Pseudospin-electron model in the self-consistent gaussian fluctuation approximation
Pseudospin-electron model with an effective many-body interaction between pseudospins via conducting electrons is investigated within the generalized random phase approximation scheme with the self-consistent inclusion of mean field type contributions (coming from the effective pseudospin intera...
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irk-123456789-1197622017-06-09T03:04:38Z Pseudospin-electron model in the self-consistent gaussian fluctuation approximation Stasyuk, I.V. Tabunshchyk, K.V. Pseudospin-electron model with an effective many-body interaction between pseudospins via conducting electrons is investigated within the generalized random phase approximation scheme with the self-consistent inclusion of mean field type contributions (coming from the effective pseudospin interaction) as well as gaussian fluctuations of the mean field (which makes it possible to obtain more accurate results in the vicinity of the critical points). Using the approach proposed here, the expressions are obtained for the pseudospin correlation function, for pseudospin mean value, as well as for the grand canonical potential. В роботі пропонується аналітична самоузгоджена схема розрахунку термодинамічних і кореляційних функцій у псевдоспін-електронній моделі при відсутності кореляцій. Отримано аналітичні вирази для псевдоспінової кореляційної функції, середнього значення оператора псевдоспіну та термодинамічний потенціал в узагальненому наближенні хаотичних фаз при врахуванні поправок типу середнього поля, що виникають внаслідок ефективної взаємодії псевдоспінів через електрони провідності, а також при врахуванні гаусових флуктуацій середнього поля. 2001 Article Pseudospin-electron model in the self-consistent gaussian fluctuation approximation / I.V. Stasyuk, K.V. Tabunshchyk // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 109-118. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.38.+i, 77.80.Bh, 63.20.Ry DOI:10.5488/CMP.4.1.109 http://dspace.nbuv.gov.ua/handle/123456789/119762 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
Pseudospin-electron model with an effective many-body interaction between
pseudospins via conducting electrons is investigated within the generalized
random phase approximation scheme with the self-consistent inclusion
of mean field type contributions (coming from the effective pseudospin
interaction) as well as gaussian fluctuations of the mean field (which
makes it possible to obtain more accurate results in the vicinity of the critical
points). Using the approach proposed here, the expressions are obtained
for the pseudospin correlation function, for pseudospin mean value, as well
as for the grand canonical potential. |
| format |
Article |
| author |
Stasyuk, I.V. Tabunshchyk, K.V. |
| spellingShingle |
Stasyuk, I.V. Tabunshchyk, K.V. Pseudospin-electron model in the self-consistent gaussian fluctuation approximation Condensed Matter Physics |
| author_facet |
Stasyuk, I.V. Tabunshchyk, K.V. |
| author_sort |
Stasyuk, I.V. |
| title |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| title_short |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| title_full |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| title_fullStr |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| title_full_unstemmed |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| title_sort |
pseudospin-electron model in the self-consistent gaussian fluctuation approximation |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119762 |
| citation_txt |
Pseudospin-electron model in the self-consistent gaussian fluctuation approximation / I.V. Stasyuk, K.V. Tabunshchyk // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 109-118. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyukiv pseudospinelectronmodelintheselfconsistentgaussianfluctuationapproximation AT tabunshchykkv pseudospinelectronmodelintheselfconsistentgaussianfluctuationapproximation |
| first_indexed |
2025-07-08T16:33:17Z |
| last_indexed |
2025-07-08T16:33:17Z |
| _version_ |
1837097181515874304 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 109–118
Pseudospin-electron model in the
self-consistent gaussian fluctuation
approximation
I.V.Stasyuk, K.V.Tabunshchyk
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received October 24, 2000
Pseudospin-electron model with an effective many-body interaction be-
tween pseudospins via conducting electrons is investigated within the gen-
eralized random phase approximation scheme with the self-consistent in-
clusion of mean field type contributions (coming from the effective pseu-
dospin interaction) as well as gaussian fluctuations of the mean field (which
makes it possible to obtain more accurate results in the vicinity of the critical
points). Using the approach proposed here, the expressions are obtained
for the pseudospin correlation function, for pseudospin mean value, as well
as for the grand canonical potential.
