wi h ozen ene gy window o 3 eV
cen e ed a he Fe mi le el. We he e o e ob ain a o al o 96 basis unc ions, aking
in o accoun he doubling due o spin.
Symme y analysis o he band s uc u e
Bands wi hou SOC
Fe
3
GeTe
2
displays a e omagne ic (FM) s uc u e whe e he magne ic momen s a
Fe si es align pa allel o he
c
axis. The space g oup symme y co esponds o SG
P63/mmc
(No.194). The Fe a oms si a ow di e en WPs labeled 2b and 4a. The
calcula ions show ha hese wo inequi alen posi ions hos a magne ic momen
pe a om o app oxima ely
1.5µB
and
2µB
, espec i ely, which is in good ag eemen
wi h he epo ed expe imen al measu emen o
1.4µB
pe a om. In he absence o
SOC (Fig.7.3a), he co ec symme y desc ip ion is gi en by a Type-1 collinea spin
space g oup (L194.1.1). This kind o g oup a ises when SOC is no aken in o accoun .
In ha si ua ion, he e a e no e ms o he ype
L·S
, which couple he spin and
eal-space o a ions, meaning ha a oms and hei magne ic local magne ic momen s
can be ans o med sepa a ely. In ha case, symme y ope a ions a e usually deno ed
by a symbol
{S|R| }
, whe e
S
ac s on spin and
R
and
a e eal-space o a ion and
ansla ion, espec i ely. The FM o de wi hou SOC espec s he con inuous o a ion
symme y o he spins abou he alignmen axis. Using he spin space g oup heo y,
one can show ha his decouples he bands in o spin-up and spin-down subse s, each
one showing he symme y o a disc e e subg oup o L194.1.1 which akes he o m
G×SZT
2,(7.4)
whe e
G
con ains only spa ial ope a ions wi h iden i y spin pa and
SZT
2
is gene a ed
by
θUx(π)
. He e,
θ
deno es TRS while
Ux(π)
is a
π
o a ion abou he
x
spin axis.
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
K M
spin
spin
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
E - EF (eV)
K M H
|K |K H
0.5
0.375
0.25
0.125
0
-0.125
-0.25
-0.375
-0.5
Sz
(ħ uni s)
a)
c) d)
b)
M
L H
K
A
ky
kz
kx
Fe
Ge
Te
Figu e 7.3: Elec onic band s uc u e along high-symme y pa hs a) wi hou SOC
and colo ed spin sec o s b) wi h SOC and
Sz
spin componen c) C ys al s uc u e as
seen om he
a
and
c
di ec ions. The a ows on he a oms ep esen he magne ic
momen s. d) BZ o he gi en s uc u e wi h he high-symme y pa h o he band
plo s ma ked in ed.
θUx(π)
is ep esen ed by
iσzK
, whe e
K
is he complex conjuga ion ope a ion. The
ope a o ac s as an analogue o TRS bu , since
(ΘUx(π))2= +1
, each sec o obeys he
symme y o a single- alued g ay g oup. The e o e, each spin sec o can be analyzed
sepa a ely as spinless e mions o a non-magne ic c ys al.
The p e iously epo ed nodal line occu s along he high-symme y line
P
joining
he poin s
K
and
H
. Since along he nodal line wo bands become degene a e, we
mus only conside 2D i eps a he endpoin s, he e o e lea ing as possible co eps
K5, K6
and
H1, H2, H3
espec i ely. Because he subduc ion o he co eps on o he
P
line mus ma ch a bo h ends o he pa h, we can disca d he
H3
symme y. This is
due o o
K5
,
K6
,
H1
and
H2
all subducing o
P3
whe eas
H3
subduces o
P1⊕P2
,
iola ing he compa ibili y ela ions. We also no e ha he e a e non-degene a e
ene gies o a gi en spin-sec o a
K
(e.g. spin-down bands a a ound
−0.6
eV) since
he e a e one-dimensional co eps o he li le g oup a ha poin . On he con a y,
he e a e only 2D co eps a Hand all he bands a e doubly degene a e.
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7.2. S udy o Fe3GeTe2
Gene a o s In a ian plane Poin g oup Co eps
{m001|0,0,1
2},Θ{¯
1|0}(kx, ky, kz= 0,1/2)
{m110|0},Θ{¯
1|0}(−k, k, kz)
{m100|0},Θ{¯
1|0}(kx= 0,1/2, ky, kz)2’/m A′, A′′
{m1¯
10|0,0,1
2},Θ{¯
1|0}(k, k, kz)
{m120|0,0,1
2},Θ{¯
1|0}(−2k, k, kz)
{m210|0,0,1
2},Θ{¯
1|0}(−k, 2k, kz)
Table 7.1: Gene a o s o he li le g oups (wi h ansla ions), in a ian coo dina es,
li le co-g oup label and co eps o he band s uc u e wi hou SOC ha p o ec
c ossings ha may lead o a nodal line. All in a ian planes sha e he same poin
g oup and co eps.
The line
P= (1/3,1/3, w)
is ela ed o
(1/3,1/3,−w)
by symme y, which im-
plies ha he a o emen ioned wo old degene acy o ms a nodal line ha closes due
o pe iodic bounda y condi ions o he BZ. This is ma kedly di e en om he mi o -
p o ec ed loops explained be o e, since hey a e con ined o a single high-symme y
line. Fo his SOC-less case, since bo h spin sec o s a e decoupled, he c ossing be-
ween up and down bands is ne e gapped and such acciden al degene acies can gi e
ise o nodal lines [103].
Apa om his, he mi o -p o ec ion also applies o his ma e ial, e en wi hou
SOC. In pa icula , he e a e six mi o planes which p o ec nodal line c ossings, as
show in Table 7.1 Fo all o he pe pendicula planes, he li le co-g oup is isomo phic
o
2′/m
, which only has wo co eps
A′
and
A′′
which di e by hei mi o eigen alue.
Bands wi h SOC
When SOC is conside ed, spin and spa ial ope a ions canno be decoupled and we
can di ec ly apply he heo y o MSGs, as explained in Chap e 2. As a consequence,
he symme y g oup is educed o he Type-III MSG
P63/mm′c′
(No.194.270), which
admi s a cose decomposi ion
M=G+θ{2110|0}G, (7.5)
whe e
G
is he uni a y subg oup gene a ed by
{6+
001|0,0,1/2}
,
{2001|0,0,1/2}
,
{¯
1|0}
and ansla ions
{1|ai}
by he ec o s
ai
de ining he s anda d hexagonal uni cell.
