Ergodicity in strongly correlated systems
We present a concise and systematic review of the ergodicity issue in strongly correlated systems. After giving a brief historical overview, we analyze the issue within the Green’s function formalism by means of the equations of motion approach. By means of this analysis, we are able to identify t...
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| Цитувати: | Ergodicity in strongly correlated systems / A. Avella, F. Mancini, E. Plekhanov // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 485–497. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1213742017-06-15T03:03:19Z Ergodicity in strongly correlated systems Avella, A. Mancini, F. Plekhanov, E. We present a concise and systematic review of the ergodicity issue in strongly correlated systems. After giving a brief historical overview, we analyze the issue within the Green’s function formalism by means of the equations of motion approach. By means of this analysis, we are able to identify the primary source of non-ergodic dynamics for a generic operator as well as to give a recipe for computing unknown quantities characterizing such a behavior within the Composite Operator Method. Finally, we present examples of nontrivial strongly correlated systems where it is possible to find a non-ergodic behavior. Представлено короткий але систематичний розгляд проблеми ергодичностi в сильноскорельованих системах. Пiсля короткого iсторичного огляду ми аналiзуємо це питання в рамках формалiзму функцiй Грiна за допомогою методу рiвнянь руху. За допомогою цього аналiзу ми можемо видiлити першоджерела неергодичної динамiки оператора а також дати спосiб розрахунку невiдомих величин, що характеризують таку поведiнку, в рамках методу композитних операторiв. Також представлено приклади нетривiальних сильноскорельованих систем де можлива неергодична поведiнка. 2006 Article Ergodicity in strongly correlated systems / A. Avella, F. Mancini, E. Plekhanov // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 485–497. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 05.30.Jp, 75.10.Jm, 71.10.-w DOI:10.5488/CMP.9.3.485 http://dspace.nbuv.gov.ua/handle/123456789/121374 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present a concise and systematic review of the ergodicity issue in strongly correlated systems. After
giving a brief historical overview, we analyze the issue within the Green’s function formalism by means of
the equations of motion approach. By means of this analysis, we are able to identify the primary source of
non-ergodic dynamics for a generic operator as well as to give a recipe for computing unknown quantities
characterizing such a behavior within the Composite Operator Method. Finally, we present examples of nontrivial
strongly correlated systems where it is possible to find a non-ergodic behavior. |
| format |
Article |
| author |
Avella, A. Mancini, F. Plekhanov, E. |
| spellingShingle |
Avella, A. Mancini, F. Plekhanov, E. Ergodicity in strongly correlated systems Condensed Matter Physics |
| author_facet |
Avella, A. Mancini, F. Plekhanov, E. |
| author_sort |
Avella, A. |
| title |
Ergodicity in strongly correlated systems |
| title_short |
Ergodicity in strongly correlated systems |
| title_full |
Ergodicity in strongly correlated systems |
| title_fullStr |
Ergodicity in strongly correlated systems |
| title_full_unstemmed |
Ergodicity in strongly correlated systems |
| title_sort |
ergodicity in strongly correlated systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121374 |
| citation_txt |
Ergodicity in strongly correlated systems / A. Avella, F. Mancini, E. Plekhanov // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 485–497. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT avellaa ergodicityinstronglycorrelatedsystems AT mancinif ergodicityinstronglycorrelatedsystems AT plekhanove ergodicityinstronglycorrelatedsystems |
| first_indexed |
2025-07-08T19:44:29Z |
| last_indexed |
2025-07-08T19:44:29Z |
| _version_ |
1837109210767163392 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 485–497
Ergodicity in strongly correlated systems
A.Avella, F.Mancini, E.Plekhanov
Dipartimento di Fisica “E.R. Caianiello” – Unità CNISM di Salerno
Università degli Studi di Salerno, I–84081 Baronissi (SA), Italy
Received June 8, 2006, in final form June 22, 2006
We present a concise and systematic review of the ergodicity issue in strongly correlated systems. After
giving a brief historical overview, we analyze the issue within the Green’s function formalism by means of
the equations of motion approach. By means of this analysis, we are able to identify the primary source of
non-ergodic dynamics for a generic operator as well as to give a recipe for computing unknown quantities
characterizing such a behavior within the Composite Operator Method. Finally, we present examples of non-
trivial strongly correlated systems where it is possible to find a non-ergodic behavior.
Key words: ergodicity, strongly correlated systems, Green’s function formalism, equations of motion
approach, composite operator method
PACS: 05.30.Jp, 75.10.Jm, 71.10.-w
1. Historical overview
The issue of ergodicity in condensed matter physics has been well known since fifties [1]. Given
two operators A and B, describing physical quantities (e.g., charge, spin, pair densities or currents),
one can study the physical response of a system described by a certain Hamiltonian H through
the generalized susceptibility
χAB = lim
h→0
∂〈A〉
∂h
, (1.1)
where h is an external field entering the Hamiltonian of the system under study in a coupling term
of the type −hB and 〈· · · 〉 stands for the statistical average in some ensemble over the perturbed
system. Kubo [1] immediately noticed that the static isolated susceptibility χI(0), defined for
an isolated system perturbed by an external field turned on adiabatically, and the isothermal
susceptibility χT , defined for a system in thermal equilibrium in presence of a time-independent
external field, may be generally different. In particular, Falk [2] has shown that the static isolated
susceptibility is just a lower bound for the isothermal one χI(0) 6 χT .
