Slow quenches in a quantum Ising chain; dynamical phase transitions and topology

2.8 MB·

by  Shraddha Sharma

Slow quenches in a quantum Ising chain; dynamical phase transitions and topology
Shraddha Sharma,1 Uma Divakaran,2 Anatoli Polkovnikov,3 and Amit Dutta1
1Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
2UM-DAE Center for Excellence in Basic Sciences, Mumbai 400 098, India
3Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA
We study the slow quenching dynamics (characterized by an inverse rate, τ−1) of a one-
dimensionaltransverseIsingchainwithnearestneighborferromagenticinteractionsacrossthequan-
tum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent tem-
poral evolution of the final wave function (reached at the end of the quenching) with the final
6 time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized “parti-
1 tion function”, we probe non-analyticities manifested in the rate function of the return probability
0 known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show
2 that DPTs survive in the subsequent temporal evolution following the quenching across two criti-
cal points of the model for a sufficiently slow rate; furthermore, an interesting “lobe” structure of
n
Fisherzerosemerge. Wehavealsomadeaconnectiontotopologicalaspectsstudyingthedynamical
u
topological order parameter (ν (t)), as a function of time (t) measured from the instant when the
J D
quenching is complete. Remarkably, the time evolution of νD(t) exhibits drastically different be-
7
havior following quenches across a single QCP and two QCPs. In the former case, ν (t) increases
D
] step-wise by unity at every DPT (i.e., ∆νD = 1). In the latter case, on the other hand, νD(t)
h essentially oscillates between 0 and 1 (i.e., successive DPTs occur with ∆νD = 1 and ∆νD = −1,
c respectively), except for instants where it shows a sudden jump by a factor of unity when two
e successive DPTs carry a topological charge of same sign.
m
PACSnumbers: 75.10.Jm,05.70.Jk,64.60.Ht
-
t
a
t
s I. INTRODUCTION (QCP)5,6, the line of Fisher zeros crosses the imaginary
t. timeaxisatinstantst∗;attheseinstantstheratefunction
a of the return probability defined as I(t)=−ln|G(t)|2/L
m The phase transition in a thermodynamic system is
shows sharp non-analyticities.
marked by non-analyticities in the free-energy density
-
d which can be detected by analyzing the zeroes of the The initial observation by Heyl et al.4 was verified
n partitionfunctioninacomplextemperatureplaneaspro- in several subsequent studies7–12 which established that
o
posedbyFisher1;asimilarproposalwasalsogivenearlier similar DPTs are observed for sudden quenches across
c
[ inthepresenceofacomplexmagneticfield2(Seealso[3]). the QCP for both integrable and non-integrable models.
When the line of Fisher zeros cross the real axis, there Subsequent works however showed that DPTs can occur

2
arenon-analyticitiesinthefree-energydensitywhichsig- following a sudden quench even within the same phase
v
naltheexistenceofafinitetemperaturephasetransition (i.e., not crossing the QCP) for both integrable13 as well
7
in the thermodynamic limit. as non-integrable models14. Furthermore, DPTs have
3
6 Inarecentwork,Heylet al.4,introducedthenotionof been explored in two-dimensional systems15,16 where
1 dynamical phase transitions (DPTs) exploiting the for- topology of the equilibrium system plays a non-trivial
0 mal similarity between the canonical partition function role15; also, the notion of a local dynamical topological
1. Z(β) = Tr e−βH, of an equilibrium system described order parameter (DTOP) which assumes integer values
0 by a Hamiltonian H (where β is the inverse tempera- andchangesbyunityateveryDPThasbeenproposed17.
6 ture) and that of the overlap amplitude or Loschmidt We note in the passing that the rate function I(t) is re-
1 overlap (LO) defined for a quantum system which is lated to the Loschmidt echo which has been studied in
v: suddenly quenched. Denoting the ground state of the the context of decoherence at zero18–26 and also at fi-
i initial Hamiltonian as |ψ (cid:105) and the final Hamiltonian nite temperatures27 and has also been useful in studies
X 0
reached through the quenching process as H, the LO of the work-statistics28 and the entropy generation29–31
ar is defined as G(t) = (cid:104)ψ0|e−iHt|ψ0(cid:105). Generalizing G(t) in quenched quantum systems. In fact, the rate func-
to G(z) defined in the complex time (z) plane, one can tion (of the return probability) discussed above in the
introduce the notion of a dynamical free energy density, context of DPTs can be connected to the singularities in
f(z) = −lim lnG(z)/Ld, where L is the linear di- the work distribution function corresponding to the zero
L→∞
mension of a d-dimensional system. work following a double quenching experiment4; in this
process, the initial Hamiltonian is suddenly quenched to
In a spirit similar to the classical case, one then looks
the final Hamiltonian at t = 0 and then quenched back
for the zeros of the G(z) (or non-analyticities in f(z))
to the initial one at a time t.
to define a dynamical phase transition. For a transverse
Ising chain, it has been observed4 that when the system Itshouldbenotedthattheperiodicoccurrencesofnon-
is suddenly quenched across the quantum critical point analyticities in the rate function for an integrable model
2
was first reported in the reference [32], in the context non-topological phase across both the QCPs separating
of a slow quenching of the transverse Ising chain though thetopologicalphaseandthenon-topologicalphases;the
the connection to DPTs and Fisher zeros remain unex- behavior of the DTOP is significantly different in these
plored. In the present work, we shall address that par- two situations. In the former case, ν (t) increases in
D
ticular issue and probe the behavior of the line of Fisher a step-like fashion by unity at every DPT while in the
zeros following slow quenches (dictated by a rate τ−1) of latter it oscillates between zero and unity except at the
integrable spin chains across their QCPs; we shall also specialinstantswhentherearetwosuccessiveDPTswith
connect this to point out the occurrences of such DPTs thesamesignofthetopologicalcharges;inthatsituation
ornon-analyticitiesmanifestedintheratefunctionofthe ν shows a discrete jump of unit magnitude.