This MSG sha es he B a ais la ice and symme ies o he uni a y subg oup
P63/m
(No,176), while also espec ing addi ional space-g oup ope a ions o No.194 combined
wi h TRS. As shown in Figu e 7.3b, he spin sec o s emain well di e en ia ed excep
a c ossing poin s o poin s whe e a gap be ween spin up and down bands is opened,
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
-1
800
600
400
200
0
80
100
60
40
20
0
-0.75 -0.50 -0.25 0 0.25 0.5 0.75 1
E - EF (eV)
Absolu e AHC (S/cm)
Weyl poiin coun
Figu e 7.8: Compu ed absolu e AHC
σAHC
xy
(black) and he ene gy dis ibu ion o he
non-ze o-chi ali y Weyl nodes ( ed) in a window ex ending 1.5 eV abo e and below
he Fe mi ene gy.
-1 -0.75 -0.50 -0.25 00.25 0.5 0.75 1
cen e ed cube
ull BZ
800
600
400
200
0
Absolu e AHC (S/cm)
E - EF (eV)
a) b) 1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
(no malized)
E - EF (eV)
KM
Figu e 7.9: a) AHC om a cube enclosing he
Γ
poin (blue), which es ima es he
con ibu ion o he SOC- ela ed gaps in (b). I shows ha he signal in he -0.5 o 0.1 eV
ange a ound he Fe mi le el can be a ibu ed o his mechanism. b) Elec onic band
s uc u e along high-symme y pa hs on he
kz= 0
plane wi h SOC and no malized
Ωxy componen o he Be y cu a u e.
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7.3. Nodal lines in hexagonal Co
he p edic ion o opological ea u es and o explain he obse ed anomalous Hall
conduc i i y. The symme y o he FM phase, wi h SOC conside ed, is desc ibed by
he MSG No.194.270 whose
{m001|0,0,1/2}
mi o plane, in pa icula , p o ec s nodal
lines on he in a ian
kx= 0,1/2
planes. We ha e ound ha Fe
3
GeTe
2
hos s many
o hese ea u es (Fig.7.4) and gi en a igo ous iden i ica ion o he nodal c ossings
h ough he discon inui ies o Wilson loops o e pa hs ha in e sec hem. We ha e
used he ace o he Wilson loop ope a o o p edic he appea ance o d um-head
su ace s a es in he p ojec ion o he nodal lines and ha e de ec ed one example in
he compu ed su ace DOS (Fig.7.5). Fu he mo e, we ha e compu ed he AHC o a
ange o chemical po en ials and de ec ed he h ee main mechanisms ha gi e ise
o he AHC esponse. Fi s ly, while he pa amagne ic phase also p o ec s nodal lines
on
kx= 0,1/2
and
ky= 0,1/2
, he FM ansi ion gaps hem and hey con ibu e o
he anomalous anspo due o being poin s whe e he Be y cu a u e concen a es
(Fig.7.6). Howe e , hey a e no he only sou ce o AHC, since i s con ibu ion is a
om he o al esponse compu ed. Secondly, we ha e iden i ied many Weyl poin s in
he band s uc u e and compu ed hei chi ali y (Fig.7.7). Thei con ibu ion o he
AHC has o be complemen ed by he wo o he sou ces, especially in he -0.6 o -0.1
eV ange, whe e Weyl c ossings a e almos absen (Fig.7.8). Finally, wi h magne ic
o de bu neglec ed SOC, he spin up and down bands a e decoupled and ee o c oss,
each se espec ing he symme ies o he single- alued g ay MSG No.194.264. These
degene acies a e li ed when SOC is b ough in o he pic u e, gi ing ise o small
gaps whe e he cu a u e is also p ominen (Fig.7.3c and 7.9a)). The calcula ions also
show ha sligh ly shi ing he chemical po en ial by app oxima ely 0.3 eV would
u he inc ease he AHC in Fe
3
GeTe
2
om a ound 200 S/cm o app oxima ely 800
S/cm, sugges ing elec on doping as a good mechanism o ob ain an enhanced e ec .
This p edic ion is suppo ed by DFT calcula ion wi h a ying numbe o elec ons
ha accoun o he e ec o elec on and hole doping. These show ha he magne ic
momen s a y only a ound 5% wi h espec o he neu al con igu a ion and ha he
elec onic band s uc u e is almos unchanged, implying ha he e ec o doping can
be app oxima ed by a simple shi in he chemical po en ial. Ou compu a ions show
ha a doping o wo elec ons pe uni cell (one pe o mula uni ) achie es an almos
op imal shi in he chemical po en ial ha leads o he ou old AHC enhancemen
(Figu e7.10).
7.3 Nodal lines in hexagonal Co
Hexagonal close-packed Co (hcp-Co) sha es he same pa en pa amagne ic g oup as
Fe
3
GeTe
2
and hence he band s uc u e is subjec ed o simila symme y cons ain s.
Expe imen ally ealized Co ilms exhibi di e en in-plane (IP) and ou -o -plane (OP)
magne ic aniso opy
A
, whe e
A(x, y)> A(z)
. As a consequence, he esul ing
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
E-EF (eV)
E-EF (eV)
-2
-1 0
+1
+2
Figu e 7.10: Absolu e AHC wi h espec o he chemical po en ial. The numbe s co -
espond o he a ia ion in elec on numbe wi h espec o he neu al con igu a ion
and he co esponding loca ion o he chemical po en ial in he AHC cu e is shown
in he inse wi h e ical ba s o ma ching colo .
hys e esis loop is no pe ec ly squa e nei he in he IP no he OP di ec ions and
he samples de elop a can ed FM o de when magne ized wi h an elec omagne ic
pulse. Mo eo e , he e a e h ee IP magne iza ion di ec ions sepa a ed by 60 deg ees
and his IP axis is andomly chosen o e e y domain when he ex e nal pulse is
applied. Howe e , no ma e he magne ized IP axis, all domains will show an OP
componen and, as a esul , mos band ea u es can be explained by he symme y
o he OP con igu a ion since his con ibu ion is consis en ac oss domains. The
symme y co esponding o he IP magne iza ion is hus “smea ed ou ” by a e aging
he inequi alen IP axes o e all domains.