We recall that the static isolated susceptibility χI(0), within the linear response theory, is
defined through the related retarded Green’s function [1]
χI(ω) = −F
[
i θ(ti − tj) 〈[A(ti), B(tj)]〉0
]
, (1.2)
where 〈· · · 〉0 stands for the statistical average on the unperturbed system and F for the Fourier
transform.
On the other hand, the isothermal susceptibility χT can be computed as
χT =
∫ β
0
〈A(−iλ)B〉0 dλ− β〈A〉0〈B〉0 . (1.3)
In fact, starting from the expression of the thermal average in the canonical ensemble (i.e., fixing
temperature) 〈A〉
〈A〉 =
1
Z
Tr
(
Ae−β(H−hB)
)
, (1.4)
c© A.Avella, F.Mancini, E.Plekhanov 485
A.Avella, F.Mancini, E.Plekhanov
where β = 1/T and Z = Tr
(
e−βH−hB
)
, we can expand e−β(H−hB) in powers of h and get
e−β(H−hB)
u e−βH
(
1 + h
∫ β
0
dλeλHBe−λH +O(h2)
)
. (1.5)
Substituting this expansion into (1.4) and retaining only the first order term in h, we get at the
numerator
Tr
(
Ae−β(H−hB)
)
u Z0〈A〉0 + hZ0
∫ β
0
dλ〈AB(iλ)〉0 (1.6)
and at the denominator
Z u Z0 (1 + hβ〈B〉0) (1.7)
and by the ratio
〈A〉 u 〈A〉0 + h
∫ β
0
dλ〈AB(iλ)〉0 − hβ〈A〉0〈B〉0 , (1.8)
where Z0 denotes the partition function of the unperturbed system. Taking the derivative after
(1.1) and exploiting the cyclic property of the trace, we obtain the isothermal susceptibility as
in (1.3).
Now, if we rewrite both expressions by means of the general formulas for the retarded Green’s
functions and the correlation functions given in the companion article [3] (see section 3), present
in this same issue, we get
χI(0) =
1
V
∑
k,l
σ(l,−1)(k)
ωl(k)
(1.9)
and
χT = β
1
V
∑
k
ΓAB(k) +
1
V
∑
k,l
σ(l,−1)(k)
ωl(k)
− β〈A〉0〈B〉0 , (1.10)
where [3,4] V is proportional to the volume of the system, the sum over l ranges over the number of
fields in the chosen basis, σ(l) are the spectral density functions, ω(l) are the poles of the propagator,
ΓAB is an unknown function appearing in case of poles with zero value.
We can immediately see that the two susceptibilities differ for the following expression
χT − χI(0) = β
1
V
∑
k
ΓAB(k) − β〈A〉0〈B〉0 . (1.11)
Now, one can check that rewriting the expression limt→∞〈AB(t)〉 by means of the general
formula for the correlation functions given in the companion article [3] (see section 3), present in
this same issue, we just get
lim
t→∞
〈AB(t)〉 =
1
V
∑
k
ΓAB(k) (1.12)
and, accordingly, the difference at the r.h.s. of (1.11) is just what enters Khintchin’s theorem [5]:
a dynamics is ergodic (i.e., phase space equilibrium averages are equal to ensemble microcanical
averages, which are much easier to compute)
〈AB〉 =
∫ ∞
0
dtAB(t) (1.13)
if an only if
lim
t→∞
〈AB(t)〉 = 〈A〉〈B〉. (1.14)
In other words a dynamic is ergodic if correlations attenuate in time. In particular, for B ≡ A,
the dynamics of A is ergodic if, during its time evolution, it has non-zero matrix elements only
between states within a zero-volume region of the phase space of the system [6].
486
Ergodicity in strongly correlated systems
It is clear now the link between ergodicity and response theory: the two definitions of sus-
ceptibility differ when the dynamics of the system is not ergodic. Let us make two little, but
important, notices: finite systems are not ergodic by definition, just due to of the inequivalence of
the ensembles; non-ergodicity at zero temperature is just the result of a degeneracy in the ground
state.
Several years later it was shown [7,8] that the difference between the two definitions of sus-
ceptibility is related to the zero-frequency anomaly exhibited by bosonic correlation functions: the
presence of undetermined constants in the bosonic correlation functions. This is exactly what the
relations derived above and the results of the companion article [3] predict establishing a definite
link between the ergodicity of the dynamics and the Green’s function formalism. It was first put in
evidence in [9] and then studied by many other authors [6–8,10–16]. There is a general belief that
this problem is of academic interest and in the last years no much attention has been paid to it. The
main reason is that the response functions, the experimentally observed quantities, are given by
retarded bosonic Green’s function which formally do not depend on such undetermined constants,
which are, therefore, considered of no physical interest. The general attitude [1,10] is to believe that
in macroscopic real systems at equilibrium at a temperature T , the fluctuations are very small and
the interaction between the system and the reservoir would introduce an irreversible relaxation and
decouple the correlation functions. Then, as suggested in [10], these constants should be always
determined using the ergodicity. This procedure is somewhat an artifice and may lead to serious
problems because it might break the internal self-consistency of the entire formulation. As remarked
in [10], the zero-frequency anomaly is a manifestation of the difficulty in extracting irreversible be-
havior from the statistical mechanics. This is true, but as long as we use the scheme of statistical
mechanics we must be careful in doing self-consistent calculations. Breaking the self-consistency
might bring to serious errors.
