Logarithmic current fluctuations in non-equilibrium quantum spin chains

0.24 MB·English

by  Tibor Antal

Logarithmic current fluctuations in non-equilibrium quantum spin chains
T. Antal,1 P. L. Krapivsky,2,3 and A. Ra´kos4
1Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA
2Department of Physics, Boston University, Boston, MA 02215, USA
3Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
4Research Group for Condensed Matter Physics of the Hungarian Academy of Sciences, H-1111 Budapest, Hungary
We study zero-temperature quantum spin chains which are characterized by a non-vanishing
9
current. For the XX model starting from the initial state |··· ↑↑↑↓↓↓ ···i we derive an exact
0
0 expression for the variance of the total spin current. We show that asymptotically the variance
2 exhibits an anomalously slow logarithmic growth; we also extract the sub-leading constant term.
Wethenarguethatthelogarithmicgrowth remainsvalidfortheXXZ modelinthecriticalregion.
n
a
PACSnumbers: 05.60.Gg, 75.10.Jm,05.70.Ln
J

9
I. INTRODUCTION eral XXZ chain in section V. A summary of our results
]
and relation to other work is given in section VI.
h
c A distinguishing feature of non-equilibrium states is
e
thepresenceofcurrents[1,2,3]. Fluctuationsofcurrents
m II. MODEL
oftenexhibituniversalbehavior[4]andshedlightonthe
- nature of non-equilibrium systems. Current fluctuations
t
a in classical systems have been extensively investigated, Most generally, we consider the quantum XXZ
t
s see e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and a re- Heisenberg spin chain with Hamiltonian
t. view[15]. Thetotalcurrentgrowslinearlywithtimeand
a current fluctuations usually exhibit an algebraic growth. H= sxsx +sysy +∆szsz (2)
m − n n+1 n n+1 n n+1
Quantum fluctuations in general, and spin current fluc-
n
- tuations in particular,are muchless understood; evenin X(cid:0) (cid:1)
d
the simplest systems quantum fluctuations often behave Here we set the coupling constants to unity in the x and
n
o verydifferentlyfromstandardstatisticalfluctuations(see y directions. The coupling constant ∆ in the z direction
c e.g. [16]). is called the anisotropy parameter. The z component of
[ In this paper we study current fluctuations in quan- the total magnetizationMz = ∞n=−∞szn is a conserved
tum spinchains. How to impose currentsin spinchains? quantityinthismodel,andouraimistostudythecorre-
2 P
v Perhapsthe simplest way is to startwith a spin chainin sponding current. In the following we shall focus on the
4 the following inhomogeneous product state time evolution of this spin chain starting from an inho-
1 mogeneousinitialstate,wherebytheleftandrighthalves
5 (1) of the infinite chain are set to different quantum states
3 |···↑↑↑↓↓↓···i andare joined at time zero. Suchinitial states provide a
.
8 This choice [17] allows one to avoid complications and particularlyconvenientframeworkto study currentsand
0 arbitrariness of coupling the chain to spin reservoirs. their fluctuations in quantum spin chains.
8 State(1)evolvesaccordingtotheHeisenbergequations We mainly consider the special case of ∆ = 0, where
0 of motion. The average magnetization profile has been themodelreducestofreefermions. Inthissystem,known
:
v computed for the simplest quantum chains, e.g. for the asthe XX model,thetime evolutioncanbe writtenina
i XX modelwheretheperturbedregionwasfoundtogrow compact form, which enables us to perform exact calcu-
X
ballistically [17]. Numerical works [18] suggest that the lations. In particular, it is possible to evaluate the scal-
r
growthis also ballistic for the XXZ chainin the critical ing limit of the magnetization profile and other physical
a
region (described by Hamiltonian (2) with ∆ <1). quantities [17, 19]. Corrections to this scaling behavior
| |
The main goal of this paper is to study the fluctua- were considered in [20, 21].
tions of spin current in quantum chains. After a brief Interestingnon-equilibriumbehaviorwasfoundin dis-
description of the model in section II, in section III we ordered spin chains [22, 23] and chains at finite temper-
present a simple derivation of the asymptotic properties atures [19, 24, 25]. An alternative method has been pro-
of non-equilibrium states in free fermion systems with posed to generate stationary currents in spin chains us-
special initial conditions. In section IV we probe fluctu- ing a Lagrange multiplier [26, 27, 28, 29, 30]. Using this
ations of the current specifically in the XX spin chain. methodfluctuationsofbasicquantitieshavebeenstudied
First we present a back-of-the-envelope calculation for in non-equilibrium steady states in [31].
the variance of the time integrated current; then we es- The relaxation from a large class of initial conditions
tablish an exact result from which we extract the long was considered in numerous studies starting from late
time asymptotical behavior. We discuss the more gen- sixties; see [32, 33, 34, 35] and [36] for a review of more
2
π
recent work. Our focus, however, is on non-equilibrium
states with a non-vanishing current.
