PHYSICAL REVIEW B 66, 094401 共2002兲
Long-time tails and anomalous slowing down in the relaxation of spatially inhomogeneous
excitations in quantum spin chains
G. O. Berim*
R.O.S.E. Informathik GmbH, Schlossstrasse 34, D-89518 Heidenheim, Germany
S. I. Berim*
Georg-Simon-Ohm-Fachhochschule Nuernberg, Postfach 9012, Nuernberg, Germany
G. G. Cabrera†
Instituto de Fı́sica ‘‘Gleb Wataghin,’’ Universidade Estadual de Campinas (UNICAMP),
C. P. 6165, Campinas 13083-970, SP, Brazil
共Received 28 May 2001; revised manuscript received 17 September 2001; published 4 September 2002兲
Exact analytic calculations in spin-1/2 XY chains show the presence of long-time tails in the asymptotic
dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for t→⬁, is given in the form
of a power law (t/ ␶ Q ) ⫺ ␯ Q , where the relaxation time ␶ Q and the exponent ␯ Q depend on the wave vector Q,
characterizing the spatial modulation of the initial excitation. We consider several variants of the XY model
共dimerized, with staggered magnetic field, with bond alternation, and with isotropic and uniform interactions兲,
that are grouped into two families, whether the energy spectrum has a gap or not. Once the initial condition is
given, the nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for t→⬁ can be obtained systematically in a form of an asymptotic series through
the stationary phase method. We found that gapped models show critical behavior with respect to Q, in the
sense that there exist critical values Q c where the relaxation time ␶ Q diverges and the exponent ␯ Q changes
discontinuously. At those points, a slowing down of the relaxation process is induced, similarly to phenomena
occurring near phase transitions. Long-lived excitations are identified as incommensurate spin density waves
that emerge in gapped systems, as a consequence of both approximate nesting of the spectrum and the
degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a
function of the wave vector Q.
DOI: 10.1103/PhysRevB.66.094401
PACS number共s兲: 75.10.⫺b, 75.10.Jm, 76.20.⫹q
I. INTRODUCTION
The spin-1/2 XY chain and its variants are among the
most widely used and quoted models in theoretical investigations on spin systems. This popularity among theorists is
due to two facts. On the one hand, this family of models
allows for exact theoretical description of many static as well
as dynamic properties共see Ref. 1–33兲, and on the other hand,
in some important cases, it provides a good description of
real systems.34 –37 As an example, we note that the dimerized
chain, with alternating antiferromagnetic bonds, has been
studied in relation with the spin-Peierls transition,38 and it is
thought to represent the spin degrees of freedom of organic
compounds that undergo the so called Peierls distortion. The
present state of the art in fabricating low-dimensional systems, with the material science technology developed after
the synthesis of the superconducting cuprate oxides, may
now tailor compounds that reveal a wealth of new magnetic
phenomena, including random spin chains, spin ladders, and
doped magnetic chains among other systems with exotic behaviors. For theorists, these systems are fascinating, with no
parallel in classical or three-dimensional physics. Being systems of low dimensionality with low values of the spin, they
are dominated by quantum effects.39 Compounds such as
PrCl3 , PrEtSO4 共Pr ethyl sulfate兲, and Cs2 CoCl4 are among
the quasi-one-dimensional systems, whose low-temperature
properties are thought to be described by the XY model.
0163-1829/2002/66共9兲/094401共12兲/$20.00
A great deal of theoretical information has been gathered
concerning equilibrium properties, with calculations of quantities such as the specific heat, the magnetic susceptibility,
and equal-time spin-spin correlation functions.1,3,8,13 As for
the dynamic properties, the quantities most thoroughly studied are the time-dependent spin-spin correlation functions
共TDCF兲
冓 冉 冊 冉
具 S ␮j 共 t 兲 S ␮l 典 ⫽ exp
冊 冔
i
i
Ht S ␮j exp ⫺ Ht S ␮l ,
ប
ប
共1兲
where H is the Hamiltonian of the system, ␮ is an index for
the spin component ( ␮ ⫽x,y,z), and 具 ••• 典 is the equilibrium average.3,7,14,23 They are important for the description
of such dynamic phenomena as magnetic resonance, magnetic neutron scattering, spin diffusion, and other relaxation
properties.24,25 It should be noted however, that their application is restricted to situations near the equilibrium state,
where the linear response theory is valid.40,41 In particular,
for the uniform XY model, an old calculation for the timedependent autocorrelation 具 S z0 (t)S z0 (0) 典 showed the absence
of spin diffusion in the limit t→⬁. 3 This behavior was
thought to be accidental and specific of the XY model for
spin 1/2 共see, for example, Ref. 25兲. However, recent research has shown that this surprising property is shared by a
whole family of integrable models, and is attributed to the
existence of a macroscopic number of conservation laws.42
66 094401-1
©2002 The American Physical Society
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
This remarkable result has been probed in magnetic resonance experiments in some one-dimensional 共1D兲 systems.36
We will turn to this point later.
In spite of the wide range of their applicability, we note
that TDCF do not give a direct description of the time evolution of the nonequilibrium process.24,41 This description is
achieved through a different method, which we briefly discuss below.
This second direction in investigating the dynamics relies
on calculation of averages of the type
具 A 典 t ⫽Tr关 ␳ 共 t 兲 A 兴 ,
共3兲
Averages of such a kind give a direct account of the nonequilibrium evolution of the physical observable A, independently of how far the initial state is from the equilibrium
state or from a stationary one. Unfortunately, calculations of
these quantities are much more involved than the calculation
of TDCF 共1兲 for linear response, and few exact results are
known.3,4,6,26 –33
Whether the dynamical process is near or far from equilibrium, most works in the literature deal with cases where
the initial state is spatially uniform. This premise is assumed
in an explicit or a nonexplicit way, and the methods developed to solve the problem are heavily based on it. In contrast, the study of spatially nonuniform excitations is practically new, in spite of its interest, both theoretical and
experimental. This problem is important for a deeper understanding of dynamical processes in many-particle spin systems with strong exchange interactions. Inhomogeneous initial states can be prepared in real systems by external actions,
for instance, strong inhomogeneous magnetic fields or acoustic waves. On the other hand, from the theoretical side, exact
results on the dynamics 共as in the case of the XY model兲, can
elucidate details of spin-spin relaxation processes in more
complicated systems.
In the present contribution, we will adopt the method
based on formulas 共2兲 and 共3兲, and will analyze in detail the
long-time evolution of the magnetization in three versions of
the XY model. The initial excitation is always prepared in the
form of a spatially inhomogeneous magnetization 共SIM兲, and
the calculation is done in exact analytic form. The three variants of the XY model that we consider are enumerated below:
共I兲 Isotropic dimerized XY model with Hamiltonian:
N
H 1 ⫽⫺ ␮ B gh
兺
j⫽1
M
S zj ⫺
兺
m⫽1
x
x
y
y
S 2m
⫹S 2m⫺1
S 2m
兲
关 J 1 共 S 2m⫺1
x
x
y
y
⫹J 2 共 S 2m
S 2m⫹1
⫹S 2m
S 2m⫹1
兲兴 .
H 2 ⫽⫺ ␮ B g
共4兲
共II兲 Isotropic XY model in a staggered magnetic field with
Hamiltonian:
兺
m⫽1
z
z
⫹h 2 S 2m
兲
共 h 1 S 2m⫺1
N
⫺J
兺 共 S xj S xj⫹1 ⫹S yj S yj⫹1 兲 .
j⫽1
共5兲
共III兲 XY model with bond alternation without magnetic
field:
M
H 3 ⫽⫺
共2兲
where ␳ (t) is the density matrix that satisfies Liouville
equation
⳵
iប ␳ 共 t 兲 ⫽ 关 H, ␳ 共 t 兲兴 .
