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VOLUME 85, NUMBER 15
PHYSICAL REVIEW LETTERS
9 OCTOBER 2000
Long-Range Correlations in the Nonequilibrium Quantum Relaxation of a Spin Chain
Ferenc Iglói 1,2 and Heiko Rieger 3
1
Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary
2
Institute for Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
3
Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany
(Received 13 March 2000)
We consider the nonstationary quantum relaxation of the Ising spin chain in a transverse field of
strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary
state by a process possessing quasi-long-range correlations in time and space, independent of the value
of h. In particular, the system exhibits aging (or lack of time-translational invariance on intermediate
time scales) although no indications of coarsening are present.
PACS numbers: 75.10.Hk, 05.50.+q, 64.60.Ak, 68.35.Rh
Nonequilibrium dynamical properties of quantum systems have been of interest recently, experimentally and
theoretically. Measurements on magnetic relaxation at
low temperatures show deviations from the classical exponential decay [1], which was explained by the effect of
quantum tunneling. On the theoretical side, among others,
integrable [2] and nonintegrable models [3] were studied in
the presence of energy or magnetic currents, as well as the
phenomena of quantum aging in systems with long-range
[4] and short-range interactions [5].
Here we pose a different question: Consider a quantum
mechanical interacting many body system described by a
Hamilton operator Ĥ without any coupling to an external
bath, which means that the system is closed. Suppose the
system is prepared in a specific state jc0 典 at time t 苷 0,
which is not an eigenstate of the Hamiltonian Ĥ. Then we
are interested in the natural quantum dynamical evolution
of this state which is described by the Schrödinger equation
and is formally given by
µ
∂
i
jc共t兲典 苷 exp 2 Ĥt jc0 典 .
(1)
h̄
Obviously the energy E 苷 具c0 jĤjc0 典 is conserved. In
particular, we want to study the time evolution of the expectation value A共t兲 of an observable  or the two-time
correlation function CAB 共t1 , t2 兲 of two observables  and
B̂, defined by
A共t兲 苷 具c0 jÂH 共t兲 jc0 典 ,
CAB 共t1 , t2 兲 苷 具c0 j 兵ÂH 共t1 兲B̂H 共t2 兲其S jc0 典 ,
(2)
where ÂH 共t兲 苷 exp共1i Ĥt兲 exp共2i Ĥt兲 is the  in the
Heisenberg picture (with h̄ set to unity) and 兵ÂB̂其S 苷
1兾2共ÂB̂ 1 B̂Â兲 is the symmetric product of two operators.
One should emphasize that in such a situation one does
not expect time translational invariance to hold, which
would manifest itself in, for instance, A共t兲 苷 A0 苷 const
and CAB 共t1 , t2 兲 苷 CAB 共t1 2 t2 兲. There will be a transient
regime in which these relations are violated and, depending on the complexity of the system, this nonequilibrium
regime will extend over the whole time axis. Then we de0031-9007兾00兾85(15)兾3233(4)$15.00
note it as quantum aging, as it can be observed, e.g., for
the Universe, which is (most probably) a closed system.
To be concrete we consider the prototype of an interacting quantum system, the Ising model in a transverse field
in one dimension defined
by the Hamiltonian #
"
L
L21
X
X
1
x
H 苷2
1h
slx sl11
slz ,
(3)
2 l苷1
l苷1
x,z
where sl are spin-1兾2 operators on site l. We consider
initial many body states that are eigenstates either of all
local slx or of all local slz operators. We will mainly
consider fully magnetized initial states, either in the x or
the z direction, which we denote with jx典 and jz典 and which
obey slx jx典 苷 1jx典 and slz jz典 苷 1jz典.
In passing we note that one obtains the zero temperature
equilibrium situation by choosing the ground state of the
Hamiltonian (3) as the initial state. This ground state has
a quantum phase transition at h 苷 1 from a paramagnetic
(h . 1) to a ferromagnetic (h , 1) phase. The former has
long-range order (LRO) along the z direction; the latter has
spontaneous symmetry breaking LRO along x. Moreover,
the nonzero temperature (T . 0) equilibrium relaxation
of (3) has been considered in [6] corresponding to an open
system coupled to a heat bath in the stationary state, which
is not related to the nonstationary closed system we consider here.
The expectation values and correlation functions we are
interested in involve the spin operators slx and slz . To
compute them, we express the Hamiltonian (3) in terms of
fermion creation (annihilation) operators [7,8] hq1 (hq )
∂
X µ
1
1
H 苷
.