Symme y analysis, DFT and wannie iza ion me hods
To pe o m he DFT calcula ions, we employed a simila p ocedu e o Fe
3
GeTe
2
in
VASP. To accoun o he co ela ions o he localized
d
elec ons in Co, we in oduced
a Hubba d-like epulsi e e m (DFT+U) in he sphe ically symme ic app oxima-
ion [110]. The epulsi e pa ame e was adjus ed o
U= 2
eV o ep oduce de
expe imen ally epo ed magne ic momen pe Co a oms o a ound 1.8
µ
B [111]. The
ene gy cu o o he plane wa e basis was se o 600 eV and he
k
poin g id was a
Gamma-cen e ed, egula Monkho s -Pack poin se o dimensions
11 ×11 ×7
. The
c ys al s uc u e ound in he li e a u e [112] was sligh ly expanded acco ding o
he expe imen al measu emen s we pe o med o be e ep oduce he exac sample
ea u es.
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7.3. Nodal lines in hexagonal Co
b)a)
E - EF (eV)
0.0
0.5
1.0
1.5
2.0
-0.5
-1.0
-1.5
-2.0 KAL H A
MKAL H A
M
Figu e 7.11: DFT elec onic band s uc u e and uni cell wi h a) OP and b) IP magne i-
za ion.
The localized Wannie unc ions as we e ob ained s a ing om a basis o s, p, d
o bi als pe Co a om o a o al o 36 Wannie unc ions gi en he doubling due o
spin. The bands we e ozen in an ene gy window o 2 eV cen e ed a ound he Fe mi
le el du ing he disen anglemen p ocedu e o exac ly ep oduce he bands in ha
ene gy ange.
Symme y analysis and DFT band s uc u e
Due o he expe imen al condi ions desc ibed abo e, we will analyze he IP and
OP con igu a ions sepa a ely. Wi h OP magne iza ion (Fig. 7.11a), he symme y
o he sys em is exac ly he same as in FM Fe
3
GeTe
2
(No.194.270). The e o e, he
same symme y-p o ec ion mechanisms o nodal lines a e p esen and he analysis is
s aigh o wa d. Fo IP magne iza ion (Fig. 7.11b) along he speci ic
[1¯
10]
di ec ion,
he symme y is educed o MSG
Cm′cm′
(No.63.464), whe e one has o ake in o
accoun ha using he hexagonal cell co esponds o a non-s anda d se ing. Because
{m001|0,0,1/2}
is no a symme y by i sel bu only combined wi h TRS, nodal lines
a e no p o ec ed and he e o e a e expec ed o be gapped wi h espec o he OP
con igu a ion.
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
E - EF (eV)
-0.4
0.4
1.91 1.84 1.77 1.70 1.64 1.57 1.50 1.36 1.22 1.15 1.08 1.02
kx (Å-1)
0.0
-0.4
-0.8
kx (Å-1)
kx (Å-1)
UL
LL
1.0
0.5
-0.5
-1.0
-2.0 -1.0 0.0
0.0 1/2
-1/2
kz
Fe mi su ace
Nodal line
a)
b) c) d)
H
LA
M
M
K
K
E
Figu e 7.12: Expe imen al da a o a nodal line along
A−L
(
kz= 1/2
). a) Band
dispe sions along pa hs pe pendicula o
A−L
showing a cone-like dispe sion whose
c ossing poin is iden i ied as a nodal line. b) Fe mi su ace map on
kz= 1/2
whe e
he con ibu ion o he uppe legs (UL) and lowe legs (LL) o he cone ea u e in a
in he Fe mi su ace is ma ked. c) Ske ch o he BZ and su ace BZ. d) Ske ch o he
band dispe sion gi ing ise o he nodal line along A−L.
Expe imen al obse a ions
ARPES expe imen s we e pe o med o s udy he elec onic band s uc u e o hcp-Co.
Figu e 7.12 shows he expe imen al da a on
kz= 1/2
. A se ies o band dispe sions
along cu s pe pendicula o he
A−L
(o
¯
Γ−¯
K
in he su ace BZ, see Fig. 7.12c)
a e p esen ed in Figu e 7.12a showing a conic band dispe sion whose c ossing poin
aises abo e he Fe mi le el and goes back down. This ea u e can also be obse ed
in he Fe mi su ace (FS) map in Figu e 7.12b, since he uppe and lowe “legs” o
he cone (UL and LL, espec i ely) con ibu e o he Fe mi su ace a
kz= 1/2
as
e idenced in he band dispe sions. This s uc u e is ske ched in Figu e 7.12d.
In Figu e 7.13, we show he expe imen al da a o he
kz= 0
plane. The FS
map e eals a lowe -like pa e n o wo in e locking bands, which a e signa u es
o a nodal ing ske ched in Figu es 7.13c-d. The band dispe sion below he Fe mi
le el along he ed pa h in Figu e 7.13a is shown in Figu e 7.13b, whe e we ha e
o e lain he OP and IP DFT calcula ions. The plo shows ha he band s uc u e and
in pa icula his nodal line a e ep oduced, as expec ed, only o he OP con igu a ion
and he in e sec ion wi h his pa h is ma ked wi h an o ange do . The e is also an
addi ional band wi h la ge spec al weigh a a ound -1.2 Å
−1
. which is no seen in
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7.3. Nodal lines in hexagonal Co
E
E1
E2
IP OP
ky (Å-1)
kx (Å-1)
ky (Å-1)
0.5
0.0
-0.5
E-EF (eV)
0.0
-0.4
-0.8
-0.8 -0.8 -0.8 -0.8 -0.80.0 0.0 0.0 0.0 0.0
0.8
-2.0 -1.0 0.0 1.0 2.0
a)
b)
c)
d)
Node
Node
Figu e 7.13: Expe imen al obse a ion o a lowe -like nodal line on
kz= 0
.a) Fe mi
su ace map. b)
¯
Γ−¯
K
band dispe sions along he ed pa h in a o di e en
kz
anging
om
kz= 0
(le ) o
kz= 1/2
( igh ). c) Ske ch o band c ossing s uc u e whe e
o med by wo bands in ed and blue. d). Ske ch o he isola ed nodal line esul ing
om c.
he heo e ical calcula ions. Howe e , his is iden i ied as a su ace s a e since i s
dispe sion does no change wi h kz.
Theo e ical analysis o he nodal lines
Gi en he p e ious symme y analysis and expe imen al ea u es, we u n o iden i y
he nodal c ossings in he DFT calcula ions and ma ch hem wi h he obse a ional
da a. We will es ic o he OP con igu a ion, since we know his is he only case whe e
a p o ec ion mechanism on he
kz= 0,1/2
holds. Fi s , we iden i y he nodal line
along
A−L