D
returnprobability. Furthermore,weaddressthequestion
concerning the fate of these DPTs when the transverse
Ising chain is slowly driven across both the QCPs of the II. TRANSVERSE ISING CHAIN AND
model; it is noteworthy that for the sudden quenching, QUENCHING OF THE TRANSVERSE FIELD
the passage through two critical points wipes out the
non-analyticities in the rate function4. We note at the In this section, we shall study the slow dynamics of a
outset that in the context of the slow quenching there transverse Ising chain and illustrate the flow of the line
are two times denoted by t˜and t, respectively, appear- of the Fisher zeros studying the temporal evolution as a
ing in the discussion here: t˜is related to the ramping functionoftimetfollowingsuchaquench. Weshallshow
of a parameter of the Hamiltonian (namely, the trans- thattheFisherzeroscrosstheimaginarytimeaxisinthe
verse field h) using the quenching protocol t˜/τ to reach complextimeplaneforaparticularmomentumleadingto
the final state |ψf(cid:105) at a desired final value of the field non-analyticitesoftheratefunctionatthecorresponding
hf; the other one, denoted by t, that appears in G(t) realtime. Letusfirstconsideraferromagnetictransverse
describes the subsequent temporal evolution of the state Ising chain described by the Hamiltonian
|ψ (cid:105) with the time-independent final Hamiltonian H .
f f
Hence, once the linear-ramping (with time t˜) of the pa- H =−J (cid:88)σxσx −h(cid:88)σz, (1)
rameteriscomplete,wesetthetimet=0,andprobethe x i i+1 i
i i
non-analyticities in I(t) as a function of t (analytically
continuingittothecomplextimeplane). Inotherwords, where σ ’s are the Pauli spin matrices satisfying the
i
the role of the slow quenching is to prepare the system standard commutation relations, J is the ferromagnetic
x
in a desired state; we then probe the time evolution of nearest neighbor interactions in the x directions (set
this state with the final Hamiltonian as a function of t. equal to unity in the subsequent discussion) and h de-
Secondly, we study Fisher zeros numerically using a fi- notesthestrengthofthenon-commutingtransversefield.
nite system, hence they do not really form a line, rather The model being translationally invariant (and hence
generate a set of closely spaced points. momentum k being a good quantum number), we use
The study of slow quenching dynamics has gained im- FouriertransformationandemployJordan-Wignertrans-
portance in recent years because of the possible Kibble- formation to map spins to spin-less fermions. Further-
Zurek (KZ) scaling33,34 of the defect density and the more noting that the ‘parity’ (of the number of Jordan-
residual energy following a quench across (or to) a QCP Wigner fermions) is conserved, one can arrive at decou-
by tuning a parameter of the Hamiltonian slowly35,36. pled 2×2 Hamiltonians for each mode k in the basis |0(cid:105)
This scaling has been verified and also modified in vari- (no fermion) and |k,−k(cid:105) (with two fermions with quasi-
ous situations37–47. (For reviews, see [48–50].) momenta k and −k, respectively) given by50
Thepaperisorganizedinthefollowingmanner: inSec
II, we discuss a transverse Ising chain driven across its (cid:18) (cid:19)
−h+cosk −isink
QCP by varying the transverse field following a linear Hk = isink h−cosk . (2)
protocol; studying the behavior of Fisher zeros we deter-
minethelocationofnon-analyticitiesintheratefunction.
Analyzing the gap in the energy spectrum (2(cid:15) =
k
Westudyboththesituationswhenthetransversefieldis (cid:113)
quenchedacrossasinglecriticalpointandboththecriti- 2 (h−cosk)2+sin2k) of the Hamiltonian, it can be
calpointsofthemodel,asshowninFigs.1and2,respec- shown that the model has two QCPs at h = ±1 where
tively, and compare the results with those of the sudden the gap vanishes for the momentum modes k = 0 and
quenching. Finally, in Sec. III, we present the rich be- k =π, respectively. These QCPs separate the ferromag-
haviorthatemergesfromthestudyofthevariationofthe netic phase (for |h|<1) from the paramagnetic phase.
DTOP (ν (t)) as a function of time (t) following a slow For all the slow quenching schemes to be analyzed in
D
quench and explore the topology of DPTs which is dic- thesubsequentdiscussions,weshallassumethatthesys-
tatedbythetopologicalpropertiesoftheequilibriumsys- tem is initially prepared in the ground state of the ini-
tem. We investigate both the situations, when starting tial Hamiltonian. We first consider a linear quenching
fromanon-topologicalphase,thesystemisquenchedtoa of the transverse field h = t˜/τ, with t˜ going from a
topologicalphase(crossingasingleQCP)ortotheother large negative value (i.e., the initial field h is large and
i
3
these two basis states.) The final state reached after
n = 0
3 0 n = 1 the ramping (for the k-th mode) can then be written as
it 2 0 nn == 2-1 |ψfk(cid:105) = vk|1fk(cid:105)+ uk|2fk(cid:105), with |uk|2 + |vk|2 = 1: here,
n = -2 |1f(cid:105) and |2f(cid:105) are the ground state and the excited states
h = -1 0 , h = 0 .5 k k
1 0 i f of the Hamiltonian H (h ) with energy eigenvalues −(cid:15)f
k f k
and (cid:15)f, respectively. Clearly |u |2 = |(cid:104)2f|k,−k(cid:105)|2 = p
0 k k k k
denote the non-adiabatic transition probability that the
system ends up at the excited state after the quench.
-1 0
Oncethequenchingiscomplete,webringinthesecond
-1 0 -8 -6 -4 -2 0 2
R time t, set equal to zero, and study the evolution of the
state |ψ (cid:105) with the final Hamiltonian H (h ); general-
fk k f
(a) izing to the complex time plane z, the LO for the k-th
mode (L =(cid:104)ψ |exp(−H (h )z)|ψ (cid:105)) is then given by
k fk k f fk
0 .4 (|vk|2 + |uk|2exp(−2(cid:15)fkz)). Here, we have rescaled the
ground state energy of the final Hamiltonian H (h ) to
k f
(t)0 .3 0, so that the excited state energy (cid:15)efxcited = 2(cid:15)fk. Sum-
I mingoverthecontributionsfromallthemomentamodes
0 .2 andconvertingsummationtotheintegral,inthethermo-
dynamic limit we obtain:
0 .1 (cid:90) π dk (cid:16) (cid:17)
f(z) = − ln |v |2+|u |2exp(−2(cid:15)fz)
2π k k k
0 .0 0
0 2 4 t 6 8 1 0 = −(cid:90) π dk ln(cid:16)(1−p )+p exp(−2(cid:15)fz)(cid:17).(3)
2π k k k
0
(b)
Wethenimmediatelyfindthezeros(i.e.,theFisherzeros)
FIG. 1: (Color online) (a) Fisher zeros as obtained from of the “effective” partition function given by:
Eq. (4) are plotted in complex z plane for different n, where
R denotes Re (z) and it is the Im (z). The line of Fisher (cid:18) (cid:19)
1 p
zSehraorspcnroosns-atnhaelyimtiacigtiineasriynathxeisrfaotreafumncotmioenntauremovbasleurevekd∗.pe(rbi)- zn(k)= 2(cid:15)f ln(1−kpk)+iπ(2n+1) , (4)
k
odicallyat(real)timest∗ (Eq.(6))calculatedatthemomenta
n
values k as shown in the panel (a). Here we have followed where n=0,±1,±2,···.