(4)
eq hq hq 2
2
q
The energy of modes, eq , q 苷 1, 2, . . . , L are given by the
solution of the following set of equations:
eq Cq 共l兲 苷 2hFq 共l兲 2 Fq 共l 1 1兲 ,
eq Fq 共l兲 苷 2Cq 共l 2 1兲 2 hCq 共l兲 ,
(5)
and we use free boundary conditions which implies for the
components Fq 共L 1 1兲 苷 Cq 共0兲 苷 0. The spin operators can then be expressed by the fermion operators as
© 2000 The American Physical Society
3233
PHYSICAL REVIEW LETTERS
Ai 苷
Bi 苷
XL
q苷1
Fq 共i兲 共hq1 1 hq 兲 ,
(7)
XL
C 共i兲 共hq1 2 hq 兲 ,
q苷1 q
and the time evolution of the spin operators follows from
the time dependence of the fermion operators: Inserting hq1 共t兲 苷 eiteq hq1 , hq 共t兲 苷 e2iteq hq into Eq. (7) one
obtains
X
关具Al Ak 典t Ak 1 具Al Bk 典t Bk 兴 ,
Al 共t兲 苷
k
(8)
X
关具Bl Ak 典t Ak 1 具Bl Bk 典t Bk 兴 ,
Bl 共t兲 苷
k
with the time-dependent contractions
X
cos共eq t兲Fq 共l兲Fq 共k兲 ,
具Al Ak 典t 苷
q
具Al Bk 典t 苷 具Bk Al 典t 苷 i
具Bl Bk 典t 苷
X
X
sin共eq t兲Fq 共l兲Cq 共k兲 , (9)
q
cos共eq t兲Cq 共l兲Cq 共k兲 .
q
For general position of the spin, l 苷 O共L兾2兲, one finds
simple formulas for the expectation values and correlation
functions involving slz operators, whereas the calculation
of those involving slx operators is a difficult task and the final result is complicated [9,10]. However, both the surfacespin autocorrelations and the end-to-end correlations are
given in quite simple form, both for the equilibrium [8]
and for the nonequilibrium case.
First we study the x-end-to-end correlations defined by
x,c
CL 共t兲
苷 具c0 j 兵s1x 共t兲sLx 共t兲其S jc0 典 ,
x,x
x,x
C̃max
共L兲 苷 C̃L 共 t 苷 th 共L兲兲兲 ~ L2a ,
0.1
0.01
0.01
0.001
0.0001
0.0001
1e-05
1e-05
0
which contain information about the existence or the absence of magnetic order in the x direction. The single time
t at which the correlations between the two spins are measured indicates the age of the system after preparation. For
the fully ordered initial state jc0 典 苷 jx典 we obtain
0.01
苷 具A1 A1 典t 具BL BL 典t 1 j具A1 BL 典t j ,
500
1000
1500
2000
0
200
400
0.1
xx
0.0001
0.0001
1e-05
where the first term on the right-hand side of Eq. (11) is the
product of surface magnetizations: m1 苷 具xjs1x 共t兲 jx典 苷
具A1 A1 典t and mL 苷 具xjsLx 共t兲 jx典 苷 具BL BL 典t . In the next
paragraph we show that limt!` j具A1 BL 典t j2 苷 0. Therex,x
fore limL,t!` CL 共t兲 苷 m21 and the stationary state, starting with jx典, has long-range order for h , 1 with m1 苷
jF1 共1兲j2 苷 1 2 h2 . Thus the surface order parameter, m1 ,
vanishes continuously at the transition point, hc 苷 1, with
a nonequilibrium exponent, b1ne 苷 1.
The connected correlations are defined via Eq. (2) as
C̃AB 共t1 , t2 兲 苷 CAB 共t1 , t2 兲 2 A共t1 兲B共t2 兲. The time depenx,x
dence of the connected
end-to-end correlations C̃L 共t兲 苷
P
j具A1 BL 典t j2 苷 j q sin共eq t兲jFq 共1兲 j2 共21兲q j2 is a result of
interference effects due to an interplay of length, L (via the
1000
1e-05
0
0.01
800
CL (t)
L = 512
h = 3.0
0.01
0.001
0.1
600
xx
CL (t)
L = 512
h = 0.8
0.001
(11)
3234
L = 128
L = 256
L = 512
0.1
0.001
0.1
2
1
xx
CL (t)
L = 512
h = 1.0
(10)
x,x
CL 共t兲
(12)
with a 苷 2兾3 for h 苷 1 and a 苷 1兾2 for h . 1 [11].
