Phase transitions in pseudospin- electron model with direct interaction between pseudospins
The analysis of thermodynamic properties of the pseudospin-electron model in the case of zero electron transfer with the inclusion of the direct pseudospin-pseudospin interaction of ferroelectric type is performed. The equilibrium conditions in the regimes µ = const and n = const are investigate...
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Інститут фізики конденсованих систем НАН України
1999
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| Цитувати: | Phase transitions in pseudospin- electron model with direct interaction between pseudospins / I.V. Stasyuk, Yu. Havrylyuk // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 487-494. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1205432017-06-13T03:05:12Z Phase transitions in pseudospin- electron model with direct interaction between pseudospins Stasyuk, I.V. Havrylyuk, Yu. The analysis of thermodynamic properties of the pseudospin-electron model in the case of zero electron transfer with the inclusion of the direct pseudospin-pseudospin interaction of ferroelectric type is performed. The equilibrium conditions in the regimes µ = const and n = const are investigated in the mean field approximation. It is shown that the interaction with electrons leads at the fixed µ value to the possibility of the first order phase transition at the change of temperature with a jump-like behaviour of 〈Sz〉. In the regime n = const there takes place an instability with respect to phase separation in the wide range of asymmetry parameter h values. Проведено аналіз термодинамічних властивостей псевдоспін-електронної моделі у випадку відсутності електронного переносу з включенням прямої взаємодії між псевдоспінами сегнетоелектричного типу. У наближенні середнього поля досліджуються умови рівноваги в режимах µ = const та n = const. Показано, що взаємодія з електронами при фіксованому значенні µ приводить до можливості фазового переходу першого роду при зміні температури зі стрибкоподібною зміною〈Sz〉. У режимі n = const має місце нестабільність щодо розділення фаз у широкому інтервалі значень параметра асиметрії h. 1999 Article Phase transitions in pseudospin- electron model with direct interaction between pseudospins / I.V. Stasyuk, Yu. Havrylyuk // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 487-494. — Бібліогр.: 9 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.487 PACS: 74.65.+n, 71.45.Gm http://dspace.nbuv.gov.ua/handle/123456789/120543 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
The analysis of thermodynamic properties of the pseudospin-electron model in the case of zero electron transfer with the inclusion of the direct
pseudospin-pseudospin interaction of ferroelectric type is performed. The
equilibrium conditions in the regimes µ = const and n = const are investigated in the mean field approximation. It is shown that the interaction with
electrons leads at the fixed µ value to the possibility of the first order phase
transition at the change of temperature with a jump-like behaviour of 〈Sz〉.
In the regime n = const there takes place an instability with respect to
phase separation in the wide range of asymmetry parameter h values. |
| format |
Article |
| author |
Stasyuk, I.V. Havrylyuk, Yu. |
| spellingShingle |
Stasyuk, I.V. Havrylyuk, Yu. Phase transitions in pseudospin- electron model with direct interaction between pseudospins Condensed Matter Physics |
| author_facet |
Stasyuk, I.V. Havrylyuk, Yu. |
| author_sort |
Stasyuk, I.V. |
| title |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| title_short |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| title_full |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| title_fullStr |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| title_full_unstemmed |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| title_sort |
phase transitions in pseudospin- electron model with direct interaction between pseudospins |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120543 |
| citation_txt |
Phase transitions in pseudospin- electron model with direct interaction between pseudospins / I.V. Stasyuk, Yu. Havrylyuk // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 487-494. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyukiv phasetransitionsinpseudospinelectronmodelwithdirectinteractionbetweenpseudospins AT havrylyukyu phasetransitionsinpseudospinelectronmodelwithdirectinteractionbetweenpseudospins |
| first_indexed |
2025-07-08T18:06:23Z |
| last_indexed |
2025-07-08T18:06:23Z |
| _version_ |
1837103043782377472 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 487–494
Phase transitions in pseudospin-
electron model with direct interaction
between pseudospins
I.V.Stasyuk, Yu.Havrylyuk
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received October 12, 1998
The analysis of thermodynamic properties of the pseudospin-electron mo-
del in the case of zero electron transfer with the inclusion of the direct
pseudospin-pseudospin interaction of ferroelectric type is performed. The
equilibrium conditions in the regimes µ = const and n = const are investi-
gated in the mean field approximation. It is shown that the interaction with
electrons leads at the fixed µ value to the possibility of the first order phase
transition at the change of temperature with a jump-like behaviour of 〈Sz〉 .
