Tuning the presence of dynamical phase transitions in a generalized $XY$ spin chain

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by  Uma Divakaran

Tuning the presence of dynamical phase transitions in a generalized XY spin chain
Uma Divakaran,1 Shraddha Sharma,2 and Amit Dutta2
1UM-DAE Center for Excellence in Basic Sciences, Mumbai 400 098, India
2Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
We study an integrable spin chain with three spin interactions and the staggered field (λ) while
the latter is quenched either slowly (in a linear fashion in time (t) as t/τ where t goes from a
large negative value to a large positive value and τ is the inverse rate of quenching) or suddenly.
In the process, the system crosses quantum critical points and gapless phases. We address the
question whether there exist non-analyticities (known as dynamical phase transitions (DPTs)) in
6 the subsequent real time evolution of the state (reached following the quench) governed by the
1 final time-independent Hamiltonian. In the case of sufficiently slow quenching (when τ exceeds a
0 criticalvalueτ1),weshowthatDPTs,oftheformsimilartothoseoccurringforquenchingacrossan
2 isolatedcriticalpoint,canoccurevenwhenthesystemisslowlydrivenacrossmorethanonecritical
point and gapless phases. More interestingly, in the anisotropic situation we show that DPTs can
n
completely disappear for some values of the anisotropy term (γ) and τ, thereby establishing the
u
existenceofboundariesinthe(γ−τ)planebetweentheDPTandno-DPTregionsinbothisotropic
J
and anisotropic cases. Our study therefore leads to a unique situation when DPTs may not occur
7
even when an integrable model is slowly ramped across a QCP. On the other hand, considering

] sudden quenches from an initial value λi to a final value λf, we show that the condition for the
h presenceofDPTsisgovernedbyrelationsinvolvingλi,λf andγ andthespinchainmustbeswept
c across λ=0 for DPTs to occur.
e
m PACSnumbers: 75.10.Jm,05.70.Jk,64.60.Ht
-
t
a I. INTRODUCTION quentlyithasbeenshownthatDPTsarenotnecessarily
t
s connectedwiththepassagethroughtheequilibriumQCP
t. and may occur following a sudden quench even within
a Inspired by the concept of non-analyticities associ-
the same phase (i.e., not crossing the QCP) for both
m ated with the free-energy density of a classical system integrable11 as well as non-integrable models12. Sub-
- at a finite temperature transition marked by the ze- sequently, these studies have been generalized to two-
d roes of the partition function in a complex temperature dimensional systems13,14 and the role of topology13 and
n plane1 (see also [2,3]), recently there has been a pro-
o the dynamical topological order parameter have been
c posal of quantum dynamical phase transitions (DPTs) investigated15. We note in the passing that the rate
[ in a quenched quantum many-body system4. Associated function I(t) is related to the Loschmidt echo which
non-analyticities are quantified in terms of the overlap has been studied in the context of decoherence17–26 and
2
v amplitude or the Loschmidt overlap (LO) defined for the work-statistics27,28. The finite temperature counter-
1 the quenched quantum system. Focussing on the sud- part of the Loschmidt echo29, namely the characteristic
5 den quenching case and denoting the ground state of function has also been useful in studies of the entropy
8 the initial Hamiltonian as |ψ0(cid:105), the Loschmidt overlap generation and emergent thermodynamics in quenched
4 is defined as G(t) = (cid:104)ψ0|e−iHft|ψ0(cid:105); here, Hf is the quantum systems30,31. In fact, the rate function (of
0 final Hamiltonian reached after the quenching process.
the return probability) discussed above in the context of
.
1 DPTs occur when the initial state is orthogonal to the DPTscanbeconnectedtotheworkdistributionfunction
0 evolved state and the LO vanishes. Generalizing G(t) to
corresponding to the zero work in a double quenching
6 G(z) defined in the complex time (z) plane, one intro- experiment4.
1 duces the corresponding dynamical free energy density,
The periodic occurrences of non-analyticities in the
v: f(z) = −limL→∞logG(z)/Ld, where L is the linear di- rate function for an integrable model was first reported
i mension of a d-dimensional system. One then looks for
X in the context of a slow quenching of the transverse field
thezerosoftheG(z)(ornon-analyticitiesinf(z)),known
in the transverse Ising chain across its QCPs32. Very re-
r as Fisher zeros. For a transverse Ising chain, it has been
a cently, associated DPTs have also been related to Fisher
observed4 that when the system is suddenly quenched
zeros crossing the imaginary axis of the complex time
across the quantum critical point (QCP)5,6, the lines of
plane33. This is believed to be in general true for an in-
Fisher zeros cross the imaginary time axis at instants of
tegrablemodelreducibletodecoupledtwolevelproblems
real time t∗; at these instants the rate function of the re-
quenched slowly across its QCP.