Key words: pseudospin-electron model, local anharmonicity, gaussian
fluctuations
PACS: 71.10.Fd, 71.38.+i, 77.80.Bh, 63.20.Ry
1. Introduction
Pseudospin-electron model (PEM) considered herein was proposed to describe
the interaction of the conducting electrons in metals or in semimetals with some two
level subsystem represented by pseudospins (e.g. anharmonic vibrations of the apex
oxygen ions in YBaCuO-type crystals [1]), as well as the proton-electron interaction
in the molecular and crystalline systems with hydrogen bonds [2].
The corresponding Hamiltonian has the following form
H = H0 +
∑
ijσ
tijc
+
iσcjσ, H0 =
∑
i
{
−µ
∑
σ
niσ + g
∑
σ
niσS
z
i − hSz
i
}
, (1)
and includes the terms describing electron transfer ( ∼ tij), energy of the subsystem
of pseudospins placed in longitudinal field ( ∼ h) and local interaction ( ∼ g) of
conducting electrons with pseudospins.
c© I.V.Stasyuk, K.V.Tabunshchyk 109
I.V.Stasyuk, K.V.Tabunshchyk
Hamiltonian (1) can be transformed, after some simplification, into the Hamil-
tonian of the electron subsystem of binary alloy type model as well as the Falicov-
Kimball model. The main difference between these models lies in the way how an av-
eraging procedure is performed (thermal statistical averaging in the case of PEM and
Falicov-Kimball model, configurational averaging for binary alloy) [3,4]. This Hamil-
tonian is also invariant with respect to the transformation µ → −µ, h → 2g − h,
n → 2−n, Sz → −Sz. It allows us to use (1) for the description of a hole-pseudospin
system as well.
In the previous papers [5–7], a self-consistent scheme was proposed for the calcu-
lation of mean values of pseudospin and electron number operators, grand canonical
potential as well as correlation functions of simplified PEM. The main idea of the
approach was based on the GRPA scheme [8] with the inclusion of the mean field
type contributions coming from the effective pseudospin interactions via conducting
electrons [5]. On the basis of this self-consistent mean field type approximation, the
energy spectrum, thermodynamics of phase transitions, the possibility of phase sep-
arations as well as the appearance of the chess-board phase were investigated. It was
shown that: in the µ = const regime (when it is supposed that the electron states of
the other structure elements, which are not included explicitly into the PEM, play
a role of a thermostat, that ensures a fixed value of the chemical potential µ), the
interaction between the electron and pseudospin subsystems leads to the possibility
of either first or second order phase transitions between different uniform phases
(bistability effect) as well as between the uniform and the chess-board ones [5,7];
in the regime n = const (this situation is more customary at the consideration of
electron systems and means that the chemical potential being the function of T , h
etc. depends now on the electron concentration), an instability with respect to the
phase separation in the electron and pseudospin subsystems can take place [5,7].
An approach, that takes into account only mean field type contributions, is
reasonable only when deviations from the average values are small or, in other
words, in the area where the effects of fluctuations are unimportant. Therefore, the
previously proposed method [5,7] makes it possible to obtain accurate results outside
the region of the critical point.
In the vicinity of the critical point the effects of the mean field fluctuations
become significant. Hence, to improve the description of the thermodynamics of our
system we should correct the approach by taking into account the contribution of
the fluctuations of the self-consistent field of pseudospins.
For this purpose, we construct a consistent scheme (using the diagram method),
to calculate the pseudospin operator mean value, the grand canonical potential as
well as the pseudospin correlation function of simplified PEM, which allows us to
take into account the gaussian fluctuations of the self-consistent mean field. Root-
mean-square (r.m.s.) fluctuations of the field are calculated in a self-consistent way.