(iii) For t $ th 共L兲 the correlations decay slower than exponentially as can be seen from the figure. (iv) For t 苷
3th 共L兲 again a sudden jump occurs as for t 苷 th 共L兲 followed by a slightly slower oscillatory decay. (v) This pattern is repeated for time t 苷 5th 共L兲, 7th 共L兲, . . . but gets
progressively smeared out by oscillations.
These features can be interpreted as follows: the elementary (tunnel) processes of the quantum dynamics of
the Hamiltonian (3) are spin flips induced by the transverse field operator slz . In this picture two spins can act
only coherently and thus give a contribution to the connected correlation function if the information about such
a spin flip process reaches the two spins simultaneously.
500
1000
1500
2000
0
1
~
with
Cxx
L (t)
L = 512
h = 0.6
500
~
slz 苷 2Al Bl ,
difference in the excitation energies: eq 2 eq21 ⬃ 1兾L),
and time, t. Evidently they vanish both for small (t ø L)
and large (t ¿ L) times, in the latter case due to random
phase factors. For intermediate time scales we obtained
through a numerical analysis of the formula the following
features of the connected correlations which can be read
from Fig. 1: (i) They are zero for times smaller than a
time th 共L兲 which is equal to the system size L for h $ 1
and increases monotonically with decreasing h for h , 1.
(ii) At t 苷 th 共L兲 a jump occurs to a value that decreases
algebraically with L:
~
(6)
~
slx 苷 A1 B1 A2 B2 . . . Al21 Bl21 Al ,
9 OCTOBER 2000
~
VOLUME 85, NUMBER 15
|Cxz
L (t)|
1000
1500
2000
L = 512
h = 1.0
0.1
0.001
0.01
0.0001
0.001
1e-05
0.0001
0
500
1000
t
1500
2000
0
500
1000
1500
2000
t
FIG. 1. Connected end-to-end correlations C̃Lxx (and C̃Lxz for
bottom left) with fixed system size L and field h as a function
of the time t calculated with Eqs. (11) and (13). The left column
shows data for decreasing field strength h 苷 1.0, 0.8, 0.6; the
broken vertical line is at t 苷 512 苷 L. The upper right figure
shows C̃Lxx for different system sizes at h 苷 1.0, the middle right
plot shows C̃Lxx at h 苷 3.0, and the lower right figure shows the
modulus of C̃Lxz for h 苷 1. Here the broken vertical line is at
t 苷 256 苷 L兾2. For the interpretation, see text.
VOLUME 85, NUMBER 15
PHYSICAL REVIEW LETTERS
Feature i tells us that signals generated in the center of
the system travel with a speed proportional to L兾th 共L兲 to
the boundary spins and reaches both simultaneously. At
x,x
this moment C̃L 共t兲 jumps to its maximum (see ii). After this, this signal is superposed by other more incoherent
signals (see iii). However, the strongest initial signal is reflected at both boundaries and reaches the opposite boundary spins simultaneously again at time t 苷 3th 共L兲 (see iv),
and so on. More and more incoherent processes occur in
the meantime, resulting in feature v.
A similar behavior can be observed for the end-to-end
correlations when starting with the state jz典, which is
X
x,z
CL 共t兲 苷
共具A1 Bk 典t 具BL Ak 典t 2 具A1 Ak 典t 具BL Bk 典t 兲 .
k
(13)
x,x
The only difference in the behavior of CL 共t兲 reported
above is (a) its long time limit vanishes for all values of
h and (b) th 共L兲, i.e., the earliest time at which the two
boundary spins are correlated is only half as big as in the
previous case. Obviously it is easier to generate and to
propagate spin flip signals when starting with a z state.
Next we study the bulk behavior of the expectation values and correlations involving slz opera-
9 OCTOBER 2000
tors. 0 First
P we introduce the shorthand notation,
t,t
关D D̃兴l,l 0 苷 k 共具Dl Bk 典t 具D̃l 0 Ai共k兲 典t 0 2 具D̃l 0 Bk 典t 0 具Dl Ai共k兲 典t 兲,
with Dl , D̃l 苷 Al or Bl and i共k兲 苷 k, 共k 1 1兲 for c 苷
z, 共x兲 and start with the nonequilibrium expectation value
c
t,t
el 共t兲 苷 具c0 jslz 共t兲 jc0 典 苷 关AB兴l,l .