In the regime n = const there takes place an instability with respect to
phase separation in the wide range of asymmetry parameter h values.
Key words: pseudospin-electron model, phase transitions, phase
separation
PACS: 74.65.+n, 71.45.Gm
1. Introduction
Pseudospin-electron model (so called Muller model [1]) is the one of theoretical
models which were proposed in connection with the investigation of characteristic
features of electron spectrum and lattice dynamics in high temperature superconduc-
tors. In this model strong Hubbard-type electron correlations are taken into account.
Pseudospin formalism is used to describe the locally anharmonic lattice vibrations.
Hamiltonian of the model is of the following form [2,3]
H =
∑
i
Hi +
∑
ijσ
tija
+
iσaiσ −
1
2
∑
ij
JijS
z
i S
z
j , (1)
Hi = Uni↑ni↓ − µ
∑
σ
niσ + g
∑
σ
niσS
z
i + ΩSx
i − hSz
i , (2)
c© I.V.Stasyuk, Yu.Havrylyuk 487
I.V.Stasyuk, Yu.Havrylyuk
where in the single site part in addition to Hubbard correlation U there are terms
that describe tunnelling splitting and asymmetry of local anharmonic potential (lon-
gitudinal field h). Hamiltonian (1) also contains terms, which describe electron trans-
fer tij and direct interaction between pseudospins Jij . The energy is accounted from
the level of chemical potential .
Most attention while investigating this model has been paid to examining the
electron states, an effective electron-electron interaction, to studying the dielectric
instabilities or ferroelectric type anomalies as well as the tendency to the spatially
modulated charge and pseudospin orderings [4,5].
This work is devoted to the study of thermodynamics of the pseudospin-electron
model in the case of zero electron transfer (tij = 0) and zero frequency of tunnelling
splitting (Ω = 0). The direct interaction between pseudospins is supposed to be a
long-ranged (Jij ∼ J/N) that makes it possible to use the mean field approximation
(MFA).
In this approximation, the Hamiltonian of the model is of the form
H =
∑
i
H̃i +
N
2
Jη2;
H̃i = −µ
∑
σ
niσ + Uni↑ni↓ + g
∑
σ
niσS
z
i − (h + Jη)Sz
i .
(3)
The interaction Jij is taken because the ferroelectric type interaction (J (~q) has
got a maximum value at ~q = 0; J(0) ≡ J > 0); the order parameter η = 〈S z
i 〉 does
not depend on the unit cell index.
We investigate the possible states and phases of the system as well as the tran-
sitions between them at the change of temperature and model parameters values.
2. Mean field approximation; Ω = 0
Thermodynamic potential of the model, calculated per one lattice site is equal
in MFA to
Ω
N
= −θ lnZi +
1
2
Jη2,
Zi = 2
[
cosh βH+e−β(U−2µ) cosh β(H − g)+2eβµ cosh β
(
H−
g
2
)]
. (4)
To investigate the equilibrium conditions we separate two regimes: µ =const
and n =const. The equilibrium for these regimes is defined by the minimum of
thermodynamical potential ((∂Ω/∂η)T,µ,h = 0) or the minimum of free energy F =
Ω+ µN ((∂F/∂η)T,µ,h = 0) respectively.