π
2
III. HYDRODYNAMIC DESCRIPTION
0
k
We begin by describing a simple method that allows
onetoobtainthelongtimeasymptoticbehaviorofafree
π
fermionsystembyemployingacontinuoushydrodynamic −2
description. This helps to avoid a lengthy exact calcu-
lation and yet the final results are asymptotically exact.
π
Specifically, a justification of this approach for the XX − -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
model is given by exact results [17, 19], which yield the
same asymptotical behavior as the hydrodynamical de- x=n/t
scription discussed below.
The Hamiltonian of the free fermion system can be FIG.1: (Coloronline)HydrodynamicdescriptionfortheXX
written in the form chain started from the |···↑↑↑↓↓↓ ···i initial condition. We
consider thescaling limit where t→∞ and x=n/t=const.
H= ǫ(k)η†η (3) Theshaded region shows theelementary excitations that are
k k present in thescaling points indexed byx.
k
X
where η† and η are creation and annihilation operators
k k with k0 < k < π k0, where k0 = arcsin(n/t). Simi-
of fermions with momentum k and ǫ(k) is the energy of −
larly, for n < 0, only the modes with sin(k) > n/t are
an excitation with wave number k.
filled, that is, the modes with π < k < π k , and
0
Inthesimplestsituation,thesystemisinitiallydivided − − −
k <k <π. For anillustrationsee Fig. 1. As eachmode
0
into two half infinite chains, each of them being in a ho-
carries a unit magnetization, the average z magnetiza-
mogeneous pure state. In this case, the elementary ex-
tion can be obtained by simply integrating through the
citations can be considered initially homogeneously dis-
filled modes m(x = n/t) = 1/2+ dk/2π. This results
tributed in each half chains. At time zero, each mode in the well known profile m(x=n/t)= 1 arcsin(x) for
starts moving with velocity v(k) = ǫ′(k). As the ex- R −π
1 < x < 1, and the magnetization keeps its initial val-
citations are entirely independent, they do not interact −
uesoutsidethisregion. Thislimitingprofilewasobtained
and keep moving with their initial velocities. This argu-
in [17] by exact calculation.
mentsuggeststhatwhetheranexcitationsispresentata
Other applications are given in Appendix A.
space-timepoint(n,t)dependsonlyontheratiox=n/t.
Moreover, keeping x fixed, a finite neighborhood of site
nbecomesasymptoticallyhomogeneousfort . This IV. FLUCTUATIONS OF THE CURRENT
→∞
physicalpictureisnotexactduetothefinitelatticespac-
ing. However, we believe that the above description be-
The local magnetization current operator for a quan-
comes asymptotically exact for any free fermion system
tum spin chain can be obtained through a continuity
in the scaling limit: n ,t , n/t = const. (See
→ ∞ → ∞ equation for the local magnetization [27]. For the XX
[25] for a rigorous derivation of this scaling limit for the
model this gives
XX model.)
Whether an excitationis presentat positionn at time j =sysx sxsy (4)
n n n+1− n n+1
t can be decided by noting that for n > 0, the modes
for the current between spin n and n+1. (We measure
which are present were initially on the left side of the
time in units of ~). The time integrated current C , i.e.,
chain with v(k)>n/t and on the right side of the chain 0
the net transported magnetization up to time t through
with v(k)<n/t. Similar argument applies for n<0.
the bond between spin 0 and spin 1, is a quantity which
Thismethodcanbeextendedtothegeneralcasewhen
islessobvioustodefineforaquantumsystemingeneral.
the two half-infinite chains are initially in mixed states,
However,inthe caseofthe setup(1), the integratedcur-
e.g., one can consider the situation when the two half-
rent C can be expressed in a simple way
infinite chains are set to different temperatures [25]. 0
As an illustration, let us calculate the magnetization C = (sz +1/2). (5)
profile in the XX model with the simplest initial condi- 0 n
n≥1
tion (1). The spectrum of the model is ǫ(k) = cos(k), X
−
that is, v(k) = sin(k). Initially, all the modes are filled The average of the integrated current C through the
0
onthe left, while the rightsideofthe chainis inthe vac- central bond grows asymptotically as πh−1ti[17]; alterna-
uum state. At time t, around site n > 0, the modes tively, this can be seen from the hydrodynamic picture
with sin(k) > n/t are filled, that is, all the modes described above.
