Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field
The pseudospin-electron model with a direct interaction between pseudospins and with transverse field is considered. In the absence of electron transfer, the equilibrium states of the model are studied in the mean field approximation. In the µ = const regime, two types of the first order phase tran...
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irk-123456789-1209822017-06-14T03:03:03Z Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field Dublenych, Yu.I. The pseudospin-electron model with a direct interaction between pseudospins and with transverse field is considered. In the absence of electron transfer, the equilibrium states of the model are studied in the mean field approximation. In the µ = const regime, two types of the first order phase transitions (with jumps of the mean values of pseudospins and average electron concentration) are revealed. At T → 0 , phase transitions of the first type disappear, on increasing the transverse field, while those of the second type persist at any large value of the field. The phase transitions of the second type are caused by simultaneous effect of pseudospin-electron interaction and transverse field. It is also shown that in the n = const regime at zero temperature, the phase separation occurs at all values of a longitudinal field h . В наближенні середнього поля досліджено рівноважні стани псевдоспін-електронної моделі з прямою взаємодією псевдоспінів та з поперечним полем за відсутности перенесення електронів. У режимі µ = const виявлено фазові переходи першого роду (зі стрибкоподібною зміною середнього значення псевдоспіна та середньої електронної концентрації) двох типів: перші у випадку T → 0 зникають зі збільшенням поперечного поля, інші залишаються, яким би великим не було це поле. Фазові переходи другого типу зумовлені одночасним впливом псевдоспін-електронної взаємодії та поперечного поля. Показано також, що в режимі n = const за нульової температури відбувається розшарування фаз в усій області значень поздовжнього поля h . 2000 Article Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field / Yu.I. Dublenych // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 815-825. — Бібліогр.: 8 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.815 PACS: 74.65.+n, 71.45.Gm http://dspace.nbuv.gov.ua/handle/123456789/120982 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The pseudospin-electron model with a direct interaction between pseudospins and with transverse field is considered. In the absence of electron
transfer, the equilibrium states of the model are studied in the mean field
approximation. In the µ = const regime, two types of the first order phase
transitions (with jumps of the mean values of pseudospins and average
electron concentration) are revealed. At T → 0 , phase transitions of the
first type disappear, on increasing the transverse field, while those of the
second type persist at any large value of the field. The phase transitions of
the second type are caused by simultaneous effect of pseudospin-electron
interaction and transverse field. It is also shown that in the n = const
regime at zero temperature, the phase separation occurs at all values of a
longitudinal field h . |
| format |
Article |
| author |
Dublenych, Yu.I. |
| spellingShingle |
Dublenych, Yu.I. Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field Condensed Matter Physics |
| author_facet |
Dublenych, Yu.I. |
| author_sort |
Dublenych, Yu.I. |
| title |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| title_short |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| title_full |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| title_fullStr |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| title_full_unstemmed |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| title_sort |
phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120982 |
| citation_txt |
Phase transitions and phase separation in a pseudospin-electron model with direct interaction between pseudospins and with transverse field / Yu.I. Dublenych // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 815-825. — Бібліогр.: 8 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT dublenychyui phasetransitionsandphaseseparationinapseudospinelectronmodelwithdirectinteractionbetweenpseudospinsandwithtransversefield |
| first_indexed |
2025-07-08T18:58:05Z |
| last_indexed |
2025-07-08T18:58:05Z |
| _version_ |
1837106292465860608 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 815–825
Phase transitions and phase separation
in a pseudospin-electron model with
direct interaction between pseudospins
and with transverse field
Yu.I.Dublenych
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received September 14, 2000
The pseudospin-electron model with a direct interaction between pseu-
dospins and with transverse field is considered. In the absence of electron
transfer, the equilibrium states of the model are studied in the mean field
approximation. In the µ = const regime, two types of the first order phase
transitions (with jumps of the mean values of pseudospins and average
electron concentration) are revealed. At T → 0 , phase transitions of the
first type disappear, on increasing the transverse field, while those of the
second type persist at any large value of the field. The phase transitions of
the second type are caused by simultaneous effect of pseudospin-electron
interaction and transverse field. It is also shown that in the n = const
regime at zero temperature, the phase separation occurs at all values of a
longitudinal field h .
