PHYSICAL REVIEW E 107, 064134 (2023)
Exci ed-s a e quan um phase ansi ions in he anha monic
Lipkin-Meshko -Glick model: Dynamical aspec s
J. Khalou -Ri e a ,1,2,*J. Gami o ,3F. Pé ez-Be nal ,2,4J. M. A ias ,3,4and P. Pé ez-Fe nández 1,4
1Depa amen o de Física Aplicada III, Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Se illa, 41092 Se illa, Spain
2Depa amen o de Ciencias In eg adas y Cen o de Es udios A anzados en Física, Ma emá icas y Compu ación,
Uni e sidad de Huel a, Huel a 21071, Spain
3Depa amen o de Física A ómica, Molecula y Nuclea , Facul ad de Física, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
4Ins i u o Ca los I de Física Teó ica y Compu acional, Uni e sidad de G anada, Fuen enue a s/n, 18071 G anada, Spain
(Recei ed 22 Feb ua y 2023; accep ed 7 June 2023; published 26 June 2023)
The s anda d Lipkin-Meshko -Glick (LMG) model unde goes a second-o de g ound-s a e quan um phase
ansi ion (QPT) and an exci ed-s a e quan um phase ansi ion (ESQPT). The inclusion o an anha monic e m
in he LMG Hamil onian gi es ise o a second ESQPT ha al e s he s a ic p ope ies o he model [Gami o e al.,
Phys.Re .E106, 044125 (2022)]. In he p esen wo k, he dynamical implica ions associa ed o his new ESQPT
a e analyzed. Fo ha pu pose, a quan um quench p o ocol is de ined on he sys em Hamil onian ha akes an
ini ial s a e, usually he g ound s a e, in o a complex exci ed s a e ha e ol es on ime. The impac o he new
ESQPT on he ime e olu ion o he su i al p obabili y and he local densi y o s a es a e he quan um quench,
as well as on he Loschmid echoes and he mic ocanonical ou -o - ime-o de co ela o (OTOC) a e discussed.
The anha moni y-induced ESQPT, despi e ha ing a di e en physical o igin, has dynamical consequences
simila o hose obse ed in he ESQPT al eady p esen in he s anda d LMG model.
DOI: 10.1103/PhysRe E.107.064134
I. INTRODUCTION
The use o oy models has been undamen al o impo -
an ad ances in all b anches o physics. These a e non i ial
models bu s ill simple enough o be sol ed analy ically and
hey can be used ei he o look in o limi ing si ua ions in
complex sys ems o o check and be e unde s and di e en
app oxima ion echniques. Some ele an examples o sol -
able models a e Ellio ’s o a ional su(3) model [1] and he
in e ac ing boson model [2–5] in nuclea physics, he Rabi
[6,7], Jaynes-Cummings [8], and Dicke models [9] in quan-
um op ics, o he Lipkin-Meshko -Glick (LMG) model in
many-body physics [10–12], jus o men ion a ew o hem.
In many cases, such models we e o iginally in oduced in
a pa icula b anch o physics and hey we e la e used in
comple ely di e en ields. In pa icula , he LMG model was
o iginally p oposed o es many-body app oxima ions such
as he ime-dependen Ha ee-Fock o pe u ba ion me hods
in nuclea sys ems [10–12], bu i has demons a ed o be
e y use ul o he s udy o quan um phase ansi ions (QPTs)
[13–16] and has been ealized expe imen ally wi h op ical
ca i ies [17], Bose-Eins ein condensa es [18], nuclea mag-
ne ic esonance sys ems [19], apped a oms [20–23], and
cold a oms [24]. Fo ins ance, he LMG model has been used
o es he possible exis ence o exci ed-s a e quan um phase
ansi ions (ESQPTs) [25] and ela ions be ween ESQPTs and
quan um en anglemen [26,27], o quan um decohe ence [28].
The ESQPT concep was in oduced in [29] and an excellen
e iew on his opic has been ecen ly published [30].
*Co esponding au ho : [email p o ec ed]
I is wo h no ing ha phase ansi ions a e well de ined o
mac oscopic sys ems, howe e , he same ideas can be applied
in mesoscopic sys ems whe e one can obse e phase ansi ion
p ecu so s e en o mode a e sys em sizes [31]. When dealing
wi h mesoscopic sys ems, he s udy o hei mean- ield o
la ge-size limi is a aluable e e ence o connec he p ecu -
so s wi h he nonanali ici ies expec ed in a QPT. Toy models,
such as he LMG model, a e simple enough o be sol ed o a
la ge numbe o pa icles, allowing o a clea connec ion wi h
he a o emen ioned la ge-size limi .
This wo k is pa o a mo e comple e s udy on he anha -
monic LMG (ALMG) model. The addi ional anha monic e m
induces, in addi ion o he al eady known ESQPT [28,32],
an anha monici y-induced ESQPT ha needs o be well un-
de s ood. In a p e ious publica ion [33], he s a ic aspec s
o bo h he g ound-s a e QPT and he wo ESQPT’s in he
ALMG model we e cha ac e ized. A mean- ield analysis in
he la ge-Nlimi was pe o med and di e en obse ables
we e used o cha ac e ize he di e en quan um phase an-
si ions in ol ed: The ene gy gap be ween adjacen le els, he
g ound-s a e QPT o de pa ame e , he pa icipa ion a io, he
quan um ideli y suscep ibili y, and he le el densi y. In his
wo k, we concen a e on he in luence o he wo ESQPTs on
he dynamics o he ALMG model. Wi h his aim, a quan-
um quench p o ocol ha consis s o an ab up change in
one o he con ol pa ame e s in he ALMG Hamil onian is
de ined. Then, he local densi y o s a es (LDOS, also known
as s eng h unc ion) oge he wi h he e olu ion o he sys em
a e he quench a e s udied using he ime e olu ion o he
su i al p obabili y, Loschmid echoes, and an ou -o - ime-
o de co ela o (OTOC).