3
The hydrodynamic approach of Sec. III does not al- In this expression, the factor 1/2 appears since correla-
low to probe fluctuations. Therefore we must return to tions between evenly spaced sites vanish.
the microscopic description. Below we shallfocus on the This argument remains valid for a more general class
variance of the total current D(t) C2 C 2. of initial conditions, where on the left (right) half of the
≡h 0i−h 0i
Letus define the left, rightandtotalmagnetizationas chain the fermions are filled up to the Fermi energy µ
L
follows: (µ ). In this case, the asymptotic state — which builds
R
up near the origin — includes fermions with momenta
M = sz, M = sz,
L n R n varyingfrom kR to kL, where kL andkR arethe Fermi
−
nX≤0 nX≥1 momenta corresponding to µL and µR. (For an illus-
M =M +M . (6) tration see Fig. 3; more details are given in Appendix
L R
A1.) The correlation function ρz(n) for these asymp-
The variance of the integrated current is equal to the
totic states can easily be calculated [27]. One finds that
variance of the left (right) magnetization: in general it behaves as ρz(n) = 1 sin2(nϕ), where
−π2n2
D(t)= M2 M 2 = M2 M 2. (7) ϕ=(kL+kR)/2[27]. Astheϕdependenceoftheasymp-
h Rit−h Rit h Lit−h Lit totic form averages out we conclude that the result (13)
Sincethetotalmagnetizationisconservedandtheinitial is unchanged for this class of initial states.
state is an M eigenstate, the fluctuation of M remains The exact evaluation of D(t) is based on (9). Follow-
zero for any time t ing the strategy of [17] we write sz in terms of the local
i
fermionic creation and annihilation operators c†,c as
(M +M )2 M +M 2 =0. (8)
h L R it−h L Rit
1
By exploiting this property we can rewrite (7) as szn =c†ncn− 2. (14)
D(t)= M M M M In the Heisenberg picture, the time dependence of these
L t R t L R t
h i h i −h i
operators, under the dynamics of the XX chain, has a
= ( sz sz szsz ). (9)
h lith mit−h l mit simple form
l≤0,m≥1
X
∞
Before presenting an exact calculation we provide a c (t)= ij−nJ (t)c , (15)
n j−n j
back-of-the-envelope derivation of our main result. The
j=−∞
X
ideaistoevaluateD(t)bysubstitutingcorrelationsin(9)
withtheirstationaryvaluesinthelocalstatewhichbuilds where J (t) are the Bessel functions. Inserting this into
n
upatthe originfort (see [17]andourFig.1). The (9) one gets
→∞
reason is that the main contribution comes from those
spins for which (l m) is not too large, and for t 1 D(t)= i−α+β−γ+δ
− ≫
these “points” are located near the origin. This leads to
l≤0,m≥1α,β,γ,δ
X X
D(t)= nρz(n), (10) Jα−l(t)Jβ−l(t)Jγ−m(t)Jδ−m(t)
×
−
nX>0 × hc†αcβihc†γcδi−hc†αcβc†γcδi . (16)
where The expectation valu(cid:0)e in the above formula is (cid:1)taken in
ρz(n)= szsz sz sz . (11) the initial state. One finds
h k k+ni−h kih k+ni
In the (homogeneous) “maximal current” stationary c†c c†c c†c c†c =
state of the XX model, the correlator ρz(n) takes the h α βih γ δi−h α β γ δi
δ δ if α ,β
same form as in the ground state [27] where it is given − α,δ β,γ ↑ ↓ (17)
by the well-known expression: (0 otherwise
1 n= odd for initial states which are product states of individual
ρz(n)= −π2n2 . (12)
spins pointing either up or down (like (1)). Here, α ,
(0 n= even ↑
β are shorthand notations for α,β with s = ,s = .
α β
ThisgivesalogarithmicaldivergenceforD(t),whichone Us↓ing identities J−k(t)=( 1)kJk(t) and [46] ↑ ↓
−
canregularizebytruncatingthesumin(10). Forafinite
J (t)J (t) J (t)J (t)
but large t the volume of the region around the origin, J (t)J (t)=t p q+1 − p+1 q
k+p k+q
which can be described by this maximal current state, 2(p q)
k≥1 −
X
grows linearly with t. Hence we choose the upper limit
in the sum in (10) to be proportional to t and obtain for the sums over l and m, one obtains
D(t) ∼t nρz(n)= 1 ln(t). (13) D(t)= t2 Jα−1(t)Jβ(t)−Jα(t)Jβ−1(t) 2. (18)
∼− 2π2 4 α β
nX=1 αX↑,β↓(cid:20) − (cid:21)
4
This expression for our initial condition (1) becomes V. XXZ SPIN CHAIN
t2 J (t)J (t)+J (t)J (t) 2
l−1 m−1 l m In a general XXZ chain (2), the coupling is nonzero
D(t)= . (19)
4 l+m 1 in the z direction. There is no explicit solution for the
l,m≥1(cid:20) − (cid:21)
X timeevolutioninthisgeneralcase. Duetothesymmetry
Thisisanexactexpressionforthevarianceofthecurrent in the XY plane, however, the z component Mz of the
which is valid at any time t 0. From this formula we total magnetization is still conserved, hence the magne-
≥
have deduced the long time asymptotic behavior tization current can be studied. It has been investigated
numerically by Gobert et. al. [18].