(14)
We note that the equilibrium (i.e., ground state) expectation value, el0 , corresponds to the energy density in the
two-dimensional classical Ising model and we use this
terminology also in this nonequilibrium situation. At
the transition point, h 苷 1, the contractions in Eq. (9)
can be expressed in terms of Bessel functions, Jn 共x兲, as
具Al Ak 典t 苷 具Bl Bk 典t 苷 共21兲l1k J2l22k 共2t兲 and 具Al Bk 典t 苷
i共21兲l1k11 J2l22k11 共2t兲 for l 苷 O共L兾2兲. Equation (14)
c
then yields el 共t兲 苷 1兾2 6 J1 共4t兲兾4t, where the 1 (2)
sign refers to c 苷 z共x兲. Thus for long times the nonequic
librium energy density approaches the limit el 苷 1兾2
algebraically ⬃t 23兾2 . It can be shown that the decay
exponent, 3兾2, is universal; its value does not depend on
the value of 0 , h , `. Moreover, the stationary value
of the energy density for a bulk spin can be calculated
exactly for all initial states [9].
The two-spin nonequilibrium dynamical and spatial connected correlations can be expressed as
z,c
C̃l,l 0 共t1 , t2 兲 苷 具c0 j 兵slz 共t1 兲slz0 共t2 兲其S jc0 典
t ,t
t ,t
t ,t
t ,t
~
Cz,z
bulk (r,t)
(15)
苷 具Al Al 0 典t2 2t1 具Bl Bl 0 典t2 2t1 2 具Al Bl 0 典t2 2t1 具Al 0 Bl 典t2 2t1 2 关AA兴l,l1 0 2 关BB兴l,l1 0 2 1 关AB兴l,l1 0 2 关BA兴l,l1 0 2 .
0
The autocorrelation function 共l 苷 l 兲 for (t1 # t2 ) is genwhich is valid both for jc0 典 苷 jx典 and jc0 典 苷 jz典. In
erally nonstationary for intermediate times, t1 兾共t2 2 t1 兲 苷
O共1兲. In the limit L ! ` at h 苷 1 it can be expressed with
Fig. 2 we show the r dependence of C̃ z,z 共r, t兲 for variBessel functions via Eq. (15):
ous times t. We see that for fixed time t the correlations
z,c
increase proportional to r 2 for distances r # t to a maxi2
C̃l,l 共t1 , t2 兲 苷 J0 共2t2 2 2t1 兲
z,z
mum value C̃max
共t兲 at r 苷 2t, which decreases with time
21
1
proportional
to
t
. For distances larger than r 苷 2t they
2
关 f共t2 1 t1 兲 6 g共t2 2 t1 兲兴 , (16)
4
drop rapidly, faster than exponentially, to zero.
The latter two features correspond perfectly to what
where f共x兲 苷 J2 共2x兲 1 J0 共2x兲, g共x兲 苷 J2 共2x兲 2 J0 共2x兲,
we observed also for the z-end-to-end correlations [see
and the 1 (2) sign refers to c 苷 x共z兲. Thus we
conclude that for intermediate times there is aging
1
n
in the z-component autocorrelation function, cont=2 , n=1,2,3,...
trary to what is reported in [5]. Asymptotically we
~ r-12
z,c
~r
0.01
have limt1 !` Cl,l 共t1 , t2 兲 苷 共ec 兲2 , and the connected
two-time correlations depend only on the time differz,c
0.0001
ence, e.g., for h 苷 1 via Eq. (16) limt1 !` C̃l,l 共t1 , t2 兲 苷
2
0
J0 共2关t2 2 t1 兴兲 2 兵J1 共2关t2 2 t1 兴兲其2 . For bulk spins this
1e-06
stationary correlation function decays algebraically as
⬃共t2 2 t1 兲22 for any value of h.
1e-08
Next we consider the spatial connected equal-time correlations, C̃ z,c 共r, t兲, which follow from Eq. (15) with t1 苷
t2 苷 t and l 苷 共L 2 r兲兾2, l 0 苷 共L 1 r兲兾2. At the criti1e-10
1
10
100
1000
10000
cal point, h 苷 1, in a similar way to the autocorrelation
r
function one gets in the limit L ! `,
∑
∏2
FIG. 2. Connected sz correlation function C̃ z,c 共r, t兲 at h 苷 1
r
z,c
given by the expression Eq. (17) for different times (t 苷 2n ,
J2r 共4t兲
C̃ 共r, t兲 苷
2t
n 苷 1, 2, 3, . . . from left to right) in a log-log plot. The two
straight lines indicate the initial r 2 dependence of C̃ z,c 共r, t兲 for
r2 2 1
fixed
t as well as the r 21 dependence of the maximum value at
2
J2r11 共4t兲J2r21 共4t兲 , (17)
4t 2
r 苷 2t.
