Adiabatic quenches through an extended quantum critical region

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by  Franco Pellegrini

Adiabatic quenches through an extended quantum critical region
Franco Pellegrini,1,2 Simone Montangero,1 Giuseppe E. Santoro,2,3,4 and Rosario Fazio2,1
1NEST-CNR-INFM & Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
2International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014 Trieste, Italy
3CNR-INFM Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy
4International Centre for Theoretical Physics (ICTP), P.O.Box 586, I-34014 Trieste, Italy
(Dated: February 4, 2008)
BygraduallychangingthedegreeoftheanisotropyinaXXZchainwestudythedefectformation
inaquantumsystemthatcrossesanextendedcriticalregion. Wediscusstwoqualitativelydifferent
8
casesofquenches,fromtheantiferromagnetictotheferromagneticphaseandfromthecriticaltothe
0
antiferromegneticphase. Bymeansoftime-dependentDMRGsimulations,wecalculatetheresidual
0
energy at the end of the quench as a characteristic quantity gauging the loss of adiabaticity. We
2
find the dynamical scalings of the residual energy for both types of quenches, and compare them

n with thepredictions of the Kibble-Zurekand Landau-Zenertheories.
a
J

9 When,bychangingsomeexternalparameters,aquan- XXZ model [28]. The parameter that will be changedto
2
tum system crosses a phase transition at a finite speed crossthedifferentphasesistheanisotropycoupling. The

it is unable to reach its equilibrium (or ground) state no study of adiabatic quenches in the XXZ model allows to
]
r matter how slow the rate of the quench is. The reason testseveralaspectsoftheproblemwhichcouldnotbead-
e
h is that the anomalous slow dynamics close to the criti- dressed previously. The system has an extended critical
t cal point prevents the system to follow adiabatically the lineinsteadofacriticalpoint(asimilarissuewasrecently
o
externaldrive. As a result,supposing that the dynamics considered in [26] for the Kitaev model). Moreover, the
.
at takes the system to a symmetry broken phase, a num- boundariesofthecriticalregionarecharacterizedbydif-
m ber of defects will appear in the final state. Dynamical ferentexponents,hence onecantestif the defectdensity
defect formation has important implications in a wide is controlled by what happens before or after the pas-
-
d spectrum of problems ranging from the study of phase sage through the critical point [29]. Finally, this model
n
transitions in the early universe [1, 2] and in superfluid allows to study defect formation in the presence of dy-
o
systems [3, 4, 5], to quantum annealing [6, 7, 8, 9, 10] namical constraints. Because of the conservation of the
c
[ and adiabatic quantum computation [11]. total magnetization, it will be impossible for the system