⳵t
M
兺
m⫽1
x
x
y
y
S 2m
⫹I 2 S 2m⫺1
S 2m
共 I 1 S 2m⫺1
x
x
y
y
⫹I 2 S 2m
S 2m⫹1
⫹I 1 S 2m
S 2m⫹1
兲.
共6兲
We will refer to them as models I, II, III, respectively. In
expressions 共4兲–共6兲, N⫽2M , ␮ B is the Bohr magneton, g is
the gyromagnetic ratio, h,h 1 ,h 2 are magnetic fields,
J,J 1 ,J 2 ,I 1 ,I 2 are exchange integrals (J 1 ⭓J 2 ⬎0 and J
⬎0), and cyclic boundary conditions are assumed. The
above restriction on J 1 ,J 2 , and J is not important in our
problem and is introduced for convenience.
To investigate SIM dynamics in the above models, we
will use a previously developed method given in Ref. 31.
This method was applied earlier by one of the authors32,33 to
the same study for the isotropic XY chain in a homogeneous
magnetic field, with Hamiltonian:
N
H 4 ⫽⫺ ␮ B gh
兺
j⫽1
N
S zj ⫺J
兺 共 S xj S xj⫹1 ⫹S yj S yj⫹1 兲 .
j⫽1
共7兲
We will refer to this model as model IV subsequently. We
remark that highly nontrivial results were obtained in Refs.
32 and 33. In particular, it was shown that in the t→⬁ limit,
some of the spatially inhomogeneous excitations do not disappear and are still time dependent. The time evolution of
SIM’s 共and also their spatial distribution兲 is probed through
z
(t) 典 of the
the computation of the Fourier components 具 S Q
magnetization as a function of the Q-wave number. The calculation yields a relaxation process in the form of a power
law
冉冊
t
␶
⫺␯
,
共8兲
where the exponent ␯ depends on the initial state. In Ref. 33
the anisotropic XY model was also studied in the limit of
strong anisotropy. It was found that the exponent ␯ in power
law 共8兲 changes discontinuously at some critical values Q c of
the wave vector Q. The values Q c depend solely on the parameters of the Hamiltonian and are not connected with the
preparation of the initial state, nor the particular component
of the magnetization that is relaxing. Moreover, in the limit
Q→Q c , the relaxation time ␶ of power law 共8兲 diverges to
␶ →⬁, with the corresponding slowing down of the process.
This phenomenon is at variance with some conventional
views for the time evolution of physical quantities in many
particle systems, according to which all quantities must be
temporally independent in the limit t→⬁ . The concept of
spin temperature, widely used in nonequilibrium magnetic
094401-2
PHYSICAL REVIEW B 66, 094401 共2002兲
LONG-TIME TAILS AND ANOMALOUS SLOWING DOWN . . .
phenomena43 is based on the assumption that the spin-spin
relaxation is much faster than the final relaxation to the lattice degrees of freedom.
Model I was also preliminary investigated by some of the
authors, and similar conclusions were attained for the dimerized XY model44 in the sense that a critical-like slowing
down of the relaxation process takes place at special points
⫾Q c of the Q space. In other words, for 兩 Q 兩 →Q c , the inverse time scale ␶ ⫺1 for some of the oscillating components
of SIM, goes to zero following the power law ␶ ⫺1
⬃円( 兩 Q 兩 ⫺Q c )円. Such behavior is not surprising for the case
Q c ⫽0, for which the corresponding value of the total magnetization S z is a constant of motion, but is unusual for Q c
z
⫽0, where the corresponding Fourier component S Q
is not
conserved.
We note that Q⫽0 is the only critical point for the uniform isotropic XY model 共model IV and limit J 1 ⫽J 2 for
model I and h 1 ⫽h 2 for model II兲. The t→⬁ behavior is
dominated by one oscillating component, and this component has no critical properties. The present paper is devoted
to elucidate this remarkable difference within a more general
context, by the extensive study of a whole family of models.
We suggest that dissimilar behaviors are due to the presence
of a gap in the spectrum of the energy excitations, the uniform isotropic model being gapless.
Our paper is organized as follows: In Sec. II, we outline
the main steps in the analytic calculation. All the models are
diagonalized by means of a modified Jordan-Wigner transformation, which maps the spin model into an equivalent
z
fermion Hamiltonian. Then, the average 具 S Q
典 t is calculated
in exact closed form, and its asymptotic behavior for t→⬁ is
obtained after taking the thermodynamic limit N→⬁. In Sec.
III, we give a detailed analysis of the results, including all
models and the long-time behavior of the magnetization.
Section IV closes the paper with final discussions.
z
To obtain exact results for 具 S Q
典 t , for each of the models
considered, we follow the steps:
共i兲 Diagonalization of the corresponding Hamiltonian using the fermion representation of the Jordan-Wigner transformation. At the end, we get a free-fermion system.
z
in terms of fermionic operators
共ii兲 Transformation of S Q
z
to get the time-dependent S Q
(t).
z
共iii兲 Averaging of S Q
(t) with the initial density matrix
␳ (0).
The last step depends on the form of initial matrix ␳ (0).
As it was shown in Ref. 31, exact solutions of this problem
may be obtained in the case when ␳ (0) is a functional of
only one component of the spin operator:
␳ 共 0 兲 ⫽F 共 S ␥ 兲
A. Diagonalization
Methods for diagonalization of Hamiltonians 共4兲–共6兲 are
well known 共see for example Refs. 9, 11, and 13兲. Minor
differences are specific for the version of the model to be
solved. In our case we have employed the following procedure:
共a兲 Use of the Jordan-Wigner transformation to change
from spin to Fermi operators (b †j ,b j ):
S xj ⫽L j 共 b †j ⫹b j 兲 /2, S yj ⫽L j 共 b †j ⫺b j 兲 /2i,
S zj ⫽b †j b j ⫺1/2,
j⫺1
共9兲
N
兺
j⫽1
共10兲
S zj exp共 iQ j 兲
␣
␣
cm
⫽b 2m
,
where ␳ (0) is the initial density matrix, and
operator in the Heisenberg representation
z
SQ
共 t 兲 ⫽exp
冉 冊 冉
z
SQ
(t)
共11兲
is the spin
共 2b †l b l ⫺1 兲 ,
L 1 ⬅1,
共 L 2j ⫽1 兲 .
共14兲
␣
␣
dm
⫽b 2m⫺1
,
m⫽1,2, . . . ,M ,
␣ ⫽⫾1,
共15兲
and of their Fourier transforms c ␣k ,d k␣ :
c ␣k ⫽
is the Fourier transform of the magnetization. In Eq. 共9兲, Q is
the wave vector characterizing the spatial inhomogeneity of
the initial state (Q⫽2 ␲ n/N,n⫽⫺N/2⫹1, . . . ,N/2). To calz
culate 具 S Q
典 t one can use the identity:
具 S Qz 典 t ⫽ 具 S Qz 共 t 兲 典 0 ⬅Tr关 ␳ 共 0 兲 S Qz 共 t 兲兴 ,
兿
l⫽1
共b兲 Introduction of two types of Fermi operators for even
and odd sites:
where ␳ (t) is the density matrix of the system, and
z
⫽
SQ
共13兲
where
L j⫽
具 S Qz 典 t ⬅ Tr关 ␳ 共 t 兲 S Qz 兴 ,
共12兲
Let us note that such initial state can actually be prepared in
real systems at low temperature, with a strong nonhomogeneous magnetic field directed along the coordinate axis ␥ .