turn probability defined as I(t) = −log|G(t)|2/L shows
In this paper, we extend the previous studies further
sharpnon-analyticitiessignalingtheoccurrenceofDPTs.
to the slow as well as sudden quenching of an inte-
The initial observation by Heyl et al.4 that DPTs are grable quantum Ising model with complicated interac-
associatedwiththesuddenquenchesacrosstheQCPhas tions across the QCPs (and also gapless phases) and es-
been verified in several studies7–10,16. However, subse- tablishthatDPTsmaycompletelydisappearinsomesit-
2
uations depending on the quenching rate (or amplitude at λ = λ in the process. Since the condition for an
c
insuddenquench)andsystemparameters. Thisisanob- adiabatic dynamics breaks in the vicinity of the QCP,
servationthat,tothebestofourknowledge,hasnotbeen one arrives at a final state (for the k-th mode) given by
reported earlier particularly for the slow quenching. We |ψ (cid:105) = v |1f(cid:105) + u |2f(cid:105), with |u |2 + |v |2 = 1; here,
fk k k k k k k
note at the outset that for the slow quenching, the final |1f(cid:105) and |2f(cid:105) are the ground state and the excited states
stateis preparedthroughthevariation ofaparameterof k k
of the final Hamiltonian H (λ ) with corresponding en-
theHamiltonianast/τ acrosstheQCPtothefinalvalue fk f
ergy eigenvalues (cid:15)f and (cid:15)f , respectively. One can de-
of time (and hence, of the parameter); on the contrary, k,1 k,2
fine the LO for the mode k as (cid:104)ψ |exp(−H z)|ψ (cid:105)
for the sudden quenches the final state happens to be fk fk fk
and the corresponding dynamical free energy4, f (z) =
the ground state of the initial Hamiltonian. In both the k
−log(cid:104)ψ |exp(−H z)|ψ (cid:105)/L, where z is the complex
casesG(t)describesthesubsequenttemporalevolutionof fk fk fk
time with z = R+it, R being the real part and t the
the system with the final time-independent Hamiltonian
imaginarypart. Summingoverthecontributionsfromall
setting the origin of time (t = 0) immediately after the
themomentamodesandconvertingsummationtothein-
quenching is complete. Let us also note that the numer-
tegral in the thermodynamic limit, one gets
ical calculations are performed for a finite system, hence
Fisherzeros donot coalesceintoa line, rather constitute (cid:90) π dk (cid:16) (cid:17)
a set of closely spaced points. f(z) = − 2π log |vk|2exp(−(cid:15)fk,1z)+|uk|2exp(−(cid:15)fk,2z)
0
Wewouldalsoliketomentionthattheslowquenching (cid:90) π dk (cid:16) (cid:17)
dynamics across or to a QCP has been studied in the = − log (1−p )exp(−(cid:15)f z)+p exp(−(cid:15)f z) ;
context of possible Kibble-Zurek (KZ) scaling34,35 of the 0 2π k k,1 k k,2
defect density and the residual energy36,37 which have (1)
been explored in various situations38–48. (For reviews,
we reiterate that t is measured from the instant the final
see [49–51].)
state |ψ (cid:105) is reached after the slow quench.
The paper is organized in the following manner: In fk
We then immediately find the zeros (i.e., the Fisher
Sec. II, we introduce the connection between Fisher ze-
zeros) of the “effective” partition function where f(z) is
ros, DPTs and the slow (as well as sudden) quenching
non-analytic as
of a generic two-level integrable model. In Sec. III, on
the other hand, we introduce a specific model, namely, a 1 (cid:18) p (cid:19)
z (k)= log( k )+iπ(2n+1) , (2)
generalized transverse Ising chain with three-spin inter- n ((cid:15)f −(cid:15)f ) 1−pk
actions and a staggered magnetic field (λ) and present k,2 k,1
its phase diagram. In Sec. IV we focus on slow quenches where n=0,±1,±2,···. It is to be emphasized that in
and show that DPTs always occur in the isotropic situa- z (k),thedifferenceoftheeigenvalues((cid:15)f −(cid:15)f ),rather
n k,2 k,1
tion even when the system is quenched across two criti- than the eigenvalues themselves, appear. The Fisher
cal points and gapless phases if the quenching is not too zeros constitute a line (more precisely, closely spaced
rapid. On the contrary, in the anisotropic case, there is points)correspondingtoeachninthecomplexzplane33.