According to the well-known relations existing between casual (C), retarded (R) Green’s func-
tions and correlation functions
<[GR(k, ω)] = <[GC(k, ω)], (1.15)
=[GR(k, ω)] = tanh
(
βω
2
)
=[GC(k, ω)], (1.16)
C(k, ω) = −
[
1 + tanh
(
βω
2
)]
=[GC(k, ω)] (1.17)
the zero-frequency excitations do not explicitly contribute to the imaginary part of the retarded
Green’s functions and, consequently, Γ does not explicitly appear in the expressions of susceptibili-
ties. At any rate, susceptibilities retain an implicit dependence on Γ through the matrix elements.
Then, the right procedure to compute both correlation functions and susceptibilities is clearly the
one that starts from the causal Green’s function, which is the only Green’s function that explicitly
depends on Γ. It is worth noticing that the value of Γ dramatically affects the values of directly
measurable quantities (e.g., compressibility, specific heat, magnetic susceptibility, . . . ) through the
values of correlation functions and susceptibilities. According to this, whenever it is possible, Γ
should be exactly calculated case by case.
If we do not have access to the complete set of eigenstates and eigenvalues of the system, which is
the rule in the most interesting cases, we have to compute correlation functions and susceptibilities
within some, often approximated, analytical framework. Now, since no analytical tool can easily
determine Γ (e.g., the equations of motion cannot be used to fix Γ as it is constant in time), one
usually assumes the ergodicity of the dynamics of ψ and simply substitutes Γ by its ergodic value
(i.e., by the r.h.s. of (1.14)):
Γerg(i, j) = 〈ψ(i)〉〈ψ†(j)〉. (1.18)
Unfortunately, this procedure cannot be justified a priori by absolutely any means. The existence
of just one integral of motion and, more generally, of any operator that has a diagonal part with
respect to the Hamiltonian [6] (i.e. by any operator that has a diagonal entries whenever written
in the basis of eigenstates of the Hamiltonian) divides the phase space into separate subspaces not
487
A.Avella, F.Mancini, E.Plekhanov
connected by the dynamics. This latter, in turn, becomes non-ergodic: time averages give different
results with respect to ensemble averages. This latter consideration also clarifies why the ergodic
nature of the dynamics of an operator mainly depends on the Hamiltonian it is subject to.
It is really remarkable that Γ is directly related to relevant measurable quantities such as
compressibility and specific heat through the dissipation-fluctuation theorem. For instance, we
recall the formula that relates the compressibility to the total particle number fluctuations
κ = β
V
N2
[〈
N̂2
〉
−N2
]
, (1.19)
where N̂ is the total particle number operator, N is its average and V is proportional to the volume
of the system. We see that a compressibility different from zero requires the non-ergodicity of the
system with respect to total particle number operator. According to this, also in the case of infinite
systems, the correct determination of Γ cannot be considered as an irrelevant issue (e.g., (1.19)
holds in the thermodynamic limit too).
In the next section, we provide some examples of violation of the ergodic condition (1.14). It is
necessary to point out, in order to avoid any possible confusion, that we are using full operators
but not fluctuation ones (i.e., we use operators not diminished of their average value, in contrast
to what is usually done for the bosonic excitations like spin, charge and pair). According to this,
the Γ can be different from zero (i.e., be equal to the squared average of the operator), and still
indicate an ergodic dynamics for the operator.
2. Examples
2.1. Two-site Hubbard model
The two-site Hubbard model is described by the following Hamiltonian
H =
∑
ij
(tij − δij µ) c†(i) c(j) + U
∑
i
n↑(i)n↓(i), (2.1)
where the summation range only over two sites at distance a from each other and the rest of
notation is standard [4]. The hopping matrix tij is defined by
tij = −2t αij , αij =
1
2
∑
k
ei k(i−j) α(k), (2.2)
where α(k) = cos(ka) and k = 0, π/a.
We now proceed to the study of the system by means of the equation of motion approach and
the Green’s function formalism [17]. A complete set of fermionic eigenoperators of H is as follows:
ψ(i) =
ξ(i)
η(i)
ξs(i)
ηs(i)
, (2.3)
where
ξ(i) = [1 − n(i)] c(i), (2.4a)
η(i) = n(i) c(i), (2.4b)
ξs(i) =
1
2
σµ nµ(i) ξα(i) + ξ(i) η†α(i) η(i), (2.4c)
ηs(i) =
1
2
σµ nµ(i) ηα(i) + ξ(i) ξ†α(i) η(i). (2.4d)
488
Ergodicity in strongly correlated systems
We define ψα(i) =
∑
j αij ψ(j) and use the spinorial notation for the field operators. nµ(i) =
c†(i)σµ c(i) is the charge (µ = 0) and spin (µ = 1, 2, 3) operator; greek (e.g., µ, ν) and latin (e.g.,
a, b, k) indices take integer values from 0 to 3 and from 1 to 3, respectively; sum over repeated
indices, if not explicitly otherwise stated, is understood; σµ = (1, ~σ) and σµ = (−1, ~σ); ~σ are the
Pauli matrices. In momentum space the field ψ(i) satisfies the equation of motion
i
∂
∂t
ψ(k, t) = ε(k)ψ(k, t), (2.5)
where the energy matrix ε(k) has the expression
ε(k) =
−µ− 2t α(k) −2t α(k) −2t −2t
0 U − µ 2t 2t
0 4t −µ+ 2t α(k) 4t α(k)
0 2t 2t α(k) U − µ
. (2.6)
Straightforward calculations, according to the scheme traced in [17], show that two correlators
∆ =
〈
ξα(i) ξ†(i)
〉
−
〈
ηα(i) η†(i)
〉
, (2.7)
p =
1
4
〈
nα
µ(i)nµ(i)
〉
−
〈
c↑(i) c↓(i)
[
c†↓(i) c
†
↑(i)
]α〉
(2.8)
appear in the normalization matrix I(k) = F
〈{
ψ(i, t), ψ†(j, t)
}〉
. Then, the Green’s functions
depend on three parameters: µ, ∆ and p. The correlator ∆ can be expressed in terms of the
fermionic correlation function C(i, j) =
〈
ψ(i)ψ†(j)
〉
; the chemical potential µ can be related to
the particle density by means of the relation n = 2 [1 − C11(i, i) − C22(i, i)]. The parameter p
cannot be calculated in the fermionic sector; it is expressed in terms of correlation functions of the
bosonic fields nµ(i) and c↑(i) c↓(i). According to this, the determination of the fermionic Green’s
functions requires the parallel study of bosonic Green’s functions.