Earlyworksontheadiabaticcrossingofaphasetransi- to reach the local ground state no matter how slow the
1
v tions dealt with classicalsystem where the external con- quench is even for a finite system.
5 trol parameter is the temperature. This problem was The one-dimensional XXZ model is defined by the
7 more recently explored also in the case of a quantum Hamiltonian
4
phase transition [12, 13]. These works stimulated an in-
4 N−1
1. t2e1n,s2e2t,h2e3o,re2t4ic,a2l5a,c2ti6v,it2y7](saened[1r4e,fe1r5e,n1c6es, 1th7,er1e8in,)1.9,T2h0e, H(t)=−J X (cid:2)σixσix+1+σiyσiy+1+∆(t)σizσiz+1(cid:3) . (1)
0 i=1
large body of results obtained so far for the adiabatic
8
0 crossing through a critical point are in agreement with describing N spin-1 interacting via a nearest-neighbour
2
: the Kibble-Zurek (KZ) theory. At the roots of the KZ Heisenberg interaction anisotropic along the z-direction,
v
mechanismthereisthehypothesisthatthedynamicsofa ∆ being the anisotropy parameter. Here σx, σy and σz
i
X systemclosetoacontinuousphasetransitioncanbecon- are the Pauli matrices. This system is invariant un-
r sidered adiabatic or impulsive depending on the vicinity der rotations around the z axis, so that the total z-
a
to the critical point. The determination of the point in component of the spin Sz is a conserved quantity. For
tot
whichthesystemstopsfollowingthe externaldriveleads time-independent couplings, the system can be exactly
to the determination of the scaling of density of defects solved, by means of the Bethe ansatz [28]. If ∆ > 1
and other observables as a function of the quench rate. the system is ferromagnetic, with all spins aligned in
In the limit of very slow quenches, defect formation can the z-direction. The low-lying excitations have a gap
be also understood by means of the Landau-Zener (LZ) ∆E =4J(∆−1) which closes for ∆→1+. For ∆<−1
theoryappliedtothe groundandthe firstexcitedstates. the system is in the antiferromagnetic N´eel phase. The
Thepassagethroughacriticalpointisnottheonlysit- two possible ground states, differing by a traslation by
uationonecanenvisageinthiscontext. Anotherparadig- onelatticespacing,aredegenerateinthethermodynamic
matic case is when the evolving many-body system is limit, i.e., for N →∞ their splitting is ∝e−cN, with c a
quantum critical in an extended region of the parameter constant. The low-lying excitations are made up by do-
space. This is the topic of the present work. The sys- mainwallsseparatingregionswiththetwodifferentN´eel
temweusetoillustrate thissituationis describedbythe phasescharacterizingthegroundstate: theyhaveatotal
2
sidered two different situations. i) An evolution from
0.6 0.006 the antiferromagnetic ground state with an initial value
J of the anisotropy ∆ ≪ −1 to the ferromagnetic region
E/0.004 II exc. at a final value ∆ i≫ 1; ii) an evolution from ∆ = 0
f i
∆
0.002 in the critical region to the antiferromagnetic region at
/J0.4 0 I exc. ∆f ≪−1.
E ∆ ≈ 1 In order to monitor the loss of adiabaticity after the
∆
quench, we consider the excess final energy of the sys-
tem relative to the ground state in the given subspace,
0.2
conveniently rescaled. More precisely
hΨ(t)|H(t)|Ψ(t)i−hΨ (t)|H(t)|Ψ (t)i
E˜ (t)= GS GS ,
res
0-2 -1 0 1 hΨ0|H(t)|Ψ0i−hΨGS(t)|H(t)|ΨGS(t)i
(2)
∆
whereΨ istheinitialwavefunction,whichwetaketobe
0
Figure 1: Excitation energy of the two lowest excited states the groundstate of the initial Hamiltonian H0 =H(tin),
of the XXZ model for N = 100 spins with open boundary and ΨGS(t) is the instantaneous ground state of H(t)
conditions in the subspace Stzot = 0. Data obtained from (in the subspace Sz =0). The denominator normalizes
tot
static DMRG simulations (m = 160 with 3 target states). the excess energy to the maximum possible attainable
Details of the spectrum in the region close to ∆ = 1 are
value, corresponding to a wavefunction |Ψ(t)i = |Ψ i
shown in theinset. 0
which does not evolve at all (totally impulsive regime).
When t → t , this quantity approaches a value E˜ =
fin res
magnetization Stzot =0,±1 and a finite gap which again E˜res(tfin), which coincides with the final number of de-
closes for ∆ →−1−. The part of the spectrum which is fects (apart from a constant factor) for tfin/τQ ≫ 1, as
relevantforourpourposesbelongstotheSz =0sector. only the z-polarizationof the spins counts, in that limit,
The excitation energies of the two lowestt-olyting excited in determining the final energy of the system. E˜res nat-
statesinthissubspaceforasystemofN =100spinsand urally takes into account only the defects formed during
openboundaryconditions(OBC)areshowninFig.1. In thequench,andrangesfromE˜res =1foratotallyimpul-
the whole region −1 ≤ ∆ ≤ 1 the spectrum is gapless. sive situation (the wavefunction does not evolve at all),
For finite sizes N and −1 ≤ ∆ ≤ 1, the gap vanishes to E˜res =0 for a fully adiabatic evolution.
linearly in N−1, ∆E ≈2πv/N, for all values of ∆ (with i) Antiferro to Ferro quench - The system is ini-
different ∆-dependent velocities v), except for ∆ = 1, tially in its ground state at ∆ = −20 (we tried differ-
where the scaling is quadratic, ∆E ≈ 1/N2. Therefore ent initial values, observing no difference), and is then
one expects a final density of defects strongly dependent quenched at finite rate to a final value of the anisotropy
on whether, during the quench, the system has crossed ∆ = 20. The residual energy E˜res(t = tfin) as a func-
this point. tion of the quench rate 1/τQ for various chain lengths
In order to study the time-dependent XXZ model we is shown in Fig. 2(A). After a saturation region occur-
have to resort to numerical simulations. The results ringforveryfastquenches,E˜res obeysa power-law,with