a clear boundary separating the DPT and the no-DPT The critical mode k (for which the gap in the spectrum
c
region; this establishes that the slow quenching of an in- vanishes at λ = λ ) remains frozen in the initial state
c
tegrable model across its QCP does not necessarily lead and hence p =1, while for modes far away from the
to DPTs. Finally, in Sec. V, we consider the sudden critical modek=pkc→0; therefore z (k)’s goes from −∞ to
k n
quenching of the staggered field and show how the pres- ∞ in the thermodynamic limit as k changes. From the
ence of DPTs following the quench is dictated by rela- continuity argument, there must exist one specific value
tionsinvolvingtheinitialandfinalvaluesofthefieldand ofk forwhichp =1/2andRe(z (k)| )vanishes;
the anisotropy parameter; it is worth mentioning that from∗ (2) we notek=tkh∗at the lines of Finsherkz=erko∗s cross the
the spin chain must be quenched across λ=0 for DPTs imaginary axis for k .
∗
to occur. Theratefunctionofthereturnprobabilityinthiscase
can be evaluated exactly in the form32,33
log|G(t)|2
II. QUENCHES OF AN INTEGRABLE MODEL I(t) = − =2 Ref(z)
AND DPT L
(cid:90) π dk (cid:32) ((cid:15)f −(cid:15)f ) (cid:33)
= − log 1+4p (p −1)sin2 k,2 k,1 t ;
Let us consider an integrable model reducible to a two 2π k k 2
0
level system for each momenta mode; the system is ini-
(3)
tially (t → −∞) in the ground state |1i(cid:105) of the initial
k
Hamiltonian for each mode. We first consider the slow
the non-analyticities in I(t) appear at the values of the
quenching case. The Hamiltonian is characterized by real time t∗s given by
n
a parameter λ which is quenched from an initial value
π
λi following the quenching protocol λ(t) = t/τ to a fi- t∗ = (2n+1) (4)
nal value λ so chosen that the system crosses the QCP n ((cid:15)f −(cid:15)f )
f k∗,2 k∗,1
3
derived by setting Re(z (k )) = 0 in Eq. (2) as the ar- In the k-space, the reduced Hamiltonian is given by
n ∗
gument of the logarithm in Eq. (3) vanishes for k = k
wonhe((cid:15)nfpk=−k(cid:15)∗f=)1./H2.owAegvaeirn,,fotrhethteimcaesein(cid:15)sftan=ts−t(cid:15)∗nfde=pe(cid:15)nfd∗, Hk =αcoskˆ1− 12(cid:20)−(1+λγe+ik) −(1+−γλe−ik)(cid:21),
Eq. (k4∗),2getsk∗s,i1mplified to k,2 k,1 k (7)
π (cid:18) 1(cid:19) where λ = h/J1, α = J3/J1 and γ = J2/J1 and ˆ1 is the
t∗ = n+ (5) 2×2 identity operator and the second part represents
n (cid:15)f 2
k∗ the2×2Landau-Zener(LZ)partoftheHamiltonian;we
shall also use the notation ∆ =(1+γe−ik) below. The
k
Weshallbrieflydwellonthecaseofsuddenquenching4 correspondingeigenvaluesofthereducedHamiltonianHk
when the parameter λ is suddenly changed from an ini- are
tial value λ to a final value λ ; in this case, the final
i f (cid:15)˜± = αcosk±(cid:15)
state |ψ (cid:105) is the initial ground state |1i(cid:105) (correspond- k k
ing to λkf) while the Hamiltonian gets mkodified to the 1(cid:112)
i = αcosk± λ2+γ2+1+2γcosk. (8)
final Hamiltonian; the LO for the given mode is given 2
by(cid:104)1i|exp(−H z)|1i(cid:105). Followingasimilarlineofargu-
k fk k The phase diagram obtained by analyzying the spec-
mentsasabove,onecanshowthatthedynamicalfreeen-
trum in Eq. (8) is shown in Fig. 1. We shall consider
ergyhasasimilarformasinEq.(1)with|u |2 →|u˜ |2 =
k k the slow as well as sudden quenching dynamics of the
p˜k = |(cid:104)1ik|2fk(cid:105)|2 and |vk|2 → |v˜k|2 = |(cid:104)1ik|1fk(cid:105)|2. There- Hamiltonian (6) by varying the parameter λ across the
fore, one finds a similar expression for the rate function QCPs and gapless phases and probe the corresponding
inEq.(3)(withpk →p˜k)whichshowsnon-analatyicities DPT scenario.
