Holog aphic Kondo and Fano esonances
Johanna E dmenge ,1,* Ca los Hoyos,2,†Andy O’Bannon,3,‡Ioannis Papadimi iou,4,§
Jonas P obs ,5,∥and Jackson M. S. Wu6,¶
1Ins i u ü Theo e ische Physik und As ophysik, Julius-Maximilians-Uni e si ä Wü zbu g,
Am Hubland, D-97074 Wü zbu g, Ge many and
Max-Planck-Ins i u ü Physik (We ne -Heisenbe g-Ins i u ),
Föh inge Ring 6, D-80805 Munich, Ge many
2Depa men o Physics, Uni e sidad de O iedo, A da. Cal o So elo 18, 33007 O iedo, Spain
3STAG Resea ch Cen e, Physics and As onomy, Uni e si y o Sou hamp on,
Sou hamp on SO17 1BJ, Uni ed Kingdom
4SISSA and INFN—Sezione di T ies e, Via Bonomea 265, I 34136 T ies e, I aly
5Rudol Peie ls Cen e o Theo e ical Physics, Uni e si y o Ox o d,
1 Keble Road, Ox o d OX1 3NP, Uni ed Kingdom
6Depa men o Physics and As onomy, Uni e si y o Alabama, Tuscaloosa, Alabama 35487, USA
(Recei ed 8 Decembe 2016; published 12 July 2017)
We use holog aphy o s udy a (1þ1)-dimensional con o mal ield heo y (CFT) coupled o an impu i y.
The CFT is an SUðNÞgauge heo y a la ge N, wi h s ong gauge in e ac ions. The impu i y is an SUðNÞ
spin. We igge an impu i y eno maliza ion g oup (RG) low ia a Kondo coupling. The Kondo e ec
occu s only below he c i ical empe a u e o a la ge-Nmean- ield ansi ion. We show ha a all
empe a u es T, impu i y spec al unc ions exhibi a Fano esonance, which in he low-Tphase is a la ge-N
mani es a ion o he Kondo esonance. We hus p o ide an example in which he Kondo esonance su i es
s ong co ela ions, and unco e a no el mechanism o gene a ing Fano esonances, ia RG lows be ween
(0þ1)-dimensional ixed poin s.
DOI: 10.1103/PhysRe D.96.021901
I. INTRODUCTION
The Kondo e ec is he sc eening o an impu i y spin by
a Landau Fe mi liquid (LFL) a low T[1,2]. A a ie y o
echniques, such as Wilson’s RG, la ge-N, CFT, and mo e
[3], ha e cap u ed many cha ac e is ic Kondo phenomena.
Ne e heless, many ques ions esis solu ion, o example
abou in e -impu i y in e ac ions, subsys em en anglemen
en opy (EE), nonequilib ium p ocesses like quan um
quenches, and mo e.
In pa icula , wha happens when he LFL is eplaced
wi h s ongly co ela ed elec ons? Fo example, how does
he Kondo e ec change in a Lu inge liquid [4–8] o he
Hubba d model [9,10]? In he la e case, expe imen s
e eal d ama ic e ec s o s ong co ela ions, such as
enhancemen o he Kondo empe a u e, TK[53]. On he
heo y side, al hough special ools like bosoniza ion [4–8]
and uncon olled mean- ield app oxima ions [9,11–17]
ha e p o ided insigh , in gene al, eliable echniques do
no ye exis o answe ques ions abou Kondo phenomena
in s ongly-co ela ed sys ems.
To add ess all o he abo e, we ha e de eloped an
al e na i e Kondo model, based on holog aphic duali y
[18–21]. Ou model eplaces he LFL by a (1þ1)-
dimensional CFT in which spin SUð2Þis eplaced by
gauged SUðNÞ, wi h la ge Nand s ong gauge in e ac ions.
Ou model has al eady e ealed no el s ong-coupling
phenomena in RG [18,19], in e impu i y in e ac ions
[19] and EE [20].
He e we ini ia e he s udy o nonequilib ium phenomena
in ou model: we compu e linea esponse (G een’s)
unc ions o a cha ged bosonic impu i y ope a o , O,in
ou model. We ha e wo main esul s.