∗
the quenching protocol h(t˜)=t˜/τ with τ =1, when the field The Fisher zeros following the slow quenching across
is quenched from h = −∞ to h = 0.5 crossing the QCP at onecriticalpointandterminatingthedrivingath =0.5
f
h=−1. Thetimetismeasuredfromtheinstantimmediately are shown in Fig. 1(a). Let us now probe the ques-
after the quenching is complete, when we set t=0. tion whether the line of Fisher zeros cross the imagi-
nary axis for a particular momenta mode k . Clearly,
∗
this happens when p = 1/2 (when the Re(z (k)) van-
k∗ n
negative) to a desired final value so that h < 1 (cho- ishes); this in fact corresponds to “infinite temperature”
f
sen to be equal to 0.5 here); hence, the system crosses state when both the levels of the two-level system are
the QCP at h = −1 where the relaxation time diverges equally populated. Furthermore, for the critical mode
leading to the breakdown of the adiabatic evolution and k = π (which does not evolve in the process of quench-
hence the final state (|ψ (cid:105)) reached is not the ground ing) p =1 while in the case of high energy modes (i.e.,
f k
state of the final Hamiltonian (H ). Considering the modes close to k ∼ 0), p → 0 (see discussion around
f k
reduced Hamiltonian (2), one can define the dynamical Eq. (8) and Fig. 3(a)), since according to KZ theory,
free energy4: f(z) = −ln(cid:104)ψf|exp(−Hfz)|ψf(cid:105)/L, where modes k > k˜ ∼ τ−1/2 move adiabatically, i.e., do not
(cid:104)ψf|exp(−Hfz)|ψf(cid:105) is the Loscmidt overlap while z is sense the passage through the QCP. Using Eq. (4), we
the complex time. It is to be noted that we have set therefore conclude that z (k → 0) → −∞ while for
n
¯h=1 throughout. z (k →π)→+∞. This immediately implies that z (k)
n n
Let us first illustrate the slow quenching scheme fol- indeed goes from ∞ to −∞, crossing the imaginary time
lowing the protocol h = t˜/τ, with the initial state is axis for a particular k as shown in Fig. 1(a). (It should
∗
|k,−k(cid:105) for very large negative h. (In this limit, the re- be noted that numerically we are dealing with a finite
duced 2×2 Hamiltonian in Eq. (2) effectively becomes systemandhenceFisherzerosdonotreallygofrom−∞
diagonalandhencethegroundstate,forthemodek,be- to +∞ but they indeed cross the imaginary axis for k .)
∗
comes |k,−k(cid:105); on the other hand, for large positive h, An exact expression for the rate function I(t), (where,
the ground state is |0(cid:105). Other than these extreme cases, we reiterate, the time t is measured after the quenching
the ground state is in general a linear superposition of is complete) can be exactly derived in the present case,
4
point at h = −1 at time t˜= −τ and that at h = 1 at
0 .9 n = 0 time t˜=τ. At the end of the ramp, the spin chain is in
n = 1
it0 .6 nn == 2-1 thestate|ψfk(cid:105)=vk|1fk(cid:105)+uk|2fk(cid:105),(with|uk|2+|vk|2 =1);
n = -2 the variation of |uk|2 as a function of k is shown in the
0 .3 Fig.3(b). Ash(t˜)isvariedthesystemdetectsboththese
critical points which are gapless at different k values (0
0 .0
and π). For τ (cid:29)1, the transition probability p is close
k
to unity for both k →0,π and falls off exponentially for
-0 .3
k (cid:29)0 or k (cid:28)π.
-0 .6 WenowproceedtoanalyzetheFisherzerosandprobe
-0 .4 -0 .2 0 .0 R 0 .2 0 .4
the non-analyticties in the rate function I(t), again set-
tingt=0whentherampingiscomplete; theLOderived
(a)
using the time evolution of |ψ (cid:105) with H (h ) general-
fk k f
izing the real t to the complex z plane. As shown in
Fig.3(b),theprofileofp clearlymarksthefactthatone
k
0
0.5 isexpectedtoobtaintwoseparatek ’sowingtothesetwo
2 ∗
10 individual critical points which can be treated indepen-
0.4
I(t)0.3 dke=nt0ofaneadchko=thπe,r.thHeerenceex,iastltthwoougihntzenrm(ke)d→iat∞e vfaolurebsokth
∗
(one close to k →0 and other close to k →π) for which
0.2
p = 1/2. For these two k values, the line of Fisher
k ∗
0.1 zeroswillindeedcrosstheimaginaryaxismarkingDPTs
at two different instants of time for each value of the in-
0.0
0 1 2 3 4 5 teger n (Eq. (6)). This is numerically verified and shown
t
in Figs. (2(a)) and (2(b)). Interestingly, we note that
(b) correspondingtoeachlobe(denotedbyn),therearetwo
distinct time instants t∗ = t± (where Fisher zeros cross
n n
FIG. 2: (Color online) (a) Lines of Fisher zeros for each theimaginaryaxis)emergingduetothepassagethrough
integer n following quenching across two QCPs when h is two QCPs with
quenched from h = −10 to h = 10 with h(t˜) = t˜/τ and
i f
τ = 1. Corresponding to each lobe denoted by n, we find
(2n+1)π
two k s close to two different critical modes (k = 0 and π); t+ =
conseq∗uently, there exist two t∗s which we shall denote by n 2(cid:15)f
t+ and t−, with t+ < t−. (b)nThe figure shows the corre- (π−k∗)
n n n n (2n+1)π
sponding non-analyticities in I(t) for different final values of t− = . (7)
h (h = 0,2,10) at different instants of times which match n (cid:15)f
f k∗
perfectly with those predicted by Eq. (6) for h = 0 and
f
Eq. (7) for hf =2,10. The DPT corresponding to the quench across the QCP
at h=−1 occurs at t+ while the instant t− is associated
n n
with the QCP at h = 1 and t− > t+. (The physical
n n
interpretation of this notation will be clear following the
(cid:90) π dk (cid:16) (cid:17) discussionofthenextsectionwhereapositive(negative)
I(t)=− ln 1+4p (p −1)sin2(cid:15)ft . (5) topologicalchargewillbeattributedtoaDPToccurring
2π k k k
0 at t(+)(t(−)).) In the case of quenching across a single
n n
As presented in Fig. 1(b), the non-analyticities in I(t) QCPinFig.1(a),weonlygetzeroscorrespondingtot∗n =
appear at the values of the real time t∗ns t+n. We can contrast this result with the case where h is
changedabruptlyfromalargepositivetoanegativevalue
acrossboththecriticalpoints,onecanstraightwayverify
(cid:18) (cid:19)
t∗ = π n+ 1 , (6) that the lines of Fisher zeros never cross the imaginary
n (cid:15)f 2 axis and hence no DPT is observed as predicted in the
k∗ earlier study by Heyl et al.4.