at the instants of real time again given by Eq. (4) when As evident from Eq. (7), the term αcosk leads to the
p˜k=k∗ =1/2. rich phase diagram of the model under consideration by
introducing gapless phases of different kinds, where the
gap in the spectrum vanishes solely due to the presence
III. GENERALIZED SPIN MODEL of αcosk. However, this term does not participate in
the dynamics. This is because of the fact that the term
In this section, we shall consider a generalized spin- αcosk is associated with the identity operator which al-
1/2 quantum XY chain with a two sublattice structure ways commuteswith the time evolutionoperator for any
in the presence of a three spin interaction (J > 0) and type of temporal evolution. The dynamics of the sys-
3
a staggered field (h) described by the Hamiltonian tem is, therefore, entirely determined by the LZ part of
Eq. (7). This in fact leads to a conspicuous behavior
as far as DPTs are concerned as we shall discuss below;
(cid:88) (cid:88)
H =−h (σiz,1−σiz,2)−J1 (σix,1σix,2+σiy,1σiy,2) furthermore, only the terms ±(cid:15)k appearing in the eigen-
values in Eq. (8) determine the instants at which DPTs
i i
(cid:88) (cid:88) occur. This is also clear from Eq. (4) that only the
− J (σx σx +σy σy )−J (σx σz σx
2 i,2 i+1,1 i,2 i+1,1 3 i,1 i,2 i+1,1 difference of eigenvalues plays a role in determining t∗s
n
i i and hence, the results will be completely independent of
(cid:88)
+ σiy,1σiz,2σiy+1,1)−J3 (σix,2σiz+1,1σix+1,2 the parameter α. Additionally, the eigenfucntions of the
i HamiltonianHk arealsoidenticaltothoseoftheLZpart.
+ σy σz σy ), (6) We note in the passing that Hamiltonian of the form
i,2 i+1,1 i+1,2
(6) has been studied extensively54,55 over decades; re-
where i is the site index and the additional subscript cently topological aspects of this kind of models have
1(2) defines the odd (even) sublattice. The parameter also been explored56.
J describes the XY interaction between the spins on
1
sublattice 1 and 2 while J describes the XY interac-
2
tion between spins on sublattice 2 and 1 such that J IV. SLOW QUENCHES: DPT-NO DPT
1
is not necessarily equal to J . In spite of the com- BOUNDARY
2
plicated nature of interactions, this spin chain is inte-
grable and exactly solvable in terms of a pair of Jordan- Let us consider a variation of the field λ = t/τ from
Wigner fermions52,53 defined on even and odd sublat- −10 to +10 so that the system is quenched from one an-
(cid:104) (cid:105)
tices as σ+ = (cid:81) (−σz )(−σz ) a†, and σ+ = tiferromagneticphasetotheothercrossingboththegap-
i,1 j<i j,1 j,2 i i,2
less phases. The probability of excitations p following
(cid:104)(cid:81)j<i(−σjz,1)(−σjz,2)(−σiz,1)(cid:105)b†i, where σiz,1 = 2a†iai −1 the quench is given by the LZ transition probkability57,58
and σiz,2 = 2b†ibi −1. The Fermion operators ai and bi pk = e−π|∆2k|2τ; here, |∆k|2 = (1+γ2 +2γcosk) which
can be shown to satisfy fermionic anticommutation rela- vanishes for k = π at the boundary between the anti-
tions. ferromagnetic (AF) phase and the gapless phase GPI for
4
6 6
γ =1 γ =0.5 1.2
4 AF 4 AF 0.2
2 2 1 2.0
λ 0o GPI GPII λ 0o GPI GPII 10.0
−2 −2 0.8
AF AF
−4 −4
−6 0 0.5 1 1.5 2 2.5 3 −6 0 0.5 1 1.5 2 2.5 3 I(t) 0.6
α α 0.4
FIG. 1: (Color online) Phase diagram of the Hamiltonian 0.2
(7) in the α − λ plane for the isotropic case γ = 1 and
an anisotropic case with γ = 0.5. For a fixed α and a 0
large magnitude of the rescaled field λ, the spin chain is 0 1 2 3 4 5
anti-ferromagnetic (AF). On the other hand, when λ is re- t