After quite cumbersome calculations, it is possible to see [17] that a complete set of bosonic
eigenoperators of H in the spin-charge channel is given by
B(µ)(i) =
B
(µ)
1 (i)
...
B
(µ)
6 (i)
, (2.9)
where
B
(µ)
1 (i) = c†(i)σµ c(i), (2.10)
B
(µ)
2 (i) = c†(i)σµ c
α(i) − c†α(i)σµ c(i), (2.11)
B
(µ)
3 (i) = dµ(i) − dα
µ(i) + d†µ(i) − d†αµ (i), (2.12)
B
(µ)
4 (i) = dµ(i) − dα
µ(i) − d†µ(i) + d†αµ (i), (2.13)
B
(µ)
5 (i) = fµ(i) − fα
µ (i) − f†µ(i) + f†αµ (i), (2.14)
B
(µ)
6 (i) = fµ(i) − fα
µ (i) + f†µ(i) − f†αµ (i) (2.15)
with the definitions:
dµ(i) = ξ†(i)σµ η
α(i), (2.16)
f0(i) = −η†(i) η(i) − d†(i) dα(i) + η†(i) η(i) ξ†α(i) ξα(i), (2.17)
fa(i) = ξ†(i) ξ(i)nα
a (i) −
1
2
i εabc nb(i)n
α
c (i). (2.18)
489
A.Avella, F.Mancini, E.Plekhanov
The field B(µ)(i) satisfies the equation of motion
i
∂
∂t
B(µ)(k, t) = κ(k)B(µ)(k, t), (2.19)
where the energy matrix κ(k) has the expression
κ(k) =
0 −2t 0 0 0 0
−4t [1 − α(k)] 0 U 0 0 0
0 0 0 U 2t 0
0 0 U 0 0 2t
0 0 8t 0 0 0
0 0 0 8t 0 0
. (2.20)
The energy spectra are given by
ω1(k) = −2t
√
2 [1 − α(k)] , (2.21)
ω2(k) = 2t
√
2 [1 − α(k)] , (2.22)
ω3(k) = −U − 4JU , (2.23)
ω4(k) = −4JU , (2.24)
ω5(k) = 4JU , (2.25)
ω6(k) = U + 4JU , (2.26)
where
JU =
1
8
[
√
U2 + 64t2 − U
]
. (2.27)
Straightforward calculations show that the correlation function has the expression
C(µ)(i, j) =
〈
B(µ)(i)B(µ)†(j)
〉
=
1
4
∑
k
6
∑
n=1
ei k(i−j)−i ωn(k)(ti−tj)
[
1 + tanh
β ωn(k)
2
]
f (n,µ)(k), (2.28)
where
f (n,µ)(0) = 0 for n = 3, 4, 5, 6, (2.29a)
f (n,µ)(π) = coth
β ωn(π)
2
σ(n,µ)(π) for ∀n. (2.29b)
Owing to the fact that zero-energy modes appear for n = 1, 2 and k = 0 [cfr. equation (2.21)], Γ
appear in the correlation functions
Γ(µ)(0) =
1
2
2
∑
n=1
f (n,µ)(0). (2.30)
One might think, as is often done in the literature, to fix this constant by its ergodic value. However,
this is not correct as we are in a finite system in the grand canonical ensemble and the ergodicity
condition does not hold. For the moment, we can state that this constant remains undetermined.
The spectral density functions depend on a set of parameters which come from the calculation
of the normalization matrix I(µ)(k) = F
〈[
B(µ)(i, t), B(µ)†(j, t)
]〉
. In particular, for the (1,1)-
component the following parameters appear:
Cα
12 =
〈
ηα(i) ξ†(i)
〉
, (2.31a)
Cα =
〈
cα(i) c†(i)
〉
, (2.31b)
d =
〈
c↑(i) c↓(i)
[
c†↓(i) c
†
↑(i)
]α〉
, (2.31c)
χα
s = 〈~n(i) · ~nα(i)〉 . (2.31d)
490
Ergodicity in strongly correlated systems
The parameters Cα and Cα
12 are related to the fermionic correlation function C(i, j) =
〈
ψ(i)ψ†(j)
〉
.