Fi s , we ind a la ge-Nmani es a ion o he Kondo
esonance [2,22,23], a signa u e o he Kondo e ec . As
expec ed, ou Kondo esonance appea s only a Tbelow he
c i ical empe a u e Tco a mean- ield ansi ion ha is
common o la ge-NKondo models [23–27]:hOibecomes
non-ze o when T≤Tc. We hus p o e unequi ocally ha
ou holog aphic model ealizes a genuine Kondo e ec , as
opposed o some o he impu i y physics, and u he mo e
show ha a la ge-NKondo esonance can su i e s ong
co ela ions essen ially in ac .
Second, a all T,O’s spec al unc ion exhibi s a Fano
esonance [28,29], which occu s when a Lo en zian es-
onance is imme sed in a con inuum o s a es (in ene gy).
A Fano esonance is cha ac e ized no only by i s posi ion,
wid h, and heigh , like a Lo en zian, bu also by an
asymme y pa ame e ,q, which measu es he ela i e
s eng h o esonan e sus non- esonan sca e ing. Ou
qinc eases as T→Tþ
c. When T≤Tc, he Fano line-
shape a ises om ou Kondo esonance, which mus be
*[email p o ec ed]‑wue zbu g.de
†[email p o ec ed]
‡[email p o ec ed]
§ioannis.papadimi [email protected]
∥[email p o ec ed]c.uk
¶[email p o ec ed]
PHYSICAL REVIEW D 96, 021901(R) (2017)
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an isymme ic due o pa icle-hole symme y (PHS), and
hence has he special alue q¼1.
Al hough Fano esonances ha e been obse ed in many
impu i y sys ems in one spa ial dimension [29–32], ou s
a ise om a quali a i ely di e en mechanism. Fo in-
s ance, in side-coupled quan um do s (QDs) [29,31,32] he
Lo en zian esonances a e he disc e e s a es on he QD,
and he con inuum comes om elec onic sca e ing s a es
in he leads. Coupling he wo, o example by a Kondo
coupling, can hen p oduce Fano esonances.
Ou model also has an impu i y coupled o a con inuum
in one spa ial dimension, i.e. he CFT. Howe e , ou model
has wo couplings: he CFT’sSUðNÞgauge coupling and
he Kondo coupling. The spec al unc ion o Oinhe i s
(0þ1)-dimensional scale in a iance om he o me , and
so exhibi s a con inuum o s a es, in con as o a QD’s
disc e e s a es. The Kondo coupling hen igge s an RG
low om ha (0þ1)-dimensional ixed poin , and c ea es
a esonance ha canno escape he con inuum, hence
p oducing a Fano line shape.
To ou knowledge, such a mechanism o p oducing
Fano esonances is no el, and mo eo e is easy o gene al-
ize o any RG low be ween (0þ1)-dimensional ixed
poin s, as ollows. Scale in a iance implies ha any spec al
unc ion will be a ea u eless con inuum, which in (0þ1)
dimensions means a powe law (o loga i hm) in equency.
A ele an de o ma ion can hen explici ly b eak scale
in a iance, igge an RG low o an IR ixed poin —in
which case we expec he con inuum o su i e—and may
also p oduce esonances. In highe dimensions, he eso-
nances would no ha e o be wi hin he con inuum, o
example he wo could be sepa a ed in momen um space.
Howe e , in (0þ1) dimensions he esonances ha e no
place o escape he con inuum, and hence mus p oduce
Fano line shapes.
In ac , such a mechanism was a wo k in some p e ious
cases, such as he la ge-NKondo model a su icien ly low
T[33], and holog aphic duals o T¼0cha ged black
b anes [34–36]. Howe e , he esul ing Fano esonances
wen uniden i ied, lea ing c ucial physics o e looked,
namely he ela i e s eng h o esonan e sus non-
esonan sca e ing, as measu ed by q. Ou esul s no only
p o ide a no el pe spec i e on hese cases, bu also p edic
Fano esonances in RG lows be ween o he (0þ1)-
dimensional ixed poin s, such as Sachde -Ye-Ki ae ixed
poin s [36–43].
Fu he esul s o ou model, including de ails o holo-
g aphic eno maliza ion use ul o holog aphic impu i y
models in gene al, will appea in [44].