(R line) on Figu e 7.14b, ma ked in ed. MSG No.194.270 has a single i ep
¯
R3¯
R4
along his high-symme y line which is subduced in o
¯
E3⊕¯
E4
a gene ic poin s
in he
kz= 1/2
plane. This means ha he wo old degene acy along R spli s in o o
bands away om he line and co esponds o a c ossing. Mo eo e ,
R= (u, 0,1/2)
is equi alen o
(−u, 0,1/2)
by symme y he e o e allowing nodal lines ex ending
om
(−1/2,0,1/2)
o
(1/2,0,1/2)
ha close due o pe iodic bounda y condi ions.
Since we know ha he conic ea u e obse ed in Figu e 7.12a mus c oss he Fe mi
le el, we can pin-poin he exac nodal along R ou o all he possible ones in ou DFT
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7. Magne ic-symme y-en o ced opological nodal lines: he cases o
Fe3GeTe2and Co
K
M
kx (Å-1)
ky (Å-1)
-2 02
-4
-2
0
2
4
E - EF (eV)
0
1
2
-1
-2
ky (1/Ang)
kx (1/Ang)
-0.5
-1
00.5
E - EF (eV)
1
1
0.5
0
-0.5
-0.7
-0.75
-0.80
-0.85
-0.9
-0.95
-1
-1
0.5 0
k ( ac ional)
0
0.2
0.4
0.6
0.8
1
WL ace / 2π
a) b)
c) d) e)
OP IP
E - EF (eV)
0.0
0.5
1.0
1.5
2.0
-0.5
-1.0
-1.5
-2.0 KAL H A
M
Figu e 7.14: Compu ed nodal lines in hcp-Co wi h OP magne iza ion. a) A selec ion
o nodal lines on
kz= 0
lying nea he Fe mi le el. b) Elec onic band s uc u e wi h
in e sec ion wi h nodal ings in ama ked in ma ching colo s. c) Dispe sion o a line
pe pendicula o
A−L
o OP (le ) and IP ( igh ) magne iza ion. The OP dispe sion
displays he cone co esponding o a nodal line obse ed in Figu e 7.12a and ma ked
wi h a ed line in b.d) Ene gy dispe sion o he g een nodal line in a.c) WL analysis
o some nodal lines on
kz= 0
, including he one in dwhich is mani es ed in he
discon inui y ma ked wi h a ed a ow. The WLs a e compu ed along
kz
o a ying
kyalong he ed pa h o he inse .
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7.3. Nodal lines in hexagonal Co
calcula ion and compu e he conic band dispe sion (Fig. 7.14c le )
1
. This does no hold
in he IP con igu a ion since, al hough he e is also a single 2D co ep
(Q)¯
GP2¯
GP2
along he
A−L
line
2
, his is he only 2D co ep o all he
kz= 1/2
plane, implying
ha he degene acy along he high-symme y line is no a c ossing as shown in
Figu e 7.14c ( igh ). The lowe -like nodal line on
kz= 0
can be ound by iden ical
me hods as he ones used o Fe
3
GeTe
2
. In Figu e 7.14a, we show a selec ion o
nodal c ossings on his plane ha lie nea he Fe mi le el, as can be seen by hei
in e sec ions wi h ma ching colo s in he high-symme y pa h o Figu e 7.14b. The
nodal ing ma ked in g een, whose compu ed dispe sion wi h
kx, ky
is shown in
Figu e 7.14d, co esponds exac ly o he ea u e obse ed in he expe imen , shown
in Figu e 7.13d. In Figu e 7.14e, we plo e olu ion wi h
ky
o WL compu ed along
kz
,
whose hi d discon inui y is due o his p o ec ed nodal ing.
Summa y
In conclusion, we ha e seen ha hcp-Co displays nodal lines p o ec ed by he
{m001|0,0,1/2}
symme y, al hough he magne ic aniso opy causes he magne iza-
ion o be can ed wi h espec o he
c
axis o he hexagonal cell. We ha e iden i ied
h ough ARPES measu emen s wo kinds o nodal line s uc u es in his compound.
Fi s , a lowe -like nodal degene acy cause by wo in e locking bands on he
kz= 0
plane. Second, a nodal line along he
A−L
high-symme y line on
kz= 1/2
. Using a
Wannie TB model ha ep oduces he low ene gy spec um om DFT, we ha e iden-
i ied hese ea u es and analyzed hem using Wilson loop calcula ions and symme y
conside a ions o con i m hei s a us as symme y-p o ec ed nodal degene acies.
Ou compu a ions ha e also de ec ed many o he nodal lines ha a e p esen in a
window o 1 eV a ound he Fe mi le el.
1
This igu e shows explici ly why he e canno be any nodal lines on
kz= 1/2
ha c oss he R
line in bo h Fe3GeTe2and OP hcp-Co due o he symme y cons ain s
2This is no a high-symme y line in MSG No.63.464.
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CHAPTER 8
Topological ansi ions in he FeSe
supe conduc o ia doping and
p essu e.
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8. Topological ansi ions in he FeSe supe conduc o ia doping and
p essu e.
In pa icula , we no e ha no ope a ion exchanges i eps so, o example,
aij
will no
ans o m o
bkl
unde any ope a ion
g∈G
. WPs posi ions a e no mixed ei he since,
by de ini ion, hey con ain all he si es ela ed by all symme y ope a ions (inside he
same uni cell). Upon aking he Fou ie ans o m
αij(k) = 1
√NX
R
e−ik·(R+ i)αij(R),(8.5)
whe e
α
is he posi ion o
αij
inside he uni cell and
N
is he o al numbe o cells,
he
R
deg ees o eedom a e changed by
k
and he Hamil onian is decoupled in o
independen blocks
H=X
k
H(k).(8.6)
The ans o ma ion p ope ies o
αij(k)
can be induced om eal space as in Equa-
ion 2.32
{R| }αij(k) = 1
√NX
T
e−ik·(T+ i){R| }αij(T)
=1
√NX
T
e−ik·(T+ i)Dα(g)jkαlk(T′),
(8.7)
whe e T′+ l=R(T+ i)+ . Changing he sum o T′
{R| }αij(k) = 1
√NX
T′
e−ik·R−1(T′+ l− )Dα(g)jkαlk(T′)
=eiRk· Dα(g)jkαlk(Rk)
(8.8)
When
{R| }
is in he li le g oup
Gk
o
k
, hen we can w i e he ep esen a ion as
a
12 ×12
ma ix. In ha case, since we can ead
Dα(g)
o he BCS, we can eadily
cons uc he ep esen a ion in ou basis, wi h ma ices
D(g)
, o he gene a o s o
G
D({2001|1/2,1/2,0})=ei
2(kx+ky)
1
6⊗−i0
0i(8.9)
D({2010|0,1/2,1/2})=ei
2(ky+kz)