Within the high density expansion method the r.m.s. of gaussian fluctuations of the
molecular field were previously considered in the case of spin models [9,10]. In the
present paper we generalize this scheme on the pseudospin-electron model (where
we have an effective many–body interaction between pseudospins via conducting
110
PEM in the self-consistent gaussian fluctuation approximation
electrons). Using the the scheme proposed here, the expression for the grand canon-
ical potential as well as a set of equations for the pseudospin mean value and r.m.s.
fluctuations parameter are obtained.
2. Mean field approximation
We perform the calculations in the strong coupling case (g ≫ t) using single-
site states as the basic ones. We rewrite the initial Hamiltonian of the simplified
PEM in the second quantized form using projecting electron annihilation (creation)
operators [5–7] aiσ = ciσP
+
i , ãiσ = ciσP
−
i (P±
i = 1
2
± Sz
i ) acting at a site with the
certain pseudospin orientation:
H = H0+Hint =
∑
iσ
{εniσ+ε̃ñiσ−
h
2
Sz
i }+
∑
ijσ
tij(a
+
iσajσ+a+iσãjσ+ã+iσajσ+ã+iσãjσ), (2)
where ε = −µ + g/2, ε̃ = −µ− g/2 are the energies of the single–site states.
Expansion of the calculated quantities in terms of the electron transfer leads to
an infinite series of terms containing the averages of the T -products of the aiσ, ãiσ
operators. The evaluation of such averages is made using the corresponding Wick’s
theorem. The results are expressed in terms of the products of the nonperturbed
Green’s functions and the averages of the products of projection operators P±
i which
are calculated by means of the semi-invariant expansion [5].
From such an infinite series we are summing up a certain partial sum of dia-
grams (in the spirit of the traditional mean field approach [5]) characterized by the
inclusion, in all basic semi-invariants, of the mean field type contributions (i.e., loop
fragments) coming from the effective many–body interaction between pseudospins
via conducting electrons.
The corresponding diagram series for pseudospin mean value has the form:
〈Sz〉 = = − −+ _1
2!
... . (3)
Here we use the following diagram notations: − Sz, − gi(ωn), wavy
line is the Fourier transform of the hopping parameter tk. Basic semi-invariants are
represented by ovals and contain the δ-symbols on site indexes.
= 〈Sz〉0 = b(h) =
Sp(Sze−βH0)
Sp(e−βH0)
, = 〈SzSz〉c0 =
∂b(h)
∂βh
. (4)
Nonperturbed electron Green’s function is equal to
= g(ωn) = 〈gi(ωn)〉, gi(ωn) =
P+
i
iωn − ε
+
P−
i
iωn − ε̃
. (5)
The full single-electron Green’s function is
Gk(ωn) = += = (g−1(ωn)− tk)
−1. (6)
111
I.V.Stasyuk, K.V.Tabunshchyk
It determines the electron spectrum.
εI,II(tk) =
1
2
(2E0 + tk)±
1
2
√
g2 + 4tk〈Sz〉g + t2k . (7)
This spectrum was investigated in detail in [5]
Expression for the loop fragment of diagram has the following form
=
2
N
∑
n,k
t2k
g−1(ωn)−tk
(
P+
i
iωn − ε
+
P−
i
iωn − ε̃
)
= β(α1P
+
i + α2P
−
i ). (8)
It creates an internal effective self–consistent field acting on the pseudospin.