duced,thesystemundergoesquantumphasetransitionsfrom
1.8
Gapless-I (GPI) phase characterized by two Fermi points
0.2
to the Gapless-II (GPII) phase characterized by four Fermi 1.6 1.5
points. The vertical line shows the direction of quenching. 1.4 2.0
After [59]. 1.2 3.0
1
I(t) 0.8
0.6
0.4
0.2
theisotropiccaseγ =1. Probingtheratefunction,wein-
deed find periodic occurrences of sharp non-analyticities 0
0 1 2 3 4 5
as expected in the case of quenching across an isolated
t
QCP(seethenumericalresultpresentedinthetoppanel
of Fig. (2)); the instants at which these non-analyticities
FIG. 2: (Color online) The rate function I(t) as obtained by
appear can be matched with those obtained from
numerically integrating Eq. (3) is plotted as a function of
(cid:113)
Eq. (5) with (cid:15)f = (1/2) λ2 +1+γ2+2γcosk = timefollowinga slow quench froma largenegative toa large
k∗ f ∗ positive value of λ. The upper panel corresponds to γ = 1
(cid:113)
(1/2) λ2 +2+2cosk for γ = 1, λ being the final where there are periodic occurrences of DPTs for all values
f ∗ f
of τ’s subject to the condition that p | < 0.5, i.e., τ >
parameter value reached after the quenching. It is worth k k=0
τ (γ)| . The lower panel corresponds to the anisotropic
mentioningthatthedynamicsiscompletelyinsensitiveto 1 γ→1
situation with γ = 0.5 for which the critical value of τ as
the fact that the system is driven across gapless phases
obtained from Eq. (9) is given by τ (γ)=1.76. As discussed
2
in the process of quenching and hence no trace of gap-
inthetext,thefigureshowsthepresenceofDPTsforτ (γ)<
1
less phases is reflected in DPTs. However, the oc- τ <τ (γ)whiletheydisappearforτ >τ (γ). Theredcolored
2 2
currences of DPTs also require the condition that the (solid line) show sharp peaks whereas black colored (dashed
minimum value of the non-adiabatic transition probabil- lines) show peaks those are rounded off.
ity p =exp(−π|∆k|2|k=0τ)=exp(−π(1+γ)2τ) must be
k=0 2 2
lessthan1/2sothatak (forwhichp =1/2)exists.
∗ k=k∗
This does not happen if the quenching is too rapid, i.e., in the (γ−τ) plane given by the equation:
τ < τ (γ) = 2log2/{π(1+γ)2} for γ (cid:54)= 1; for γ = 1,
1
τ (γ)| = log2/(2π). One therefore does not indeed 2log2
1 γ→1 exp(−π(1−γ)2τ/2)=1/2; τ =τ (γ)= .
observe DPTs even in the isotropic case for too rapid 2 π(1−γ)2
quenching processes. (9)
For a fixed γ, if τ exceeds τ (γ), DPTs disappear. This
2
is verified numerically and shown in the lower panel of
We now move to the more interesting situation which
Fig.(2)whereweevaluateI(t)bynumericallycalculating
arises in the anisotropic case (γ (cid:54)= 1); in this case,
|∆k|2 =(1+γ2+2γcosk), assumes the minimum value pks and using the values (cid:15)f1,k = −(cid:15)fk and (cid:15)f2,k = (cid:15)fk (cor-
at the boundary between AF and the GPI phase for the responding to λ = 10) in Eq. (3); we show that DPTs
f
mode k =π and is given by |∆ |2 =(1−γ)2, and hence occurringforτ <τ (γ), disappearwhen τ exceeds τ (γ).
k 2 2
themaximumvalueofthenon-adiabatictransitionprob- Referring to the situation γ = 1, when maximum value
ability pmax = exp(−π(1−γ)2τ/2). As emphasized be- of p (for k = π) is equal to unity and τ (γ)| → ∞,
k k 2 γ→1
fore, DPTs can occur only when p = 1/2. If the max- DPTs periodically appear for all values of τ > τ (γ).
k 1
imum possible value of pmax is less than 1/2, no DPT On the contrary, for γ (cid:54)=1, there always exists a critical
k
can appear even when the system is quenched across the τ (γ) and DPTs appear only when τ (γ) < τ < τ (γ).