A0 0
0B0
0 0 B
,
A=



0 0 0 e−iπ/4
0 0 e−i3π/40
0e−iπ/40 0
e−i3π/40 0 0




B=



0 0 0 ei3π/4
0 0 eiπ/40
0ei3π/40 0
eiπ/40 0 0




(8.10)
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8.1. Su ace s a es in FeTe0.55Se0.45
D({4+
001|1/2,0,0})=ei
2kx

C0 0
0F0
0 0 J
, C =



0 0 ei3π/40
000e−i3π/4
ei3π/4000
0e−i3π/40 0




F=



0 0 e−iπ/40
000eiπ/4
e−iπ/4000
0eiπ/40 0




, J =e−π/40
0eiπ/4⊗
1
2
(8.11)
D{¯
1|0}) =
1
3⊗



0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0




(8.12)
The e a e many s a egies o ob ain all he in a ian e ms in a Hamil onian. In
his wo k, we lis ed he coupling in o de o inc easing dis ance and sequen ially
in oduced he e ms in he exp ession. Fo each dis ance, we selec a coupling and
ob ain al he symme y ela ed hopping e ms by applying he gene a o s o
G
un il
he e m is in a ian . As in he
k·p
model, one hen has wo addi ionally impose
TRS and he mi ici y. We p esen he inal exp ession o he Hamil onian in
k
space
H(k) = 

ϵa0 0
0ϵb0
0 0 ϵc
⊗
1
3+ ∆(k) + ∆†(k),(8.13)
whe e
∆(k) =




















0 0 10000 0 +
2(kx) 0 +∗
2(ky) 0
000 10 0 0 0 0 −
2(kx) 0 −∗
2(ky)
0 0 0 0 0 0 0 0 −
2(ky) 0 −∗
2(kx) 0
0 0 0 0 0 0 0 0 0 +
2(ky)′0 +∗
2(kx)
0 0 0 0 0 0 ′
10 3(kx) 0 ∗
2(ky) 0
000 0000 ′
10 3(kx) 0 ∗
3(ky)
0 0 0 0 0 0 0 0 3(ky) 0 ∗
3(kx) 0
0 0 0 0 0 0 0 0 0 3(ky) 0 ∗
3(kx)
0 0 0 0 0 0 0 0 0 0 40
0 0 0 0 0 0 0 0 0 0 0 4
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0




