II. MAIN STEPS OF SOLUTION
The quantity of our interest is the time-dependent z component of SIM,
共 ␥ ⫽x,y,z 兲 .
d k␣ ⫽
M
1
冑M
1
冑M
兺
m⫽1
␣
exp共 ⫺i ␣ km 兲 c m
,
M
兺
m⫽1
␣
exp关 ⫺i ␣ k 共 m⫺1/2兲兴 d m
,
共16兲
k⫽2 ␲ m 1 /M , with m 1 ⫽⫺M /2⫹1, . . . ,M /2, where we
have adopted the compact notation
c ⫺ ⫽c, c ⫹ ⫽c † ,
冊
i
i
z
exp ⫺ Ht .
Ht S Q
ប
ប
d ⫺ ⫽d,
094401-3
d ⫹ ⫽d † .
共17兲
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
共c兲 After the above steps, Hamiltonians 共4兲–共6兲 are transformed into quadratic forms in terms of operators (c ␣k ,d k␣ ).
Diagonalization of these quadratic Hamiltonians is standard,
but depends on the specific model under consideration. For
Hamiltonians 共4兲 and 共5兲 it can be done with the help of the
Bogoliubov transformation given below:
c k␣ ⫽u k ␩ ␣k ⫹ v k␣ ␤ ␣k ,
␣ ␣
␣
d ␣k ⫽ v ⫺
k ␩ k ⫺u k ␤ k ,
共18兲
␣
␣
where ( ␩ ␣k , ␤ ␣k ) are new Fermi operators, v ⫺
k ⫽( v k ) * , and
u k is a real function of k. We are using the same convention
共17兲 for ␣ ⫽⫾. For Hamiltonian 共4兲, the functions (u k , v k )
are given by
u k⫽
1
冑2
,
v k␣ ⫽
1
冑2
exp共 i ␣␪ k 兲 ,
␦ ⫽J 2 /J 1 ⭐1.
u k ⫽ 关共 P k ⫺ ␻ d 兲 /2P k 兴 1/2,
␻ 1 ⫽J 1 /2ប.
Note that here and in the following, we neglect, as usually
done,1,4,9,13 the boundary term of the order 1/N that appeared
in a chain with cyclic boundary conditions. The thermodynamic limit N→⬁ will be taken at the end of the calculation.
z
B. Calculation of ŠS Q
‹t
Using formulas 共10兲 and 共13兲–共16兲, one can easily obtain
the following expression,
z
⫽⫺M ␦ Q,0⫹
SQ
共20兲
is real 共independent of ␣ ) and
␤ ␣k 共 t 兲 ⫽ ␤ ␣k exp共 i ␣ ⌰ (i)
k t 兲,
␻ e ⫽J/ប.
共21兲
z
SQ
共 t 兲 ⫽⫺M ␦ Q,0⫹
␣
␣
⫹ ␤ ␣k ⫺ ␤ ⫺k
d k␣ ⫽ 共 ␩ k␣ ⫹ ␩ ⫺k
兲 /2.
共22兲
共 i⫽1,2,3 兲 共23兲
␻ (3)
k ⫽⫺ 共 I 2 /ប 兲 cos共 k/2 兲 ,
⌰ (1)
k ⫽⫺ ␻ 0 ⫺ ␻ 1 R k ,
⌰ (2)
k ⫽⫺ ␻ s ⫺ P k ,
⌰ (3)
k ⫽ 共 I 1 /ប 兲 cos共 k/2 兲 ,
where
共25兲
for models I and II, where
(i)
(i)
⍀⫹
i 共 k,Q 兲 ⫽ ␻ p ⫺⌰ q ,
(i)
(i)
⍀⫺
i 共 k,Q 兲 ⫽ ␻ p ⫺ ␻ q ,
i⫽1,2
and functions (u p , v p ) are defined by formulas 共19兲 and 共20兲
respectively.
In the same way, for model III, we obtain
except for constant terms that are not important for the dynamics. The index i⫽1,2,3 refers to models 共I兲, 共II兲, and
共III兲, respectively. The dispersion relations of Eq. 共23兲 are
given by
␻ (2)
k ⫽⫺ ␻ s ⫹ P k ,
兺k 兵 共 u p u q ⫹ v *p v q 兲 exp关 i 共 ⍀ ⫺i 共 k,Q 兲 t 兴
⫹
⫹ 共 u q v p ⫺u p v q 兲 exp关 ⫺i⍀ ⫹
i 共 k,Q 兲 t 兴 ␤ p ␩ q 其
As a final result, all the three Hamiltonians can be represented in the diagonal form
␻ (1)
k ⫽⫺ ␻ 0 ⫹ ␻ 1 R k ,
i⫽1,2
⫹
⫹
⫹ 共 u p v q* ⫺u q v *
p 兲 exp关 i⍀ i 共 k,Q 兲 t 兴 ␩ p ␤ q
␣
␣
c k␣ ⫽ 共 ⫺ ␩ ␣k ⫹ ␩ ⫺k
⫹ ␤ k␣ ⫹ ␤ ⫺k
兲 /2,
兺k 关 ␻ (i)k ␩ †k ␩ k ⫹⌰ (i)k ␤ †k ␤ k 兴 ,
共24兲
⫺
⫹
⫻␩⫹
p ␩ q ⫹ 共 u p u q ⫹ v p v q* 兲 exp关 ⫺i⍀ i 共 k,Q 兲 t 兴 ␤ p ␤ q
共d兲 Following Ref. 11, diagonalization of Hamiltonian 共6兲
is achieved through
H i ⫽ប
兺k 共 c ⫹p c q ⫹d ⫹p d q 兲 ,
to get
␻ d ⫽ ␮ B g 共 h 1 ⫺h 2 兲 /2ប,
P k ⫽ 共 ␻ 2d ⫹ ␻ 2e cos2 k/2兲 1/2,
␻ s ⫽ ␮ B g 共 h 1 ⫹h 2 兲 /2ប,
␩ k␣ 共 t 兲 ⫽ ␩ k␣ exp共 i ␣ ␻ (i)
k t 兲,
v k ⫽ 关共 P k ⫹ ␻ d 兲 /2P k 兴 1/2sgn共 cos k/2兲 ,
where
P k ⫽ 冑␻ 2d ⫹ ␻ 2e cos2 共 k/2兲 ,
where p⫽k⫺Q,q⫽k⫹Q, with definitions common for all
z
, we first exmodels. To obtain the time dependence of S Q
press relation 共24兲 in terms of the canonical ␩ ’s and ␤ ’s
through transformations 共18兲 for models I and II, and relation
共22兲 for model III. Then, we insert the time evolution of such
operators
For Hamiltonian 共5兲, one gets
v k␣
␻ 0 ⫽ ␮ B gh/ប,
共19兲
where
1⫺ ␦
tan共 k/2兲 ,
tan ␪ k ⫽
1⫹ ␦
R k ⫽ 冑1⫹ ␦ 2 ⫹2 ␦ cos k,
z
SQ
共 t 兲⫽
⫹
(3)
(3)
␩⫹
兺k 兵 ␤ ⫺q
p exp关 i 共 ␻ p ⫹⌰ q 兲 t 兴
(3)
⫹ ␩ ⫺p ␤ q exp关 ⫺i 共 ␻ (3)
p ⫹⌰ q 兲 t 兴 其 .
具 S Qz 典 t ,
共26兲
To calculate the time dependent average
we use the
identity 共11兲, which means that, equivalently, one can calcuz
(t) in the Heisenberg picture
late the average of operator S Q
in relation to the initial density matrix ␳ (0).
