Quantum relaxation after a quench in systems with boundaries

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by  Ferenc Iglói

Quantum relaxation after a quench in systems with boundaries
Ferenc Igl´oi1,2,∗ and Heiko Rieger3,†
1Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.Box 49, Hungary
2Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
3Theoretische Physik, Universita¨t des Saarlandes, 66041 Saarbru¨cken, Germany
(Dated: January 21, 2011)
Westudythetime-dependenceofthemagnetization profile,m(t),ofalargefiniteopenquantum
l
1
Ising chain after a quench. We observe a cyclic variation, in which starting with an exponentially
1
decreasing period the local magnetization arrives to a quasi-stationary regime, which is followed
0
byan exponentially fast reconstruction period. Thenon-thermalbehaviorobservedat near-surface
2
sites turns over to thermal behavior for bulk sites. Besides the standard time- and length-scales a

n non-standard time-scale is identified in thereconstruction period.
a
J

0 Recent experimental progress in controlling ultracold pending on the parameters of the quench [7, 16, 17].
2 atomic gases in optical lattices has opened new per- Somequantumsystemsdonotthermalizecompletelyand

spectives in the physics of quantum systems. In these displayadifferentbehaviorforcorrelationfunctionsoflo-
]
h measurements the coupling in an interacting system can cal and for non-local operators, such that the former do
c be tuned very rapidly, commonly denoted as “quench”, notexhibiteffectivethermalbehavior[16]. Aninteresting
e
for instance by using the phenomenon of Feshbach reso- issuenotbeingaddressedsofaristhecharacterizationof
m
nance and the couplings to dissipative degrees of free- the non-stationary, that means not time-translation in-
-
dom (such as phonons and electrons) are very weak. variant,quantum relaxationfollowinga quench: Prepar-
t
a As a consequence one can study coherent time evolu- ingthequantumsysteminanon-eigenstateofitsHamil-
t
s tion of isolated quantum systems. Among the fasci- tonian, how is thermalization achieved during the time-
.
t nating new experiments we mention the collapse and evolution? How do correlations develop in time towards
a
m revival of Bose-Einstein condensates[1], quenches in a the stationary (i.e. time translation invariant) state, is
spinorcondensate[2],realizationofone-dimensionalBose there a time dependent correlationlength, etc.?
-
d systems[3] and measurements of their non-equilibrium Another importantissue concerns quantum relaxation
n relaxation[4]. and potential thermalization in the presence of bound-
o Concerning the theoretical side of quantum quenches aries. Theoretical studies of non-equilibrium quantum
c
herethefirstinvestigationshadbeenperformedonquan- relaxation have focused on bulk sites up to now, but all
[
tum XY and quantum Ising spin chains[5–7] before the realsystemshaveafiniteextentandtheyareboundedby

2 experimental work has been started. The new experi- surfacesandthe physicalpropertiesinthe surfaceregion
v
mental results in this field have triggered intensive and areconsiderablydifferentfromthoseinthebulk[18]. Ob-
4
systematic theoretical researches, which are performed viously an interesting question is whether the time and
6
6 ondifferentsystems, suchas1DBose gases[8], Luttinger length scales characterizing the stationary relaxation in
3 liquids[9]andothers[10]. Besidesstudiesonspecificmod- the bulk is altered in the vicinity of the boundary, and
1. elstherearealsofield-theoreticalinvestigations,inwhich how thermalization is achieved there.
1 relation with boundary critical phenomena and confor- In this paper we will address these two issues: The
0 mal field-theory are utilized[11, 12]. non-stationary quantum relaxation after a quench and
1
One fundamental question of quantum quenches con- the effect of boundaries. For this we focus on a compu-
:
v cerns the nature of the stationary state of this non- tationally tractable model for a quantum spin chain and
i equilibrium quantum relaxation including the issue of studytherelaxationofprofilesofobservablesintheearly
X
thermalization and potential descriptions by Gibbs en- timestepsaswellastheirbehaviorinthelong-timelimit.
r
a sembles. For non-integrable systems exact thermal- We also address the behavior in large,but finite systems
ization of stationary states was conjectured[13], how- and study the consequences of the recurrence theorem.
ever the numerical results on specific systems are The system we consider in this paper is the quantum
controversial[13–15]. On the other hand integrable sys- Ising chain defined by the Hamiltonian:
temsaresensitivetotheinitialstatesandtheirstationary
L−1 L
states are thermal-like being in a form of a generalized
Gibbs ensemble[8]. H=−Xσlxσlx+1−hXσlz , (1)
l=1 l=1
Thermalization includes generically (i.e. away from
critical points) an exponential decay of correlationfunc- in terms of the Pauli-matrices σx,z at site l. In the
l
tions in the stationary state on length and time scales non-equilibrium process the strength of the transverse
that can be related to the correlationlength and time of field is suddenly changed from h (t < 0) to h (t 0).