2 1 2
QCPs and gapless phases. We therefore find a boundary We would like to emphasize that these observations are
5
below, in this case also whether DPTs are present or
absentdependonsomeconditionsinvolvingλandγ both
2 0
t for γ =1 and (cid:54)=1. Referring to the Hamiltonian (7), we

1 find that the ground state and the excited state, i.e., the
t
1 6 adiabatic basis states, for a given λ (say, λ ) is given by
2 i
1 2 N O -D P T s θ θ
|1i(cid:105)=cos k(1,0)T −sin k(0,1)T
t k 2 2
8 D P T s |2i(cid:105)=sinθk(1,0)T +cosθk(0,1)T, (10)
k 2 2
p
4 2 lo g 2 / where tanθ = −|∆ |/λ which clearly does not depend
k k
on α. When the field λ is suddenly changed to λ , the
lo g 2 /2 p N O -D P T s i f
excitationprobabilityisgivenbyp˜ =|u˜ |2 =|(cid:104)1i||2f(cid:105)|2.
0 k k k k
1 .0 0 .8 0 .6 0 .4 0 .g2 0 .0 Asdiscussedbefore,thenecessaryconditionforthepres-
ence of a DPT requires p˜ | = 1/2. Using Eqs. (10),
k k=k∗
we immediately find
FIG. 3: (Color online) The phase diagram in the γ-τ plane p˜ = |u˜ |2 =|(cid:104)1 (λ )||2 (λ )(cid:105)|2 =sin2[(θi −θf)]/2
showing the DPT and the no-DPT regions following a slow k k k i k f k k
quenchasdiscussedinFig.2. Theuppercurvecorrespondsto = 1(cid:20)1− λfλi+|∆k|2 (cid:21); (11)
the condition presented in Eq.(9) i.e., τ2(γ)=2log2/(π(1− 2 (cid:112)(λ2+|∆ |2)(cid:113)(λ2 +|∆ |2)
γ)2) which diverges in the isotropic case (γ = 1). The lower i k f k
curvedenotedbyτ (γ)=2log2/(π(1+γ)2)isobtainedfrom
1
therequirementpk=0 =1/2andτ1(γ)|γ=1 =log2/2π. Itisto it should be noted that p˜k depends on λi, λf and γ but
be noted that the values of both τ (γ) and τ (γ) are zoomed never on α.
1 2
byafactorof10forbettervisibility. TheDPTsexistfortheγ TopredictthepresenceofDPTs,itissufficienttoana-
andτ valueslying inthe regionboundedbyτ2(γ) andτ1(γ). lyze|u˜k=0|2and|u˜k=π|2;thenecessaryconditionforDPT
would then be |u˜ |2 >1/2 and |u˜ |2 <1/2. (We re-
k=π k=0
call that the |∆ | is minimum for the mode k = π and
k
all appropriately supported by the behavior of the lines hence probability of excitation is maximum for that par-
ofFisherzeros,e.g.,onecanverifythatthelinesofFisher ticular mode). If these conditions are satisfied, from the
zeros never cross the imaginary axis in the no-DPT re- argumentofcontinuityoneconcludesthattheremustex-
gion in the anisotropic case. All these conditions are ist a k for which |u˜ |2 =1/2, ensuring the existence
∗ k=k∗
summarised in Fig. (3). ofDPTs. Wenotethatthisisthemostgenericcondition
Interestingly, the relation (9) does not depend on the forDPTstooccuraslongasonecansharplydefineak .
∗
parameter α which plays no role in the temporal evo- Using (11), one can show that p˜ becomes equal to 1/2
k
lution of the system. Let us also addresss the question for a mode k only when
whathappenswhenαisquenchedkeepingλandγ fixed.