2470-0045/2023/107(6)/064134(13) 064134-1 ©2023 Ame ican Physical Socie y
J. KHALOUF-RIVERA e al. PHYSICAL REVIEW E 107, 064134 (2023)
The p esen pape is o ganized as ollows. In Sec. II, he
ALMG model is in oduced, i s algeb aic s uc u e e iewed,
and he ele an ma ix elemen s o he calcula ions in he
u(1) basis a e explici ly gi en. Sec ion III is de o ed o he
analysis o a quan um quench p o ocol. Pa icula ly, he ime
e olu ion o he su i al p obabili y when he sys em unde -
goes a quan um quench is discussed o unde s and how his
quan i y is in luenced by he p esence o he ESQPTs in he
sys em. In Sec. IV, he ESQPTs impac on he e olu ion o an
OTOC is explo ed. Finally, some conclusions a e p esen ed in
Sec. V.
II. THE MODEL
The LMG model can be used o desc ibe one-dimensional
spin-1/2 la ices wi h in ini e- ange in e ac ions [10–12]. Fo
an a ay o Nsi es, he Hamil onian is w i en in e ms o
collec i e spin ope a o s ˆ
Sβ=N
i=1ˆsi,β wi h β=x,y,zand
whe e ˆsi,β is he βcomponen o he spin ope a o o a
pa icle in si e i. The e o e, he usual LMG Hamil onian is
w i en as
ˆ
H=(1 −ξ)(S+ˆ
Sz)+2ξ
SS2−ˆ
S2
x,(1)
wi h S=N/2. The ope a o ˆ
Sxcan be w i en in e ms
o he usual ladde ope a o s ˆ
S+and ˆ
S−, de ined as
ˆ
S±=ˆ
Sx±ıˆ
Sy, and ξ∈[0,1] is a con ol pa ame e ha
d i es he sys em om one phase o he o he one. In-
deed, om an algeb aic poin o iew, he Eq. (1)LMG
Hamil onian p esen s a u(2) algeb aic s uc u e wi h wo
limi ing dynamical symme ies: u(2) ⊃u(1) and u(2) ⊃
so(2) [34]. Each dynamical symme y is associa ed wi h
a di e en phase o he physical sys em. Fo ξ=0 he
sys em educes o he u(1) dynamical symme y and his
phase is usually e e ed o as he no mal (o symme ic)
phase, whe eas o ξ=1 heso(2) dynamical symme y is
ealized and he co esponding phase is called he de o med
(o b oken-symme y) phase [34].
Inspi ed by he wo ks in Re s. [35–37], we ha e included
in he Eq. (1) Hamil onian a second-o de Casimi ope a o o
u(2), S2
z,
ˆ
H=(1 −ξ)(S+ˆ
Sz)+2ξ
SS2−ˆ
S2
x
+α
2S(S+ˆ
Sz)(S+ˆ
Sz+1).(2)
Again, he Hamil onian depends on he ξcon ol pa am-
e e which d i es he sys em be ween phases. In addi ion,
a new con ol pa ame e , α, is in oduced. The pu pose o
his wo k is o explo e he in luence o his new e m and
he co esponding con ol pa ame e on he dynamics o he
sys em. I is wo h no icing ha o α=0, he o iginal Hamil-
onian, Eq. (1), is eco e ed, and o αdi e en om ze o, he
ξ=0 limi is ans o med om a unca ed one-dimensional
ha monic oscilla o o an anha monic oscilla o . Tha is he
eason why Hamil onian (2) is e e ed o as he anha monic
LMG model. Mo eo e , we obse e ha he so(2) limi is no
longe eco e ed o ξ=1 unless αis ze o.
The Hilbe space o his sys em has dimension 2N,bu
due o he conse a ion o he o al spin, [ ˆ
S2,ˆ
H]=0, we
can ocus on he sec o o maximum i ep o he sys em, so
he o al spin quan um numbe S=N/2 h ough he wo k.
This leads o a d as ic educ ion o Hilbe space dimen-
sion ha now becomes N+1. Howe e , he basis o he
Hilbe space gi en by he subalgeb a u(1), |S,Mzwi h Mz=
−N/2,...,0,...,N/2 ( he p ojec ion o he o al spin Son he
zdi ec ion), is used along his wo k. The ma ix elemen s o
Hamil onian (2)in heu(1) basis a e gi en by
S,M
z|ˆ
Sz|S,Mz=MzδM
z,Mz,
S,M
z|ˆ
S2
z|S,Mz=M2
zδM
z,Mz,
ˆ
S2
x=1
4(ˆ
S2
++ˆ
S2
−+ˆ
S+ˆ
S−+ˆ
S−ˆ
S+),
S,M
z|ˆ
S+ˆ
S−+ˆ
S−ˆ
S+|S,Mz=NN
2+1−2M2
zδM
z,Mz,
S,M
z|ˆ
S2
+|S,Mz=N
2N
2+1−Mz(Mz+1)N
2N
2+1−(Mz+1)(Mz+2)δM
z,Mz+2,
S,M
z|ˆ
S2
−|S,Mz=N
2N
2+1−Mz(Mz−1)N
2N
2+1−(Mz−1)(Mz−2)δM
z,Mz−2.(3)
In addi ion, Hamil onian (2) conse es pa i y (−1)S+Mzand
he ope a o ma ix can be spli in o wo blocks, he i s one
including e en pa i y s a es and he second one wi h odd pa i y
s a es, wi h dimensions N/2+1 and dimension N/2 o an
e en N alue.