0
≥
an equilibrium system at an effective temperature de- The Hamiltonian in Eq.(1) can be expressed in terms
2
0 0
TABLE I: Decay exponent of the off-diagonal (longitudinal) -1 l=O1->6O -5 l=O1-6>D
magnetization in theinitial (equilibrium) period. -2 ll==3428 --1150 ll==3428
bulk h01=/8hc h01>/2hc log m(t)l----6543 ll==lll===116891240628 log m(t)l----33225050 ll==lll===116891240628
-40
boundary 1/2 3/2 -7 (a) -45 (b)
-8 -50
0 200 400 600 800 1000 1200 0 100 200 300 400 500 600
t t
0 -1.8
of free fermions[19], which is used in studies of its non- -5 l=D1->6O -2 l=1D2->8D
-10 l=32 -2.2 l=112
enqeutiizliabtrioiunm,σpzr,owpheircthieiss[6a,l1o6c]a.lTopheerabtuolrk,htraasnnsovner-tsheemrmaga-l m(t)l---221505 llll====46898406 m(t)l ---222...864 llll====98646048
behavior[5,1l1,17],whereasthebulk(longitudinal)mag- log --3350 ll==111228 log -3-.32 ll==3126
netization,σx,whichisanon-localoperator,haseffective --4450 (c) --33..64 (d)
l -50 -3.8
thermal behavior[17]. Here we concentrate on the latter 0 200 400 600 800 1000 1200 0 100 200 300 400 500 600
t t
quantity and study the time-dependence of its profile,
m (t)=lim Ψ(0) σx(t)Ψ(0) , where Ψ(0) is the FIG.1: (Coloronline)Relaxationofthelocalmagnetization,
grlound statbe→o0f+tbhhe 0init|iall H|am1iltiobnian (1)|in0 thibe pres- lqougemnclh(t)w,itahtpdaiffraemreentterpso:sait)iohn0s=in0a.0Lan=d h25=60c.h5ai(nOa→fterOa)
enceofanexternallongitudinalfieldb. Accordingto[20] b) h0 = 0.5 and h = 1.5 (O → D) c) h0 = 1.5 and h = 0.5
this can be written as the off-diagonalmatrix-elementof (D → O)d) h0 =1.5 and h=2.0 (D → D).
the Hamiltonian (1):
ml(t)=hΨ0(0)|σlx(t)|Ψ1(0)i. (2) tumcorrelationsintime signalizedbythe reconstruction
of the value of the local observable. In the following we
(0)
Here Ψ is the first excitedstate (which is the ground analyze the different regimes of the relaxation.
| 1 i
state of the sector with odd number of fermions) of the In the free relaxation regime: t<t =l/v only inco-
l
initial Hamiltonian (t < 0). In the ordered phase, h0 < herent quasi-particles pass the reference point resulting
(0)
h = 1 where m (t < 0) > 0, Ψ is asymptotically inanexponentialdecayofthemagnetization(cf. Fig.1):
c l | 1 i
degenerate with the ground state, Ψ(0) . For h h
the magnetizationvanishes as ml(t<| 00)i∼L−x w0ith≥thec ml(t)≡m(t)≈A(t)exp(−t/τ), t<tl , (3)
systemsize fort<0. The decayexponent, x, isdifferent
at the critical point, h = hc, and in the paramagnetic with an oscillating prefactor, A(t). In the regime h>hc
phase, h > hc, as well in the bulk (l/L = O(1)) and at and h0 < hc we have A(t) ∼ cos(at + b), thus m(t)
the boundary (l/L 0), see Table I. changes sign. On the other hand in the other parts of
To calculate the→magnetization profile in Eq.(2) we the phase diagram m(t) is always positive, i.e. A(t)
∼
haveusedstandardfree-fermionictechniques[19,21]. For [cos(at+b)+c], with c > 1. The characteristic time-
the surface site, l = 1, most of the calculations are an- scale, τ = τ(h,h0), is the relaxation or phase coherence
alytical, whereas for l > 1 numerical calculations have time, which is extracted from the numerical data. The
been made for large finite systems up to L=384. exponential form of the decay in Eq.(3) indicates ther-
Wehaveperformedquenchesforvariouspairsoftrans- malization, at least for bulk sites, which is in agreement
versefields,h andhandcalculatedthetime-dependence with the similar decay of the autocorrelationfunction.