The parameter χα
s can be expressed in terms of the bosonic correlation function C(µ)(i, j) =
〈
B(µ)(i)B(µ)†(j)
〉
. In order to use the standard procedure of self-consistency, we need to cal-
culate the parameter d. For this purpose we should open both the pair channel and a double
occupancy-charge channel (i.e., we will need the static correlation function 〈n↑(i)n↓(i)n
α(i)〉).
The corresponding calculations are reported in [17] where it is shown that these two channels do
not carry any new unknown Γ. The self-consistence scheme closes; by considering the four channels
(i.e., fermionic, spin-charge, pair and double occupancy-charge) we can set up a system of coupled
self-consistent equations for all the parameters. However, Γ(µ)(0) has not been determined yet: we
have not definitely fixed the representation of the Green’s functions.
In conclusion, the standard procedure of self-consistency is very involved and is not capable of
giving a final answer because of the problem in fixing the Γ. We will now approach the problem by
taking a different point of view. The proper representation of the Green’s functions should satisfy
the condition that all the microscopic laws, expressed as relations among operators must also hold
at macroscopic level as relations among matrix elements. For instance, let us consider the fermionic
channel. We have seen that there exists the parameter p, not explicitly related to the fermionic
propagator, that can be determined by opening other channels. However, we know that at the end
of the calculations, if the representation is the right one, the parameter p should take a value such
that the symmetries are conserved. By imposing the algebra constraints [4] and by recalling the
expression for ∆ we get three equations
n = 2(1 − C11 − C22), (2.32a)
∆ = Cα
11 − Cα
22 , (2.32b)
C12 = 0. (2.32c)
This set of coupled self-consistent equations will allow us to completely determine the fermionic
Green’s functions. Calculations show [17] that this way of fixing the representation is the right one:
all the symmetry relations are satisfied and all the results exactly agree with those obtained by
means of Exact Diagonalization. We do not have to open the bosonic channels; the fermionic one
is self-contained.
Next, let us consider the spin-charge Green’s functions. In the spin-charge sector we have the
parameters Cα, Cα
12, χ
α
s , d and two Γ
b0 =
1
4
2
∑
i=1
f
(i,0)
11 (0), (2.33)
bk =
1
4
2
∑
i=1
f
(i,k)
11 (0), k = 1, 2, 3. (2.34)
Since we are in lack of an external applied magnetic field, bk takes the same values for any value
of k.
The parameters Cα and Cα
12 are known, since the fermionic correlation functions have been
computed. The parameters χα
s and d can be computed by means of the equations
d =
1
4
〈
nα
µ(i)nµ(i)
〉
− p, (2.35)
χα
s = 〈~n(i) · ~nα(i)〉 . (2.36)
The Γ are fixed by the algebra constraints
C
(µ)
11 (i, i) = 〈nµ(i)nµ(i)〉 . (2.37)
By recalling (2.28) and (2.29) we have
bµ = 〈nµ(i)nµ(i)〉 −
1
4
6
∑
i=1
[
1 + coth
β ωi(π)
2
]
σ
(i,µ)
11 (π) (2.38)
491
A.Avella, F.Mancini, E.Plekhanov
with
〈nµ(i)nµ(i)〉 =
{
n+ 2D for µ = 0,
n− 2D for µ = 1, 2, 3,
(2.39)
D = 〈n↑(i)n↓(i)〉 is the double occupancy and can be calculated by means of the fermionic cor-
relation functions D = n − 1 + C11. Equations (2.35) and (2.38) constitute a set of coupled
self-consistent equations which will completely determine the Green’s function in the spin-charge
channel. Calculations show that this way of fixing the representation is the right one: all the sym-
metry relations are satisfied and all the results exactly agree with those obtained by means of
Exact Diagonalization.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
n
U = 4
T = 0 b
0
b
k
T = 1 b
0
b
k
-4 -3 -2 -1 0 1 2 3 4
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
U
T = 0.01
n = 0.6 b
0
b
k
n = 0.8 b
0
b
k
n = 0.9 b
0
b
k
Figure 1. (left) b0 and bk are plotted as functions of n for U = 4 and T = 0 and 1. U and T are
expressed in units of t. (right) b0 and bk are plotted as functions of U for T = 0.01 and n = 0.6,
0.8, and 0.9. U and T are expressed in units of t.
b0 and bk are plotted as functions of n and U in figure 1 for various temperatures. It is worth
noting that they assume their ergodic values (i.e. n2 and 0, respectively) only in some regions of
the parameter space: (at zero temperature) at n = 1 (both b0 and bk) and at n = 0.5 (b0 only).
In these regions, the grand-canonical ensemble is equivalent to the microcanonical one and the
underlying ergodicity of the charge and spin dynamics emerges.
It is worth noting that b0 is directly related to the compressibility by means of the following
relation [17]
κ =
2
kBT
1
n2
[
b0 − n2
]
. (2.40)
According to this, if we erroneously set the value of b0 to the ergodic one (i.e., n2) we would get a
constant zero compressibility.