A comple e mean- ield analysis o he semiclassical limi
o Hamil onian (2) has been ca ied ou using spin cohe en
s a es in Re . [33], e ealing o α<0 a second-o de g ound-
s a e QPT as well as wo c i ical lines co esponding o wo
ESQPTs and bo h ma ked by a high densi y o s a es. A
ecen ly published wo k by Nade and collabo a o s ocuses
on a gene al LMG Hamil onian ha can be easily connec ed
wi h ou ALMG ealiza ion [38]. One o hese high densi y o
s a es c i ical lines was al eady known o he LMG model
064134-2
EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 107, 064134 (2023)
0 0.2 0.4 0.6 0.8 1
ξ
-0.2
0
0.2
0.4
0.6
0.8
ε
ξ1
ξ2
εgs(ξ1)
εc1(ξ2)
ε =dεgs(ξ)/dξ|ξ1
(ξ−ξ1)+εgs(ξ1)
εc1=ξ
εc2=0.4
FIG. 1. Illus a ion o he angen me hod discussed in he ex o α=−0.6. Ene gy spec um o he sys em in he plane ε×ξwhe e he
wo ESQPT c i ical lines a e highligh ed wi h ed and yellow dashed lines. The dashed blue line is he angen o he g ound-s a e cu e εgs(ξ)
a he poin ξ1. This shows schema ically he g aphical de e mina ion o he c i ical quench ξ1→ξ2 o a gi en ini ial s a e. In gene al, he
in e sec ion o he angen line wi h he c i ical lines p o ides he c i ical ξ2 alue o which he sys em eaches he ESQPT c i ical ene gy a e
he quench. The dashed blue line s ands o he angen o he highes exci ed-s a e cu e.
[28,32]. He e, we pay heed o he o he one, ha we call
anha monici y-induced ESQPT c i ical line [33]. Pa icula ly,
i is wo h explo ing whe he his c i ical line is o a simila
na u e as he o he one and o wha ex en i has an impac on
he sys em dynamics. Fo his pu pose, he dynamics o he
sys em is s udied by means o he su i al p obabili y once he
sys em unde goes a quan um quench and an ou -o - ime-o de
co ela o (OTOC).
III. QUENCH DYNAMICS
The e olu ion o he sys em desc ibed by Hamil onian (2)
a e a quan um quench should be sensi i e o he p esence o
ESQPTs [28,38–41]. We explo e he ESQPT in luence on he
sys em dynamics wi h a quan um quench p o ocol, s a ing
om an eigens a e o he Hamil onian, ypically he g ound
s a e, and ollowing he sys em e olu ion once a con ol pa-
ame e in ˆ
His ab up ly modi ied. The quenching b ings he
sys em o an exci ed s a e ha e ol es wi h ime. The analysis
o he ensuing sys em dynamics is a aluable ool o de ec
and explo e ESQPTs in physical sys ems [28,39]. Le us jus
no e ha , om a ma hema ical poin o iew, his quenching
analysis can be pu in ela ion o he su i al p obabili y o a
pa icula ealiza ion o he Loschmid echo.
The Hamil onian in Eq. (2) depends on wo con ol pa am-
e e s, ξand α. In gene al, o nega i e α alues, he e exis wo
di e en ESQPTs and each one o hem has a c i ical ene gy
line ma ked by a high le el densi y [33,38]. Since we a e in e -
es ed in cha ac e izing bo h ESQPTs, a ixed alue o α<0is
selec ed and he ime e olu ion o he sys em is explo ed a e
an ab up change in he con ol pa ame e ξ. The aim is o
s udy how he sys em dynamics is modi ied by he exis ence
o wo c i ical lines. In he ollowed quan um quench p o ocol,
he sys em is ini ially p epa ed in a ce ain no malized eigen-
s a e |0o ˆ
H1=ˆ
H(ξ1). A ime =0 a quan um quench
akes place, changing ξ om ξ1 o ξ2. Thus, he Hamil onian
o he sys em is now gi en by ˆ
H2=ˆ
H(ξ2) and he ini ial
s a e, |0, is no longe an eigens a e o ˆ
H2and, consequen ly,
e ol es wi h ime in a non i ial way. The p obabili y ampli-
ude o inding he e ol ed s a e, |0( ), in he ini ial s a e,
|0, can be e alua ed easily. The exp ession o his p oba-
bili y ampli ude, deno ed as a( ), is a( )=0|0( ).The
su i al p obabili y, F( ), also called nondecay p obabili y o
ideli y, is gi en by he absolu e squa e o a( ),
F( )=|a( )|2=|0|0( )|2=|0|e−ıˆ
H2 |0|2.(4)
Since ou goal is o e ince he e ec on he sys em dynam-
ics o he ex e nal quench when eaching one o he ESQPTs’
c i ical lines, he de e mina ion o sui able ξ2 alues is e y
impo an , since he quenched sys em has o each he co -
esponding c i ical ene gies. This can be achie ed using he
me hod o he angen , de eloped in Re . [40]. In Fig. 1,a
ypical e olu ion o he ene gy le els, ε, o he Hamil onian
in Eq. (2) is plo ed as a unc ion o he con ol pa ame e
ξ, o a alue o α=−0.6. In his igu e, he e is a change
in he g ound s a e a a ound ξ=0.2 ha co esponds o he
g ound-s a e QPT. In addi ion, wo lines o high le el densi y
in he exci a ion spec a a e immedia ely appa en (sepa a i-
ces, see Re . [33]). These lines ma k he c i ical ene gy o
064134-3
J. KHALOUF-RIVERA e al. PHYSICAL REVIEW E 107, 064134 (2023)
he ESQPTs and sepa a e he phases in such ansi ions. In
he case shown in his igu e, he sepa a ices occu a he
c i ical ene gies εc1=ξ(yellow dashed line) and εc2=0.4
( ed dashed line). A de ailed discussion on his s uc u e,
including hei dependence o he con ol pa ame e s in he
mean- ield limi , can be ound in Re . [33], whe e he s a ic
p ope ies o he ALMG model a e p esen ed. F om Fig. 1,
i is clea ha o analyzing he h ee phases one has o s a
om he de o med phase ξ>ξ
c=0.2. Due o he s uc u e o
ou Hamil onian, changing ξ om an ini ial alue ξ1implies
ha he sys em is exci ed along a s aigh line angen o he
ene gy line a ξ1. Thus, i he ini ial s a e is he g ound s a e
|0=|gs o a pa icula ξ1 alue (ξ1>0.2), one needs o
ind he alue o he ξpa ame e , ξ2, o which he angen
o he ini ial ene gy le el ε1(ξ)a ξ1c osses he c i ical line
εc(ξ)a ξ2in he plane ε×ξ. This is illus a ed in Fig. 1 o
he case in which he eigens a e |0=|gsis he g ound s a e
o H1=H(ξ1). I is wo h no icing ha , wi hin he ange o
alues de ined o ξ, using his me hod i is no possible o
c oss bo h ESQPTs lines om a gi en ini ial s a e. Indeed,
o hose alues o ξabo e he alue o he c i ical ξc o he
QPT, i is only possible o each he i s ESQPT, εc1=ξ,(i
can be seen plo ing he angen o he g ound-s a e line). I is
wo h men ioning ha i one uses he same angen me hod
s a ing om he symme ic phase (ξ<ξ
c=0.2), one can
each he second ESQPT c i ical line, εc2=0.4, (yellow line),
bu i would be impossible o explo e p ope ly i s impac
on he dynamics o he sys em since one is o ced o mo e
o e he i s ESQPT c i ical line ( ed dashed line). Conse-
quen ly, he angen me hod om he sys em g ound s a e is
sui able o he s udy o he i s ESQPT ( ed dashed line), bu
no he second one (yellow dashed line).