derived by setting Re(z (k ))=0 in Eq. (4) because the The slow quenching scheme analyzed here brings in
n ∗
argument of log in Eq. (5) vanishes when k = k and another time scale in the problem, namely, the inverse
∗
t=t∗. quenching rate τ. A pertinent question at this point
n
We then consider the case when the transverse field is would be how does the scenario depicted in Figs. 1 and
quenched from a large negative value to a large positive 2 gets altered when τ is varied. Let us first consider
value with a rate τ−1, crossing both the QCPs in the the case of crossing a single QCP at h = −1 ( Fig. 1),
process; the results are presented in Fig. 2. The field is where gap vanishes for the critical mode k =π; expand-
variedash(t˜)=t˜/τ sothatthesystemcrossesthecritical ing around this mode, one can rewrite the reduced 2×2
5
which is essential for observing a DPT. In Fig. 3(a), we
plot numerically obtained |u |2 as a function of k; in-
1 .0 k
t=0.001 deed an analytical expression for the same can be de-
t=1
0 .8 t= 1 0 rived using the Landau-Zener (LZ) transition formula
2|k for non-adiabatic transition probability51–54 which can
|u0 .6 be approximated as |uk∼π|2 = exp(−π(π−k)2τ). From
these observations, it is straightforward to conclude that
0 .4 itisthevalueofk andhence,thoseoft∗,relatedthrough
∗ n
Eq. (6), which get modified when τ is changed. On the
0 .2 other hand, we would like to emphasize that the behav-
ior of Fisher zeros and non-analyticities in I(t) do not
0 .00 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 change qualitatively; the non-analyticities only occur at
k
different instants of time depending upon the value of τ.
This claim will be further illustrated in Fig. 5(b). The
(a)
existence of DPTs for τ →0 is expected since for a large
amplitude sudden quench across a single QCP a k will
∗
always be present as shown in Fig. 3(a)4.
1 .0 We now proceed to analyze the situation presented in
t= 0 .0 0 1 Fig. 2; in this case the system crosses both the criti-
t=1
2|u|k0 .8 t=10 hca=l p1oi(nwtsithatcrhiti=cal−m1o(dweitkh=cr0i)t.icaAlsmaordeesuklts=boπt)hatnhde
0 .6
modes k =0 and k =π are temporally frozen leading to
|u |2 =|u |2 =1 (see Fig. 3(b)); the corresponding
0 .4 k=0 k=π
LZformulaisp =|u |2 =e−πτsin2k. Therefore,foreach
k k
0 .2 value of τ, we find two values of k (with p = 1/2)
∗ k=k∗
yielding two real instants given by t+ and t− as defined
n n
0 .0
0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 before. We therefore again conclude that only the in-
k stants of real time at which DPTs occur will depend on
the inverse rate τ. Referring to Eq. (6), we note that
(b)
(cid:113)
t+ or t− depends on (cid:15)f = 2 (h −cosk )2+sin2k ,
n n k∗ f ∗ ∗
FIG.3: (Coloronline)Numericallyobtainedvaluesoftheadi- where k assumes two values (close to k =0 and k =π).
∗
abatictransitionprobability(ortheprobabilityoftheexcited However, in the limit of small τ(→ 0), (i.e., in the sud-
state) p =|u |2 =|(cid:104)ψ |2k(cid:105)|2, for quenching across a single
k k fk f den limit) as presented in Fig. 3(b), one does not find
QCP (Fig. (a)) and two QCPs (Fig. (b)) plotted as a func-
a value of k and no DPT is expected. Furthermore, it
tion of k for several values of τ. The horizontal line denotes ∗
can be shown in the following straightforward manner
p =1/2. Fig. (a)confirmsthatthe valueofk depend onτ
k ∗
that if quenching is fast (i.e., τ <ln2/π), the DPTs dis-
andthereexistsavalueofk eveninthelimitτ →0. Onthe
∗
other hand, as shown in Fig. (b), there are two k for each τ appear for passage across two critical points: analyzing
∗
leading to two instants of time t+ and t− where DPTs occur the profile of p = |u |2 = e−πτsin2k, we note that the
n n k k
foreachlobedenotedbynasshowninFig. 2. Thevaluek minimum value of p (that occurs at k = π/2) should
∗ k
depend on τ and in the limit τ → 0, the situation becomes at least be less than 1/2, for DPTs to be present. This
sviamluileasrotfoktahnedsuhdendceennqouekncehxinisgtsc.ase as |uk|2 > 1/2 for all implies that pk→π/2 =pmkin =e−πτc =1/2; here τc refers
∗ to a critical τ and DPTs exist for τ >τ ; otherwise, the
c
situation is identical to the sudden quenching across two
QCPs where DPTs get wiped out.