1
Our main interest lies in the so-called critical region
D(t)= (lnt+C) (20)
2π2 ( 1 < ∆ < 1); we shall also discuss the isotropic ferro-
−
magnetic (∆=1) and anti-ferromagnetic(∆= 1) spin
withconstantC =2.963510026.... Theleadingtermco- −
chains. TheXX modelbelongstothecriticalregion,and
incides withour heuristicargument(13). The derivation
the behaviorinthe entirecriticalregionis believedto be
of (20) is relegated to appendix B.
similar to the behavior of the XX model. In particular,
We also evaluated expression (19) numerically and
the magnetization profile plausibly scales linearly with
plotted it in Fig. 2 for t < 50. We observe a logarith-
time: m(n,t) M(n/t) [18]. On the other hand, due
mic increase in time. The numerical estimate for the →
to the existence of kink-like ground states [37, 38], the
constant, C 2.9633, is in a good agreement with the
≈ magnetizationprofile is expected to become frozenwhen
exact result.
∆ >1[18]. Algebraicscalingm(n,t) M(n/ta)seems
In addition to this logarithmic growth, one observes | | →
to emerge for the XXX model (∆ = 1) in the scaling
oscillations with decreasing amplitude. We found that regime n , t with n/t|a k|ept finite. Numeri-
the formula → ∞ → ∞
callythe exponentisa=0.6 0.1[18],sothenon-trivial
±
1 cos2t part of the profile is sub-ballistic.
D(t)= lnt+C (lnt+C′) , (21)
2π2 − t For the XX spin chain, we have obtained the correct
(cid:20) (cid:21)
currentfluctuations in the leading order from the simple
with C′ 1.95 (and C given exactly in (B23)) gives a formula(10)whentheupperlimitofthesumwaschosen
≈
very good fit to the numerical data even for relatively to grow linearly with time. The reason for this choice
short times. Since the mean current is 1/π, in aver- of upper limit is that the front and the whole profile
age one fermion crosses the origin in time π. Hence the “moves”linearlywithtime,hencethecutoffmustbehave
cos2t oscillations can be interpreted as a consequence similarly. We shall use (10) also for the XXZ chain.
of the quantum nature of the magnetization: each pass- We shall assume that the upper bound moves linearly,
ing fermion causes a bump in the fluctuations. Similar namely as vt, in the critical region ( 1 < ∆ < 1). The
−
arguments were used to explain oscillations in the mag- actual value of the ‘velocity’ v is unknown, but it does
netization profile in the same system [21]. not affect the leading order term anyway.
The next issue is whether one can use the equilibrium
0.4 spin correlations ρz(n) in the presence of current. For
the XX chain we know [17, 27] that current does not
affect the z component of the correlations significantly
0.3
(may introduce a modulation), only the x and y compo-
nents. Here we boldly assume the same for the XXZ
) model, at least for its large distance behavior. Thus we
D(t 0.2 use the equilibrium correlationsin (10). The asymptotic
formulae for ρz(n) are [39, 40]
0.1 δn−2 0<∆<1
−
[1+( 1)n](2π2)−1n−2 ∆=0
ρz(n)=− − (22)
0 (−1)nAn−4π2δ−δn−2 −1<∆<0
0 10 20 30 40 50 ( 1)nBn−1√lnn δn−2 ∆= 1
− − −
t where weused the shorthand notation
FIG. 2: (Color online) Shown is the variance D(t) vs. time 1
t. Red curve is a result of the exact numerical evaluation δ = (23)
4πarccos(∆)
of (19), blue curve shows the result (20) of our asymptotical
analysis. On top of the logarithmic growth we find a sub- We now insert (22) into equation(10). The amplitude
leading oscillating term. Eq. (21) gives a fit, which is almost A=A(∆)hasbeenguessedrelativelyrecently(see[39]),
indistinguishable from thenumerical data (red line).
yet we do not need this result. Indeed, the leading oscil-
lating term in ρz(n) has the exponent a = 4π2δ varying
5
in the range 1 < a < 2 when the anisotropy parameter The current simply “slides” through the system ballis-
varies in the 1 < ∆ < 0 range. Because this leading tically with no disturbance, hence the tiny fluctuations.
−
termoscillates,weformpairsandfindthattheoscillating Conversely, if an impurity is present at the origin, the
terms in (10) yield the contribution variance grows linearly with time [43]. Similarly, in
stochastic particle systems [4, 5, 6, 7, 8, 9] the noise –
An−a+1 A(n+1)−a+1 aAn−a (24)
whichis intrinsically presentinthese models – generates
− ≃
algebraic fluctuations.
that decays faster than n−1. Hence the oscillating term
We have argued that current fluctuations in the in-
provides merely a constant contribution to the variance
homogeneous XXZ model are also logarithmic (in the
D(t)whilethesub-leadingn−2termresultsintheleading
criticalregion). Our arguments are heuristic and a more
logarithmically diverging contribution. Therefore
rigorousderivationisakeychallengeforfuturework. In-
D(t)=δlnt (25) triguingly,fluctuationsseemmoretractablethane.g. the
average magnetization profile in the XXZ chain, which
This prediction implies that the logarithmic behavior of is completely unknown.