II. HOLOGRAPHIC KONDO MODEL
We i s b ie ly e iew some essen ial ea u es o he
CFT and la ge-Napp oaches o he Kondo model, and how
ou model builds upon and ex ends hem.
The CFT app oach [45] is based on s-wa e educ ion
o LFL e mions abou he impu i y, plus linea iza ion
o he dispe sion ela ion. In/ou -going s-wa es become
ela i is ic le / igh -mo ing e mions, ψLand ψR, in he
adial di ec ion, . Re lec ing ψR o <0and elabeling
ψR→ψLleads o ψLalone on he en i e axis, wi h
impu i y a ¼0. The ψL o m a (1þ1)-dimensional
chi al CFT wi h SUð2Þ1×Uð1Þspin and cha ge Kac-
Moody cu en s, espec i ely. In he Hamil onian, he
Kondo in e ac ion is δð ÞgKSAJA, wi h coupling cons an
gK, impu i y spin SA, and spin cu en JA,A¼1,2,3.
An an i e omagne ic coupling, gK>0, is ma ginally
ele an , and igge s an RG low o an IR chi al CFT
cha ac e ized by a phase shi o ψLand impu i y
sc eening [45].
The la ge-Napp oach begins by eplacing spin
SUð2Þ→SUðNÞ, ollowedbyN→∞wi h λK≡
NgK ixed [3,23,46,47]. We will only conside SAin
o ally an isymme ic SUðNÞ ep esen a ions o ank Q,
and in oduce Ab ikoso pseudo e mions χ ia SA¼
χ†TAχ, wi h SUðNÞgene a o s TA,A¼1;…;N2−1.
Doing so in oduces an auxilia y Uð1Þac ing only on
χ, bu wi h cha ge ixed by p ojec ing on o s a es wi h
χ†χ¼Q.A la geN,SAJA¼−O†O=2wi h O≡ψ†
Lχ
[19], which is cha ged unde bo h he cha ge and
auxilia y Uð1Þ’s.
Ou holog aphic model [18] begins by gauging
SUðNÞ, hus in oducing he ’ Hoo coupling, λ.We
hen add deg ees o eedom o make he gauge heo y a
(1þ1)-dimensional CFT wi h spa se ope a o spec um
when Nand λbo h →∞, bu whose de ails o he wise
a e i ele an . The heo y is hen holog aphically dual o
Eins ein-Hilbe g a i y in (2þ1)-dimensional an i-de
Si e space, AdS3[48]. The cha ge Uð1ÞKac-Moody
cu en is dual o a Uð1ÞChe n-Simons gauge ield, A,
he auxilia y Uð1Þis dual o a Maxwell ield aon an
AdS2de ec a ¼0, and Ois dual o a complex scala
ield Φalso in AdS2, cha ged unde bo h Aand a.As
long as he s ess-ene gy enso is ini e, a la ge Nwe
can neglec back eac ion o A,a,Φ(dual o undamen al
ields) on he geome y (dual o adjoin ields). When
T>0, he bulk me ic is hus he BTZ black b ane,
ds2¼1
z2ðh−1ðzÞdz2−hðzÞd 2þd 2Þ;
wi h hðzÞ¼1−z2=z2
Hwhe e zH¼1=2πT, and uni AdS
adius. The ields aand Φa e localised o he asymp-
o ically AdS2subspace a ¼0, wi h induced me ic
gmn (m; n ¼z, ). We desc ibe he dynamics o A,a, and
Φby he simple quad a ic ac ion [18],
S¼−N
4πZBTZ
A∧dAþSAdS2;ð1aÞ
JOHANNA ERDMENGER e al. PHYSICAL REVIEW D 96, 021901(R) (2017)
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SAdS2¼−NZx¼0
dzd ffiffiffiffiffiffi
−g
p1
4 mn mn
þðDmΦÞ†ðDmΦÞþM2Φ†Φ;ð1bÞ
wi h ield s eng h ¼da, co a ian de i a i e DmΦ¼
ð∂mþiAm−iamÞΦ, and mass-squa ed M2. A he ho i-
zon z¼zHwe equi e egula i y o all ields. A he
bounda y z¼0,a’s leading mode, a∼Q=z, is ela ed o
Q:Q≠0b eaks χ’s PHS, so he PHS alue Q¼0is
dual o he PHS alue Q¼N=2, and inc easing jQj
co esponds o inc easing jQ−N=2j.