From Eqs. 共25兲 and 共26兲, one sees that this problem reduces to calculation of averages of the type 具 ␩ †p ␩ q 典 0 ,
具 ␤ †p ␤ q 典 0 , . . . , including all the combinations of canonical
operators. This is achieved going back to the spin represen094401-4
PHYSICAL REVIEW B 66, 094401 共2002兲
LONG-TIME TAILS AND ANOMALOUS SLOWING DOWN . . .
tation, in order to employ the initial condition 共12兲. One is
then led to computation of spin averages including the
‘‘string’’ operators 共14兲. This technique has been previously
used by some of the authors in Refs. 32, 33, and 44, and is
described in detail in Ref. 45. Putting everything together in
Eq. 共25兲 for models I and II, and taking the continuum limit
z
N→⬁, we get expressions for 具 S Q
典 t in the form of integrations over the Brillouin zone:
1
具 S Qz 典 t ⫽
4␲
冕
␲
⫺␲
共a兲 For model I, we get the formula
具 S Qz 典 t ⫽
冕
␲
⫺␲
兵 F ⫹ 共 k,Q 兲 cos关 t⍀ ⫹
1 共 k,Q 兲兴
⫹F ⫺ 共 k,Q 兲 cos关 t⍀ ⫺
1 共 k,Q 兲兴 其 dk,
z
z
F ⫾ 共 k,Q 兲 ⫽A ⫾ 共 k,Q 兲 具 S Q
典 0 ⫹B ⫾ 共 k,Q 兲 具 S Q⫹
␲典 0 ,
A ⫾ 共 k,Q 兲 ⫽1⫿
共27兲
where i⫽1,2, for the two models treated here. Quantities in
Eq. 共27兲 are
共 1⫹ ␦ 2 兲 cos Q⫹2 ␦ cos k
,
R pR q
B ⫾ 共 k,Q 兲 ⫽⫾
x
F ⫾ 共 k,Q 兲 ⫽⫾4u q 共 u 2p ⫺ 兩 v p 兩 2 兲关 i 共 e ⫺ip/2⫹e iq/2兲 具 ␧ Q
典 0 Imv q
具 S Qz 典 t ⫽
for ␥ ⫽x, and
1
具Sz 典
4␲ Q 0
冕
␲
⫺␲
dk„⫺ 关 ␻ 2e cos共 p/2兲 cos共 q/2兲 / P p P q 兴
⫻cos共 q/2兲兴 / P p P q 其 cos关 t⍀ ⫺
2 共 k,Q 兲兴 ….
A ⫾ 共 k,Q 兲 ⫽1⫿ 关 1⫺2u 2p u 2q ⫺2 兩 v p v q 兩 2 ⫺4u p u q Re共 v *
p v q 兲兴 ,
B 共 k,Q 兲 ⫽⫾4iu p u q Im共 v *
p vq兲,
for ␥ ⫽z.
Substituting the specific values 共19兲 and 共20兲 for (u p , v p )
into formula 共27兲 one obtains the corresponding expressions
for models I and II. In the same way, analogous results may
be obtained for model III.
Below, we will summarize these results together with the
ones obtained in Ref. 33 for model IV. We note that exact
solutions can be obtained for the special forms of the initial
density matrix ␳ (0) given by Eq. 共12兲.
共1兲 In the case ␥ ⫽x 共the case ␥ ⫽y is the same by symmetry兲, we have
共28兲
具 S Qz 典 t ⫽
1
具Sz 典
2␲ Q 0
冕
␲
⫺␲
cos关 t⍀ 共 k,Q 兲兴 dk,
⍀ 共 k,Q 兲 ⫽ ␷ Q cos共 k/2⫹ ␣ Q 兲 ,
tan ␣ Q ⫽
I 1 ⫹I 2
tan共 Q/2兲 ,
I 1 ⫺I 2
ប ␷ Q ⫽ 冑I 21 ⫹I 22 ⫺2I 1 I 2 cos Q.
sin 共 k/2兲
⫺␲ P p P q
2
⫺
⫻ 兵 cos关 t⍀ ⫹
2 共 k,Q 兲兴 ⫺cos关 t⍀ 2 共 k,Q 兲兴 其 dk,
共35兲
Let us note that at the specific value Q⫽ ␲ , the time depenz
dence of 具 S Q
典 t is given by the simple formula:
具 S ␲z 典 t ⫽ 具 S ␲z 典 0 J 0
␲
共34兲
where
for models I, III, IV, and
冕
共33兲
共c兲 For model III, we have the compact expression:
⫾
i
x
␻ ␻ sin共 Q/2兲 e iQ/2具 ␧ Q⫹
␲典 0
␲ e d
共32兲
2
2
⫻cos关 t⍀ ⫹
2 共 k,Q 兲兴 ⫹ 兵 1⫹ 关 ␻ d ⫹ ␻ e cos共 p/2 兲
z
z
F ⫾ 共 k,Q 兲 ⫽A ⫾ 共 k,Q 兲 具 S Q
典 0 ⫹B ⫾ 共 k,Q 兲 具 S Q⫹
␲典 0 ,
具 S Qz 典 t ⫽⫺
i 共 1⫺ ␦ 2 兲 sin Q
.
R pR q
共b兲 For model II, we have the more explicit form
x
⫹ 共 e ⫺ip/2⫺e iq/2兲 具 ␧ Q⫹
␲ 典 Rev q 兴
具 S Qz 典 t ⫽0
共31兲
with
兵 F ⫹ 共 k,Q 兲 cos关 t⍀ ⫹
i 共 k,Q 兲兴
⫹F ⫺ 共 k,Q 兲 cos关 t⍀ ⫺
i 共 k,Q 兲兴 其 dk,
1
4␲
冉
冊
I 1 ⫹I 2
t ,
ប
where J 0 (x) is the Bessel function of first kind.
共d兲 For model IV, the result
共29兲
for model II, where
具 S Qz 典 t ⫽ 具 S Qz 典 0 J 0 共 ⍀ Q t 兲 ,
共36兲
was found in Ref. 33 with ⍀ Q ⫽2(J/ប)sin Q/2.
N
具 ␧ xk 典 0 ⫽
兺
j⫽1
具 S xj S xj⫹1 典 0 exp共 ik j 兲
共30兲
is the Fourier transform of the short-range order correlation
of the x component.
共2兲 In the case ␥ ⫽z, we have different expressions for all
the cases.
III. ANALYSIS OF RESULTS
In this section, we first discuss general properties of the
time evolution of the magnetization, which can be inferred
directly from Eqs. 共27兲–共36兲. In addition, we will study in
detail the asymptotic behavior at long times, in the limit t
→⬁. We will show that the evolution of SIM’s displays
094401-5
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
interesting unusual features, depending on the wave vector Q
characterizing the inhomogeneous initial state.
z
A. General properties of ŠS Q
‹t
Let us first consider model I in the case ␥ ⫽z for the
initial density matrix, since it presents a specific feature that
is absent in the other models studied here. From Eqs. 共27兲,
共31兲, and 共32兲, one sees that the value of the magnetization
具 S Qz 典 t at t⬎0, depends on the initial values of the two Fouz
z
rier components 具 S Q
典 0 and 具 S Q⫹
␲ 典 0 . This means that, if
z
only one component of S Q exists at t⫽0, for example, for
wave vector Q 0 , i.e.,
具 S Qz 典 0 ⫽ 具 S Qz 0 典 0 ␦ Q,Q 0 ,
then a twofold response develops for t⬎0, with components
具 S Qz 0 典 t and 具 S ␲z ⫹Q 0 典 t , whose time evolution is described by
z
and
Eq. 共27兲, with F ⫾ (k,Q)⫽A ⫾ (k,Q 0 ) 具 S Q
典 ␦
0 0 Q,Q 0
z
⫾
⫾
F (k,Q)⫽B (k,Q 0 ⫹ ␲ ) 具 S Q 典 0 ␦ Q,Q 0 ⫹ ␲ , respectively. In
0
contrast, for models II–IV, if there is only one initial comz
ponent 具 S Q
典 , only one component 具 S Qz 典 t of the same wave
0 0
vector Q⫽Q 0 exists at subsequent times, t⬎0.