the variance is universal. The amplitude diverges at For free fermion systems, current fluctuations were
∆ = 1; analytically δ (4π)−1[2(1 ∆)]−1/2 as ∆ 1. found to be asymptotically Gaussian [43] and therefore
→ − ↑
This divergence is not very surprising since the isotropic the variance provides a complete characterization of the
Heisenberg ferromagnet apparently exhibits a truly dif- fullcurrentstatistics. Moreover,accordingto[44]thisin-
ferent behavior. dicatesthattheentanglement(betweentheleftandright
In the other extreme ∆ 1, the amplitude δ ap- halves)is simply proportionalto the varianceof the cur-
proaches the finite value (4π↓2)−−1. However, as indicated rent (with a factor 3/π2). In order to check this relation
by the last formula in (22), the oscillating asymptotic wecomparedourresultsforD(t)tonumericalresultsfor
is Bn−1√lnn for the isotropic anti-ferromagnet [41]. A the time dependent entanglement entropy (for the same
calculationsimilarto(24)givesn−2√lnnaftercanceling model and initial condition) presented in Fig. 19 of [18],
theoscillations. Thisleadstothefollowingquitesurpris- and we found a good agreement in the leading order.
ing behavior: Little is known about higher moments of current fluc-
tuations for interacting fermions. There is no reason to
dn
D √lnn (lnt)3/2 (26) believethattheyareGaussianandthefullcurrentstatis-
∼ n ∼ tics is very difficult to probe (both theoretically and ex-
Z
perimentally) for interesting interacting fermion systems
Note that for the isotropic anti-ferromagnet the magne-
[45]. (Even for classical interacting particles, the deriva-
tization is non-trivial in the interval that grows slower
tionof the full countingstatistics is usually a formidable
than linearly with time. (The natural guess is the diffu-
challenge, see e.g. [6, 8, 9, 11, 12, 13, 14].) For the
sive√tgrowth.) However,theupperlimitintheintegral
XXZ model, however, it could be possible to compute
in (26) would affect only the pre-factor.
higher order cumulants by employing the heuristic ap-
Thus our tentative predictions for the variance of the
proachwhichwehaveappliedtocomputingthevariance.
time integrated current are: (i) The enhanced logarith-
mic growth D (lnt)3/2 for ∆ = 1; (ii) The universal
∼ −
logarithmicbehaviorD(t)=δlnt withknownpre-factor
VII. ACKNOWLEDGMENTS
(23) in the critical region 1<∆<1.
−
We thank Alexander Abanov, Deepak Dhar, Viktor
VI. DISCUSSION Eisler, Ro´bert Juh´asz, Eduardo Novais, Pierre Pujol,
Zolt´an Ra´cz and Gunter M. Schu¨tz for illuminating dis-
cussions. We also gratefully acknowledge financial sup-
Wehavestudiedthefluctuationsofthetimeintegrated
portfromNIHgrantR01GM078986(T.A.),theHungar-
magnetization current in the quantum XX chain that
ian Scientific Research Fund OTKA PD-72607 (R. A.),
evolves starting from an inhomogeneous non-stationary
and Jeffrey Epstein for support of the Programfor Evo-
initial state (1). We have derived an exact formula (19)
lutionary Dynamics at Harvard University.
for the variance D(t) of the current. We have shown
that the variance increases logarithmically in the long
time limit (20), which is consistent [42] with numerical
APPENDIX A: HYDRODYNAMIC
evaluation of the exact formula (see Fig. 2). In addition
DESCRIPTION: EXAMPLES
to this logarithmicgrowth,we haveobservedoscillations
with decreasing amplitude. We have argued that this
logarithmic leading order behavior remains unchanged 1. XX model
for a more general class of initial conditions (where the
magnetization on the two half-chains is not saturated). In section III we demonstrate how the hydrodynamic
These small logarithmic fluctuations reflect the ideal approach works for the XX model in the simplest case,
conductor nature of the integrable XX quantum chain. i.e.,whenallthespinspointup(down)ontheleft(right)
6
π
side at t = 0. One can easily recover the exact results
k
L
foraslightlymorecomplicatedinitialstateaswell. Con-
sider a product state composed of a ground state (in an
π
appropriateexternal field) with mL (mR) magnetization 2
on the left (right) side of the chain. That is, the modes k
R
k < k < k with k = π(1/2+m ) are
− L(R) L(R) L(R) L(R) k 0
filled on the left (right) side. Without any restriction, in
the following we assume that mL >mR. −kR
There are two qualitatively different cases. (1) If the π
sign of m and m are equal the scaling profile consists −2
L R
ofthreesegments. Hereweassumethatm >0,m >0
L R k
but the case mL <0,mR <0 is entirely analogous. Ap- − πL
plyingtheargumentofsectionIIIherethemagnetization − -1 v v 0 v v 1
R L L R
− −
profile takes the following scaling form:
y =n/t
mL+mR 0<x<v
2 L FIG.3: (Coloronline)HydrodynamicdescriptionfortheXX
m(x)= m2R + 41 − arcs2iπn(x), vL <x<vR (A1) chain started from an initial condition where the left (right)
 mR vR <x, side of the chain is in the ground-state with magnetization
mL (mR). We consider the scaling limit where t → ∞ and
with v v(k ) = cos(πm ) and v v(k ) = y = n/t = const. The shaded region shows the elementary
L L L R R
cos(πm ).≡and m( x)=m +m m(x).≡ excitations that are present in the scaling points indexed by
R L R
− − y.