0
ofthelocalmagnetizationatdifferentsites,l L/2. The In the quasi-stationary regime: tl <t<T tl , T =
≤ −
results depend primarily on whether the system before L/v,twotypesofquasi-particlesreachthereferencepoint
and after the quench is in the ordered (O) or disordered l: type 1 passed l only once at a time t′ < t and type
(D) phase,see Fig.1 for differentcombinations of O and 2 passed it twice at two times t′ < t′′ < t with a reflec-
D. One can identify different time regimes that can be tion at the nearby boundary between t′ and t′′. These
interpretedintermsofquasi-particles,whichareemitted two types interfere, resulting in a comparatively slow re-
at t =0, travel with a constant speed, v =v(h,h ), and laxation (cf. Fig. 1). Deep inside the ordered phase the
0
are reflected at the boundaries. quasi-particles can be identified with kinks moving with
As argued in Ref.[11] only those quasi-particles are a speed v [22] andin the regime tl t T halfof the
± ≪ ≪
quantum entangled that originate from nearby regions quasi-particles reaching the site l are of type 1 (flipping
in space, others are incoherent. When the latter arrive the spin at l once) and half of them type 2 (flipping it
at a reference point l they cause relaxation of local ob- twice), leading to a quasi-stationary relaxation.
servables (such as magnetization). Here we extend this The magnetization profiles for fixed times t<T/2 are
picture by noting that in a system with boundaries the shown in Fig.2 for the same quenches as in Fig.1. For
samequasi-particlecanreachthe pointl twice (ormore) sufficientlylargelthequasi-stationarymagnetizationhas
at different times after reflections. This induces quan- anexponentialdependence,suchthatcomparingitsvalue
3
0 0
TABLE II: Correction to the quasi-stationary behavior for -2 -10
thesurfacemagnetization indifferentdomainsofthequench. -20
hh<>hh00 t−t−3/12hcc0oos<(sa(ahttc++bb)) L−t−3/12/2[c[coos(sa(hat0t++>bb)h)+c+cc],L−c3/>2]1 log m(t)l-1---0864 tttttt======tt==O11223349494838->86420864O (a) log m(t)l-----7654300000 tttt====tttt====1111O24792469-48260482>D (b)
-12 -80
0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200
l l
0 -2
at two sites, l1 and l2, we have -10 -2.5
-3
ml1(t1)/ml2(t2)≈exp[−(l1−l2)/ξ] , (4) log m(t)l----54320000 ttt===tt==112D49494->86420O log m(t)l --43--..5455 t=tttt====1D24792->48260D
with oscillating prefactors. t=288 -5.5 t=144
-60 t=336 (c) -6 (d) t=168
In the limits L and t one can define a -70 t=384 -6.5 t=192
→ ∞ → ∞ 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200
quasi-stationarylimiting value whichwillbe denoted by, l l
m . For the surface site we have the exact result
l FIG. 2: (Color online) Non-equilibrium magnetization pro-
files, logm(t), at different times after a quench with param-
(1 h2)(1 h2)1/2 l
m = − − 0 , h ,h<1, (5) eters given in Fig.1 for L=384. From theasymptotic values
1 0
1 hh of the slopes one can measure the correlation length.
0
−
and zero otherwise. Note that the non-equilibrium sur-
face magnetization has different type of singularities for on h close to h = 1. The typical values are in the range
ahn→alyz1e−d(thh0e c<or1r)ecatinodnftoerrmh,0∆→(t,1L−)(=hm<1(1t)). Wme1,haanvde aco(hn,stha0n)t≈an0d.8h6a−s0n.o88h. dFeoprehnd≥en1cteh.e speed is practically
−
its asymptotic behavior is summarized in Table II in the Approximate periodicity with T startsfort>T,when
different domains of h and h0. These corrections are in quasi-particles start to be reflected second time and the
power-law form, which signals that the relaxation of the spin-configuration of the system becomes approximately
surface magnetization has non-thermal behavior. equivalent to that at t T.
For l > 1 we observe that ml is monotonously de- The time- and length−scale, as defined in Eq.(3) and
creasing with l and thus ml > 0 for h0,h < 1 and zero Eq.(4), respectively, as well as the characteristic quasi-
otherwise. The correction terms are identical with those
particlespeedv(h,h )=ξ/τ,canbeextractedwithhigh
0
given in Table II so that a finite distance, l, the local numerical accuracy from our data for the magnetization
magnetization has non-thermal behavior. profiles, typically with a precision of 3 4 digits. Com-
Inthereconstruction regime: T tl <t<T moreand plementary calculations of the autocor−relation function
−
more quasi-particles of type 2 reach the reference point, G (t) = Ψ(0) σx(t)σx(0)Ψ(0) , and the equal-time cor-
which implies, within a kink-picture, that incoherent l h 0 | l l | 0 i
relation function, C (r) = Ψ(0) σx (t)σx(t)Ψ(0) show
spin flips in the past are progressivelyreversedby quasi- t h 0 | l+r l | 0 i
thattheyyieldthesamecorrelationtimeandlength,but
particlesreturningtothesitelafterreflection. Formono-
with less accuracy. Based on our results for the profiles
disperse quasi-particles (velocity v) one would expect a
we have conjectured possibly exact results about the re-
T-periodicity and thus m (t) = m (T t), i.e an expo-
l l
− laxation time, as discussed below.