When the initial condition is prepared with ␥ ⫽x, the avz
erage 具 S Q
典 t vanishes identically for models I, III, and IV, for
any t⬎0. This result may be understood intuitively, without
any calculation, if one notes that the part of the Hamiltonian
that describes the coupling of spins with the magnetic field
commutes with the other one describing the exchange interactions 关in the case of model III, this statement is trivial for
we only have the exchange part兴. Denoting the latter as H ex ,
we get the time evolution
冋
具 S Qz 典 t ⫽Tr ␳ 共 0 兲 exp
冉
冊 冉
i
i
z
H t SQ
exp ⫺ H ex t
ប ex
ប
冊册
, 共37兲
which is governed by the exchange only. Inside the trace in
Eq. 共37兲, we now perform a unitary transformation consisting in a rotation by ␲ around the x axis. Under this operation, the spin operators transform as follows:
S xj →S xj ,
S yj →⫺S yj ,
S zj →⫺S zj .
The exchange Hamiltonian H ex is invariant under this transformation because it is a sum of products of the form S xj S xj⫾1
and S yj S yj⫾1 . The initial density matrix ␳ (0) is also invariant
since it only depends on the S x component of the spins ( ␥
⫽x). So, the right-hand side of Eq. 共37兲 will change its sign,
z
yielding 具 S Q
典 t ⫽⫺ 具 S Qz 典 t i.e., 具 S Qz 典 t ⫽0. Another important
consequence of Eq. 共37兲 is that the time evolution of SIM’s
is independent of the value of the external field h 共valid for
any ␥ ), and is determined solely by the exchange term.
In the case of model II, the exchange part does not commute with the total Hamiltonian H 2 . This leads, in general,
z
to a nonvanishing 具 S Q
典 t , for t⬎0, which gives results prox
portional to 具 ␧ Q⫹ ␲ 典 0 , where 具 ␧ xk 典 0 is the Fourier transform
of the spin-spin correlation function 具 S xj S xj⫹1 典 0 evaluated at
the initial condition 关see relation 共30兲兴. If the initial state is
prepared with a single Q vector in the form 具 ␧ xk 典 0
x
⫽具␧Q
典 ␦ , then 具 S ␲z ⫹Q 0 典 t is the only component that
0 0 k,Q 0
exists for t⬎0.
For Q⫽Q 0 ⫽0 共homogeneous initial state兲, and for any
␥ , we get the 共evident兲 result
z
z
具 S Q⫽0
典 t ⫽ 具 S Q⫽0
典0 ,
for models I, II, IV. This is a direct consequence of the conservation of the total z component of the magnetization. In
the case of model III, when ␥ ⫽z, we get the closed result:
z
z
具 S Q⫽0
典 t ⫽ 具 S Q⫽0
典 0J 0共 ␯ 0t 兲 ,
where ប ␯ 0 ⫽ 兩 I 1 ⫺I 2 兩 and J 0 is the Bessel function of zero
order.
Special important limits are ␦ ⫽1(J 1 ⫽J 2 ) in model I and
h 1 ⫽h 2 in model II, that map onto model IV. In both of these
z
cases, the corresponding formulas for 具 S Q
典 t reduce to Eq.
共36兲, which was obtained in Ref. 33.
Unfortunately, it is difficult to get more information from
formula 共27兲, 共29兲, 共32兲, and 共33兲, because they are rather
involved. So, in the following, we will consider the limiting
case t→⬁, which will be studied using the stationary phase
method.
z
B. Long-time behavior of ŠS Q
‹ t for the initial condition ␥ Äz
1. Model I
According to the stationary phase method,46 the long-time
z
evolution of 具 S Q
典 t is dominated by the contribution of the
stationary points of the functions ⍀ ⫾
1 (k,Q). The number of
these points depends on the value of the wave vector Q. For
convenience of further discussion, let us introduce here the
so-called critical values of Q:
Q c 1 ⫽arccos ␦ ,
Q c 2 ⫽ ␲ ⫺arccos ␦ ,
Q c 3 ⫽0.
共38兲
The above values are determined in a standard manner
through the stationary phase method, being the locus where
stationary points become degenerate.46 Let us first examine the role of ⍀ ⫹
1 (k,Q). It is easy to show that it has five
nondegenerate stationary points for Q in the interval
Q c 1 ⬍ 兩 Q 兩 ⬍Q c 2 :
k 1 ⫽0, k 2 ⫽⫺k 3 ⫽ ␲ ,
k 4 ⫽⫺k 5 ⫽ ␸ Q ,
␸ Q ⫽arccos共 ⫺ ␦ ⫺1 cos Q 兲 .
共39兲
In turn, for Q c 2 ⬍ 兩 Q 兩 ⭐ ␲ and Q c 3 ⬍ 兩 Q 兩 ⬍Q c 1 , one gets three
nondegenerate points at
k 1 ⫽0, k 2 ⫽⫺k 3 ⫽ ␲ .
Exactly at the critical values, we obtain:
共i兲 For Q⫽⫾Q c 1 , one nondegenerate point k 1 ⫽0
and two degenerate ones k 2 ⫽⫺k 3 ⫽ ␲ „关 ⍀ ⫹
1 (⫾ ␲ ,Q) 兴 ⬙
⫹
IV
⫽关⍀⫹
(⫾
␲
,Q)
⫽0,
⍀
(⫾
␲
,Q)
⫽0
,
where
deriva兴
关
兴
⵮
1
1
tions are taken with respect to k….
094401-6
PHYSICAL REVIEW B 66, 094401 共2002兲
LONG-TIME TAILS AND ANOMALOUS SLOWING DOWN . . .
共ii兲 For Q⫽⫾Q c 2 , two nondegenerate points k 2 ⫽⫺k 3
⫽ ␲ , and one degenerate at k⫽k 1 ⫽0„关 ⍀ ⫹
1 (0,Q) 兴 ⬙
⫹
IV
⫽关⍀⫹
1 (0,Q) 兴 ⵮ ⫽0,关 ⍀ 1 (0,Q) 兴 ⫽0….
共iii兲 For Q⫽Q c 3 ⫽0 , there is no time evolution, since the
z
magnetization 具 S Q⫽0
典 t is a constant of the motion
⍀ 4共 Q 兲 ⫽ ␻ 1
␶ ⫺1
4 ⫽␻1
z
z
具 S Q⫽0
典 t ⫽ 具 S Q⫽0
典0 .
␺ Q ⫽arccos共 ⫺ ␦ cosQ 兲 .