Le us i s examine he εc1=ξc i ical line ( ed dashed
line), ha can be eached using he angen me hod om he ξ1
g ound s a e. On he one hand, he ene gy o he co esponding
ini ial g ound s a e is εgs(ξ1) and he equa ion o he angen
line a ξ1 o he cu e desc ibed by he g ound s a e o
he sys em in he ε×ξplane eads ε =m(ξ−ξ1)+εgs(ξ1),
whe e mis he slope o he angen o he g ound-s a e cu e a
ξ1. On he o he hand, he line o he i s ESQPT ( ed dashed
line) is εc1=ξ. The e o e, bo h lines c oss a
ξ2=ξc1=mξ1−εgs(ξ1)
m−1,(5)
whe e εgs(ξ1)=gs|ˆ
H1|gs/N(g ound-s a e ene gy pe pa -
icle a ξ1) and he slope mo he angen line is ob-
ained making use o he Hellman-Feynman heo em in
Eq. (2). Indeed, m=gs|ˆ
H|gs/N=dεgs(ξ)/dξ|ξ=ξ1, whe e
ˆ
H=2
S(S2−ˆ
S2
x)2−(S+ˆ
Sz).
A simila analysis can be pe o med o he anha monici y-
induced c i ical line. Howe e , as we no iced abo e, he
angen o any poin along he g ound-s a e line wi h ξ>ξ
c
ne e c osses he second c i ical line (dashed yellow line) o
he ange o alues o ξconside ed in his model. Hence, o
explo e his sepa a ix one should s a om a mo e app op i-
a e ˆ
H1eigens a e. In pa icula , we ha e selec ed he highes
exci ed s a e (deno ed as |∗). As wi h he g ound s a e, he
mos exci ed s a e o ou sys em is well-de ined in he he mo-
dynamic limi by a cohe en s a e [42]. Then, ou ini ial s a e is
now |0=|∗o ˆ
H1. Le us deno e he slope o he angen
o he ene gy line o he highes s a e a ξ1as m2. Then, his
angen line will each he anha monici y-induced ESQPT line
gi en by εc2=ε0which is a cons an . In he α=−0.6, he
alue o ε0=0.4 was compu ed wi h a mean- ield o malism
[33]. The e o e, he alue o he c i ical ξ,ξc2, eads
ξc2=m2ξ1+ε0−ε∗(ξ1)
m2
,(6)
whe e ε∗(ξ1)=∗|ˆ
H1|∗/Nand m2=∗|ˆ
H|∗/N=
dε∗(ξ)/dξ|ξ=ξ1is he slope o he co esponding angen
line.
Once a way o c ossing bo h ESQPT lines is a ailable, he
dynamic e olu ion o he sys em and he e ec o c ossing an
ESQPT line can be examined. This can be accomplished com-
pu ing he su i al p obabili y F( )Eq.(4). Resul s o F( )as
a unc ion o ime a e shown in Fig. 2 o N=300, α=0(le
column) and −0.6 (cen e and igh columns) and di e en
ini ial s a es (ei he he g ound s a e |0=|gsin he le
and cen al columns o he mos exci ed s a e |0=|∗in
he igh column) o selec ed ξ alues. The α=0 case in
he le mos panels is included o he sake o comple eness
and e e ence. The panels in his column depic he ime
e olu ion o he su i al p obabili y o dec easing alues o
ξ2, s a ing always om he g ound s a e |gs o ξ1=0.6.
The calcula ed ξ2a he c ossing wi h he ESQPT is ξc=0.3.
In gene al, he su i al p obabili y has a egula oscilla o y
beha io excep in he egion close o he ESQPT c i ical
ene gy, ξ2=0.3, whe e he sys em unde goes an ESQPT and
he su i al p obabili y suddenly d ops down o ze o and s a s
o oscilla e andomly wi h small ampli udes. Once he c i ical
ene gy o he ESQPT is c ossed, he su i al p obabili y
s a s o oscilla e in a egula way again. This phenomenon
was epo ed o he i s ime in Re . [28]. In he cen al and
igh mos columns he same obse able is plo ed including a
nonze o anha monic e m (α=−0.6).
In he panels o he second column o Fig. 2, he su i al
p obabili y is depic ed o dec easing alues o ξ2and s a ing
always om he g ound s a e o a Hamil onian wi h ξ1=0.6
and α=−0.6. Due o he nega i e α alue, he sys em un-
de goes wo ESQPTs, displayed in he spec um by means o
c i ical lines wi h a no ewo hy accumula ion o ene gy le els
(see Fig. 1). One o he wo c i ical lines ( ed dashed line)
can be aced back o he ESQPT al eady p esen in he α=0
case [25]. Howe e , he second one (yellow dashed line) is
linked o he p esence o he anha monic e m in he Hamil-
onian [33]. The na u e and physical in e p e a ion o he
anha monici y-induced ESQPT is di e en om he al eady
known ESQPT associa ed wi h he g ound-s a e QPT. Hence,
in p inciple, he e is no a-p io i eason o bo h beha ing in
he same way. Howe e , as we see i we compa e he esul s
o he c i ical ξc alues in he di e en columns, he esul s
ob ained o he α=0 and he anha monic cases a e simila .