Key words: pseudospin-electron model, phase transitions, phase
separation
PACS: 74.65.+n, 71.45.Gm
1. Introduction
In high temperature superconductors one can observe a series of physical phe-
nomena which exist only separately in other types of crystals. One of the reasons for
it is the strong electron correlation of the Hubbard type in the conductivity bands
formed mainly by the superconducting Cu2-O2 planes. Another significant feature of
the high-Tc superconductors is the presence of strongly anharmonic elements in their
structure. That can lead to various instabilities. The amplification of an effective at-
traction between electrons as a result of their interaction with anharmonic phonons
was considered as one of the possible mechanisms of the high-Tc superconductivity
yet in the first publications on this subject [1–3].
c© Yu.I.Dublenych 815
Yu.I.Dublenych
In the best studied high-Tc superconductor, YBa2Cu3O7−δ, the strongly anhar-
monic elements are the so-called apical oxygen ions O4 (see e. g. [4]). To describe
their vibrations, a pseudospin-electron (or Müller) model was proposed [5]. The pseu-
dospin variable Sz
i = ±1/2 takes into account two possible positions of an ion O4 in a
double potential well. The model includes electron-electron and pseudospin-electron
interactions as well as the electron hopping. The energy of the double potential
well asymmetry is considered as a longitudinal field h, and the energy of tunnelling
splitting as a transverse field Ω.
In [6] it has been shown that the electron transfer leads to the effective interaction
between pseudospins. Therefore, a simpler model was proposed without electron
transfer but with a direct pseudospin interaction. In [7] the phase transitions in
such a model were investigated in the mean field approximation and without a
transverse field. The authors showed that in the µ = const regime a first order
phase transition with a jump-like behaviour of 〈S z〉 is possible, and in the n = const
regime an instability with respect to phase separation exists in a wide range of
asymmetry parameter h values. A similar investigation was also performed for the
two-sublattice model with direct interactions of ferroelectric type [8]. However, in
the mentioned works jumps of ions O4 in double potential well were not taken into
account. The aim of the present paper is to investigate the effect of the transverse
field (describing the mentioned jumps) on phase transitions and phase separation in
the model.
2. Thermodynamic equilibrium conditions and thermodynami c
functions in the mean field approximation
Let us consider a pseudospin-electron system described by the Hamiltonian
H =
∑
i
Hi −
1
2
∑
ij
JijS
z
i S
z
j , (1)
where
Hi = Uni↓ni↑ − µ(ni↓ + ni↑) + g(ni↓ + ni↑)S
z
i − hSz
i − ΩSx
i (2)
is the single-site Hamiltonian.
Here Jij describes the direct interaction between pseudospins S z
i and Sz
j , U is
the Hubbard correlation, g is the pseudospin-electron interaction constant and h is
the longitudinal field or the energy of double potential well asymmetry. The term
with the transverse field Ω is included.
In our investigation we will use the mean field approximation:
Sz
i S
z
j ≈ −η2 + η(Sz
i + Sz
j ), (3)
where η = 〈Sz
i 〉.
In this approximation the Hamiltonian has the following form:
H =
∑
H̃i +
N
2
Jη2, (4)
816
Phase transitions and phase separation. . .
where
H̃i = Uni↓ni↑ − µ(ni↓ + ni↑) + g(ni↓ + ni↑)S
z
i − (h + Jη)Sz
i − ΩSx
i , (5)
and J =
∑
i
Jij =
∑
j
Jij.
The vectors |ni↑, ni↓, S
z
i 〉 form a complete basic set for the Hamiltonian H̃i:
|1〉 = |0, 0, 1/2〉, |1̃〉 = |0, 0,−1/2〉,
|2〉 = |1, 1, 1/2〉, |2̃〉 = |1, 1,−1/2〉,
|3〉 = |0, 1, 1/2〉, |3̃〉 = |0, 1,−1/2〉,
|4〉 = |1, 0, 1/2〉, |4̃〉 = |1, 0,−1/2〉.
(6)
In this basis the Hamiltonian H̃i is not diagonal, but it is easy to find a basis in
which H̃i will be diagonal [6]. As a result we obtain the following set of eigenvalues:
λ1,1̃ = ∓1
2
√
h̃2 + Ω2,
λ2,2̃ = −2µ+ U ∓ 1
2
√
(h̃− 2g)2 + Ω2,
λ3,3̃ = −µ∓ 1
2
√
(h̃− g)2 + Ω2 = λ4,4̃, (7)
where h̃ = h+ Jη.