The la ge-NKondo in e ac ion −λKO†O=2is classically
ma ginal, hence Ohas UV dimension Δ¼1=2, which
ixes M2and hence Φ’s nea -bounda y expansion,
Φ∼ffiffiffi
z
pðαlog zþβÞ. In oducing he Kondo in e ac ion
amoun s o adding a bounda y e m ∝−λKO†O=2 o S,
which changes Φ’s bounda y condi ion om α¼0 o α¼
−λKβ[18,49,50]. Fo mo e de ails abou he bounda y
e ms, see [19,44]. A holog aphic scaling analysis e eals
ha λK uns loga i hmically, λK¼1=log ðT=TKÞ, di e g-
ing a he dynamically-gene a ed Kondo empe a u e,
TK≡Λe−1=λK=ð2πÞ, wi h λKe alua ed a he UV cu o ,
Λ. A holog aphic an i e omagne ic UV Kondo coupling,
λK>0, is hus ma ginally ele an , b eaks con o mal
in a iance, and igge s an RG low.
As men ioned abo e, ou model has a mean- ield phase
ansi ion [18]:hOi¼0(Φ¼0) when T>T
cand hOi≠0
(Φ≠0) when T≤Tc. Condensa e o ma ion hOi≠0
b eaks he cha ge and auxilia y Uð1Þ’s o he diagonal,
and signals he Kondo e ec , including a phase shi o ψL,
dual o a Wilson line o A, and impu i y sc eening, dual o
educ ion o lux be ween z¼0and z¼zH. We e e
o he T>T
cand T≤Tcphases as “unsc eened”and
“sc eened,” espec i ely. In [18–21] we compu ed Tc
nume ically. Below we ob ain an exac o mula o Tc.
III. FANO RESONANCES
I a e a ded G een’s unc ion o complex equency ω,
GðωÞ, has a pole a ωp,GðωÞ∼Z
ω−ωp, wi h complex esidue
Z¼ZRþiZI, hen nea he pole he spec al unc ion
ρðωÞ≡−2ImðGðωÞÞ will ha e a Fano esonance [28,29]
(se ing ImðωÞ¼0),
ρFanoðωÞ¼ ðω−ω0þqΓ=2Þ2
ðω−ω0Þ2þðΓ=2Þ2;ð2Þ
wi h posi ion ω0¼ReðωpÞ, wid h Γ¼2jImðωpÞj, and
asymme y pa ame e q¼−ZR=ZIþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZ2
R=Z2
I
p.Fano
esonances a e an isymme ic when q¼1, meaning ρðωÞ
is odd unde PHS, and symme ic when q¼0(an an i-
esonance) o ∞(a Lo en zian), meaning ρðωÞis e en.
Fano esonances a ise when a Lo en zian esonance is
imme sed in a con inuum (in ene gy), due o in e e ence
be ween he wo. The asymme y pa ame e qcon ains
key dynamical in o ma ion, speci ically, q2∝ he a io o
p obabili ies o esonan and non- esonan sca e ing.
In ou model, he AdS2subspace inhe i s scale in a i-
ance om AdS3, o in dual ield heo y language, he
impu i y inhe i s scale in a iance om he CFT, so ρðωÞo
impu i y ope a o s mus be a ea u eless con inuum. Ou
ma ginally- ele an Kondo coupling hen b eaks scale
in a iance and p oduces a esonance, while Q≠0b eaks
PHS. We will show ha ρðωÞo O hen indeed gene ically
exhibi s asymme ic Fano esonances.
IV. SPECTRAL FUNCTIONS
We compu e GðωÞholog aphically by sol ing o
linea ized luc ua ions abou solu ions o he unsc eened
and sc eened phases [44,51,52]. A all T, we ind ha he
Kac-Moody cu en ’sGðωÞis una ec ed by he impu i y.