The su i al p obabili y is oscilla o y and egula excep once
ξ2is close o ξc, he c i ical alue o he i s o second ES-
QPT. In all cases, when ξ2=ξc, he quenched sys em eaches
he c i ical ene gy and he su i al p obabili y suddenly d ops
down o ze o and oscilla es andomly wi h a small ampli ude
( ed cu es). This is simila o wha happens in he α=0 case.
Once ξ2is smalle han ξc, a pe iodic oscilla o y decaying
064134-4
EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 107, 064134 (2023)
0.25
0.5
0.75
1
F( )
ξ2=0.5
α=0, |ψ0>=|gs>
ξ2=0.5
α= −0.6, |ψ0>=|gs>
ξ2=0.6
α=−0.6, |ψ0>=|ψ∗>
0.25
0.5
0.75
1
F( )
ξ2=0.4 ξ2=0.3 ξ2=0.5
0.25
0.5
0.75
1
F( )
ξc=0.300 ξc=0.241 ξc=0.255
0 10203040
0.25
0.5
0.75
1
F( )
ξ2=0.1
010203040
ξ2=0.1
010203040
50
ξ2=0.1
FIG. 2. Su i al p obabili y F( ) as a unc ion o ime ( ) o a sys em size N=300. The le mos column includes F( ) esul s o α=0
and he middle and igh mos columns o α=−0.6. The ini ial s a e o he le mos and middle columns is he ξ1=0.6 Hamil onian g ound
s a e, |0=|gs, and he ini ial s a e o he igh mos column is he ξ1=0.7 Hamil onian highes exci ed s a e, |0=|∗, o each he
second c i ical line o he ene gy spec um. Di e en quan um quenches a e shown o di e en alues o ξ2. The e a e some c i ical alues
o ξ2,ξc, o which he sys em is se led in he c i ical ene gy o an ESQPT ( hi d ow) and he su i al p obabili y d ops down o ze o (wi h
small andom luc ua ions).
beha io is obse ed in F( ). As explained abo e, he quench
om he g ound s a e ne e eaches he second ESQPT line.
Fo ha pu pose, one has o s a om a di e en ini ial s a e.
Thus, o explo e how F( ) is a ec ed by he second ESQPT,
he quan um quench is pe o med using as an ini ial s a e he
highes exci ed s a e o he sys em, |∗, o a gi en alue
o ξ1. In his way, he second c i ical line (yellow dashed
line) o he ESQPT is accessible a e he quench. In he
panels o he igh column o Fig. 2, he su i al p obabili y
o dec easing ξ2 alues is plo ed o an ini ial s a e equal
o he highes exci ed s a e o he Hamil onian wi h ξ1=
0.7 and α=−0.6. Fo his pa ame e selec ion, he second
c i ical line is eached a ξc=0.255. In his column, again,
esul s a e e y simila o he ones ob ained in he p eceding
cases. The ideli y F( ) oscilla es egula ly while ξ2>ξ
c2,
bu when he ξ2pa ame e ge s close o he c i ical alue,
ξc2=0.255, he su i al p obabili y d ops down o ze o and
andomly oscilla es wi h a small ampli ude. Once he c i ical
line is c ossed, F( ) eco e s an oscilla o y decaying pe iodic
beha io , bu a a ce ain ime, his pe iodic oscilla o y be-
ha io becomes dis o ed. The eason o his phenomenon
is ha he angen line o he highes exci ed-s a e cu e a
ξ1in he plane ε×ξ emains e y close o he c i ical line
εc2=0.4 a e he quench o lowe alues o ξ2up o 0. One
should no e ha when s a ing om he highes exci ed s a e
he i s ESQPT c i ical line is no accessible a e he quench
(see Fig. 1).
I we deno e he eigens a es o ˆ
H(ξi,α) wi h i=1,2as
|ψj(ξi)wi h j=0,1,...,N/2, hen we can w i e he ini ial
s a e |0in he basis o ˆ
H2=ˆ
H(ξ2,α) eigen unc ions as
|0=jCj|ψj(ξ2)and hen
F( )=0|e−ıˆ
H2 |0
2
=
j
|Cj|2e−iEj 
2
=dEe−iE ρ0(E)
2
,(7)
whe e Ejis he ene gy o he j h ˆ
H2eigens a e and
ρ0(E)=j|Cj|2δ(E−Ej), called he s eng h unc ion o
local densi y o s a es (LDOS) [43,44], is he ene gy dis ibu-
ion o |0weigh ed by he Cjcomponen s.
F om Eq. (7) i is clea ha he ideli y F( ) is he abso-
lu e alue o he LDOS Fou ie ans o m squa ed and his
quan i y can p o ide some clues on he F( ) ime dependence
o he quench a he c i ical alues ξc, deno ed in ed in
he hi d ow o Fig. 2.InFig.3we plo he LDOS o he
same cases included in Fig. 2, hence in he i s column, we
show he LDOS o he g ound s a e o ˆ
H1=ˆ
H(ξ1=0.6,α =
0) o di e en ˆ
H2cases, all o hem wi h α=0. In he
064134-5

J. KHALOUF-RIVERA e al. PHYSICAL REVIEW E 107, 064134 (2023)
0.00 0.25 0.50
0.0
0.1
0.2
0.3 2=0.6
0.0 0.2 0.4
0.00
0.05
0.10
0.15 2=0.5
0.0 0.2
0.00
0.01
0.02
0.03
2= 0.255
0.0 0.2
0.00
0.02
0.04
2=0.1
0.0 0.2 0.4
0.00
0.25
0.50
0.75 2=0.5
0.0 0.2
0.0
0.1
0.2 2=0.3
0.0 0.1 0.2
0.000
0.025
0.050
0.075
2= 0.241
0.0 0.2
No malized ene gy
0.00
0.05
0.10
2=0.1
0.0 0.5
0.0
0.2
0.4
0.6
2=0.5
0.0 0.5
0.0
0.1
0.2
LDOS | j(2)| 0(1)|
2
2=0.4
0.0 0.5
0.00
0.02
0.04
0.06
2=0.3
0.0 0.5
0.00
0.05
0.10
2=0.1
FIG. 3. LDOS |ψj(ξ2)|ψ0|2as a unc ion o he no malized exci a ion ene gy εj(a b. uni s) o sys ems wi h ξ1=0.6andα=0.0(le
column), ξ1=0.6andα=−0.6 (middle column), and ξ1=0.7andα=−0.6 ( igh column) (N=300 in all cases). The chosen s a es a e he
g ound s a e |ψ0=|gs(ξ1)(le and middle columns) and he mos exci ed s a e wi h e en pa i y |ψ0=|∗(ξ1)( igh column), exp essed
in all cases in he basis o eigens a es o he Hamil onian ˆ
H(ξ2,α), being ξ2 he quench pa ame e . The cases ha co espond o a c i ical alue
o ξ2( hi d ow) a e plo ed using ed colo .