Hamiltonian
(cid:32) (cid:33)
−h−1+ (π−k)2 −i(π−k)
H = 2 . (8)
k i(π−k) h+1− (π−k)2
2 III. TOPOLOGICAL ASPECTS:
Analyzing the Hamiltonian (8), we shall now probe the
non-adiabatic transition probability (i.e., the probability TopologicalpropertiesofDPTsarerelatedtotopologi-
oftheexcitedstate|2f(cid:105))givenbyp =|u |2,attheendof calstructureofequilibriumphasediagram,whichforthe
k k k
thequenchingprocess; itisobviousthatthemodek =π present model in turn, is determined by the sign of the
is temporally frozen as the off-diagonal term vanishes, quantity (h+cosk) /(h+cosk) =(h+1)/(h−1);
k=0 k=π
and hence |u |2 =1. On the other hand, modes away clearly it is negative in the topological (ferromagnetic
k=π
from k = π evolve adiabatically and hence |u |2 = 0. phase with |h|<1) while positive in the non-topological
k(cid:28)π
Fromthecontinuityargument,onethenimmediatelyex- (paramagnetic) phase, |h| > 1. We now proceed to in-
pectsavalueofk =k forwhichp =|u |2 =1/2 vestigate the behavior of the DTOP; to define it in the
∗ k=k∗ k=k∗
6
present context, we refer to the LO: and the corresponding dynamical phase φdyn =
k
−(cid:82)tds(cid:104)ψ (s)|H |ψ (s)(cid:105) = −2|u |2(cid:15)ft with |ψ (cid:105) =
Lk =(cid:16)|vk|2+|uk|2exp(−2i(cid:15)fkt)(cid:17). (9) vk|10fk(cid:105)+ufkke−2i(cid:15)fkfs|2ffkk(cid:105). Let us nowk dekfine the gefokmet-
ric phase as φG =φ −φdyn, so that
Letusfirstconsiderthesituationofquenchingfromthe k k k
non-topological(paramagnetic)tothetopological(ferro-
magnetic) phase across a single QCP (take for example,
the situation illustrated in the Fig. 1(a)). In a spirit
similar to the Ref. [17], we can also express the LO as
L =|r |exp(iφ ). Inthepresentcaseofslowquenching,
k k k
(cid:32) (cid:33)
−|u |2sin(2(cid:15)ft)
φ =tan−1 k k , (10)
k |v |2+|u |2cos(2(cid:15)ft)
k k k
(cid:32) (cid:33)
−|u |2sin(2(cid:15)ft)
φG =tan−1 k k +2|u |2(cid:15)ft. (11)
k |v |2+|u |2cos(2(cid:15)ft) k k
k k k
Note that for the mode k = 0, we have |v |2 = 1, and DPTthereforeitsufficestoexplorethequantity∂φG/∂k
k=0 k
theexcitationprobability|u |2 =0;ontheotherhand, for the momentum k :
k=0 ∗
forthecriticalmodek =π, |v |2 =0and|u |2 =1.
k=π k=π
The geometric phase therefore satisfies the periodicity
relation: ∂φGk (cid:12)(cid:12) =2tan((cid:15)f t)∂|vk|2(cid:12)(cid:12) +2∂|uk|2(cid:12)(cid:12) ((cid:15)f t). (15)
∂k k∗ k∗ ∂k k∗ ∂k k∗ k∗
φG−φG =0 mod 2π (12)
π 0 While the first term diverges at every DPT, the second
termprovidesalinearlyvaryingcontributionwithaslope
Asthelatticemomentumgoesfrom0toπ,φG goesfrom
k determined by the sign of its coefficient (∂ |u |2)| . (In
−π to π thus completing a full circle (See Fig. 4). Fo- (cid:72) k k k∗
fact, it can be shown that the term ∂ φ varies sym-
cusing on the mode k , we find that k k
∗ metrically between positive and negative values in the
(cid:16) (cid:17) intermediate time between two DPTs while (cid:72) ∂ φdyn in-
φG| =tan−1 −tan((cid:15)f t) +2|u |2(cid:15)f t, (13) k k
k k∗ k∗ k∗ k∗ creases or decreases monotonically with time depending
on the sign of (∂ |u |2)| ). It is therefore clear that the
k k k∗
with|u |2 =1/2;thisshowsthatφG| isfixedtozeroor signofthechangeinν ataDPTwillbedeterminedby
k∗ k k∗ D
πforallvaluesoftexceptatDPTsgivenbythecondition the sign of the coefficient of the linearly increasing term
n(cid:15)fka∗tte∗ns =bet(wne+en10/2a)nπdwπhbereetwφeke(nt)twisoilDl-PdeTfisnaedt;kφ=Gk kal.ter- (c∂okn|nuekc|t2e)d|k∗t(o=an−(in∂kd|evxk|t2h)|eko∗r)e.m(iTnhrisef.ha[s17b].e)enPloosoisteivlye
∗
We study the behavior of DTOP or the winding num- sign of (∂k|uk|2)|k∗ corresponds to a jump ∆νD = +1
ber while negative sign yields ∆νD =−1 at every DPT.
Fig. 5 shows the variation of ν as a function of time
D
1 (cid:73) π ∂φG following a quench from large negative h to the topolog-
ν = k , (14)
D 2π ∂k ical phase (e.g., h = 0.5); ν (t) increases in a step-like
0 f D
fashion at every DPT with ∆ν = 1. Let us compare
D
close to DPTs (discussed in the previous section) focus- with Fig. 2 of Ref. [17], where a sudden quench from
ing on the quenching across a single QCP first. In the the topological phase to the trivial phase of a p-wave su-
context of sudden quenches, the DTOP has been found perconducting chain was studied. As discussed above,
to appropriately characterize a DPT17. We address the the sign of ∆ν at a DPT is given in terms of the sign
D
question whether the same is true in the case of slow of (∂ |u |2|)| ; this is negative in the situation stud-
k k k∗
quenches and show that it is indeed the case. ied in Ref. [17] where |u |2 = 1 and |u |2 = 0.