nential increase in t with a growth rate similar to the
The relaxation time τ(h,h ) is divergent at two
initial decay rate. Indeed we find 0
points: i) at the stationary point, h = h , where
0
ml(t)≡m(t)≈B(t)exp(t/τ′), T −tl <t<T , (6) τh(−h1,,hw0h)i∼ch(cha−nhb0e)−d2erainveddiip)efrotrusrmbaatlilvhe,lyw.hFeorrehτ(h=,h00)th∼e
0
which is practically position independent and where the two singularities merge at h=0: τ(h,h =0) h−3.
0
∼
growthrate ofτ′(h,h ) depends onthe conditions ofthe To obtain information about τ(h,h ) away from the
0 0
quench, being approximately proportional to τ(h,h ): singularitieswe considera quench fromthe fully ordered
0
τ/τ′ = 0.883 0.002. It turned out to be useful to initial state (h = 0) first. A quench into the disor-
0
±
measure the cross-over time, t˜= T/2, which is defined dered phase (h 1) yield to high numerical accuracy
≥
as the crossing point of the two asymptotic regimes: τ(h 1,h = 0) = π/2, i.e. independent of h. For
0
≥
Aexp( t˜/τ) = Bexp(t˜/τ′), where A and B are aver- a quench into the ordered phase (h 1) we introduce
− ≤
aged prefactors. During the cross-over time the quasi- τ˜(h,h = 0) = h3τ(h,h = 0) to get rid of the singu-
0 0
particles travela distance, L/2,thus their speed is given larity at h = 0. In the limit h 0 we obtain τ˜(h =
→
by: v(h,h ) = L/2t˜, which can be measured accurately. 0,h = 0) = 3π/2, and for h > 0 we consider the ratio:
0 0
Wehavenoticed,thatforh<1thespeedisproportional yτ(h) = ∆τ˜(h)/∆τ˜(0) with ∆τ˜(h) = τ˜(h) τ˜(1) and
−
to h: v(h,h ) = ha(h,h ), where a(h,h ) is practically compare it with a similar expression for the correlation
0 0 0
independent of h and has just a very week dependence length yξ(h) = ∆ξ˜(h)/∆ξ˜(0) with ∆ξ˜(h) = ξ˜(h) ξ˜(1),
0
−
4
where ξ˜(h) = ξ(h)h2. The two ratios yτ(h) and yξ(h), systemsandwhichobey presumablyexactrelationscon-
as shown in Fig.3a, are almost indistinguishable. Since jectured on the basis of the numerical data. In a finite
ξ(h)= 1/log((1+√1 h2)/2)isknownexactly[7],the system an exponentially fast reconstruction of the local
− −
relaxationtimeforaquenchfromanorderedinitialstate magnetizationisobserved,involvingatime-scale,τ′,and
(h = 0) can therefore be estimated very accurately, if characterizing an approximately periodic dynamics.
0
not exactly, by the relation yτ(h)=yξ(h).
Several results for observables displaying thermal be-
haviorin the bulk are expected to be valid also in other,
1.2 1.2 even non-integrable spin chains: Absence of thermaliza-
1 1
tion at the boundaries, identity of correlation time and
0.8 0.8
y(h) 000 ...0246 yyyξξτ’ τ-y(h,h)0 000 ...0246 hhhhhh000000======000000......124680 lneennTgthtihaislliynwofianrskfitnhriateecsoabnnesdetnrsuescmutipio-pninoifirntneifidtnebitsyeyststhyeesmteHsmuasnn.gdaarinanexNpoa--
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 tional Research Fund under grant No OTKA K62588,
h h
K75324 and K77629 and by a German-Hungarian ex-
FIG. 3: (Color online) Left: The ratios yξ = ∆ξ˜(h)/∆ξ˜(0) change program (DFG-MTA).
and yτ = ∆τ˜(h)/∆τ˜(0) for a quench from h0 = 0 as a func-
′
tion of h. The curve yξ derives from the exactly known
form for ξ(h), see text. Right: The ratios yτ(h,h0) =
∆τ˜(h,h0)/∆τ˜(0,0) = τ(h,h0)·h(h−h0)2/π −(1−h0)2/2 ∗ Electronic address: [email protected]
for various h0 as a function of h. † Electronic address: [email protected]
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−
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0 0 0
−
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