.
(8.14)
131
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8. Topological ansi ions in he FeSe supe conduc o ia doping and
p essu e.
The ma ix elemen s depend on eal pa ame e s o he model as ollows:
ϵa=a+lcos kz, ϵb=b+mcos kz,
ϵc=c+ (cos kx+ cos ky)(e−ikzξ+eikzξ′)+ncos kz,
1=δcos kx
2cos ky
2, ′
1=δ′cos kx
2cos ky
2,
±
2(s)=±i e−ikz(zc−1) cos s
2+q′e−ikzzccos s
2,
3(s)= ′e−ikz(zc−1) cos s
2+q′e−ikzzccos s
2,
4=τe2ikz(zc−1) cos kx
2cos ky
2
(8.15)
In he exp essions abo e,
zc
and
1−zc
a e he chalcogen posi ions in he uni cell
along he caxis and he es o pa ame e s co espond o:
•a, b, c: onsi e ene gies o aij, bij, cij o bi als
•l, m, n
: couplings be ween o bi als
aij, bij, cij
in adjacen cells in he
z
di ec ion
•ξ, ξ′
: hopping be ween
cij
o bi als along la ice ec o s
(−1,0,−1)
,
(0,−1,−1)
,
(0,1,−1) and (1,0,−1) o ξand hei opposi es o ξ′
•δ, δ′
:
a1j
-
a2j
and
b1j
-
b2j
couplings, espec i ely, o la ice ec o s
(0,−1,0)
,
(0,0,0),(1,−1,0) and (1,0,0)
•τ
: coupling o
c1j
and
c2j
o ec o s
(−1,−1,2)
,
(−1,0,2)
,
(0,−1,2)
and
(0,0,2).
• , ′
: nea es -neighbo s hyb idiza ion e ms among
aij
and
bij
, espec i ely,
wi h
c1j
sepa a ed by
(0,0,−1)
and
(1,0,−1)
, and wi h
c2j
by
(0,−1,1)
and
(0,0,1)
The pa ame e s can be i ed o ep oduce he DFT band dispe sion along
Γ−Z
(see
Table 8.2) and he esul is shown in Figu e 8.4a. Adjus men s o he DFT i ha e
o be made in o de o co ec ly ep oduce he highly-co ela ed phase de ec ed by
expe imen s. In going, o Figu e 8.4b, one has o adjus he
d
band dispe sion by
a eno maliza ion ac o o 3 [159,160,177], which amoun s o dec easing he
l, m
pa ame e s. Fu he mo e, we inc ease he s eng h o he SOC coupling be ween
dxz, dyz
and
pz
o bi als [178], ob aining Figu e 8.4c. A coupling e m is conside ed o
include SOC i ei he
•Couples spin-up and down o bi als
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8.1. Su ace s a es in FeTe0.55Se0.45
Pa ame e s o i
a−2.33 δ5.30 n1.625
b−4.40 δ′9.25 ξ−1.0
c−6.13 τ−14.0ξ′0
0.16 l−0.049 q0.4
′0.19 m−0.049 q′−0.01
Table 8.2: Pa ame e s o he model o ep oduce he DFT band dispe sion a ound he
Fe mi le el along Γ−Z.
EF -E (eV)
0.2
0.3
0.0
a) b) c) d)
Figu e 8.4: Fi o he TB model dispe sion o DFT adjus ed o expe imen al e idence.
a) Fi o DFT dispe sion b) Adjus ed o accoun o
d
band eno maliza ion ac o
o 3 [159, 160, 177] c) Wi h SOC coupling be ween
dxz
and
pz
inc eased [178] d)
Compa ison wi h expe imen al measu emen s.
•
Couples spin-up wi h spin-up wi h a di e en ampli ude han spin-down and
spin-down
Since he second op ion is o bidden by TRS, gi en he change o basis in Equa ion 8.4
and he o m o he Hamil onian,
, ′, q, q′
a e he only ou SOC-enabled hopping
ampli udes. We also ind ha he size o he
pz
-
d
gap is mainly go e ned by
q, q′
,
since i couples
aij, bij
o
cij
, which ha e
dxz, dyz
and
pz
cha ac e , espec i ely.
Figu e 8.4d shows he compa ison wi h he expe imen al esul , showing ha he
pz
band comple ely disappea s. Since his si ua ion canno be ob ained wi hou
subs an ial change in he i ed pa ame e s, his is in line wi h co ela ions in FeSe
being e y s ong and no being cap u ed by a ee-elec on model co ec ly.
Summa y
We ha e shown ha , acco ding o he DFT calcula ions, FeTe
0.55
Se
0.45
shows a band
in e sion ha may be he cause o he opological ansi ion in o a s ong
Z2
s a es,
which hos s Di ac su ace s a es a ound gamma in a slab con igu a ion ini e along
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8. Topological ansi ions in he FeSe supe conduc o ia doping and
p essu e.
he
c
axis. We ha e pe o med a symme y analysis o cons uc an EBR-based TB
model ha ep oduces he low ene gy spec um, especially along he
Γ−Z
line,
whe e he in e sion occu s. Howe e , due o he s ong eno maliza ion, e idenced
by he ARPES measu emen s, he DFT pic u e is no accu a e. To ma ch ou ab-ini io
calcula ions wi h he obse ed band ea u es, we show ha one can empi ically adjus
he band dispe sion by a eno maliza ion ac o o 3, in line wi h p e ious esul s.
All in all, ou wo k p o ided s ong e idence o he non- i ial band opology in
FeTe
0.55
Se
0.45
and he heo e ical calcula ions helped iden i y he p ecise eason o
his phase ansi ion in he doped compound, o igina ing om a band in e sion along
Γ−Zand a gap opening h ough dand po bi al hyb idiza ion.
8.2 Enginee ing opological phases in FeSe ia
uniaxial s ain
Ou wo k on FeTe
0.55
Se
0.45
helped us unde s and he impo ance o he chalcogen
posi ion and he sensibili y o he low ene gy spec um along
Γ−Z
o d i ing opo-
logical phase ansi ions in his pa icula sys em h ough band in e sions. Inspi ed
by hese obse a ions, i is na u al o ask he ques ion o whe he a simila e ec can
be achie ed by sligh ly modi ying he la ice s uc u e h ough ex e nal p essu e. In
he ollowing sec ions, we show ha uniaxial s ain can indeed achie e opologically
non- i ial phases. Mo eo e , he di ec ion and s eng h o his ex e nal pe u ba ion
can selec wo di e en weak and s ong TI phases.
Me hods
Elec onic s uc u e calcula ions we e pe o med wi hin densi y unc ional heo y
(DFT) wi h he Vienna Ab-ini io Simula ion Package (VASP) so wa e [74,75], choosing
P ojec o Augmen ed Wa e (PAW) pseudo-po en ials [101] wi h he Pe dew-Bu ke-
E nzenho (PBE) implemen a ion o he Gene alized G adien App oxima ion (GGA)