In he unsc eened phase, we ind ha all cha ged GðωÞ
anish, i.e. GOOðωÞ≡hOðωÞOð−ωÞi ¼ 0, while
GO†OðωÞ≡hO†ðωÞOð−ωÞi ¼ N
λK1−1
λKDðωÞ;
DðωÞ≡H−1
2þiQ −iω
2πTþH−1
2−iQ
þln2T
TK;
wi h Ha monic numbe H½x, and λKe alua ed a Λ. The
o m o GOO†ðωÞis he same, bu wi h Q→−Q. Scale
in a iance in (0þ1)-dimensions and Δ¼1=2imply a
i ial UV con inuum: limω→∞ρO†OðωÞ¼0.
Fo gi en Qand T,GO†OðωÞhas poles in ωwhen
DðωÞ¼0. Figu e 1shows ou nume ical esul s o he
FIG. 1. Posi ions o poles in he complex ω=ð2πTÞplane
o Q¼1=2. Blue and pu ple deno e lowes and nex -lowes
poles, espec i ely, o GO†OðωÞ(ReðωÞ>0) and GOO†ðωÞ
(ReðωÞ<0), o T=Tc om 100 down o 1.001. Red and o ange
deno e he same o GO†OðωÞ o T=Tc om 1 down o 0.2.
A ows indica e mo emen o poles as Tdec eases.
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posi ions o he lowes (closes o ω¼0) and nex -lowes
poles o GO†OðωÞand GOO†ðωÞin he complex ω=ð2πTÞ
plane, o Q¼1=2. O he Qgi e simila esul s. As
T→Tþ
c, he lowes pole mo es owa ds he o igin, a i es
he e a Tc, and when T<T
c, c osses in o he ImðωÞ>0
egion, signaling ins abili y (no shown). We hus iden i y
Tcas he Twhe e Dðω¼0Þ¼0,
Tc¼1
2TKexp −2ReH1
2þiQ:
Figu e 2shows he no malized spec al unc ion
¯
ρO†OðωÞ≡−2λ2
K
NImGO†OðωÞ e sus eal ω=ð2πTÞ o Q¼
1=2and T=Tc¼16, 8, 4, 2. We ind a Fano esonance,
as ad e ised, wi h asymme ic minimum and maximum.
Nume ically, ω0≈ReðωpÞand Γ≈2jImðωpÞj,asin(2),
whe e ωpis GO†OðωÞ’s lowes pole. As T→Tþ
c,qg ows:
q≈1.7a T¼16Tcwhile q≈4a T¼2Tc.
Fo Tjus abo e Tc,T≳Tc, expanding in Tabou Tc
and in ωabou ω¼0gi es, o GO†OðωÞ’s lowes pole,
ωp
2πT¼−iT=Tc−1
ψ0½1
2þiQ;Z¼−iN
λ2
K
2πTc
ψ0½1
2þiQ;ð3Þ
wi h digamma unc ion ψ½x. The esonance heigh hus
g ows as ðT=Tc−1Þ−1and he wid h sh inks as T=Tc−1.
I is he e o e no ela ed o a Kondo esonance, which
g ows loga i hmically as T→Tþ
K[22]. Indeed, a la ge N
we expec he Kondo esonance only in he sc eened phase
[23]. Ou esonance is p esumably a bound s a e o ψL
and χ, he alding he nascen sc eened phase.
The Zin (3) gi es q ha depends only on Q, shown in
Fig. 3. (An i)symme ic alues q¼1,0,∞occu when
Q→0,∓∞, espec i ely. Indeed, Fig. 4shows ha e en
o ela i ely modes Q¼1, he esonance is nea ly
Lo en zian, he minimum ha ing p ac ically anished.
In he sc eened phase, we ha e nume ical esul s o
GO†OðωÞ[18–21,44]. Figu e 1shows ou nume ical esul s
o he posi ions o he lowes and nex -lowes poles in
GO†OðωÞ o Q¼1=2. O he Qgi e simila esul s. A
T¼Tc he poles a e coinciden wi h hose o GO†OðωÞand
GOO†ðωÞin he unsc eened phase. As Tdec eases below
Tc,GO†OðωÞ’s lowes pole, ωp, mo es s aigh down he
ImðωÞaxis.
FIG. 2. The no malized spec al unc ion ¯
ρO†OðωÞ e sus eal
ω=ð2πTÞ o Q¼1=2and, om sho es o alles , T=Tc¼16
( ed), 8 (g een), 4 (o ange), and 2 (blue).