we present have been obtained by means of the time- an exponent which is approximately 0.25 (an extrapola-
dependent Density Matrix Renormalization Group algo- tionto the thermodynamiclimitwouldgiveanexponent
rithm (t-DMRG) with a second order Trotter expansion 0.251±0.004, see the inset of Fig. 2(A)). The origin of
ofH [30,31](see[32]forareview). Weconsideredchains this power-lawcanbe undestoodbymeansofaLZargu-
up to N = 200 with open boundary conditions. For the ment [12], supposing that the loss of adiabaticity is en-
slowest quenches we had to restrict N to smaller values tirely due to the closingofthe gapat∆=1 (see Fig.1).
in order to obtain reliable results. The smallest Trotter The probability of getting excited by this point only is
time-steps were chosen to be δt = 10−4J, and the trun- given by Pex(τQ,N) = e−γτQ(∆EN)2, where γ is a con-
cated Hilbert space dimension in the DMRG was up to stant related to the slope of the two approaching eigen-
m=200. We checked that in all the cases presented the values(groundandexcitedstates),and∆EN =2π2J/N2
resultsdonotdependontheTrotterdiscretizationδ and is the finite-size smallest gap at the ∆ = 1 point, corre-
t
on the DMRG-truncation of the states. sponding to a spin-wave of momentum k = π/N (for
The fully polarized ferromagnetic ground state (i.e., OBC). The density of defects for large τQ can be esti-
allspins upordown)isa goodeigenstateforeveryvalue matedbyevaluatingthetypicallengthL˜ǫ(τQ)ofadefect-
of ∆. It is therefore relevant to consider only quenches free region (in units of the lattice spacing a), ǫ being a
that start either from the antiferromagnetic or from the small quantity of our choice, denoting the probability of
critical region. The anisotropy parameter ∆ is changed getting excited. Requiring Pex(τQ,L˜ǫ) = ǫ immediately
in time according to ∆(t) = t/τ . In this work we con- implies that L˜4 = τ γ′/logǫ−1. Consequently for the
Q ǫ Q
3
1
N (A)
30
50
70
100
0.5 200
-2
10
s s
re 0.28 re
˜E ˜E
nt N
e0.27
0.25 n 8
po 10
x0.26 12
E 16
0.25
0 1/N 0.04
-4
10
0.1 1 10 100 0.001 0.01 0.1
1/τ 1/τ
Q Q
Figure 3: (Color online) Final energy after a quench in XXZ
1/τ (B) chains of different lengths (see legend) as a function of the
Q
1 quench velocity 1/τQ (only ’slow’ quenches plotted). Data
2 fromt-DMRGsimulations(m=20,δtfrom10−3J to10−2J).
4 The dashed lines represent LZ prediction starting from the
10
0.4 ground state (lower curves) or from the first excited state
(uppercurves,see text), for thevarious lengths.
)
t
(
s
e
r
˜E
find is however more complicated than expected. The
0.2
wavefunctionisnotfrozenthroughoutthecriticalregion.
Nevertheless at ∆=1 there is a clear kink in the depar-
ture from the adiabaticity which dominates the density
ofdefects(andtheresidualenergy)inthefinalstate. For
0 the slowest quenches the dependence of the residual en-
-10 -5 0 5 10
∆ ergy on the quench rate crosses-over from a power-law
to an exponential. As it has been pointed [12, 14], this
Figure 2: (Color online) (A) Final excess energy after a regime is described by the LZ theory. Here a good esti-
quench for XXZ chains of various lengths N (see legend) as mateoftheresidualenergycanbe obtainedbyassuming
a function of the quench rate 1/τQ. The dashed lines rep- thatthewholebehaviourisdeterminedbywhathappens
resent power-law fits of the data for the various N’s. Data
at∆=1 wheretherearetwoexcitedstatesthatbecome
tfroom10−t-2DJM).RGInsseitm:uelaxtpioonnsen(tms ofrfotmhe20potwoe3r0-l,awδt fifrtosmfo1r0−va4rJ- degenerate at ∆ ≥ 1. Lower and upper bounds to the
residual energy can be simply obtained by considering
ious chain lengths N as a function of 1/N. The dashed
line is a linear fit to the exponent, extrapolating to 1/4 the the LZ transition probability from the ground state to
thermodynamic limit. (B) Evolution of the excess energy the first excited state (lower bound) and the transition
of a state during quenches for N = 100 spins in the region fromthe firstto the secondexcited state (upper bound).
−10 < ∆ < 10 for various rates 1/τQ. Data from t-DMRG AsitcanbeseenfromFig.3the actualratesliebetween
simulations (m=30, ∆t=10−3J).
these two curves.
ii)Critical to Antiferro quench-Anotherinterest-
ing situation which can be studied with the XXZ model
residual energy we get:
is the adiabatic quench from the gapless phase, for ex-
E˜ ∼ 1 ∝τ−41 , (3) ample at ∆ = 0, to a final point deep inside the N´eel
res L˜ (τ ) Q phase. In this case the critical region terminates with
ǫ Q
a Berezinskii-Kosterlitz-Thouless point at ∆ = −1, and
in very good agreement with the numerical data. Thus the applicability of the KZ theory needs to be tested.
it seems that the scaling of the point ∆ = 1 dominates This situation is also relevant for the cases of strongly
over the rest of the critical region in a fairly wide re- interacting bosons in one dimension driven from the su-
gion of quench rates. To understand the importance of perfluid to the Mott phase [18].
this point, and to see how adiabaticity is lost during the In our simulations we let ∆ to evolve linearly from
quench,welookedattheevolutionoftheresidualenergy, ∆ = 0 to ∆ = −6. The residual energy E˜ as a
i f res
Eq. 2, for various quench rates, see Fig. 2(B). What we function of the quench rate 1/τ is shown in Fig.4. The
Q
4
1
Normale Superiore. This work has been developed by
N
using the DMRG code released within the ‘Powder with
50
70 Power’project (www.qti.sns.it).
100
200
0.1
s 1.1
e
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ThisresearchwaspartiallysupportedbyMIUR-PRIN, J. of Comp. and Theor. Nanosc. (in press), cond-
EC-Eurosqip and CRM “Ennio De Giorgi” of Scuola mat/0603842.