o he exchange-co ela ion unc ional [54]. The e ec o spin-o bi coupling (SOC)
was included in all calcula ions. The ene gy cu o o he plane wa e basis we se o
600 eV. As
k
poin g id we employed a Gamma-cen e ed, egula Monkho s -Pack
poin se o dimensions
11 ×11 ×10
, p o iding a sampling o uni o m densi y in
all di ec ions o he B illouin zone (BZ). A gaussian b oadening o
0.1
eV has been
used h oughou he calcula ions. S uc u al op imiza ion o he ion posi ions was
pe o med using he conjuga e g adien me hod implemen ed in VASP un il all o ces
exe ed on he a oms had a modulus smalle han
10−3
eV/Å. To elax he s ained
s uc u es we se he espec i e componen s o he s ess enso o ze o be o e he
g adien s ep.
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8.2. Enginee ing opological phases in FeSe ia uniaxial s ain
P4/nmm
No.129
Pmmn
No.59
a1
a2
a3
Figu e 8.5: A omic s uc u e o FeSe. S ain is applied along he
a1
di ec ion o he uni
cell ( ed a ows) o he uncomp essed e agonal s uc u e belonging o SG No.129.
Once enough uniaxial p essu e is applied, he sys ems adop s an o ho hombic sym-
me y desc ibed by SG No.59. The leng h o he
a2
axis is exagge a ed o be e display
he o ho hombic symme y. The pa ame e s o he e agonal and o ho hombic
phases a e speci ied in in he appendix o supl. in .
In o de o cons uc a basis se o maximally localized Wannie unc ions o
any o he c ys al s uc u es, we employed Wannie 90 [66] using a s a ing basis
o
d
o bi als o Fe and
p
o Se. This esul s, a e he wannie iza ion p ocess, in a
o al o 32 basis unc ions, aking in o accoun he doubling due o spin. A ozen
disen anglemen window o app oxima ely 2 eV cen e ed a he co esponding Fe mi
le el is chosen o ep oduce he low ene gy spec um and he gaps whe e su ace
s a es appea .
Elec onic s uc u e a ambien p essu e
In egula condi ions, FeSe c ys allizes in a e agonal s uc u e whose symme y is
desc ibed by he space g oup (SG)
P4/nmm
, No.129 in he Belo -Ne ono a-Smi no a
(BNS) no a ion [78]. The g oup is gene a ed by he iden i y, la ice ansla ions
and he elemen s
{4+
001|1/2,0,0}
,
{2010|0,1/2,0}
wi h in e sion
{¯
1|0,0,0}
, in Sei z
no a ion. The con en ional, p imi i e uni cell con ains 2 Fe a oms in he 2a Wycko
posi ion (WP) and 2 Se a oms in he 2c WP. P e ious s udies [130,149] ound ha
he band s uc u e, especially along he
Λ
high-symme y line joining
Γ = (0,0,0)
and
Z= (0,0,1/2)
, in ac ional coo dina es, is e y sensi i e o he
z
posi ion o
he Se a oms inside he uni cell, one a
z
and he o he a
−z
. The e o e, doping
FeSe wi h iso-elec onic Te impu i ies loca ed a he chalcogen posi ions modi ies
he band dispe sion along
Λ
and i has been shown o esul in a opological phase
ansi ion due o band in e sion [149]. The in oduc ion o Te modi ies he e o e he
coupling be ween Te/Se a oms in adjacen cells in he
a3
di ec ion bo h by changing
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8. Topological ansi ions in he FeSe supe conduc o ia doping and
p essu e.
M X RZAZ
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
E - EF (eV)
Figu e 8.6: Elec onic band s uc u e o he expe imen ally de e mined e agonal
c ys al s uc u e o FeSe. The bands up o he g een, ed and blue se s co espond o
illings o 20, 24 and 28 elec ons espec i ely. The s a es a
Γ
and
Z
a e labeled by
i eps o hei li le g oups, acco ding o he symme y o SG 129.
he
z
coo dina e o chalcogen a oms in he uni cell and inc easing he o e lap due o
mo e ex ended pzo bi als [149].
We i s pe o med DFT calcula ions o he undoped FeSe c ys al s uc u e as
measu ed expe imen ally, shown in Figu e 8.6. Con a y o doped c ys als, one can
check ha he s uc u e a ambien p essu e does no ha e a gap o a illing o 24
elec ons, nea he Fe mi le el because he op ed band and he lowes blue one a
Z
ha e
¯
Z9
and
¯
Z7
, espec i ely, which subduce o
¯
Λ6
and
¯
Λ7
along he
GM −Z
line
and hus canno hyb idize since hey ha e di e en symme y [179]. Consequen ly,
one canno de ine a no ion o opology nea he Fe mi le el. No e ha h oughou
his wo k “ illing” e e s o he band index a he han he ac ual numbe o elec ons
ob ained by in eg a ing he densi y o s a es (DOS) up o he chemical po en ial. The
epo ed sensi i i y o he dispe sion o he down-c ossing
pz
-dominan band and
he ene gy o he la e
d
bands sugges ha modi ying he band s uc u e h ough
p essu e may achie e a gap opening and a opological phase ansi ion.
E ec o s ain
To s udy he e ec o uniaxial s ain on he o iginal FeSe s uc u e, we mimicked
uniaxial s ain by ixing one o he cell di ec ions o a educed alue and elaxing he
emaining la ice pa ame e s and a omic posi ions un il he o ce on e e y a om is
less han
1
meV / Å. The p essu e co esponding o one s ain con igu a ion can be
ex ac ed om he componen o s ess enso in he
a1
di ec ion, which p o ides an
app oxima e alue necessa y o he expe imen al ealiza ion.
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8.2. Enginee ing opological phases in FeSe ia uniaxial s ain
S uc u e SG 24 elec ons SIs TI 28 elec ons SIs TI
acomp ession 1% 59 NLC z2w,3= 0 S ong NLC z2w,3= 1 S ong
z4= 3 z4= 1
ccomp ession 3% 129 SEBR z2w,3= 1 Weak LCAO z2w,3= 0 T i ial
z4= 2 z4= 0
cexpansion 1% 129 NLC z2w,3= 0 S ong - - -
z4= 3