Now, we can introduce the following mean field Hamiltonian:
HMF=
∑
i
{ε(ni↑+ni↓) + ε̃(ñi↑+ñi↓)− ySz
i }, (9)
where y = h + α2 − α1 is a mentioned effective field. Summation of the diagram
series (3) is equivalent to the averaging with the Hamiltonian (9). The result can be
expressed in the form
〈Sz
l 〉 = b(y), (10)
b(y) =
1
2
tanh
{
β
2
y + ln
1 + e−βε
1 + e−βε̃
}
. (11)
Diagram equation for the pseudospin correlation functions 〈S zSz〉q (obtained
within the framework of GRPA with the insertion of the mean field type contribu-
tions into all zero-order semi-invariants) is as follows [5]:
−= (12)
This equation differs from the one for the Ising model in RPA by the replacement of
the exchange interaction by the electron loop Π
q
= , which describes
an effective many–body interaction between pseudospins via conducting electrons:
Π
q
=
2
N
∑
n,k
Λ2
nt̃n(k)t̃n(k+q), Λn=
g
(iωn+µ)2−g2/4
, t̃n(k)=
tk
(1− gntk)
. (13)
The first term in (12) is the second-order semi-invariants renormalized due to the
inclusion of “single-tail” parts, and is thus calculated by means of HMF:
−= + 1
2!
− −... (14)
112
PEM in the self-consistent gaussian fluctuation approximation
Finally, the solution of the equation (12) in the analytical form is:
〈SzSz〉q =
1/4− 〈Sz〉2
1 + 2
N
∑
n,k
Λ2
nt̃n(k)t̃n(k + q)(1/4− 〈Sz〉2) , (15)
and is different from zero only in a static case (ωn = 0) (this is due to the fact that
the pseudospin operator commutes with the Hamiltonian).
In the same approximation, the grand canonical potential has the diagrammatic
representation:
β∆ΩMFA =
+ − −1
2!
_ − ... 1
3!+
+ ...+1
2
_ 1
3
_ − (16)
The corresponding analytical expression is:
ΩMFA = − 2
Nβ
∑
n,k
ln(1−tkg(ωn))−
2
Nβ
∑
n,k
g(ωn))t
2
k
g−1(ωn))− tk
− 1
β
ln Sp(e−βHMF). (17)
All quantities can be derived from the grand canonical potential by differentiating
dΩMFA
d(−h)
= 〈Sz〉, d〈Sz〉
d(βh)
= 〈SzSz〉q=0, (18)
that shows the thermodynamical consistence of the proposed approximation [5].
3. Self–consistent gaussian fluctuation approximation
In constructing a higher order approximation, we use MFA as the zero–order
one. This means that all “single-tail” parts of diagrams are already summed up and
all semi-invariant are calculated using the distribution with the Hamiltonian HMF
(9). We represent this graphically by thick ovals:
− ...+ _1
2!= − (19)
As the simplest approximation that goes beyond the MFA, we use the approach
that takes into account the so-called “double-tail” diagrams (such an approxima-
tion was used in [9,10] where the magnetization of the ordinary Ising model was
considered). The corresponding diagram series for the pseudospin mean value is:
〈Sz〉 = + _1
2!= 1
2
_ + 1
2
_
2 +... 1
1!
_ (20)
113
I.V.Stasyuk, K.V.Tabunshchyk
The diagram equation for pseudospin correlator 〈S zSz〉q within the approximation
developed here is given by (12), but now zero-order correlators are renormalized,
also, due to the “double-tail” parts, and thus the corresponding diagram series is:
Ξ = +...= + 1
2
_ 1
1!
_
2
_1
2!