This finishes the analysis of the stationary points that dominate the long-time behavior of Eq. 共31兲. The corresponding
z
asymptotic development of 具 S Q
典 t for t→⬁, can be represented as a sum of several oscillating components with
Q-dependent frequencies and amplitudes:
具 S Qz 典 t ⬃ 兺 S i 共 Q,t 兲 ,
共40兲
i
⬁
S i 共 Q,t 兲 ⫽
兺
l⫽0
冉冊
t
a i,l 共 Q 兲
␶i
⫺ ␯ i (2l⫹1)
cos关 t⍀ i 共 Q 兲 ⫹ ␣ i,l 兴 ,
共41兲
where ␶ i ⬅ ␶ i (Q) are functions of Q and the exponent ␯ i
assumes the values 1/2 or 1/4 共the latter value will be discussed in detail below兲. The number of components S i (Q,t)
depends on Q. For Q c 1 ⬍ 兩 Q 兩 ⬍Q c 2 共excluding Q⫽⫾ ␲ /2,
which is a degenerate point where ⍀ 1 and ⍀ 2 coincide兲,
there are four components. For 0⭐ 兩 Q 兩 ⬍Q c 1 and Q c 2 ⬍ 兩 Q 兩
⬍ ␲ 共excluding accidental degeneracies兲, there are only three
terms in relation 共40兲. The frequencies ⍀ i (Q) and the inof Eq. 共41兲 are given
verse of the characteristic times ␶ ⫺1
i
below:
⍀ 1 共 Q 兲 ⫽ ␻ 1 共 1⫹ ␦ 2 ⫹2 ␦ cos Q 兲 1/2,
␶ ⫺1
1 ⫽ ␻ 1 ␦ 共 1⫹ ␦ cos Q 兲
兩 ␦ ⫹cos Q 兩
3
RQ
,
Q⫽⫾Q c 2
⍀ 2 共 Q 兲 ⫽ ␻ 1 共 1⫹ ␦ 2 ⫺2 ␦ cos Q 兲 1/2,
␶ ⫺1
2 ⫽ ␻ 1 ␦ 共 1⫺ ␦ cos Q 兲
⍀ 3共 Q 兲 ⫽ ␻ 1
␶ ⫺1
3 ⫽␻1
␦
2 共 1⫺ ␦ 2 兲
Q⫽Q c 3 ⫽0
冉
兩 ␦ ⫺cos Q 兩
3
R Q⫹
␲
,
R ␺ Q ⫺Q ⫺R ␺ Q ⫹Q
2
2 共 1⫺ ␦ 2 兲
R ␸ Q ⫺Q ⫹R ␸ Q ⫹Q
2
冊
,
共 R ␸ Q ⫺Q ⫺R ␸ Q ⫹Q 兲 sin Q sin ␸ Q ,
Q c 1 ⬍ 兩 Q 兩 ⬍Q c 2 .
The function ⍀ ⫺
1 (k,Q) vanishes at Q⫽Q c 3 ⫽0 and Q
⫽⫾ ␲ . For any other wave number Q, it has two nondegenerate stationary points:
k 6 ⫽⫺k 7 ⫽ ␺ Q ,
␦
冉
Q⫽⫾Q c 1
冊
,
共42兲
We display typical Q dependences of the frequencies ⍀ i (Q)
in Figs. 1 and
and of the inverse of the relaxation times ␶ ⫺1
i
2, respectively, for particular values of ␦ . Slowing down of
the relaxation process occurs at points where the ␶ ⫺1
i ’s vanish.
At a generic Q point, the first nonvanishing terms in Eq.
共41兲 have the form
S i 共 Q,t 兲 ⬇a i,0共 Q 兲
冉冊
t
␶i
⫺1/2
cos关 t⍀ i 共 Q 兲 ⫹ ␣ i,0兴共 i⫽1, . . . ,4兲 ,
共43兲
with the amplitudes
c
z
s
z
a i,0共 Q 兲 ⫽a i,0
共 Q 兲具S Q
共 Q 兲 具 S Q⫹
典 0 ⫹ia i,0
␲典 0 ,
showing that the value at Q depends on the two initial comz
z
ponents 具 S Q
典 0 and 具 S Q⫹
␲ 典 0 . Explicit formulas for the coefc
s
ficients a i,0(Q) and a i,0
(Q), as well as the phases ␣ i,0 , are
given in Ref. 45.
The dynamical process described by relations 共40兲 and
共41兲, with the explicit formulas 共42兲, is remarkable, since it
exhibits long-time tails in the relaxation of SIM’s to the spatially homogeneous state. In contrast to the exponential relaxation, which is characterized by a single parameter that
yields the time scale or the relaxation rate, the power law
relaxation given by relations 共41兲 and 共43兲 is characterized
by two parameters: ␶ ⫺1
i , which determines the inverse time
scale of the process, and the exponent ␯ i , which determines
the relaxation rate. In general, no conservation laws or longlived hydrodynamic modes seem to be associated with the
above long-time tails.
In the neighborhood of critical points, the relaxation of
some of the components of relation 共40兲 begins to stop, with
the corresponding relaxation time ␶ i diverging, as shown in
Fig. 2. Exactly in the limit, the corresponding exponent ␯ i
jumps discontinuously from 1/2 to 1/4. This slowing down
of the relaxation process is very similar to the critical slowing down found in phase transition phenomena.47
Let us summarize below the singular behavior of the relaxation time at special points 共they are displayed in Fig. 2兲:
共a兲 ␶ ⫺1
1 →0 when Q→⫾Q c 2 , for any ␦ ;
共b兲 ␶ ⫺1
2 →0 when Q→⫾Q c 1 , for any ␦ ;
⫺1
共c兲 ␶ 4 →0 when Q→⫾Q c 1 ,⫾Q c 2 , for any ␦ ;
共d兲 ␶ ⫺1
3 →0 when Q→0,⫾ ␲ , for any ␦ .
All the ␶ ⫺1
vanish according to the law:
i
共 R ␺ Q ⫺Q ⫹R ␺ Q ⫹Q 兲 兩 sin Q 兩 sin ␺ Q ,
␶ ⫺1 ⬃円共 兩 Q 兩 ⫺Q c 兲 円for兩 Q 兩 →Q c .
We now give the behavior of components of relation 共40兲 at
the critical points:
094401-7
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
FIG. 1. Frequencies of the asymptotic components of the magnetization in Eq. 共41兲, as functions of the wave number Q, for several values of
the parameter ␦ . The different branches ⍀ i , for
i⫽1,2,3,4, are indicated by the numbers. We also
display the critical points ⫾Q c 1 and ⫾Q c 2 by
solid squares. Note that the ⍀ 4 branch is tangent
to the ⍀ 1 and ⍀ 2 branches at Q c 2 and Q c 1 , respectively, but it is only defined between critical
points, Q c 1 ⬍ 兩 Q 兩 ⬍Q c 2 . The complete ⍀ 4 curve
is shown as a guide for the eye.
共I兲 In the limit Q→⫾Q c 1 , the components S 2 (Q,t) and
S 4 (Q,t) merge into a single component with ␯ ⫽1/4:
S 2 共 ⫾Q c 1 ,t 兲 ⫽a 2,0共 ⫾Q c 1 兲
冉冊
t
␶c
共II兲 Analogously, for Q→⫾Q c 2 , the components S 1 (Q,t)
and S 4 (Q,t) join together with ␯ ⫽1/4 and the time dependence:
⫺1/4
cos关 t⍀ 2 共 Q c 1 兲 ⫹ ␲ /8兴 ,
共44兲
S 1 共 ⫾Q c 2 ,t 兲 ⫽a 1,0共 ⫾Q c 2 兲
冉冊
t
␶c
⫺1/4
cos关 t⍀ 1 共 Q c 2 兲 ⫹ ␲ /8兴 ,
where
with
冑24
a 2,0共 ⫾Q c 1 兲 ⫽
8␲
冑24
4
4
z
⌫ 共 1/4兲关共 1⫹ ␦ 兲 具 S ⫾Q
c
1
a 1,0共 ⫾Q c 2 兲 ⫽
典0
8␲
z
⌫ 共 1/4兲关共 1⫺ ␦ 兲 具 S ⫾Q
典
c 0
⫹i 冑1⫺ ␦
⫹i 冑1⫺ ␦ 2 具 S ␲z ⫾Q 典 0 兴 ,
c
1
2
2
␶ ⫺1
c ⫽3 ␻ 1 ␦ / 共 1⫺ ␦ 兲 ,
where ⌫(x) is the gamma function. The other components
follows the law 共43兲 with Q⫽⫾Q c 1 .
2
2
具 S ␲z ⫾Q c
2
典0兴.
The other components follows the law 共43兲 with Q
⫽⫾Q c 2 .
共III兲 The component S 3 (Q,t) is critical at Q⫽Q c 3 ⫽0,
with the relaxation time ␶ 3 ⬃1/兩 Q 兩 diverging in the vicinity
of Q⫽0. Note that the frequency ⍀ 3 vanishes in the limit
094401-8
PHYSICAL REVIEW B 66, 094401 共2002兲
LONG-TIME TAILS AND ANOMALOUS SLOWING DOWN . . .