The equation for the order parameter is obtained from these conditions in the
form
η =
1
2
sinhβH + e−β(U−2µ) sinhβ(H − g) + 2eβµ sinh β(H − g
2
)
cosh βH + e−β(U−2µ) cosh β(H − g) + 2eβµ cosh β
(
H − g
2
) . (5)
488
Phase transitions in pseudospin-electron model
The average number of electrons n =
〈
1
N
∑
iσ
niσ
〉
is determined as follows:
n = −
1
N
(
∂Ω
∂µ
)
T,h
=
4
Zi
[
e−β(U−2µ) cosh β(H − g) + eβµ cosh β(H −
g
2
)
]
. (6)
The equation (5) and expression (6) form a set of equations for the parameter η
and for the chemical potential.
After eliminating the chemical potential, the equation for the order parameter
transforms to the following form:
η =
1
2
(
n
2
−
δ
2
)
thβ(H − g) +
1
2
δthβ
(
H −
g
2
)
+
1
2
(
2− n
2
−
δ
2
)
thβH, (7)
where
δ =
(2− n)ay
b+ ay
=
√
(1− n)2 + n(2− n)e−βU b
a2
− 1
e−βU b
a2
− 1
, (8)
and the b/a2 ratio, is given by the expression
b
a2
=
cosh βg + cosh β(2H − g)
1 + cosh β(2H − g)
. (9)
The equations (5) and (7) determine the values of η parameter, which correspond
to the extremuma of thermodynamical potential Ω and free energy F respectively.
From the set of all possible roots of it, it is necessary to take into consideration only
those which provide the minimum values of Ω or F .
3. The equation of state η = η(h) and phase diagrams at
T = 0 (regime n = const )
The equation for the parameter η, obtained in the regime of fixed number of
electrons, can be solved analytically in the limit T = 0.
Typical example of η(h) dependence is shown in figure 1, which corresponds
to the n value in the interval 0 6 n 6 1. At some regions of values of h, where
η(h) function possesses S-like behaviour and has got three or more values, the first
order phase transitions with the jumps of parameter take place at the change of h.
Phases η = −1
2
(phase 1), η = 1
2
− n (phase 2), η = 1
2
− n
2
(phase 3), η = 1
2
(phase 4)
occur between phase transition points and outside them. At the change of parameter
values the regions, where metastable phases occur, can overlap. The disappearing of
some phase transitions takes place and therefore some intermediate phases are not
realizable. In case 1 6 n 6 2 the dependence of η(h) is generally similar. The phase
3 and phase 2’ at η = 3/2− n, which now appears instead of phase 2, may play the
role of intermediate phases.
489
I.V.Stasyuk, Yu.Havrylyuk
η
Figure 1. The dependence of η parameter on h field at T = 0 (in the case g > J
2 ;
J
2 − nJ
4 < U < g − nJ
4 ; 0 < n < 1)
The analysis shows that the states described by 1̄ (z1 6 h 6 z2), 2̄ (z3 6 h 6 z4),
3̄ (z5 6 h 6 z6) -lines are unstable while the states 1 . . . 4 correspond to the minimum
values of the free energy. The intersections of F1, F2, F3, F4 lines determine the points
of the first order phase transition between relevant phases. Some of these phase
transitions will not be realizable, if the relevant points of crossing lie above some
other Fk-line.
The values of field h = hik, at which phase transitions i ↔ k occur, are the
following:
h12 =
nJ
2
; h23 = U − J
2
+ 3n
4
J ;
h34 = 2g − U − J
2
+ Jn
4
; h13 =
nJ
4
+ n
2−n
U ;
h24 = g − 1−n
2
J ; h14 = ng
(10)
at 0 6 n 6 1, and
h12 = g − 1−n
2
J ; h23 = 2g − U − J + 3n
4
J ;
h13 = U + Jn
4
; h12 = 2g − J
2
+ Jn
4
− 2−n
n
U ;
h2′4 = 2g − J + Jn
2
J ; h14 = ng
(11)
at 1 6 n 6 2.
Some examples of phase diagrams (n, h) where the areas of occurrence of the
phases 1 . . . 4 at T = 0 are shown, are presented in figure 21. Their emerging de-
pends on the values of U and g constants. In addition, they show the possibility of
transforming the phases at the change of electron concentration n. To study this
problem more attentively, we investigated the dependence µ(n) described by the
equation (6).