(2) A qualitatively different case is when m > 0 and
L
m <0. The argumentofsectionIII leadsto thefollow-
R
ing asymptotic profile:
δ. That is, the front still moves ballisticly, but it can go
arbitrarilyslowly. By inverting the function ω′(k˜)=x/2
mL+mR 0<x<v
2 L
m(x)= mm2RR++ 4112 −− aarrccss2πiiπnn((xx)), vvRL <<xx<<v1R (A2) k1,2 =arccos−x2±p(1−δx2)(δ2−x2) (A5)
m v <x,
R R
with 0 < x < δ, after some elementary calculations, one
Asanillustrationseefigure3. Hereweassumedv <v , obtains the magnetization profile as well
L R
as the opposite case is entirely similar. In all cases m(x) 1 1 1 x
has the symmetry property m(−x)=mL+mR−m(x). m(x)=−2 + 2π(k2−k1)=−π arcsin δ . (A6)
We mention that the special case of mR = mL was (cid:16) (cid:17)
−
considered in detail in [17, 25]. Surprisingly, the dimerization only rescaled the asymp-
totic magnetization profile, and the result agrees with a
homogeneous chain with coupling δ at each bond. That
2. Dimerized XX model means that the weaker links behave as bottlenecks and
theygovernthedynamicsofthechain. Theaveragetime-
integrated current through the origin is tδ/π, which was
In all the above example the speed of the front was
alsofound to be linear intime in [18]. The value oftheir
one, but it is not necessary. Consider, as an example,
current agrees with our result (see figure 4).
the dimerized XX model [36]
H = J (sx sx +sy sy ) (A3) APPENDIX B: LARGE TIME ASYMPTOTIC OF
− m m m+1 m m+1
D(t)
X
withJ =1foroddm,andJ =δ <1forevenm. The
m m
spectrum of the model is Here we study the large time behavior of (19). As
the summation in (19) goes to infinity, we use a general
ω(k˜)= 1 1+δ2+2δcosk˜, (A4) larget asymptotic formula[46] ofBesselfunctions which
±2 is valid for all m<t:
q
π
where k˜ is a sublattice wave number going from π to cos tf(µ)
π (an excitation with sublattice wave number k˜ ha−s mo- J (t)= 2 − 4 +O 1 , (B1)
m πt (1h µ2)1/4 i t
mentum k = k˜/2). The two signs in (A4) indicate two r − (cid:18) (cid:19)
branches with k ( π/2,π/2). For the simplest ini- where
∈ −
tial condition with m = m = 1/2, the fastest mode
L R m
present on the left [at k˜ −= (π arccosδ)] has speed f(µ)= 1 µ2 µ cos−1µ, µ= . (B2)
± − − − t
p
7
8
Using (B2) we expand f(λ) and f(µ) to give
7
π(l m) l2 m2
6 tf(λ) tf(ν λ)= − + − +tO(λ4,µ4).
− − − 2 2t
5
Whenl,m √t,wecankeeponlythefirstleadingterm
+ 4 of this sum≪. In this limit (B4) reads
M
3
2 π(l m)
J (t)J (t)+J (t)J (t)= cos − . (B6)
2 l−1 m−1 l m πt 2
(cid:20) (cid:21)
1
Now in (19) we write = , and
l,m≥1 n≥2 l+m=n
0 divide the summation through n into four parts
0 5 10 15 20 25 30 35 40 P P P
[t1/2−ǫ] [t1/2+ǫ] [t] ∞
t
= + + + (B7)
FIG. 4: (Color online) Shown is the average transferred to- nX≥2 nX=2 n=[t1X/2−ǫ]+1 n=[t1X/2+ǫ]+1 n=X[t]+1
tal magnetization M+ = hPn≥0(Snz +1/2)i as a function of where [] denotes the integer part. The corresponding
timestartedfromthe|···↑↑↑↓↓↓···iinitialconditioninthe ·
terms in (19) will be referred to as
dimerized XX model. The data points show the result of
a time-dependent DMRG calculation published in [18], solid
D(t)=D (t)+D (t)+D (t)+D (t) (B8)
linesindicatethepredictionofthehydrodynamicdescription. 1 2 3 4
The coupling constants for the weak link are 1, 0.8, 0.6, 0.4,
respectively. Wechoosetheexponentin(B7)intherange
0.2 (from above to below).
0 < ǫ < 1/2; later we shall take the ǫ 0 limit. Since
t1/2−ǫ t1/2 as t , in D (t) we ca→n use the simple
1
≪ → ∞
formula (B6), and immediately perform one summation.