k=0 k=π
As we are integrating in Eq. 14 the full derivative of On the contrary, for a slow quenching of the trans-
a periodic function (modulo 2π), the integral remains verse Ising chain starting from large negative h across
constant unless there is some discontinuity in the phase, the QCP at h = −1 (with the critical mode k = π),
which should be manifested in the δ-function type con- we find the value of ν increases at every DPT. Here,
D
tribution to the derivative of the geometric phase. We |u |2 = 1 and |u |2 = 0 ; from continuity one ex-
k=π k=0
can anticipate that such discontinuities develop only at pectsak forwhich|u |2 =|v |2 =1/2andhence,
∗ k=k∗ k=k∗
7
3 .5
10 4 n

9 3 3 .0 d
I(t)
8
2 2 .5
7
)
*t/(t)0 456 - 011 n , I(td 12 ..50
3
-2 1 .0
2
1 -3 0 .5
0 -4
0 0.5 1 1.5 2 2.5 3 3.5 0 .0
k 0 1 2 t 3 4 5 6
(a)
FIG.4: (Coloronline)VariationofφG asafunctionofk and
k
t/t∗ (where t∗ =π/2(cid:15)f ) following a slow quench h(t˜)=t˜/τ
0 0 k∗
(with τ = 1) from the initial value of the field h = −10 to
i
the final value h = 0.5 across the QCP at h = −1. As the n t
f , = 0 .5
lattice momentum k goes from 0 to π, φG varies from −π to 4 d
+π. Itistobenotedthatbetweent∗ andk3t∗ thereisasingle I(t), t = 0 .5
wfuilnldwiningdrinesgualtnindgheinncνeDνD==12w.hile 0between03t∗0 and 5t∗0 two , I(t) 3 nI(dt ),, tt==55
d
n
2
∆ν (t )=sgn(∂ |u |2|)| is positive. To establish that
D c k k k∗
this is indeed the case, we reverse the quenching scheme
to h(t˜) = t˜/τ with t˜starting from a large positive value 1
andendinginthetopologicalphase. Thespinchainthen
crosses the QCP at h=1 where the gap vanishes for the
mode k = 0 and hence |u |2 = 0 and |u |2 = 1, 0
k=π k=0 0 1 2 3 4 5 6 7 8
leading to a negative ∆ν (t∗) = sgn(∂ |u |2|)| ; one
D n k k k∗ t
can straightway verify numerically that ∆ν is indeed
D
negative in this case (see Fig. 6). Furthermore, one can (b)
investigate the variation ∂ν /∂t as function of time: ν
D D
is independent of time always except at DPTs (t∗) when
n FIG. 5: (Color online) (a) νD along with the rate function
there is a diverging (δ-function) peak. I(t)areplottedasafunctionoftimetfollowingaslowquench
Equipped with above observations, we can now inter- h(t˜) = t˜/τ (with τ = 1) from h = −10 to h = 0.5 so that
i f
pretEq.(14)asageneralizedGauss’slaw; asthesystem the system crosses the QCP at h=−1. We find an increases
evolves in time following a quench across a single QCP, inνD byafactorofunitywheneverthereisaDPT.(b)Using
two different values of τ (0.5 and 5), we show that the time
one can assume that a positive topological charge enters
thesystemateveryt∗. Ontheotherhand, ifthesystem instants at which DPTs occur depend on τ. However, νD(t)
n indeed shows a jump of unit magnitude at every DPT for all
isinitiallychosentobeinthenon-topologicalphasewith
τ as explained in the text.
a large positive h (i.e., the situation depicted in Fig. 6),
we can infer that a negative change enters the system at
every DPT.
times t(+) and t(−) with t(−) >t(+). From Fig. (7), it is
Let us now address the question how does the above n n n n
situation gets altered when the field is quenched from a evidentthatatt(n+), ∆νD =1(inotherwords, apositive
large negative to a large positive value so that the sys- chargeentersthesystemasshowninFig.8);ontheother
tem is swept across both the QCPs at h=±1, from one hand, the DPT at t(−) is associated with ∆ν = −1 or
n D
topologically trivial phase to the other. In this case as a negative topological charge. This makes ν oscillate
D
well ν exhibits a discrete change at every DPT, how- between 0 and 1. This pattern follows up to 5th lobe
D
ever,thereexistfurtherintricaciesaspresentedinFig.7. and there is a discontinuity at the 6th lobe: from Fig. 8,
To analyze this remarkable behavior, we plot the flow of we find that t− >t+. Therefore, we have two successive
6 7
Fisher zeros following a double quench as shown in the DPTs both with ∆ν = +1 (i.e., with identical topo-
D
Fig. 8. We note that corresponding to each lobe denoted logical charges) and consequently, ν jumps from 1 to 2.
D
by n, there are two DPTs occurring at two instants of In general, this type of change would first happen when
8
connected regions. As discussed in Ref. [4] this prevents
analytic continuation of a boundary partition function
0 .5
defined as Z (z) = (cid:104)ψ|exp[−zH]|ψ(cid:105) into the real time-
ψ
0 .0 axis beyond the critical time t∗. On the other hand, for
t) the quench across two critical points the time domain is
(
, I-0 .5 splitintoalternatingintervals. Theanalyticcontinuation
d ofZ(z)isonlypossibletotheintervalsreachablewithout
n
-1 .0 crossing Fisher zeros, i.e. 0 < t < t+, t− < t < t+ and
0 0 1
so on. As we discussed above such intervals where an-
-1 .5 alytic continuation is possible correspond to the sectors
n with zero topological charge. As it is clear from Figs. 7
-2 .0 d
I(t) and8suchpiecewiseanalyticcontinuationispossiblefor
finitetimeuntilthezerowithpositivetopologicalchange
-2 .5
0 2 4 6 8 1 0 from the n+1-st branch of Fisher zeros crosses the zero
t
with negative topological charge from the n-th branch,
i.e. t+ < t−. Beyond this point no analytic continua-
n+1 n
tion of the boundary partition function is possible.
FIG. 6: (Color online) This shows that when the system
quenched across the QCP at h = 1 to the topological phase Finally, we address the question how does this topo-
(h = 0.5) starting from a large positive h (h = 10), logical structure depend on τ; as argued in the previous
f i
sgn(∂k|uk|2|k∗), and hence ∆νD is indeed negative. section, τ determines the time instants at which DPTs
occur. Hence there will not be any qualitative change in
the topological pattern, i.e., ∆ν = 1 ( or −1) at every
D
DPT as illustrated in Fig. 5(b). However, in the case
3 .5 of crossing across two QCPs the values of n for which
n d t+ <t− and hence we have two successive DPTs with
3 .0 I(t) n+1 n
∆ν =1 may depend on τ.