FIG. 2. Inverse of the relaxation times
( ␻ 1 ␶ i ) ⫺1 of the power-law decay of Eq. 共41兲, as
functions of the wave number Q for the same
values of the parameter ␦ displayed in the previous figure. The numbers label the different
branches, in close correspondence with Fig. 1.
Solid squares are used to display ( ␻ 1 ␶ 4 ) ⫺1 ,
which vanishes at the critical points. The arrows
in the upper figure indicate values out of the
scale.
Q→0. The criticality of Q c 3 has a different connotation,
since it is related to the conservation of the total z component
of the spin,
peared in Eq. 共33兲 of model II, can be mapped onto the
corresponding ⍀ ⫾
1 (k,Q) and ␻ 1 R k of model 共I兲, with the
substitution:
N
s
S Q⫽0
⫽
兺
j⫽1
␦→
S zj ,
which is valid for all the model with axial symmetry along
the z axis. The divergence of ␶ 3 near Q⬇0 is connected to
the stability of hydrodynamic excitations of long wavelength.
共IV兲 Note the exception of Q→⫾ ␲ for the component
S 3 (Q,t), because the amplitude a 3,0(Q) goes to zero in this
limit.
2. Model II
冑␻ 2d ⫹ ␻ 2e ⫺ 兩 ␻ d 兩
,
冑␻ 2d ⫹ ␻ 2e ⫹ 兩 ␻ d 兩
␻ 1→
兩␻ 兩
1
冑␻ 2d ⫹ ␻ 2e ⫹ d .
2
2
共45兲
To avoid numerous definitions, we use the previous symbols
␦ and ␻ 1 throughout this section, with the meaning given
above in relation 共45兲. With these variables, the stationary
and
and critical points, the inverse characteristic times ␶ ⫺1
i
the frequencies ⍀ i (Q) coincide with those of model I.
z
The expression for 具 S Q
典 t has the form 共41兲 with the following values of the first nonvanishing amplitudes a i,k (Q)
共at a generic Q point兲:
When applying the same method to get the asymptotic
behavior of Eq. 共33兲 for long times, one sees that the work is
facilitated by the development of the preceding section. In
fact, the frequencies ⍀ ⫾
2 (k,Q) and the functions P k that ap094401-9
z
a 1,1共 Q 兲 ⫽ 具 S Q
典0
␦ 共 1⫺ ␦ 兲 2 共 1⫺cos Q 兲
4
冑8 ␲
RQ
,
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
z
a 2,0共 Q 兲 ⫽ 具 S Q
典0
冑
2 ␦ 共 1⫺cos Q 兲
,
2
␲
R Q⫹
␲
冑
典 冑
2 1⫹ ␦ cos Q
,
␲ 1⫹ ␦
z
a 3,0共 Q 兲 ⫽ 具 S Q
典0
z
a 4,0共 Q 兲 ⫽ 具 S Q
x
a 2,0共 Q 兲 ⫽ 具 ␧ Q⫹
␲典 0
0
x
a 3,0共 Q 兲 ⫽⫺ 具 ␧ Q⫹
␲典 0
2 ␦ ⫹cos Q
,
␲ 1⫹ ␦
x
a 4,0共 Q 兲 ⫽ 具 ␧ Q⫹
␲典 0
Note that the component with frequency ⍀ 1 (Q) decays to
zero as t ⫺3/2 共exponent ␯ ⫽1/2 and l⫽1), i.e., more rapidly
if compared with the similar component of model I. We now
summarize the behavior at the critical points:
共i兲 At the critical point Q⫽⫾Q c 1 , the components
S 2 (Q,t) and S 4 (Q,t) are degenerate and decay to zero with
exponent ␯ ⫽1/4, following the law 共44兲 with
冑24
4
a 2,0共 ⫾Q c1 兲 ⫽
2␲
⌫ 共 1/4兲
␦
1⫹ ␦
.
冉冊
t
␶c
⫺3/4
cos关 t⍀ 1 共 Q c 2 兲 ⫹ ␲ /8兴 ,
共46兲
a 2,0共 ⫾Q c 1 兲 ⫽
␦
1
.
共 3/2兲 3/4⌫ 共 3/4兲
2␲
1⫹ ␦
冑
2
共 t/ ␶ Q 兲 ⫺1/2cos共 t ␷ Q ⫺ ␲ /4兲 ,
␲
⫺1
⫽␷Q .
where ␷ Q is given by formula 共35兲 and ␶ Q
z
C. Long-time behavior of ŠS Q
‹ t for the initial condition ␥ Äx
z
As it was mentioned in Sec. II B, 具 S Q
典 t , for the initial
condition ␥ ⫽x, vanishes identically for models I, III, IV, and
only has a nonzero value for model II. For the latter, the
z
asymptotic expression of 具 S Q
典 t for t→⬁, has the form of Eq.
共41兲 with the quantities ⍀ i and ␶ i defined through formulas
共42兲 and 共45兲. The coefficients a i,l (Q) for the first nonvanishing amplitudes in Eq. 共41兲, at generic Q points, are
冑␦ 共 1⫺ ␦ 兲
2
,
共 1⫺e iQ 兲 /R Q
2 冑2 ␲
2 冑␦
冑2 ␲ 共 1⫹ ␦ 兲
共 1⫺e iQ 兲共 1⫹ ␦ cos Q 兲 ,
共 1⫺e iQ 兲共 1⫹ ␦ ⫺1 cos Q 兲 ,
冑24
2␲
⌫ 共 1/4兲
冑␦
1⫹ ␦
共 1⫺ ␦ ⫿i 冑1⫺ ␦ 2 兲 .
冉 冑 冊
1
共 3/2兲 3/4⌫ 共 3/4兲 冑␦ 1⫿i
2␲
1⫺ ␦
.
1⫹ ␦
IV. FINAL DISCUSSION
The long-time behavior of Eq. 共34兲 is much simpler than
those displayed by models I and II, for the phase function
⍀(k,Q) in Eq. 共35兲 has only one stationary point at k⫽k 1
⫽⫺2 ␣ Q . In the asymptotic limit t→⬁ , it may be represented in the form
x
a 1,1共 Q 兲 ⫽ 具 ␧ Q⫹
␲典 0
冑2 ␲ 共 1⫹ ␦ 兲
2
共 1⫺e iQ 兲 /R Q⫹
␲,
共ii兲 In analogous form, at Q⫽⫾Q c 2 the components
S 1 (Q,t) and S 4 (Q,t) become degenerate with the time evolution of Eq. 共46兲, with ␯ ⫽1/4 and l⫽1, and the amplitude
a 1,1共 ⫾Q c 2 兲 ⫽
3. Model III
具 S Qz 典 t ⫽ 具 S Qz 典 0
2 冑␦
4
with
a 1,1共 ⫾Q c 2 兲 ⫽
冑2 ␲
where 具 ␧ xk 典 0 is the correlation function defined in Eq. 共30兲.
We remark that the first nonvanishing term for the component with frequency ⍀ 1 has ␯ ⫽1/2 and l⫽1, yielding a
t ⫺3/2 power law for the time decay.