We will also use the scaled variable λ = l/t. For m > t
Thus the first sum becomes
the Bessel function J (t) becomes exponentially small
m
thus we can safely neglect those terms in (19). Hence in [t1/2−ǫ]
the double sum we consider only the range λ,µ≤1. D1(t)=π−2 1 , (B9)
Based on (B1) the asymptotic form of a Bessel func- n 1
n even −
tionwithashiftedindex canbe obtainedbysimply Tay- X
lor expanding f(µ) in the argument of the cosine (the which in the t limit is simply
→∞
term coming from the expansion of the denominator is
(1/2 ǫ)ln(t)+γ +ln(2)
negligible), which gives D (t)= − E , (B10)
1 2π2
J (t)= 2 µcosa− 1−µ2sina +O 1 (B3) where γE =0.5772... is the Euler constant.
m−1 πt (1 µ2)1/4 t Now we consider the contribution from the region
r −p (cid:18) (cid:19) t1/2−ǫ n t1/2−ǫ. Using cos2() 1 we find that
where we introduce the shorthand notation ≤ ≤ · ≤
π π [t1/2+ǫ]
a=tf(λ) , b=tf(µ) . 1 2ǫ
− 4 − 4 D2(t)≤ n 1 ≈ π2 ln(t) (B11)
n=[t1X/2−ǫ]+1 −
Now using the asymptotic formulas (B1) and (B3) in
(19) we obtain forfixedpositiveǫandlarget. Henceonecanseethatfor
small ǫ the contribution from D (t) becomes negligible
2
(2/πt)T (as compared to D (t).
J (t)J (t)+J(t)J (t)= , 1
l−1 m−1 l m (1 λ2)1/4(1 µ2)1/4 In D (t) and D (t) we can replace summation by in-
3 4
− −
(B4) tegration, since t1/2+ǫ as t . We take the
→ ∞ → ∞
where square of (B5) and drop rapidly oscillating terms (like
sinacosa) while sin2 and cos2 are replaced by 1/2; this
T = (1+λµ)cosa cosb+ 1 λ2 1 µ2 sina sinb results in replacement of T2 by (1+λµ)/2. Overall, we
− −
λ 1 µ2cosa sinb pµ 1 pλ2 sina cosb. (B5) find that
− − − −
From herpeafter we omit notingpthe relative O(1/√t) cor- 1 [t] 1
D (t)=
rections. 3 2π2 (n 1)2
When λ,µ 1, we can replace (1 λ2)1/4(1 µ2)1/4 n=[t1X/2+ǫ]+1 −
≪ − −
by one in (B4), and the expression for T simplifies to 1+λµ
(B12)
× (1 λ2)(1 µ2)
T =cos(a b)=cos[tf(λ) tf(µ)]. l+Xm=n − −
− −
p
8
and where the first integral is convergent in the limit t
→
,ǫ 0. Therefore the contribution gathered in the
∞ →
[2t] t1/2+ǫ n t region is
1 1
≤ ≤
D (t)=
4 2π2 (n 1)2
n=X[t]+1 − D (t)= (1/2−ǫ)lnt+C3 (B18)
1+λµ 3 2π2
. (B13)
× (1 λ2)(1 µ2)
l+Xm=n − − where
p
Note that there is only an exponentially small contribu- 1 dν 1
tion from n>2t terms. C = dx[F(ν,x) 1]=0.34929294... (B19)
3
ν −
Notice that the second sum in (B12-B13) can be re- Z0 Z0
placed by
Byasimilarargumentthecontribution(B13)fromthe
1 1 t n 2t region remains finite in the t limit:
F(ν,x)dx, (B14) ≤ ≤ →∞
n
Z0 C
4
D (t)= (B20)
where we used the shorthand notation 4 2π2
1+ν2x(1 x) with
F(ν,x)= − , (B15)
[1 ν2x2][1 ν2(1 x)2]
− − − 2 dν ν−1
tand intrtohdeuficersdtpνsu=m inn/t(B=12-λB+13µ) caanndaxlso=beλ/reνp.laFceodr C4 =Z1 ν Z1−ν−1dxF(ν,x)=1.34385423... (B21)
→ ∞
by an integral, which leads to
Combining the contributions (B10), (B11), (B18), and
(B20) from the three regions of (B7), and taking now
1 1 dν 1
D (t)= dxF(ν,x) (B16) the ǫ 0 limit, we obtain
3 2π2 Ztǫ−1/2 ν Z0 →
1
It is useful to rewrite (B16) as D(t)= (lnt+C) (B22)
2π2
1 1 dν 1 where
D (t)= dx[F(ν,x) 1]
3 2π2 Ztǫ−1/2 ν Z0 −
(1/2 ǫ)lnt C =γE +ln2+C3+C4 =2.963510026... (B23)
+ − , (B17)
2π2
[1] T. Ligget, Interacting particle systems (Springer-Verlag, [11] T. Sasamoto, J. Phys.A 38, L549 (2005).