2.2. Tight-binding model
A narrow-band Bloch system in presence of an external magnetic field is described by the
following Hamiltonian
H =
∑
ij
(tij − µ δij) c
†(i) c(j) − h
∑
i
n3(i), (2.41)
where n3(i) is the third component of the spin density operator and h is the intensity of the ex-
ternal magnetic field. The indices i and j run on an infinite d-dimensional lattice. Straightforward
calculations show that the causal Green’s function G
(µ)
C (i, j) = 〈T [nµ(i)nµ(j)]〉 and the correla-
tion function C(µ)(i, j) = 〈nµ(i)nµ(j)〉 of the charge-spin operator nµ(i) = c†(i)σµ c(i) have the
492
Ergodicity in strongly correlated systems
following expressions
G
(µ)
C (k, ω) = −i (2π)d+1a−d δ(d)(k) δ(ω) Γ(µ) −Q(µ)(k, ω), (2.42)
C(µ)(k, ω) = (2π)d+1 a−d δ(d)(k) δ(ω) Γ(µ) +
[
1 + tanh
β ω
2
]
=
[
Q(µ)( k, ω)
]
, (2.43)
where δ(d)(k) is the d-dimensional Dirac delta function. Q(µ)(k, ω) comes from the proper fermionic
loop and is the Fourier transform of
Q(µ)(i, j) = Tr [σµGC(i, j)σµGC(j, i)] . (2.44)
Here GC(i, j) =
〈
T
[
c(i) c†(j)
]〉
is the causal fermionic function and has the expression
GC(k, ω) =
2
∑
n=1
σ(n)
1 + e−β En(k)
[
1
ω − En(k) + iδ
+
e−β En(k)
ω − En(k) − iδ
]
(2.45)
with
E1(k) = −µ− 2d t α(k) − h, (2.46)
E2(k) = −µ− 2d t α(k) + h, (2.47)
σ(1) =
(
1 0
0 0
)
, σ(2) =
(
0 0
0 1
)
, (2.48)
where
α(k) =
1
d
d
∑
i=1
cos(ki a). (2.49)
Γµ is fixed by the algebra constraints [4] which requires
Γ(µ) = 〈nµ(i)nµ(i)〉 −
ad
(2π)d+1
∫
ddk dω
[
1 + tanh
β ω
2
]
=
[
Q(µ)(k, ω)
]
. (2.50)
The loop Q(µ)(k, ω) can be calculated by means of (2.45). Calculations show
ad
(2π)d+1
∫
ddk dω
[
1 + tanh
β ω
2
]
=[Q(µ)(k, ω)] = 〈n〉 − 〈n↑〉
2
− 〈n↓〉
2
for µ = 0, 3, (2.51)
= 〈n〉 − 2 〈n↑〉 〈n↓〉 for µ = 1, 2. (2.52)
By recalling the algebra constraints (2.39), equation (2.50) gives for the Γ
Γ(0) = 〈n〉
2
, (2.53)
Γ(1,2) = 0, (2.54)
Γ(3) = 〈n3〉
2
(2.55)
in accordance with the ergodic nature of the spin and charge dynamics in this system.
It is worth noting that the compressibility of this system can be computed by means of the
general formula (1.19) that holds in the thermodynamic limit too and gives
κ =
1
〈n〉
2
β
2
ad
2(2π)d
2
∑
n=1
∫
ddk
1
Cn(k)
, (2.56)
where Cn(k) = cosh2
(
β En(k)
2
)
. We can see that an ergodic charge dynamics can lead to a non-
ergodic value of the Γ relatively to the total number operator, which is an integral of motion. Also
in the infinite systems the decoupling inspired by the requirement of ergodicity cannot always be
applied.
493
A.Avella, F.Mancini, E.Plekhanov
2.3. Heisenberg chain
We will now study [18,19] the ergodicity of the dynamics of the operator Sz
i , the z-component
of the spin at site i, in the 1D anisotropic extended Heisenberg model described by the following
Hamiltonian:
H = −Jz
∑
i
Sz
i S
z
i+1 + J⊥
∑
i
(Sx
i S
x
i+1 + Sy
i S
y
i+1) + J ′
∑
i
SiSi+2, (2.57)
where Sx
i , Sy
i and Sz
i are the x, y and z components of the spin-1/2 at site i, respectively. The
model (2.57) is taken on a linear chain with periodic boundary conditions. We take the interaction
term parameterized with Jz ferromagnetic (Jz > 0) and the next-nearest-neighbor interaction
term, which is parameterized with J ′, isotropic. In order to frustrate ferromagnetism, we have
considered only the case with J ′ > 0, that is, with an antiferromagnetic coupling between next-
nearest neighbors. According to this, only chains with even number of sites have been studied in
order to avoid topological frustration that would be absent in the thermodynamic limit. Since it
is possible to exactly map all the results obtained for J⊥ > 0 to those for J⊥ < 0 by means of a
simple canonical transformation, we have restricted our study only to positive values of J⊥.
We have numerically diagonalized the Hamiltonian (2.57) for chains of size L ranging between
6 and 18 by means of Exact Diagonalization (ED) (divide and conquer algorithm) and for chains
of size L ranging between 20 and 26 by means of Lanczos Diagonalization (LD). We have system-
atically taken into account translational symmetry and classified the eigenstates by the average
value of Sz =
∑
i S
z
i , which is a conserved quantity. Whenever we have used ED, all eigenvalues
and eigenvectors of (2.57) have been calculated up to machine precision and, therefore, we have
been able to determine the exact dynamics of the system for all temperatures. On the contrary,
when we have used LD, we have been limited to the zero-temperature case since only the ground
state can be considered exact in LD.