FIG. 3. Asymme y pa ame e q e sus Q, o T≳Tc.
FIG. 4. The no malized spec al unc ion, ¯
ρO†OðωÞ, e sus eal
ω=ð2πTÞ o Q¼1and, om sho es o alles , T=Tc¼1.04
(g een), 1.02 (o ange), and 1.01 (blue).
FIG. 5. The no malized spec al unc ion ¯
ρO†OðωÞ e sus eal
ω=ð2πTÞ o Q¼1=2and, om alles o sho es , T=Tc¼0.998
(blue), 0.991 (o ange), and 0.964 (g een).
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F om ou expe ience wi h he unsc eened phase, we
expec ωp o p oduce a Fano esonance in he no malized
spec al unc ion, ¯
ρO†OðωÞ. C ucially, ReðwpÞ¼0,soωp
p ese es PHS, ReðωÞ→−ReðωÞ, so we expec an an i-
symme ic Fano esonance a ReðωÞ¼0. Mo eo e ,
jImðωpÞjinc eases as Tdec eases, and so should he wid h
Γ. Figu e 5con i ms ou expec a ions: ¯
ρO†OðωÞ’s only
signi ican ea u e is a Fano esonance a ReðωÞ¼0
wi h q¼1, meaning pe ec ly an isymme ic minimum
and maximum, and whose Γinc eases as Tdec eases.
Addi ionally, he heigh dec eases, and indeed ou nume ics
sugges limT→0¯
ρO†OðωÞ¼0.
Figu e 6shows ou nume ical esul s o he posi ion
o ωp e sus small λ2
K
N2hOi2=ð2πTÞ, o equi alen ly, Tjus
below Tc,T≲Tc, o Q¼1=2. Figu e 6also shows a
linea i demons a ing ha .1
ωp∝−ihOi2:ð4Þ
Ou model’s mean- ield beha io hOi∝ðTc−TÞ1=2 hen
implies Γ∝Tc−T o T≲Tc.
The beha io in (4) is in ac iden ical o ha in a LFL
a la ge N. In a LFL, he Kondo esonance is o mally
de ined in he LFL e mion spec al unc ion, and a la ge N
appea s only in he sc eened phase, wi h Γ∝hOi2[23].Fo
T≲Tc, he mean- ield beha io hOi∝ðTc−TÞ1=2 hen
implies Γ∝Tc−T. C ucially, in he sc eened phase he
Kondo esonance also appea s in o he spec al unc ions,
due o ope a o mixing induced by he symme y b eaking
[23]. In pa icula , a Kondo esonance should p oduce a
pole in GO†OðωÞp ecisely o he o m in (4)2Ou esul (4)
hus p o es he exis ence o a Kondo esonance in ou
model when T≲Tc, wi h de ining ea u es essen ially
in ac despi e he s ong in e ac ions.
V. CONCLUSION
In a holog aphic model desc ibing he in e ac ion o a
magne ic impu i y wi h a s ongly co ela ed CFT a la ge
N, we disco e ed a no el mechanism o p oducing Fano
esonances, namely ia RG lows be ween (0þ1)-dimen-
sional ixed poin s. The o igin and consequences o such
Fano esonances, in exis ing cases ha ha e gone uniden-
i ied and in no el cases, dese e u he s udy, pa icula ly
o he physics con ained in he asymme y pa ame e q.
ACKNOWLEDGMENTS
We would like o hank Ian A leck, Na han And ei, Pie s
Coleman, Ma io Flo y, Hen ik Johannesson, And ew
Mi chell, Max New zella, and Philip Phillips o help ul
con e sa ions and co espondence. C. H. is suppo ed
by he Ramon y Cajal ellowship RYC-2012-10370, he
As u ian G an No. FC-15-GRUPIN14-108 and he
Spanish na ional G an No. MINECO-16-FPA2015-
63667-P. A. O’B. is a Royal Socie y Uni e si y Resea ch
Fellow. J. P. is suppo ed by he Cla endon Fund and S
John’s College, Ox o d, and by he Eu opean Resea ch
Council unde he Eu opean Union’s Se en h F amewo k
P og amme (ERC G an ag eemen 307955).
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