second and hi d columns we depic he LDOS o ini ial s a es
ha a e he g ound s a e o ˆ
H1=ˆ
H(ξ1=0.6,α =−0.6) and
he mos exci ed s a e o ˆ
H1=ˆ
H(ξ1=0.7,α =−0.6). The
LDOS o he c i ical quench alues a e depic ed in ed. I
can be clea ly seen ha , o all columns, in he c i ical con ol
pa ame e cases he LDOS is nonze o a he ESQPT c i ical
ene gy and has a clea local minimum a his ene gy alue.
The Fou ie ans o m o such LDOS p oduces he pa icula
ime dependence shown in he panels o he Fig. 2 hi d ow.
Ano he quan i y o in e es , inspi ed on Loschmid ’s ob-
jec ions o Bol zmann H heo em, is he Loschmid echo
[45,46]. This quan i y, conside ed a p obe o he sensibili y o
a sys em dynamics unde pe u ba ions, is used o benchma k
he eliabili y o quan um p ocesses [47]. I was shown o
be a alid QPT de ec o [48] and, mo e ecen ly, i has been
used o check he in luence o he ESQPT on he dynamics
o he LMG model [49]. Conside an ini ial wa e unc ion,
|ψ, which e ol es a ime unde a Hamil onian ˆ
H1,|ψ( )=
e−iˆ
H1 |ψ. We can e e se he ime e olu ion wi h ano he
Hamil onian ˆ
H2,eiˆ
H2 e−iˆ
H1 |ψ. The squa ed o e lap o he
esul an s a e wi h he ini ial s a e |ψis he Loschmid echo
(LE), deno ed as M( )[46,50],
M( )=|ψ|eiˆ
H2 e−iˆ
H1 |ψ|2.(8)
064134-6
EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 107, 064134 (2023)
Ano he physical in e p e a ion o his quan i y is possible,
since Eq. (8) is he dis ance be ween he same ini ial s a e once
i is e ol ed o a ime wi h wo di e en Hamil onian ope -
a o s. One o he p ope ies o g ound-s a e and exci ed-s a e
QPTs is ha , nea he c i ical egion, s a es a e qui e sensi i e
o pe u ba ions. A way o quan i y his e ec is compu ing
he LE o he eigens a es o he sys em ˆ
H1=ˆ
H(ξ,α) wi h a
ime- e e sal unde ˆ
H2=ˆ
H(ξ+δ, α),
Mj( )=|ψj(ξ,α)|eiˆ
H(ξ+δ,α) |ψj(ξ,α)|2,(9)
whe e |ψj(ξ,α)is he j h eigens a e o ˆ
H1and δis a small
pe u ba ion. The LE, as well as i s long- ime a e age alue,
de ec s he ESQPT in he LMG model wi hou anha monici y
[49].
In Fig. 4we plo Mj( ) o se e al eigens a es o a sys em
wi h ξ=0.3 and α=−0.6. The o al numbe o bosons
is N=300, he sys em has been pe u bed wi h δ=0.01,
and only s a es wi h e en pa i y a e conside ed. Resul s a e
shown o j=0,20,48,82,103, and 120. The wo s a es
wi h ene gies closes o ESQPTs c i ical ene gies (j=48 and
103) a e plo ed in ed. As expec ed, he g ound s a e j=0
pe o m small oscilla ions wi h a single equency a ound a
alue close o one, wi h a maximum alue equal o one. O he
s a es a om he c i ical egion, as j=20,82, and 120,
ha e a mo e complex oscilla ion pa e n, no ha monic, wi h a
la ge ampli ude and wi hou eaching uni y in he conside ed
ime ange. Howe e , in he case o eigens a es close o he
c i ical ene gy, j=48 and 103, M( ) is only one o =0
and he oscilla ions o he LE a e o a much mo e i egula
na u e, some hing simila o wha happens o he ideli y F( )
in Fig. 2.
As shown in Re . [49], he ime a e aged alue o he LE
o he j h eiens a e, Mj, is a con enien p obe o de ec an
ESQPT. This quan i y is de ined as
Mj=lim
T→∞
1
TT
0
d M( )=
kcδ
jk
4,(10)
whe e cδ
jk a e he coe icien s o he j h eigen unc ion o he
Hamil onian ope a o ˆ
H(ξ+δ, α), exp essed in he basis o
eigens a es o ˆ
H(ξ,α): |ψj(ξ+δ, α)=kcδ
jk |ψk(ξ,α).
The LE ime a e aged alue is equal o he in e se o he
pa icipa ion a io (PR) [51]o |ψj(ξ+δ, α)compu ed using
he basis {|ψk(ξ,α)}.InFig.5we plo he ime a e aged
LE e sus he no malized exci a ion ene gy o all e en pa i y
s a es o he sys em s udied in Fig. 4. The s a es included in
Fig. 4ha e been ma ked using ed pluses o c i ical ones
(j=48 and 103) and blue c osses o o he s (j=20,82,
and 120). Mjhas local maxima loca ed o he eigens a es
close o he c i ical ene gies, as i was obse ed in he LMG
model wi hou anha monici y [49]. Hence, he ime a e aged
LE de ec s he new ESQPT associa ed o he anha monic e m
in he LMG Hamil onian and con i ms ha his quan i y is a
good ESQPT p obe.