D
)
(t 2 .5
, I
d 2 .0
n IV. CONCLUDING COMMENTS
1 .5
We have studied the slow quenching dynamics of a
1 .0
transverse Ising chain across its QCPs and analyzed the
0 .5 Fisher zeros and hence DPTs which are reflected in the
non-analyticitiesoftheratefunction. Weemphasizethat
0 .0
0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 although the non-analyticities in the dynamical free en-
t ergy leading to periodic occurrences of DPTs have been
reported in the context of slow quenching of the Hamil-
FIG. 7: (Color online) ν along with the rate function I(t) tonian (1)32, they have not been connected to the Fisher
D
are plotted as a function of time t following a slow quench zeros. Furthermore, we show that these DPTs survive
h(t˜) = t˜/τ from h = −10 to h = 10, so that the system evenwhenthesystemisquenchedacrossboththeQCPs
i f
crosses both the QCPs at h=−1 and h=+1. We find that of the model when the line of Fisher zeros crosses the
νD oscillates between 0 and 1, i.e., νD = ±1 at successive imaginary axis for two characteristic momenta values.
DPTs. HoweverthereisajumpinνDbyunityforsomevalues Thisisinsharpcontrastwiththesuddenquenchingcase
of t when there are two successive DPTs with ∆ν = 1 (or
D when an abrupt passage across two QCPs wipes out the
when a positive charge crosses a negative charge as shown in
non-analyticities. In fact, for a sufficiently slow quench-
Fig. 8); between two jumps the alternating pattern persists.
ing, twoQCPsofthemodelappeartobeindependentof
each other. Secondly, in this case, it is essential to cross
the QCP to observe the DPTs, otherwise the transition
t(n−) >t(n++)1, i.e., second DPT (with a negative topologi- probability pk is always less than 1/2 for all values of k
calcharge)correspondingtothelobenhappensatalater and hence a DPT can not occur.
(real) time compared to the first DPT (with a positive We have also analyzed the behavior of the DTOP
charge) associated with (n+1)-th lobe. (i.e., ν (t)). The slow quenching scheme discussed in
D
Finally,letusdiscusstheissueconcerningtheanalytic this paper, allows to study both cases: the quenching
continuation of the boundary partition function to the across a single QCP (i.e., from a non-topological phase
real time axis. Comparing Figs. 1(a) and 2(a) we see a to the topological phase) or two QCPs (i.e., from one
verydifferenttopologypatternoftheFisherzerosforthe non-topological phase to the other) depending upon the
quenchesacrossoneandtwocriticalpoints. Intheformer final value of the transverse field. In the former case,
case Fisher zeros slice the complex time space into dis- the DTOP increases (or decreases depending upon the
9
direction of the quenching) step-wise by unity at those Acknowledgments
instants of time where DPTs occur. This can be inter-
pretedaschargesofthesamesignenteringthesystemat
everyDPT.Onthecontrary,inthelattercasetheDTOP WeacknowledgeJun-ichiInouefordiscussionsandSei
oscillates between 1 and 0; this implies that a DPT with Suzukiforextensivediscussions,criticalcommentsonthe
a positive topological charge ∆ν = 1 at real time t+ manuscript and collaboration in related works. Special
D n
is followed by a DPT with a negative topological charge thanks to Utso Bhattacharya for his critical comments.
∆ν =−1 at t−. Furthermore there are sharp increases SS acknowledges CSIR, India and also DST, India, and
D n
byafactorofunitywheneverthereisacrossingofFisher AD and UD acknowledge DST, India, for financial sup-
zerosbelongingtoneighboringlobesi.e.,t− >t+ ,there port. AD and SS acknowledge Abdus Salam ICTP for
n n+1
aretwosuccessiveDPTswithtopologicalchargesofsame hospitality where the initial part of the work was done.
signs; this is indeed remarkable. A.P. was supported by AFOSR FA9550-13-1-0039, NSF
DMR-1506340, ARO W911NF1410540
1 M.E. Fisher, in Boulder Lectures in Theoretical Physics 26 S. Sharma, A. Russomanno, G. E. Santoro and A. Dutta,
(University of Colorado, Boulder, 1965), Vol. 7. EPL 106, 67003 (2014).
2 C. Yang and T. Lee, Phys. Rev. 87, 404 (1952). 27 P. Zanardi. H. T. Quan, X. Wang and C. P. Sun, Phys.
3 W. van Saarloos and D. Kurtze, J. Phys. A 17, 1301 Rev. A 75, 032109 (2007).
(1984). 28 A. Gambassi and A. Silva, arXiv: 1106.2671 (2011); P.
4 M.Heyl,A.Polkovnikov,andS.Kehrein,Phys.Rev.Lett., Smacchia and A. Silva, Phys. Rev. E 88, 042109, (2013).
110, 135704 (2013). 29 R. Dorner, J. Goold, C. Cormick, M. Paternostro and V.
5 S.Sachdev,Quantum Phase Transitions (CambridgeUni- Vedral, Phys. Rev. Lett. 109, 160601 (2012).
versity Press, Cambridge, UK, 1999). 30 S.SharmaandA.Dutta,Phys.Rev.E92,022108(2015).
6 S. Suzuki, J-i Inoue and Bikas K. Chkarabarti, Quantum 31 A. Russomanno, S. Sharma, A. Dutta and G. E. Santoro,
Ising Phases and Transitions in Transverse Ising Models J. Stat. Mech., P08030 (2015).
(Springer, Lecture Notes in Physics, Vol. 862 (2013)). 32 F. Pollmann, S. Mukerjee, A. G. Green, and J. E. Moore,
7 C. Karrasch and D. Schuricht, Phys. Rev. B, 87, 195104 Phys. Rev. E 81, 020101(R) (2010).
(2013). 33 T. W. B. Kibble, J. Phys. A 9, 1387 (1976).
8 N. Kriel, C. Karrasch, and S. Kehrein Phys. Rev. B 90, 34 W. H. Zurek, Nature 317, 505 (1985).
125106 (2014) 35 W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett.