NewYork, 1985). [12] P.L.Ferrari andH.Spohn,Commun.Math. Phys.265,
[2] B. Schmittmann and R. K. P. Zia, in Phase Transitions 1 (2006).
andCriticalPhenomena,Vol.17,editedbyC.Domband [13] A. R´akos and G. M. Schu¨tz, Markov Processes and Re-
J. Lebowitz (Academic, New York,1995). lated Fields 12, 323 (2006).
[3] G. M. Schu¨tz, in Phase Transitions and Critical Phe- [14] S.ProlhacandK.Mallick,J.Phys.A41,175002(2008).
nomena, Vol. 19, edited by C. Domb and J. Lebowitz [15] B. Derrida, J. Stat. Mech. P07023 (2007).
(Academic, London, 2000). [16] Y.Castin,inUltra-coldFermiGases,editedbyM.Ingus-
[4] M. Pr¨ahofer and H. Spohn, Phys. Rev. Lett. 84, 4882 cio,W.Ketterle,andC.SalomonIOSPress,Amsterdam,
(2000). 2007 Vol. 164 of Proceedings of the International School
[5] P.A.FerrariandL.R.G.Fontes,Ann.Probab. 22,820 of Physics Enrico Fermi, p. 289.
(1994). [17] T. Antal, Z. R´acz, A. R´akos, and G. M. Schu¨tz, Phys.
[6] B. Derridaand J. L.Lebowitz, Phys.Rev.Lett. 80, 209 Rev. E59, 4912 (1999).
(1998). [18] D. Gobert, C. Kollath, U. Schollw¨ock, and G. Schu¨tz,
[7] J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 Phys. Rev.E 71, 036102 (2005).
(1999). [19] Y. Ogata, Phys.Rev.E 66, 016135 (2002).
[8] K.Johansson, Commun. Math. Phys. 209, 437 (2000). [20] D. Karevski, Eur. Phys.J. B 27, 147 (2002).
[9] T.BodineauandB.Derrida,Phys.Rev.Lett.92,180601 [21] V. Hunyadi, Z. R´acz, and L. Sasv´ari, Phys. Rev. E 69,
(2004). 066103 (2004).
[10] R. J. Harris, A. R´akos, G. M. Schu¨tz, J. Stat. Mech. [22] S.AbreitandD.Karevski,Eur.Phys.J.B30,77(2002).
P08003 (2005). [23] T. Platini and D. Karevski, Eur. Phys. J. B 48, 255
9
(2005). [37] F.C.Alcaraz,S.R.Salinas,andW.F.Wreszinski,Phys.
[24] T. Platini and D. Karevski, J. Phys. A 40, 1711 (2007). Rev. Lett.75, 930 (1995).
[25] Y.Ogata, Phys.Rev. E 66, 066123 (2002). [38] T. Matsui, Lett. Math. Phys.37, 397 (1996).
[26] T. Antal, Z. R´acz, and L. Sasv´ari, Phys. Rev. Lett. 78, [39] S. Lukyanov,Phys. Rev.B 59, 11163 (1999).
167, (1997). [40] T. Giamarchi, Quantum physics in one dimension (New
[27] T. Antal, Z. R´acz, A. R´akos, and G. M. Schu¨tz, Phys. York, Oxford University Press, 2004).
Rev.E 57, 5184 (1998). [41] I. Affleck,J. Phys.A 31, 4573 (1998).
[28] J. Cardy and P.Suranyi,Nuc.Phys. B 565, 487 (2000). [42] Thelongtimeasymptoticsofcurrentfluctuationsforthe
[29] D.S. Kosov, J. Chem. Phys. 120, 7165 (2004). freefermioncasewerealsoconsideredinRef.[43].Forthe
[30] Z. R´acz, J. Stat. Phys.101, 273 (2000). second cumulant, a logarithmic growth was found with
[31] V.Eisler,Z.R´acz,andF.vanWijland,Phys.Rev.E67, prefactor twice larger than in Eq. (20).
056129 (2003). [43] K. Sch¨onhammer, Phys.Rev.B 75, 205329 (2007).
[32] T. Niemeijer, Physica 36, 377 (1967). [44] I. Klich and L. Levitov, Quantum noise as an entangle-
[33] J. A. Tjon, Phys. Rev.B 2, 2411 (1970). ment meter, arXiv:0804.1377 (2008).
[34] E. Barouch and M. Dresden, Phys. Rev. Lett. 23, 114 [45] Ya. M. Blanter and M. Bu¨ttiker, Phys. Rep. 336, 1
(1969). (2000).
[35] F.Igl´oiandH.Rieger,Phys.Rev.Lett.85,3233(2000). [46] I.S.GradshteynandI.M.Ryzhik,TableofIntegrals,Se-
[36] G.O.Berim,S.I.Berim,andG.G.Cabrera,Phys.Rev. ries, and Products, 7nd ed.(Elsevier, Amsterdam,2007).
B 66, 094401 (2002).