It is known that the dependence of particle concentration on chemical potential is
one of the factors that determines thermodynamical equilibrium of the system. The
state with homogenous distribution of particles is unstable at
(
∂n
∂µ
)
T
< 0. Hence,
the phase separation into the regions with different concentrations takes place.
Let’s take, for instance, the case U > g, 0 < h < g; the corresponding phase
diagram in plane (n, h) is shown on figure 2a. One can see that in phases 1 and 4
1The (n, h) and (U, h) phase diagrams for all possible cases are given in [6]
490
Phase transitions in pseudospin-electron model
a)
b)
2U-g+J/23g-2U-J/2
4/J(g-U)
2’
2
3
1
n
h
4
1
2
J/2
U+J/2
U-J/2 2gg 2g-J/2
2g-U+J/2
2g-U-J/2
2-4/J(g-U)
3
Figure 2. (n, h) - phase diagrams at T = 0; a) U > g; b) g − J
4 < U < g;
the chemical potential increases (or remains constant) with the increase of n, while
within the phase 2 there is a possibility of descending dependence of µ on n. That
-g/2
U-g/2
n
0 21
1
2
1
µ
Figure 3. The dependence of chem-
ical potential on the electron con-
centration n (T = 0). U > g;
g > J/2; 0 < h < J/2.
means the instability of homogenous state of
the system. In the case illustrated on figure 3
the phase 2 splits at T = 0 into phase 4 (n =
0, η = 1
2
) and into phase 1 (n = 1, η− 1
2
) with
weight coefficients 1− n and n respectively.
Phase separated states are bordered by
binodal lines. For example, at T = 0 and
U > g they are:
n = 0, (0 < h < g)
n = 1, (0 < h < 2g)
n = 2, (g < h < 2g).
(12)
Such boundaries surround the regions of
occurring of the intermediate phases 2, 2’,
and 3. The mentioned phases might be stable
only if there were some factors that maintain
the space homogeneity of electron concentra-
tion.
491
I.V.Stasyuk, Yu.Havrylyuk
Figure 4. The n dependence of chemical potential µ. The parameter values:
g/J = 1.0; U/J = 1; h/J = 0.25; [1 – T/J = 0.25, 2 – T/J = 0.14, 3 –
T/J = 0.03, 4 – T/J = 0.24, 5 – T/J = 0.26]. µ is given in dimensionless units:
µ → µ/J .
4. Phase transition at µ = const
The dependence of the order parameter on field h and temperature θ, at the
constant value of chemical potential, is determined according to equation (5).
Here we consider the case of non-zero temperature. Among all possible solutions
η = f(θ, h, µ) of equation (5), let us separate the zero ones. In (θ, h)-plane they
define the curves which are described by an equation
sinh
(
βh
2
)
+ e−β(U−2µ) sinh β
(
h
2
− g
)
+ 2eβµ sinh β
(
h
2
−
g
2
)
= 0. (13)
From this equation one can get
h = θ ln
ξ1
ξ2
, (14)
where
ξ1 = 1 + e−β(U−g−2µ) + 2eβ(
g
2
+µ),
ξ2 = ξ1|g→−g .
(15)
After substituting this expression for field h into equation (10) we get an equation
η = 1
2
thβJη
2
, that defines the order parameter along the line h = θ ln ξ1
ξ2
and has got
the standard form of molecular field equation. In addition to zero solution, non-zero
ones occur at θ < θc = J
4
. One can make a conclusion that at the temperatures
θ < θc the relation (14) has the meaning of equation that describes the line of phase
equilibrium (first order phase transition curve), and temperature θc corresponds to
the critical point. Critical value of hc is given by the expression
hc =
J
4
ln
1 + e−
4
J
(U−g−2µ) + 2e
4
J
( g
2
+µ)
1 + e−
4
J
(U+g−2µ) + 2e−
4
J
( g
2
−µ)
. (16)
492
Phase transitions in pseudospin-electron model
The feature of this phase transition is that the curve of phase equilibrium gen-
erally is not parallel to the temperature axis. The occurrence of a bent in the cooc-
currence curve testifies to the possibility of the first order phase transition at the
change of temperature with a jump of the order parameter η if the value of field h
is within the interval placed between 2H ∗ and hc values.