In this case, we have the opportunity to exactly compute Γ in terms of the exact eigenvalues
En and eigenstates |n〉 of the system. As a matter of fact, it reads as [4]
Γ =
1
Z
∑
n,m
En=Em
e−βEn〈n|Sz
i |m〉〈m|Sz
i |n〉. (2.58)
As already discussed above, the dynamics of an operator (e.g., Sz
i ) is ergodic whenever (1.14) is
satisfied, or equivalently, (2.58) is equal to its ergodic value:
Γerg = 〈Sz
i 〉
2 =
1
Z2
∑
n,m
e−β(En+Em)〈n|Sz
i |n〉〈m|Sz
i |m〉. (2.59)
The dynamics of a finite system is hardly ergodic, since (2.58) and (2.59) are unlikely to coincide.
In the thermodynamic limit, the sums in (2.58) and (2.59) become series and no conclusion can
be drawn a priori. Since we have diagonalized the Hamiltonian (2.57) numerically (i.e., only for
finite systems) and since L→ ∞ is the most interesting case, we have analyzed our results through
finite-size scaling in order to speculate on the properties of the bulk system.
If the ground state of (2.57) is N -fold degenerate then, at T = 0, (2.58) and (2.59) read as
follows:
Γ =
1
N
N
∑
n,m=1
|〈n|Sz
i |m〉|2, Γerg =
(
1
N
N
∑
n
〈n|Sz
i |n〉
)2
,
respectively.
Thanks to the translational invariance enjoined by the system, 〈Sz
i 〉 is independent of i and
proportional to the z-component of the total spin operator average 〈Sz
tot〉. It is easy to show that,
even if there is a finite magnetic moment per site in any of such N degenerate ground states, 〈Sz
i 〉
at T = 0 is always zero in the absence of magnetic field. Indeed, if a ground state with non-zero
494
Ergodicity in strongly correlated systems
〈Sz
i 〉 = M exists, also another ground state with 〈Sz
i 〉 = −M exists. Thus, at zero temperature,
Γerg is always zero and the only quantity of interest is Γ. A finite value of this latter implies
non-ergodicity. Obviously, if N = 1 then both values coincide. Therefore, a non-ergodic phase
corresponds to degenerate ground states with finite magnetization.
In the studied range of coupling constants (see figure 2) we have found two non-ergodic phases
(NE-I and NE-II), two ergodic ones (E-I and E-II) and a weird phase (W). Our computational
facilities limit the range of chain sizes that we can analyze such that we could not establish, by
means of finite-size scaling, whether the weird phase (W) is ergodic or not. In the non-ergodic
phases (NE-I and NE-II), we were able not only to perform the finite-size scaling, but also to write
down an analytic expression for Γ as a function of the chain size L. The weird phase (W) has
exhibited a strong dependence of the ground state upon the particular values of the couplings. On
the contrary, the other phases exhibit ground states that are independent of the particular values
of the coupling constants.
Figure 2. Zero-temperature ergodicity phase diagram in the J
′−J⊥ plane. Due to the symmetry
of the Hamiltonian only the upper half is shown (see in the text). Only two ergodic phases (E-I
and E-II) have been found in the reported parameter space. The others are either non-ergodic
(NE-I and NE-II) or impossible to conclusively analyze (W). The latter phase might shrink to
a transition line in the bulk limit.
In the standard Heisenberg model (J ′ = 0 and J⊥ = Jz) at T = 0 the dynamics is non-ergodic
for ferromagnetic coupling (J⊥ = Jz < 0) as the system has a L+ 1 degenerate ground state
Γ =
1
12
+
1
6L
. (2.60)
It is clear from (2.60) that Γ remains non-ergodic in the thermodynamic limit as well. This point
(J ′ = 0 and J⊥ = Jz) becomes a line in our phase diagram and is denoted as NE-I (see figure 2).
In fact, the next-nearest-neighbor interaction J ′ may frustrate (J ′ > 0) or favor (J ′ < 0) the
ferromagnetism. In the latter case, the ground state remains unchanged for any value of J ′ < 0.
Therefore, we expect the line denoting the phase NE-I to extend also to negative J ′ as well. If, on
the contrary, J ′ is positive and large enough to frustrate the system in such a way that the ground
state loses its ferromagnetic character, the ergodicity is restored. This occurs at a finite critical
J ′ ∼ 0.25Jz. For values of J ′ larger than the critical one, we find a non-degenerate ground state
with 〈Sz
tot〉 = 0.
If J⊥ 6= Jz the rotational invariance is broken so that states with the same 〈S2
tot〉, but different
〈Sz
tot〉, are not degenerate anymore. In the non-ergodic region (NE-II) of the phase diagram (see
figure 2), the ground state is just doubly degenerate (not L+1 degenerate as in (NE-I)): one ground
state corresponds to a configuration with all spins up and the other corresponds to a configuration
with all spins down. Hence, the value of Γ in this phase is 1/4 and does not depend either on
495
A.Avella, F.Mancini, E.Plekhanov
the Hamiltonian couplings or on the number of sites in the chain. It is clear that this phase also
extends to negative values of J ′. This kind of ground state puts the frustration introduced by
next-nearest-neighbor interaction up to J ′ ∼ 0.3Jz (see figure 2).
The ergodic region (E-I) of the phase diagram (see figure 2) has Γ = 0 for all sizes of the system
and values of the couplings: the unique ground state belongs to the sector with 〈Sz
tot〉 = 0. On the
contrary, the other ergodic phase (E-II) has non-zero values of Γ for values of L not multiples of
four. The ground state in this phase has average total spin equal to one and, therefore, Γ = 1/L2.
We obviously conclude that (E-II) phase is ergodic in the thermodynamic limit.