Table 8.3: Symme y and opology o di e en FeSe s uc u es unde uniaxial s ain.
The i s column indica es he a io be ween he expe imen al and compu ed leng h
o he
a1
,
a2
and
a3
ec o s. The hi d and six h columns a e he i ep decomposi ion:
spli linea combina ion o a omic o bi als (LCAO), spli EBR (SEBR) and non-linea
combina ion o EBRs (NLC). The columns 4-5 and 7-8 con ain he SIs and he TI
classi ica ion o illings o 24 and 28, espec i ely.
M X RZAZ
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
E - EF (eV)
Figu e 8.7: Elec onic band s uc u e o c ys al unde s ain along he
a1
di ec ion
which educes i s leng h by 1% and d i es a ansi ion o an o ho hombic phase (SG
No.59). The bands up o he g een, ed and blue se s co espond o illings o 20, 24
and 28 elec ons espec i ely. The s a es a
Γ
and
Z
a e labeled by hei li le g oup
i eps.
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8. Topological ansi ions in he FeSe supe conduc o ia doping and
p essu e.
In ensi y (a.u.)
M
M
M
M
a) b)
E-EF(eV)
0.22
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Figu e 8.8: Densi y o s a es o ini e geome y in he
a3
di ec ion compu ed using
he Wannie TB model. Su ace spec um along he di ec ion
M−Γ−M
o he
c ys al a) comp essed in he
a1
di ec ion b) expanded in he
a3
di ec ion. The inse s
show a zoom in o he su ace Di ac cone due o he s ong Z2-odd opology.
Comp ession along aaxis
Following he abo e desc ibed me hod, we ind ha a con ac ion along
a1
o
a2
a o s a opological phase ansi ion by inc easing he band dispe sion o he Se
pz
dominan band, simila o he e ec p e iously achie ed by Te doping [149]. Figu e
8.7 shows he band s uc u e when he la ice pa ame e along he
a1
di ec ion is
educed by 1%, co esponding o a p essu e o 0.89 GPa. The symme y is lowe ed o
an o ho hombic phase desc ibed by SG
Pmmn
(No.59), which is essen ial o he
band s uc u e o show a gap o a illing o 24 elec ons, as shown in Figu e8.7. In
pa icula , he subduc ion o i eps om SG No.129 o No.59.
¯
Γ6,¯
Γ7→¯
Γ5¯
Γ8,¯
Γ9→¯
Γ6(8.16)
and he ac ha now he e is only one i ep along he
Λ
line mean ha he dispe si e
band, wi h mo e p onounced dispe sion han in he uns ained s uc u e, can now
hyb idize and gap ou wi h he la e one. The disconnec ed bands o a illing o
20 elec ons is a non-linea combina ion (NLC) o Elemen a y Band Rep esen a ions
(EBRs) and he nex ou bands up o 24 elec ons a e i ial. The e o e, he gap a 24
elec ons is opological since o e all he 24 bands canno be decomposed as a sum o
EBRs wi h in ege , posi i e coe icien s, meaning he e is no i ial a omic limi ha
ep oduces his se o bands. The illing o 28 elec ons (blue) is again a NLC because
he ou bands ma ked in blue co espond o a a b anch o he EBR induced om
i ep ¯
Ag¯
Ago he si e-symme y g oup ¯
1a WP 4c.
We can u he e ine he opological classi ica ion using he i ep decomposi ion
o ind he SIs [25,180], which can be ob ained om he in e sion eigen alues a ime-
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8.2. Enginee ing opological phases in FeSe ia uniaxial s ain
M X RZAZ
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
E - EF (eV)
M X RZAZ
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
E - EF (eV)
a) b)
Figu e 8.9: Elec onic band s uc u e o FeSe unde uniaxial s ain along he
c
axis. a)
Reducing
a3
by 3%. b) Expanding
a3
by 1%. The symme y is s ill e agonal and
desc ibed by SG No.129. The bands up o he g een, ed and blue se s co espond o
illings o 20, 24 and 28 elec ons espec i ely. The s a es a
Γ
and
Z
a e labeled by
hei li le g oup i eps.
e e sal-in a ian momen a (TRIMs). Fo a illing o 24, he alues
z2w,3= 0 mod 2
and
z4= 3 mod 4
o SG No.59 indica e ha s ain induces a ansi ion o a s ong
TI p o ec ed by TRS. A s ong TI canno be de o med in o a s ack o 2D quan um
spin Hall insula o s and displays su ace s a es on any o he su aces. Likewise, he
illing o 28 shows
z2w,3= 1
and
z4= 1
which, al hough he weak index is odd, also
indica es a s ong TI phase because he
z4
index is also odd. Figu es 8.8a shows he
densi y o s a es o he o ho hombic phase along he
M−Γ−M
pa h compu ed
om he Wannie TB model o he ini e slab in he
a3
di ec ion, whe e we iden i y
a D iac su ace cone, expec ed gi en he classi ica ion as s ong TI o ha illing.
Comp ession and expansion o caxis
Ou calcula ions also show ha a ia ion o he
a3
leng h is also an e ec i e me hod
o induce opological ansi ions, while p ese ing he e agonal symme y o he
undis u bed s uc u e. In pa icula , we p edic wo ways o achie ing wo di e en
opological phases. Fi s , comp ession o he
c
axis by 3% (Figu e 8.9a) d i es he
sys em in o a weak TI phase a a illing o 24 ( ed bands), co esponding o 1.3 GPa
o uniaxial p essu e. This is because he i s 16 bands a e i ial while he nex 8
co espond o a b anch o a spli EBR (SEBR) and hus canno be an EBR by hemsel es.
Compu ing he SIs yields
z4= 2
, which implies ha his is a weak TI, and
z2w,3= 1
, so
he phase is equi alen o a se ies o laye s o Quan um Spin Hall insula o s (QSHIs)
s acked in he
a3
di ec ion. Thus, his s uc u e should hos opological su ace
s a es only on he aces pe pendicula o
a1
and
a2
only. A second way o ob ain a
opological phase is by expanding he
c
axis by as li le as 1% (Figu e 8.9b), which
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