At the temperatures below the critical value crossing the phase coexistence curve
leads to the jump of electron concentration between values that correspond to phases,
which are involved in phase separation (this corresponds to the break point on the
dependence Ω(µ)). At the same time these values are points of binodal lines, which
are determined according to the Maxwell rule from the plot of function µ(n). An
illustrative sample of this is presented in figure 4. The plots presented are obtained
using numerical calculations based on formulas (5) and (6).
5. Conclusions
The investigation performed shows that pseudospin-electron model with a long-
range interaction possesses some features that make it different from the ordinary
Ising model. They are as follows:
– The possibility of the first order phase transition occurring at the change of
temperature and at fixed value of the field (regime µ =const)
– Instability with respect to phase separation in the wide range of parameter h
(regime n =const) with the emerging of the regions with different electron concen-
tration and different orientation of pseudospins.
The results obtained give a more substantiated interpretation of the behaviour of
the pair correlation function and dielectric susceptibility of the model in the case t 6=
0, J = 0 investigated in GRPA [5,7]. It is possible to conclude that the divergence of
susceptibility χss|µ=const and other related quantities at certain values of temperature
and h ∼ g (at U = ∞) corresponds to the point of the high-temperature phase
instability. The temperature of this instability is situated below the temperature of
the first order phase transition that leads to the jump of 〈S z〉.
When the model is used to describe the thermodynamics of oxygen subsystem in
YBa2Cu3O7-crystals, the results of this work might be used as a basis for describ-
ing the bistability phenomena in apex oxygen sublattices as well as for studying
the experimentally observed spatial nonuniformities and phase-separated states in
monocrystalline specimens [8,9].
This work was partially supported by the Ministry of Ukraine for Science and
Technology (project No. 2.4/171). The authors express their gratitude to O.Velychko
for the assistance in the preparing the manuscript.
References
1. Muller K.A. Phase transitions. // 1988 (Special issue).
493
I.V.Stasyuk, Yu.Havrylyuk
2. Frick M., von der Linden W., Morgenstern I., Raedt H. Local anharmonic vibrations,
strong correlations and superconductivity. A quantum simulation study. // Z. Phys.
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4. Stasyuk I.V., Shvaika A.M. Electron spectrum and dielectric susceptibility of the Hub-
bard model with local lattice anharmonicity. // Acta Physica Polonica A, 1994, vol. 85,
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5. Stasyuk I.V., Shvaika A.M., Danyliv O.D. Dielectric instability and charge ordering in
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rect interaction between pseudospins. // Preprint of the Institute for Condensed Mat-
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Фазові переходи у псевдоспін-електронній моделі з
прямою взаємодією між псевдоспінами
І.В.Стасюк, Ю.Гаврилюк
Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
Отримано 12 жовтня 1998 р.
Проведено аналіз термодинамічних властивостей псевдоспін-елек-
тронної моделі у випадку відсутності електронного переносу з вклю-
ченням прямої взаємодії між псевдоспінами сегнетоелектричного
типу. У наближенні середнього поля досліджуються умови рівноваги
в режимах µ = const та n = const. Показано, що взаємодія з електро-
нами при фіксованому значенні µ приводить до можливості фазо-
вого переходу першого роду при зміні температури зі стрибкоподіб-
ною зміною 〈Sz〉 . У режимі n = const має місце нестабільність щодо
розділення фаз у широкому інтервалі значень параметра асиметрії
h .
Ключові слова: псевдоспін-електронна модель, фазові переходи,
розділення фаз
PACS: 74.65.+n, 71.45.Gm
494
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