+ 1
2
_ (21)
The contribution, which correspond to the “double-tail” fragment of the diagram,
can be written in the following analytical form (using the notation (13)):
X = =
22
N3
∑
n,n′
∑
k,k′
∑
q
Λ2
nt̃n(k)t̃n(k−q)〈SzSz〉qΛ2
n′ t̃n′(k′)t̃n′(k′+q), (22)
〈SzSz〉q =
Ξ
1 + 2
N
∑
n,k
Λ2
nt̃n(k)t̃n(k+q)Ξ
. (23)
Since the pseudospin correlator (23) is frequency independent, in the expression (22)
we have two independent sums over internal Matsubara frequencies that allows one
(using decomposition into simple fractions) to sum over all internal frequencies:
2
N
∑
n,k,k′
Λ2
nt̃n(k)t̃n(k
′) =
2β
N
∑
k,k′
tktk′(ε− ε̃)2
[εI(tk)− εII(tk)][εI(tk′)− εII(tk′)]
×
{
n[εI(tk)]− n[εI(tk′)]
εI(tk)− εI(tk′)
+
n[εII(tk)]− n[εII(tk′)]
εII(tk)− εII(tk′)
− n[εI(tk)]− n[εII(tk′)]
εI(tk)− εII(tk′)
− n[εII(tk)]− n[εI(tk′)]
εII(tk)− εI(tk′)
}
.(24)
Here n(ε) = 1
1 + eβε
is a Fermi distribution.
Let us now return to the problem of summation of the diagram series (20) and
(21). By means of the procedure described in [10], and using notations (11), (22) we
can write:
〈Sz〉 = b(y) +
1
1!
b(y)[2]
X
2
+
1
2!
b(y)[4]
(X
2
)2
+
1
3!
b(y)[6]
(X
2
)3
+ · · ·
=
1√
2πX
+∞
∫
−∞
exp
(
− ξ2
2X
)
b(y + ξ)dξ, (25)
Ξ = b(y)[1] +
1
1!
b(y)[3]
X
2
+
1
2!
b(y)[5]
(X
2
)2
+
1
3!
b(y)[7]
(X
2
)3
+ · · ·
=
1
X
√
2πX
+∞
∫
−∞
exp
(
− ξ2
2X
)
ξb(y + ξ)dξ. (26)
Therefore, the contribution of diagram series with “double-tail” parts corresponds
to the average with the Gaussian distribution where X can be interpreted as the
114
PEM in the self-consistent gaussian fluctuation approximation
root-mean-square (r.m.s.) fluctuation of the mean field around the mean value of
y. Thus we obtain a self-consistent set of equations (25), (22) for pseudospin mean
value and r.m.s. fluctuations parameter.
The diagram series for the grand canonical potential within the approximation
accepted here is:
β∆Ω = β∆ΩMFA + 1
2
_ { 1
3
_− +...1
2
_ {
− 1
2
_
− −1
2
_ 1
1!
_ _1
2!
1
2
_
2 −...
(27)
The grand canonical potential written in this form satisfies the stationary conditions:
dΩ
d〈Sz〉 = 0,
dΩ
dX
= 0, (28)
which are equivalent to the equations (25), (22). The consistency of the approxi-
mations used for the pseudospin mean value, pseudospin correlation function and
grand canonical potential can be checked explicitly using the relations:
dΩ
d(−h)
= 〈Sz〉, d〈Sz〉
d(−hβ)
∣
∣
∣
X=const
= 〈SzSz〉q=0. (29)
In the limit of vanishing fluctuations, our results go over into the ones obtained
within the mean field approximation.
In the analytical form, the first term in the diagram series for the grand canonical
potential (27) is equal to
1
2
ΞX =
1
2
1√
2πX
+∞
∫
−∞
exp
(
− ξ2
2X
)
ξb(y + ξ)dξ. (30)
The bracketed diagram series can be presented as follows:
−1
2
{
1
2
Ξ2
( 2
N
∑
n,k
Λ2
nt̃n(k)
2
)2
− 1
3
Ξ3
( 2
N
∑
n,k
Λ2
nt̃n(k)
2
)3
+ · · ·
}
=
= −1
2
{
Ξ
2
N
∑
n,k
Λ2
nt̃n(k)
2 − ln
(
1 + Ξ
2
N
∑
n,k
Λ2
nt̃n(k)
2
)
}
. (31)
The remainder of the diagram in the series (27) can be written as
−
{
b(y)[1]
X
2
+
1
2!