IV. ESQPTs AND OTOC
Ou -o - ime-o de co ela o s (OTOCs), ha appea ed o
he i s ime in he 1960s in he con ex o supe conduc i i y
[52], a e a ou -poin empo al co ela ion unc ion able o
measu e he en anglemen sp ead in a quan um sys em om
FIG. 4. Loschmid echoes o a sys em wi h ξ=0.3, α=−0.6,
N=300, and a pe u ba ion ac oss he con ol pa ame e ξo
δ=0.01. F om op o bo om we display Mj( ) o he j h s a e
wi h e en pa i y: 0, 20, 48, 82, 103, and 120. The s a es close o
he ESQPTs a e plo ed in ed.
he deg ee o noncommu a i i y in ime be ween ope a o s.
Since hen, a e a long pe iod o ela i e inac i i y, he e
has been a emendous enzy a ound his concep on a ious
on s [53]. They e u ned o he limeligh wi h he p oposal o
OTOCs as a iable quan um chaos indica o , due o i s expo-
nen ial inc ease a ea ly imes in ce ain sys ems [54–57], and
o diagnose he sc ambling o quan um in o ma ion [58–61].
Besides, OTOCs a e sensi i e p obes o quan um phase an-
si ions [62–68]. Despi e he ac ha he expe imen al access
o ou -o - ime-o de co ela o s is hinde ed by he unusual
064134-7
J. KHALOUF-RIVERA e al. PHYSICAL REVIEW E 107, 064134 (2023)
FIG. 5. Time-a e aged o M( ) e sus he no malized exci a ion
ene gy ε o he same sys em in oduced in Fig. 4. The highligh ed
s a es ( ed plus symbols o ansi ion s a es and blue c osses o
o he s) co espond o he s a es s udied in Fig. 4.
ime o de ing o i s cons i uen s ope a o s ha p ecludes he
measu emen using local ope a o s, se e al app oaches using
di e en expe imen al pla o ms ha e success ully p o ided
OTOC esul s [23,69–75].
Gi en wo ope a o s, ˆ
Wand ˆ
V, i is possible o p obe he
sp ead o ˆ
W( ) wi h ˆ
V h ough he expec a ion alue o he
squa e commu a o
Cw, ( )=[ˆ
W( ),ˆ
V(0)]†[ˆ
W( ),ˆ
V(0)],(11)
whe e ˆ
W( )=eıˆ
H ˆ
We
−ıˆ
H is he ope a o ˆ
Win he Heisen-
be g’s ep esen a ion [53,76–79]. The expec a ion alue is
usually compu ed in he canonical ensemble. Howe e , in
ecen wo ks, i has also been compu ed o e gi en ini-
ial s a es o o e he sys em eigens a es (mic ocanonical
OTOC) [77,78]. The squa ed commu a o Eq. (11) can be
ew i en as Cw, ( )=Aw, ( )−2Fw, ( ). The i s e m is
a wo-poin co ela o , Aw, ( )=ˆ
W†( )ˆ
V†(0) ˆ
V(0) ˆ
W( )+
ˆ
V†(0) ˆ
W†( )ˆ
W( )ˆ
V(0)and he ou -o - ime o de appea s in
Fw, ( ), he eal pa o a ou -poin co ela o ,
Fw, ( )=[ˆ
W†( )ˆ
V†(0) ˆ
W( )ˆ
V(0)].(12)
Wi hou loss o gene ali y, i we conside ope a o s ha a e
uni a y, hen Eq. (11) eads Cw, ( )=2−2Fw, ( ).
In a ecen LMG model s udy, he ESQPT e ec s on he
mic ocanonical OTOC and he OTOC ollowing a quan um
quench we e explo ed o ˆ
W=ˆ
V=ˆ
Sx/S[64]. The ime
e olu ion o he OTOC a e a sudden quench was analyzed
and i was concluded ha he equilib ium alue ( he long
ime a e age alue) o his obse able can be used as a good
ma ke o he ESQPT because i beha es as an o de pa-
ame e , able o dis inguish be ween he phases below and
abo e he ESQPT, espec i ely. Ou goal he e is o analyze
how he OTOC beha es once he ALMG sys em goes h ough
he anha monici y-induced ESQPT line. This s udy is o
ele ance since he physical na u e o his ESQPT is di e en
om he one o he al eady known ESQPT o he usual LMG
model. Mo eo e , he possibili y o using an OTOC as an
o de pa ame e o bo h ESQPTs is conside ed.
We ha e used in ou analysis he mic ocanonical OTOC
[77,80], de ined as
Fn( )=[n|ˆ
W†( )ˆ
V†(0) ˆ
W( )ˆ
V(0)|n],(13)
whe e he s a e |nis he n h eigens a e o he Hamil onian
Eq. (2), whose ene gy is En. This s a e is compu ed o a gi en
se o Hamil onian pa ame e s, ξand α.
Following Re . [64], we ha e i s selec ed ˆ
W=ˆ
V=ˆ
Sx/S
as he OTOC ope a o s. The eason behind his elec ion is
wo old. On he one hand, he expec a ion alue o he ˆ
Sx
ope a o is known o be an o de pa ame e o he QPT in he
LMG model, and i has also been shown in p e ious wo ks
ha i beha es as an o de pa ame e o he ESQPT [81]. On
he o he hand, he ˆ
Sxope a o is ela ed wi h he b eaking o
pa i y symme y in he spec um eigens a es [43]. Howe e ,
he ob ained esul s (no shown) indica e ha in his case he
Fn( ) equilib ium alue only de ec s he occu ence o he i s
ESQPT, independen ly o i s na u e, and no he second one.