Figure 3. Finite-size scaling in the case of T = 0 for different points in the phase diagram of
figure 2. Symbols on panel a): + corresponds to (NE-II), N corresponds to (NE-I), � corresponds
to (E-I) and © to (E-II) regions of figure 2, respectively. On panel b) different examples from
(W) region are shown. Hamiltonian couplings are shown in the legend. All energies are expressed
in units of Jz.
The values of Γ in these four phases (NE-I, NE-II, E-I and E-II) exhibit perfect finite-size scaling
as shown in figure 3a). This has allowed us to make definite statements in the thermodynamic limit
as well.
The weird phase (W) (see figure 2) is characterized by a quite strong size dependence, as shown
in figure 3b) where a tentative finite-size scaling of Γ in different points of the phase is presented.
This region manifests a diverging finite-size scaling within the range of sizes that we were able
to handle. In this case, the behavior of Γ as a function of L strongly depends on the particular
choice of the Hamiltonian couplings and is highly non-monotonous when increasing L, according
to the strong dependence on L of 〈Sz
tot〉 in the ground state. In this critical region the eigenvalues
of (2.57) present many-level crossings, which means that the maximum value of L we were able to
reach (Lmax = 26) is not large enough to perform a sensible finite-size scaling analysis. However,
we expect that this phase becomes ergodic in the thermodynamic limit, although still different
from the ergodic phases E–I and E–II.
We can summarize our findings in the thermodynamic limit at zero temperature as follows:
Γ =
1
12 if J⊥ = ±Jz and J ′ . 0.25Jz ,
1
4 if |J⊥| < Jz and J ′ . 0.3Jz ,
??? in the weird phase (W) (see figure 2),
0 otherwise.
(2.61)
496
Ergodicity in strongly correlated systems
3. Conclusions
In conclusion, we have analyzed the issue of ergodicity, after a brief historical overview, within
the Green’s function formalism by means of the equations of motion approach. We have identified
the primary source of non-ergodic dynamics for a generic operator in the appearance of zero-
frequency anomaly in its correlation functions and have given a recipe for computing the unknown
quantities characterizing such a behavior within the Composite Operator Method. Finally, we
have presented examples of non-trivial strongly correlated systems where it is possible to examine
a non-ergodic behavior: two-site Hubbard model, tight-binding model, Heisenberg chain.
References
1. Kubo R., J. Phys. Soc. Jpn., 1957, 12, 570.
2. Falk H., Phys. Rev., 1968, 165, 602.
3. Mancini F., Avella A., Condens. Matter Phys., 2006, 9, No. 3, 569.
4. Mancini F., Avella A., Adv. Phys., 2004, 53, 537.
5. Khintchin A.I. Mathemathical Foundations of Statistical Mechanics. Dover Publ. Inc., New York, 1949.
6. Suzuki M., Physica, 1971, 51, 277.
7. Morita T., Katsura S., J. Phys. C., 1969, 2, 1030.
8. Kwork P.C., Schultz T.D., J. Phys. C, 1969, 2, 1196.
9. Stevens K.W.H., Toombs G.A., Proc. Phys. Soc., 1965, 85, 1307.
10. Callen H., Swendsen R.H., Tahir-Kheli R., Phys. Lett. A, 1967, 25, 505.
11. Fernandez J.F., Gersch H.A., Proc. Phys. Soc., 1967, 91, 505.
12. Ramos J.G., Gomes A.A., Nuovo Cimento A, 1971, 3, 441.
13. Huber D.L., Physica A, 1977, 87, 199.
14. Aksenov V.L., Konvent H., Schreiber J., Phys. Stat. Sol. (b), 1978, 88, K43.
15. Aksenov V.L., Schreiber J., Phys. Lett. A, 1978, 69, 56.
16. Aksenov V.L., Bobeth M., Plakida N.M., Schreiber J., J. Phys. C, 1987, 20, 375.
17. Avella A., Mancini F., Saikawa T., Eur. Phys. J. B, 2003, 36, 445.
18. Bak M., Avella A., Mancini F., Phys. Stat. Sol. (b), 2003, 236, 396.
19. Plekhanof E., Avella A., Mancini F., cond–mat/0606152;
Plekhanof E., Avella A., Mancini F., Phys. Rev. B (in press).
Ергодичнiсть в сильноскорельованих системах
А.Авелла, Ф.Манчiнi, Є.Плеханов
Фiзичний факультет iм. Е.Р.Каiанiелло, Унiверситет Салерно, I–84081 Баронiссi, Салерно, Iталiя
Отримано 8 червня 2006 р., в остаточному виглядi – 22 червня 2006 р.
Представлено короткий але систематичний розгляд проблеми ергодичностi в сильноскорельованих
системах. Пiсля короткого iсторичного огляду ми аналiзуємо це питання в рамках формалiзму фун-
кцiй Грiна за допомогою методу рiвнянь руху. За допомогою цього аналiзу ми можемо видiлити пер-
шоджерела неергодичної динамiки оператора а також дати спосiб розрахунку невiдомих величин,
що характеризують таку поведiнку, в рамках методу композитних операторiв. Також представлено
приклади нетривiальних сильноскорельованих систем де можлива неергодична поведiнка.
Ключовi слова: ергодичнiсть, сильноскорельованi системи, формалiзм функцiй Грiна, метод
рiвнянь руху, метод композитних операторiв
PACS: 05.30.Jp, 75.10.Jm, 71.10.-w
497
498
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