9 F.Andraschko,J.Sirker,Phys.Rev.B89,125120(2014). 95, 105701 (2005).
10 E. Canovi, P. Werner, and M. Eckstein, Phys. Rev. Lett. 36 A. Polkovnikov, Phys. Rev. B 72, 161201(R) (2005).
113, 265702 (2014). 37 J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
11 M. Heyl, Phys. Rev. Lett. 115, 140602 (2015) . 38 B. Damski, Phys. Rev. Lett. 95, 035701 (2005).
12 T. Palmai, Phys. Rev. B 92, 235433 (2015). 39 R.W.CherngandL.S.Levitov,Phys.Rev.A73,043614
13 S.VajnaandB.Dora,Phys.Rev.B89,161105(R)(2014). (2006).
14 S. Sharma, S. Suzuki and A. Dutta, Phys. Rev. B . 92, 40 V.Mukherjee,U.Divakaran,A.Dutta,andD.Sen,Phys.
104306 (2015). Rev. B 76, 174303 (2007); U. Divakaran, V. Mukherjee,
15 S. Vajna and B. Dora, Phys. Rev. B 91, 155127 (2015). A.Dutta,andD.Sen,J.Stat.Mech.P02007(2009);U.Di-
16 M. Schmitt and S. Kehrein, Phys. Rev. B 92, 075114 vakaran, A. Dutta, and D. Sen, Phys. Rev. B 78, 144301
(2015) (2008); V. Mukherjee and A. Dutta, EPL 92 , 37004
17 J. C. Budich and M. Heyl, Phys. Rev. B (2015). (2010).
18 H.T. Quan, Z. Song, X.F. Liu, P. Zanardi, and C.P. Sun, 41 T. Caneva, R. Fazio, and G. E. Santoro Phys. Rev. B 76,
Phys.Rev.Lett. 96, 140604 (2006). 144427(2007);F.Pellegrini,S.Montangero,G.E.Santoro,
19 D.Rossini,T.Calarco,V.Giovannetti,S.Montangero,R. and R. Fazio, Phys. Rev. B 77, 140404(R) (2008).
Fazio, Phys. Rev. A 75, 032333 (2007). 42 K.Sengupta,D.Sen,andS.Mondal,Phys.Rev.Lett.100,
20 F. M. Cucchietti, et al, Phys. Rev. A 75, 032337 (2007); 077204(2008);D.Sen,K.Sengupta,S.Mondal,Phys.Rev.
C.CormickandJ.P.Paz,Phys.Rev.A77,022317(2008). Lett. 101, 016806 (2008).
21 LorenzoCVenutiandP.Zanardi,Phys.Rev.A81,022113 43 S Deng, G Ortiz, L Viola, EPL 84 , 67008 (2008); Phys.
(2010);LorenzoC.Venuti,N.T.Jacobson,S.Santra,and Rev. B 80, 241109(R) (2009).
P. Zanardi, Phys. Rev. Lett. 107, 010403 (2011). 44 A. Dutta, R. R. P. Singh, and U. Divakaran, EPL 89,
22 S. Sharma, V. Mukherjee, and A. Dutta, Eur. Phys. J. B, 67001 (2010); T. Hikichi, S. Suzuki, and K. Sengupta,
85, 143 (2012). Phys. Rev. B 82, 174305 (2010).
23 V. Mukherjee, S. Sharma, A. Dutta, Phys. Rev. B 86, 45 R. Barankov and A. Polkovnikov, Phys. Rev. Lett.
020301 (R) (2012). 101, 076801 (2008); C. De Grandi, V. Gritsev and A.
24 T. Nag, U. Divakaran and A. Dutta, Phys. Rev. B 86, Polkovnikov, Phys. Rev. B 81, 012303 (2010).
020401 (R) (2012). 46 M. Thakurathi, W. DeGottardi, D. Sen, S. Vishveshwara,
25 B. Dora, F. Pollmann, J. Fortgh, G. Zarand, Phys. Rev. Phys. Rev. B 85, 165425 (2012).
Lett. 111, 046402 (2013); R. Sachdeva, T. Nag, A. Agar- 47 E. Canovi, E. Ercolessi, P. Naldesi, L. Taddia, and D.
wal, A. Dutta, Phys. Rev. B 90, 045421 (2014). Vodola, Phys. Rev. B 89, 104303 (2014).
10
n = 0 -1
3 . 0 n = 1 -1
+ 1
n = 2
t
i
n = 3 -1
2 . 4 t - + 1
6 -1
n = 4
+ 1 -
t
n = 5 -1 5
1 . 8 t + + 1 +
n = 6 6 + 1 t 5
n = 7 -1
+ 1
1 . 2 n = 8
-1
n = 9 + 1
-1
0 . 6 h = - 1 0 , h = 1 0 + 1
i f -1
t
= 1 t - + 1
0 -1
0 . 0 t + + 1
0
- 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3
R
FIG. 8: (Color online) The flow of Fisher zeros following quenching from h = −10 to h = 10 across two QCPs with the
i f
protocol h(t˜) = t˜/τ with τ = 1. Each lobe is characterized by the integer n which increases from bottom to top with the
lowestlobecorrespondington=0. ThereexisttwoDPTsassociatedwitheachn;theDPTatt+ isassociatedwithapositive
n
topological charge (+1) while t− denotes a DPT with a negative topological charge (−1). This explains the oscillation of ν
n D
between 0 and 1. However, we note that t+ < t− and hence there are two successive DPTs both with positive topological
6 5
charge occurring at t+ and t+, respectively; this leads to a jump in ν (t) by a factor of unity (as shown in Fig. 7) following
5 6 D
which the oscillating pattern persists.
48 A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- 1965).
tore, Rev. Mod. Phys. 83, 863 (2011). 52 N.V.VitanovandB.M.Garraway,Phys.Rev.A53,4288
49 J. Dziarmaga, Advances in Physics 59, 1063 (2010). (1996); N. V. Vitanov, Phys. Rev. A 59, 988 (1999).
50 A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, 53 S. Suzuki and M. Okada, in Quantum Annealing and Re-
T.RosenbaumandD.Sen,QuantumPhaseTransitionsin lated Optimization Methods, Ed. by A. Das and B. K.
Transverse Field Spin Models: From Statistical Physics to Chakrabarti (Springer-Verlag, Berlin, 2005), p. 185.
QuantumInformation (CambridgeUniversityPress,Cam- 54 C. De Grandi, A. Polkovnikov, Quantum Quenching, An-
bridge, 2015). nealingandComputation,Eds.A.Das,A.ChandraandB.
51 C. Zener, Proc. Roy. Soc. London Ser A 137, 696 (1932); K. Chakrabarti, Lect. Notes in Phys., vol. 802 (Springer,
L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Heidelberg 2010).
Non-relativisticTheory,2nded.(PergamonPress,Oxford,