We decided o explo e o he possibili ies such as ˆ
W=ˆ
Sy/S,
ˆ
V=ˆ
Sx/So ˆ
W=ˆ
S+/S,ˆ
V=ˆ
S−/S. In bo h cases we ob ain
he expec ed esul s, wi h equilib ium alues sensi i e o he
anha monici y-induced ESQPT in he symme ic phase and o
he wo ESQPTs in he b oken symme y phase.
Nume ical solu ions o he ime e olu ion o he OTOC
Eq. (13) wi h ˆ
V=ˆ
S−/Sand ˆ
W=ˆ
S+/Sa e p esen ed in
Fig. 6. These a e esul s o a selec ed se o posi i e pa i y
s a es o a sys em wi h size N=300 ha a e ob ained by he
diagonaliza ion o he Hamil onian Eq. (2). The ime e olu ion
o he mic ocanonical OTOC is depic ed o di e en ini ial
s a es and ξ=0.5 wi h ei he α=0 (le -column panels) o
α=−0.6 ( igh -column panels). Despi e he di e en ope -
a o s included in he OTOC, a qui e simila phenomenology
o ha poin ed ou in Re . [64] is obse ed. Howe e , i is
wo h emphasizing ha i we kep ˆ
V=ˆ
W=ˆ
Sx/S, once he
i s c i ical ene gy is c ossed, he ime a e age alue o he
OTOC is ze o as Fn( ) oscilla es a ound ze o.
The beha io o he mic ocanonical OTOC, Fi( ), depends
on he egion o he spec um in which he sys em is loca ed.
Pa icula ly, Fi( ) de elops a egula beha io , wi h small
ampli ude oscilla ions a ound a posi i e alue. This alue de-
c eases un il he Fi( ) oscilla es a ound ze o, when he c i ical
ene gy alue is eached. Fo he s a es close o he ESQPT
c i ical ene gy ( ed colo cu es), no only Fi( ) oscilla es
a ound ze o, bu i also beha es in a highly i egula way, as in
Re . [64]. This is a ea u e sha ed by bo h columns in Fig. 6,
hough in he igh column panels he second and ou h panel
co espond o c i ical ene gies o he wo ESQPTs ha a ise
in his case.
Le us now o discuss in mo e de ail he le column
(α=0). Recall ha in his case he e is jus one ESQPT
loca ed in he mean- ield limi a ene gy ε=ξ, i s alue
o hese plo s is ε=ξ=0.5. We ha e selec ed he g ound
s a e and ou o he posi i e pa i y eigens a es, i=0, 30, 58,
100, and 140 in Figs. 6(a),6(c),6(e),6(g), and 6(i), espec-
i ely. The s a e wi h he close ene gy o he c i ical ESQPT
ene gy is i=58—Fig. 6(e)—whe e he ESQPT p ecu so s
a e clea ly mani es ed. In he cases wi h ene gies below he
064134-8
EXCITED-STATE QUANTUM PHASE TRANSITIONS IN … PHYSICAL REVIEW E 107, 064134 (2023)
FIG. 6. Time e olu ion o he mic ocanonical OTOC, Fi( ), o selec ed posi i e pa i y eigens a es o an ALMG model wi h a sys em size
N=300. In all panels ξ=0.5, he le column panels e e s o Fi( ) o α=0 and he igh column panels include esul s o α=−0.6.
OTOCs o di e en ini ial s a es a e shown: Fo he le column om op o bo om; (a) |i=0(g ound s a e), (c) |i=30,(e)|i=58,(g)
|i=100,and(e)|i=140. Fo he igh column om op o bo om: (b) |i=0(g ound s a e), (d) |i=74, ( ) |i=95,(h)|i=115,(j)
|i=140. The e a e some ene gies in which he eigens a e is se led a he c i ical ene gy o an ESQPT. These a e he cases o panels (e) in
he le column and (d) and (h) in he igh column, highligh ed using a ed colo .
c i ical ene gy Fi( )>0. Howe e , as can be seen in Fig. 6(e),
once he c i ical ene gy o he ESQPT is eached, he OTOC
oscilla es andomly a ound ze o. Fo ene gies la ge han he
c i ical ene gy—le Figs. 6(g) and 6(i)— he Fn( ) display
high and low equency oscilla ions a ound a ze o mean alue.
The e o e, he s eady-s a e alue o Fn( ) will be equal o ze o
o hese s a es. As one goes up in ene gy in he spec um, he
same kind o oscilla o y beha io is obse ed, wi h smalle
ampli udes.
The e a e some new ea u es a ising in he igh column
panels, ha include Fi( ) esul s o he α=−0.6 anha -
monic case. As p e iously men ioned, in his case he e a e
wo c i ical ESQPT lines ha in he mean- ield limi lie a
ε=1+α=0.4 and ε=ξ=0.5. We show he esul s o
he wo eigens a es wi h local minimum PR alues i=74 and
115 in Figs. 6(d) and 6(h).Again, heFi( ) OTOC oscilla ions
a he c i ical lines a e ma kedly i egula . These wo s a es
ha e been highligh ed using ed colo . The o he h ee alues
included in he igh column o Fig. 6a e i=0 (g ound s a e),
95, and 140. In he egion be ween he wo c i ical lines,
he en elope o Fi( ) has a sine-like oscilla o y beha io
a ound ze o, so i s s eady-s a e alue equals ze o. Once he
second c i ical line is c ossed and he sys em ene gy inc eases,
Fi( ) p esen s again an oscilla o y beha io a ound posi i e
alues, as can be clea ly seen in Fig. 6. I is wo h poin ing
ou ha he cha ac e is ic imes o he di e en mic ocanon-
ical OTOCs span a wide ange o equencies. In pa icula ,
Fig. 6( ) exhibi s a much longe pe iod (smalle equency)
han he es o he panels. The oscilla o y equency o he
ou -poin co ela o can be aced back o ene gy di e ences
be ween pai s o s a es o di e en pa i y [68]. The e o e,
whene e di e en pa i y eigens a es a e degene a e, he
s a iona y alue o he OTOC has a nonze o con ibu ion. This
occu s a ene gies less han he c i ical ene gy o he i s
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