We make a summary of the critical behavior at singular
points:
共i兲 At the critical point Q⫽⫾Q c 1 , the components
S 2 (Q,t) and S 4 (Q,t) merge into one component with time
evolution given by Eq. 共44兲, with ␯ ⫽1/4 and l⫽0, with the
amplitude
共ii兲 At Q⫽⫾Q c 2 , the components S 1 (Q,t) and S 4 (Q,t)
merge into a single component with the evolution 关 ␯ ⫽1/4
and l⫽1 in Eq. 共41兲兴
S 1 共 ⫾Q c 2 ,t 兲 ⫽a 1,1共 ⫾Q c 2 兲
2 冑␦ 共 1⫺ ␦ 兲
We have studied the relaxation to the homogeneous state,
of an initial excitation that has been prepared with a nonhomogeneous magnetization profile along the magnetic chain
共SIM兲. The time evolution of this excitation is probed
z
(t) 典 of
through the calculation of the Fourier component 具 S Q
the magnetization, which is analyzed as a function of the
wave vector Q. We use periodic boundary conditions for the
spin Hamiltonian and take the thermodynamic limit (N
→⬁) before studying asymptotic long times. As remarked in
Ref. 4, the order of the limits is very important. All the
models treated here show long-time tails in the relaxation of
具 S Qz (t) 典 , which is apparent from the asymptotic study for t
→⬁, developed in the previous sections. This behavior
manifests itself in the form of a power law decay in the
long-time evolution of SIM’s. This slow relaxation is a remarkable result for its own sake and is probably due to the
absence of dissipation in the models. Since the systems are
isolated at zero temperature, the dynamics is exclusively
driven by quantum fluctuations.
Our calculation also shows striking differences, when one
compares the behaviors of models I and II on the one hand,
with models III and IV, on the other. The dissimilarity lies in
the presence of critical values Q c i for the wave vector Q,
where we get a slowing down of the relaxation process. Near
094401-10
LONG-TIME TAILS AND ANOMALOUS SLOWING DOWN . . .
PHYSICAL REVIEW B 66, 094401 共2002兲
critical points, the time scale ␶ i , giving the relaxation of
z
(t) 典 , diverge
some specific oscillating components of 具 S Q
⫺1
with the law ␶ ⬃円( 兩 Q 兩 ⫺Q c i )円, for 兩 Q 兩 →Q c i , reducing the
damping of these excitations. Such behavior is not surprising
z
for the case Q c 3 ⫽0, for which the corresponding value S Q⫽0
is an integral of the motion, but is unusual for the case Q c i
and the spectral properties is currently under research.48
Our results are all exact, with closed analytical forms,
from where the asymptotic regime for t→⬁ is obtained. This
has been done using the fermion representation of the
Jordan-Wigner approach for spin chains. Other procedures to
solve the same systems are available, the most conspicuous
being the Bethe ansatz method, which applied to the XY
model might hint at ‘‘hidden’’ conservation laws of the several versions of the model treated here. However, it is difficult to conceive a similar analytic calculation of the relaxation properties, using Bethe ansatz techniques.
We remark that integrable models display other unusual
dynamical phenomena, with anomalies in the transport
properties.42 For the Heisenberg model, the absence of spin
diffusion has been probed by bosonization techniques49 and
numerical calculations.50 Reference 50 presents exact numerical computations for the spin-1/2 XXZ chain at infinite
temperature (T⫽⬁), which include, as particular cases, the
isotropic Heisenberg model and the XY regime, which are
gapless, and the Heisenberg-Ising model, with a gap in the
spectrum and long-range order for the ground state. Numerical data are inconclusive due to finite-size effects, but there
are strong hints of a crossover from nondiffusive to diffusive
behavior when we go from the gapless region to the gapped
one. Concerning the low-temperature properties, a calculation by Sachdev and Damle51 in the gapped region, shows
that the diffusive behavior holds in the presence of longrange order. It is however puzzling to admit that low-energy
properties of the spectrum may affect the T⫽⬁ behavior.
From a more fundamental point of view, we have studied
the time evolution of a closed quantum many-body system
共at zero temperature兲, which is prepared in an arbitrary nonhomogeneous initial state. The dynamics is solely given by
the Schrödinger equation for the wave function or by the
Liouville equation for the density operator 共which includes
the more general situation of a mixed ensemble for the initial
state兲. In the thermodynamic limit and for long times (N
→⬁ first than the limit t→⬁), we get irreversibility in the
form of a power law relaxation for the magnetization, despite
the dynamics being unitary. No internal interactions are
present, since all the Hamiltonians are reduced to the freeparticle form. The irreversibility can be entirely ascribed to
interference effects, which in our calculation are handled using the stationary phase method at asymptotic long times.
The infinite number of degrees of freedom precludes quantum recurrences, which may be observed in finite chains or
systems with a discrete spectrum.52,53 Due to constructive
interference, gapped models develop long-lived collective
excitations with the texture of spin density waves 共charge
density waves for the fermion versions兲 that persist for
longer times. We believe that those structures are universal
features of gapped one-dimensional models.
Finally, our simple quantum systems are good candidates
for aging effects, a concept that has been coined to specify
nonequilibrium dynamics that depends on the initial conditions and relaxes very ‘‘slowly.’’54 But a complete characterization of quantum aging requires the calculation of two-time
correlation functions.53
z
⫽0, when S Q
is not conserved. In this sense, the criticality at
⫾Q c 1 , ⫾Q c 2 has different implications in the theory. The
position of the critical values ⫾Q c 1 and ⫾Q c 2 depends only
on parameters of the Hamiltonian, and in the limits Q→
⫾Q c 1 ,⫾Q c 2 , the exponent ␯ jumps discontinuously, the relaxation rate becoming slower than for excitations with 兩 Q 兩
⫽Q c 1,2. This means that the critical components will be the
only surviving ones at long enough times. These long-lived
excitations may be pictured as spin density waves 共charge
density waves for the fermion model兲 with incommensurate
wave vector Q c 1,2, which develop in models that undergo a
Peierls instability.38 In contrast, the dynamics of models III
and IV does not have such anomalies. There is no gap in the
spectrum, and in the language of particles, the systems are
always metallic and display no critical points 共except the
point Q⫽0). In the asymptotic regime t→⬁, only one oscillating component exists which displays no critical behavior.
Peculiar properties of the energy spectrum seem to determine the critical relaxation phenomena. In fact, models I and
II display an energy gap between the ground and excited
states, whose size is monitored by the parameter ␦ 关for
model II through the transformation 共45兲兴. The existence of
critical points is directly related to the presence of the gap,
since the latter changes the curvature of the dispersion relation for the energy. If one looks at formula 共25兲 for the Fouz
(t) of the magnetization, one realizes that
rier component S Q
processes that contribute with the ⫾⍀ ⫹
i frequency 共the one
that is critical兲 come from transitions between both branches
of the spectrum 共interband transitions兲, where a particle with
momentum q⫽k⫹Q is destroyed in a given band, and a
particle with momentum p⫽k⫺Q is created in the other
one, with a net momentum transfer ⌬k⫽⫾2 兩 Q 兩 . So the
criticality at Q c 1,2 is not directly related to the one-particle
spectrum, but to a big density of states for those interband
transitions that exchange the same momentum ⌬k⫽⫾2 兩 Q 兩
coherently 共constructive interference兲. The above is a consequence of a nesting effect, which is achieved when Q
⫽Q c 1,2, with the onset of a spin density wave for the S z
magnetization. In fact, the spectrum can be linearized in the
neighborhood of the inflection points, yielding parallel linear
branches for the upper and lower bands, which are connected
by p⫽k⫺Q c and q⫽k⫹Q c . This is accompanied by the
degeneracy of some stationary points.
We have found that the above anomalous behavior in the
relaxation is not an exclusive property of the Peierls transition. In fact, other 1D gapped spin models, as the XY chain
with anisotropic interactions, present the same slowing down
of the relaxation at special points 共Ref. 33 treats the Ising
limit兲. A general connection between this critical behavior
094401-11
PHYSICAL REVIEW B 66, 094401 共2002兲
G. O. BERIM, S. I. BERIM, AND G. G. CABRERA
*Present address: Department of Chemical Engineering, The State
University of New York, Buffalo, NY 14260.
†
Email address: cabrera@ifi.unicamp.br
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