In this case, the initial ground state of the Hamiltonian
λ λ +|∆ |2
Hk (which is also the ground state of the LZ part of the (cid:112) f i (cid:113) k =0 =⇒ λfλi+|∆k|2 =0,
Hamiltonian (7)), only evolves through an overall phase (λ2+|∆ |2) (λ2 +|∆ |2)
i k f k
accumulation. SincetheLZpartisunaltered,thereisno
(12)
non-trivialdynamicsforanymodek,andhence,noDPT
where |∆ |2 is evaluated at the corresponding value of
is expected; the evolved state is never orthogonal to the k
k. To illustrate the main point in a transparent manner,
state |ψ (cid:105). It is also noteworthy that in the anisotropic
case,thefkdefectdensityshowsanexponentialdecay(with we choose λf = −λi = λ, (or the other way round i.e.,
λ = −λ = −λ) for which Eq. (12) assumes a simpler
τ) as opposed to the standard power-law KZ scaling59. f i
form:
−λ2+|∆ |2
k =0 =⇒ λ2 =|∆ |2.
V. SUDDEN QUENCHES: CONDITIONS FOR (cid:112)(λ2+|∆ |2)(cid:112)(λ2+|∆ |2) k
k k
DPTS (13)
We shall analyze the condition given in (13) for the
Inthissection,weshallconsiderasuddenquenchingof modes k = 0 and k = π for both γ = 1 and γ (cid:54)= 1. In
theparameterλfromaninitialλ toafinalvalueλ . We the former case (γ = 1), |u˜ |2 = 1 as the off-diagonal
i f k=π
address the questions whether DPTs are always present terms of the LZ part of the Hamiltonian (7) vanish for
in the subsequent temporal evolution and how does the k = π so that this mode is temporally frozen. On the
situationgetalteredintheanisotropiccaseincomparison other hand, the condition that |u˜ |2 ≤ 1/2, demands
k=0
to the isotropic case. Remarkably, as we shall illustrate λ2 ≤ ∆2| = 4. This implies that whenever the field
k k=0
6
λ is quenched from a value λ ≥ −2 to a final value
i
λ ≤+2, DPTs will indeed appear. Otherwise, they are
f
absent. Thisisnumericallyverifiedasshownintheupper 1 .8
l = - 2, l
= 2
panel of Fig. 4. 1 .6 l i= - 2.5, fl
Proceeding to the anisotropic case, we find from = 2 .5
i f
Eq. (13) that the condition |u˜ |2 ≥ 1/2, leads to 1 .4 a = 1, g = 1
k=π )
λ ≥ (1−γ) while the requirement |u˜k=0|2 ≤ 1/2, yields I(t 1 .2
λ≤(1+γ). Therefore, for a sudden quench from −λ to
+λwithagivenγ,onefindsarangeofλdictatedbythe 1 .0
condition(1−γ)≤λ≤(1+γ)forwhichDPTswouldap-
0 .8
pear as numerically verified in the lower panel of Fig. 4.
This condition immediately reduces to the isotropic case 0 .6
for γ = 1, where, as shown above, the magnitude of λ
0 .4
should be less than 2 to observe DPTs.
Referring to Eq. (12), we find that for DPTs to occur, 0 .2
the quantity λ λ must be negative; that implies that
i f 0 .0
the spin chain must be quenched across λ = 0. In that 0 2 4 6 8 1 0 1 2 1 4
sense, the line λ = 0 is special; this is in congruence t
with the observation reported in the Ref. [60] where it
has been shown the Loschmidt echo when studied as a
function of λ shows a dip only at λ=0, thereby detect-
ing only a special point of the phase diagram. There-
1.8 λ=-0.4, λ=0.4
fore for a generic situation, the condition for DPT to i f
occur would be |u˜ |2 > 1/2 and |u˜ |2 < 1/2 along 1.6 λi=-0.5, λf=0.5
k=π k=0 λ=-1.5, λ=1.5
with the condition the system is quenched across λ=0; t) 1.4 λi=-1.8, λf=1.8
for a quench from an initial value −λ to a final value ( i f
i I 1.2 γ = 0.5
λ , Eq. (12) then leads to a more generic condition
f
(1 − γ) ≤ (cid:112)|λ |λ ≤ (1 + γ) for DPTs to occur. (If 1.0
i f
the quenching is from +λ to −λ , the condition gets
i f 0.8
(cid:112)
modified to (1 − γ) ≤ λ |λ | ≤ (1 + γ).) This has
i f 0.6
alsobeennumericallyverified. Whatneedstobeempha-
sized is that whether DPTs are present following a sud- 0.4
denquenchiscompletelyindependentofthefactwhether
0.2
the system is quenched across a QCP or not; therefore,