3235
VOLUME 85, NUMBER 15
PHYSICAL REVIEW LETTERS
9 OCTOBER 2000
TABLE I. Power law dependencies of correlations on and behind the front.
amax
C̃Lxx 共t
C̃Lxz 共t
zc
CL 共t
zc
苷 aL兲
苷 aL兲
苷 aL兲
C̃ 共t 苷 ar兲
1
1兾2
1兾2
1兾2
a 苷 amax
h苷1
21
22兾3
L
L21兾4
L25兾4
r 24兾3
L
L21兾2
L21
r 21
Eq. (13)]: spins that are separated by a distance r can be
correlated only after the first signal from spin flip processes
in between them reaches simultaneously the two spins, i.e.,
for times t larger than r兾2 (for h 苷 1 and jc0 典 苷 jz典. The
first feature, that correlations for distances smaller than
2t are diminished only algebraically, is different from the
faster decay of end-to-end correlations and is characteristic for bulk spins. For r # 2t the correlation function
C̃ z,c 共r, t兲 obeys the characteristic scaling form
C̃ z,c 共r, t兲 苷 t 21 g共r兾t兲
(18)
with g共x兲 ~ x 2 for x ø 1. The scaling parameter r兾t appearing in the scaling function g共x兲 is reminiscent of the
fact that space and time scales are connected linearly at the
critical point in the transverse Ising chain since the dynamical exponent is z 苷 1. Away from the critical point we
have to evaluate our expressions [9] for C̃ z,c 共r, t兲 numerically. The results show similar features as the case h 苷 1
and will be presented elsewhere [9].
For completeness we finally list our results for the maximum value for connected spin-spin correlations since they
decay algebraically with various new exponents; a detailed
derivation will be given in [9]. We confine ourselves to
h $ 1 since here the time th of maximum correlation is
fixed, whereas for h , 1 the value of th depends on h
and has to be determined numerically, which renders the
precise determination of the decay exponents difficult. We
define the ratio a 苷 t兾L and amax 苷 th 共L兲兾L and consider equal time correlations for fixed values of a. In the
picture of a propagating front that separates a region in
the space-time diagram for the chain in which spins are
uncorrelated from a region in which they are correlated,
one observes quasi-long-range correlations on the front,
the latter being defined by the ratio t兾L 苷 amax . For
distances smaller than the distance of maximum correlation or times larger than th the correlations decay slower
than exponential in time, e.g., algebraically for bulk spins
[C̃ zc 共t, r 苷 fixed兲 ⬃ t 22 ]. When we vary both space and
time with fixed ratio t兾L or t兾r we get power laws, as long
as we stay behind the front (i.e., t $ th ). For a . amax
we again observe power laws, but with different exponents;
they are listed in Table I.
To conclude, we studied a novel type of dynamically
produced long-range correlations in a quantum relaxation
process in a quantum spin chain. Starting with a homogeneous initial state the quantum mechanical time evolution according to the Schrödinger equation drives the
system into a stationary state, which has algebraically de3236
a . amax
a 苷 amax
21兾2
L
···
L25兾8
r 22兾3
h.1
a . amax
L21
···
L21
r 21
caying time-dependent autocorrelations but no critical fluctuations. However, during the relaxation process spin-spin
correlations build up upon arrival of a front of coherent
signals, which afterwards decay algebraically in the bulk.
On the front and behind it for a fixed ratio of space and
time scales one observes quasi-long-range order. This does
not depend on any external parameter like the transverse
field. This type of algebraic correlation needs not to be
triggered by some tuning parameter and is therefore reminiscent of phenomena in self-organized criticality [12].
The scenario we have reported here is a result of quantum
interference and one may expect that a similar one holds
for other quantum systems, too. At this point one should
mention the possibility of coarsening in quantum systems
as reported, for instance, in [13], which is different from
the scenario we have reported here.
This study has been partially performed during F. I.’s
and H. R.’s visits in Saarbrücken and Budapest, respectively, financed by a binational DAAD/MÖB project.
F. I.’s work has been supported by the Hungarian National
Research Fund under Grants No. OTKA No. TO23642,
No. TO25139, and No. MO28418 and by the Ministry of
Education under Grant No. FKFP 0596兾1999.
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[10] For the equilibrium system the equal-time correlations can
be written in the form of a r 3 r determinant; for the
nonequilibrium problem it involves the sum of O共r!兲 determinants [9].
[11] These exponents follow from the fact that in the q sum
of j具A1 BL 典j there are ⬃L2兾3 (⬃L3兾4 ) terms of O共1兾L兲 at
h 苷 1 (h . 1) due to constructive interference.
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