b(y)[3]
(X
2
)2
+ · · ·
}
=−1
2
+∞
∫
−∞
{
1−erf
( |ξ|√
2X
)}
sign(ξ)b(y+ξ)dξ,
115
I.V.Stasyuk, K.V.Tabunshchyk
where we use the relation (26) and definition of the erf function. Finally, the diagram
series for a grand canonical potential Ω can be written in the following analytical
form:
Ω = ΩMFA+
1
2
1√
2πX
+∞
∫
−∞
e−
ξ2
2X ξb(y+ξ)dξ − 1
2
+∞
∫
−∞
{
1−erf
( |ξ|√
2X
)}
sign(ξ)b(y+ξ)dξ
− 1
2
Ξ
2
N
∑
n,k
Λ2
nt̃n(k)
2 +
1
2
ln
(
1 + Ξ
2
N
∑
n,k
Λ2
nt̃n(k)
2
)
. (32)
To sum over the Matsubara frequency, the relation (24) should be used in the limit
of k′ → k.
4. Conclusion
A consistent method is presented, which takes into account the corrections due
to the gaussian fluctuation of self-consistent field, in order to calculate thermody-
namic and correlation functions of the pseudospin-electron model with an effective
interaction. The diagram series and corresponding formulae are obtained for the
pseudospin mean value 〈Sz〉, for a pseudospin correlation function 〈S zSz〉q as well
as for the grand canonical potential Ω. The possibility exists to investigate their
behaviour under the changes of thermodynamical and model parameters. The pa-
rameter X (22) (r.m.s. fluctuation of the mean field) is calculated by means of the
self-consistent renormalization of the correlation function (12), (21). From the ex-
pression for a grand canonical potential, calculated within the scheme presented
here, the equations for 〈Sz〉 and X parameters satisfying the stationary conditions
(28) are obtained.
This approach can be used to investigate the role of gaussian fluctuations in the
phase separation phenomena as well as in the transition into a modulated phase in
PEM.
It should be noted that the analytical scheme presented in our paper can be easily
reduced to the Onyszkiewicz type approach (which was successfully used in the case
of spin models with direct spin interaction [11,12]) where the renormalization (20)
is performed with the use of the simplest possible pseudospin correlation function
involving gaussian fluctuations of the mean field:
+...= + 1
2
_ 1
1!
_
2
_1
2!
+ 1
2
_
(33)
Within the framework of this approximation, grand canonical potential satisfies the
stationary conditions (28) and can be written as:
Ω = ΩMFA +
1
4
ΞX − 1
2
+∞
∫
−∞
{
1− erf
( |ξ|√
2X
)}
sign(ξ)b(y + ξ)dξ, (34)
116
PEM in the self-consistent gaussian fluctuation approximation
X = Ξ
(
2
N
∑
q,n
Λ2
nt̃
2
n(k)
)2
. (35)
This method is more suitable for numerical calculations than the above presented
but it takes into account the restricted class of diagrams.
References
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117
I.V.Stasyuk, K.V.Tabunshchyk
Самоузгоджене врахування гаусових флуктуацій у
псевдоспін-електронній моделі
І.В.Стасюк, К.В.Табунщик
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 24 жовтня 2000 р.
В роботі пропонується аналітична самоузгоджена схема розрахунку
термодинамічних і кореляційних функцій у псевдоспін-електронній
моделі при відсутності кореляцій. Отримано аналітичні вирази для
псевдоспінової кореляційної функції, середнього значення операто-
ра псевдоспіну та термодинамічний потенціал в узагальненому на-
ближенні хаотичних фаз при врахуванні поправок типу середнього
поля, що виникають внаслідок ефективної взаємодії псевдоспінів че-
рез електрони провідності, а також при врахуванні гаусових флукту-
ацій середнього поля.
Ключові слова: псевдоспін-електронна модель, локальний
ангармонізм, гаусові флуктуації
PACS: 71.10.Fd, 71.38.+i, 77.80.Bh, 63.20.Ry
118
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