the passage through a QCP is never a necessary criteria. 0.0
0 2 4 6 8 10 12 14 16 18 20
All these above conditions are summarized in Fig. 5
t
VI. CONCLUSION
FIG.4: (Coloronline)ThepresenceandtheabsenceofDPTs
following a sudden quenching of λ from −λ to +λ for α =
We have explored the possibility of DPTs following 1. The upper panel corresponds to γ = 1 where there are
slow as well as sudden quenches of a model Hamiltonian periodic occurrences of DPTs when λ = 2 while DPTs get
with a rich phase diagram with two gapless phases. We rounded off when λ = 2.5. The lower panel corresponds to
find some worth mentioning results not reported before. the anisotropic situation with γ = 0.5 where the presence of
DPTs are wiped out when λ = 1.8 (> (1+γ)) or λ = 0.4
The term of the reduced Hamiltonian that results into
(< (1 − γ)) while these are prominently present when for
these gapless phases do not participate in the dynamics
λ = 1.5 (= (1+γ)) and λ = 0.5 (= (1−γ)) as discussed
and hence the passage through the gapless phases is not
in the text. It should also be noted that the position of the
reflected in the behavior of DPTs those may occur fol-
maxima (or non-analyticities) depend on the magnitude λ.
lowingthequenchbothinisotropicandanisotropiccases.
Consequently,fortheslowquenchesintheisotropiccase,
thereareperiodicoccurrencesofDPTsasexpectedinthe
case of a slow passage of an integrable model through provides a unique example of a situation where DPTs
an isolated QCP if the quenching is not too rapid (i.e., could be absent even when an integrable model is slowly
for τ > τ (γ)). On the contrary, in the anisotropic ramped across a QCP.
1
case, one finds a region in which DPTs exist bounded Concerning the sudden quenches we find that even in
by two limiting quenching rates τ (γ) and τ (γ) in the theisotropiccasethepresenceofDPTsisnotguaranteed;
1 2
γ−τ planeassummarizedinFig.3; intheisotropiccase neither the situation is like a sudden quench through a
τ (γ)| = 2log2/π and τ (γ)| → ∞. This model single QCP as in the case of slow quenches. Rather both
1 γ→1 2 γ→1
7
in the isotropic and anisotropic cases, one finds restric-
tionsonthevaluesofλ andλ dependingontheparam-
i f
eter γ determined from Eqs. (12) and (13). It is never
4 .0 g 2 £ l £ g 2 l
( 1 - ) ( 1 + ) , = - 1 important whether the spin chain is quenched across the
f i
3 .5 ( 1 - g ) 2/ 2 £ l £ ( 1 + g ) 2/ 2 , l = - 2 QCPintheprocessofquenching;however,itshouldnec-
f i essarily be swept through λ = 0, i.e., either λ or λ
i f
3 .0
should be negative for DPTs to appear. We have illus-
f
l 2 .5 N O - D P T s tratedthesedifferentscenariosinFig.5. Thisisremark-
D P T s able that DPTs can be made to appear (or disappear)
2 .0 in the same model by tuning either the anisotropy term
γ or the inverse quench rate τ for slow quenches, and λ
1 .5 i
and λ for sudden quenches such that the system must
f
1 .0 be driven across λ=0.
D P T s
0 .5
0 .0
1 .0 0 .8 0 .6 g 0 .4 0 .2 0 .0
Acknowledgments
FIG.5: (Coloronline)Thephasediagramintheλ −γ plane
f
We acknowledge Jun-ichi Inoue for discussions and
showing the regions where DPTs will occur following sudden
quencheswithλ =−1(redtriangles)and=−2(bluecircles) Anatoli Polkovnikov and Sei Suzuki for collaboration in
i
toafinalvalueλ . Inboththecases,DPTsoccurwhenλ lies relatedworks. ShraddhaSharmaacknowledgesCSIR,In-
f f
withintherange(1−γ)2/|λ |and(1+γ)2/|λ |aselaborated dia and also DST, India, and AD and UD acknowledges
i i
in the text. In the isotropic situation (γ = 1), the condition DST, India, for financial support. AD and SS acknowl-
gets simplified to 0≤λf ≤4/|λi|. edgeAbdusSalamICTPforhospitalitywheretheinitial
part of the work was done.
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