The partition function per unit cell is equal to
Zi =
∑
r
(
e−βλr + e−βλr̃
)
e−β
Jη2
2 , (8)
and the total partition function is
Z = ZN
i . (9)
Now we can write the Gibbs thermodynamic potential per unit cell:
ω = −Θ ln
∑
r
(
e−βλr + e−βλr̃
)
+
1
2
Jη2. (10)
From the equilibrium condition in the regime µ = const
(
∂ω
∂η
)
T,µ,h
= 0 (11)
we obtain the equation for the mean value of η having a meaning of the order
parameter:
η =
1
∑
r
(e−βλr+e−βλr̃)
{
h̃
2
√
h̃2+Ω2
(
e−βλ1−e−βλ
1̃
)
+
h̃−2g
2
√
(h̃−2g)2+Ω2
(
e−βλ2−e−βλ
2̃
)
+
h̃−g
√
(h̃−g)2+Ω2
(
e−βλ3−e−βλ
3̃
)
.(12)
817
Yu.I.Dublenych
In the case U = 2µ, the curve η(h) is symmetrical with respect to the point
(g, 0).
The average number of electrons per unit cell can be obtained by differentiating
the expression (10) with respect to µ :
n = 2− 2 · e
−βλ1 + e−βλ
1̃ + e−βλ3 + e−βλ
3̃
∑
r
(e−βλr + e−βλr̃)
. (13)
3. Phase diagrams and phase separation at zero temperature
First, let us consider the case T → 0 and µ = const. Having found partially the
limit we obtain the equation
η =
1
e−βλ1 + e−βλ2 + 2e−βλ3
h̃
2
√
h̃2 + Ω2
e−βλ1 +
h̃− 2g
2
√
(h̃− 2g)2 + Ω2
e−βλ2
+
h̃− g
√
(h̃− g)2 + Ω2
e−βλ3
. (14)
The form of this equation in the limit T → 0 depends on which of the values λ1,
λ2, λ3 is minimal.
1. λ1 = min, η = η1(h) =
h̃
2
√
h̃2+Ω2
, or h̃ = 2Ωη√
1−4η2
,
ω = ω1(h) = − Ω
2
√
1−4η2
+ J
2
η2, n = 0;
2. λ2 = min, η = η2(h) =
h̃−2g
2
√
(h̃−2g)+Ω2
, or h̃ = 2Ωη√
1−4η2
+ 2g,
ω = ω2(h) = −2µ+ U − Ω
2
√
1−4η2
+ J
2
η2, n = 2;
3. λ3 = min, η = η3(h) =
h̃−g
2
√
(h̃−g)2+Ω2
, or h̃ = 2Ωη√
1−4η2
+ g
ω = ω3(h) = −µ − Ω
2
√
1−4η2
+ J
2
η2, n = 1.
The curves ηi(h) for cases 1, 2 and 3 are parallel and their centers of symmetry are the
points (0,0), (2g,0), (g,0), respectively. If Ω < J
2
, they have a “reverse motion”, i.e.,
the part of a curve where the derivative η ′(h) < 0. For the values of h corresponding
to a reverse motion the dependence ηi(h) is many-valued.
The curve ηi(h) as the solution of equation (14) at T → 0 is composed of parts
of the curves ηi(h) (figure 1). The transition from curve 2 to curve 3 goes along the
straight line
h = (µ− U)
√
4Ω2
g2 − 4(µ− U)2
+ 1 +
3g
2
− Jη, (15)
818
Phase transitions and phase separation. . .
Figure 1. Order parameter η as a function of the field h (solid line) and the curves
ηi(h) (dotted lines). T = 0, J = 1, µ = −0.01, U = 0.1, g = 0.28, Ω = 0.55. Thick
lines correspond to thermodynamically stable states.
and from curve 3 to curve 1 along the straight line
h = µ
√
4Ω2
g2 − 4µ2
+ 1 +
g
2
− Jη. (16)
If U > g, the inequalities − g
2
+ U < µ < g
2
+ U and −g
2
< µ < g
2
are the
necessary and sufficient conditions of the existence of transitions 2→3 and 3→1,
respectively. However, in the case U < g those inequalities are necessary but not
sufficient conditions of the existence of those transitions, because if the value h̃ =
h + Jη for the transition 3→1 is smaller than that for the transition 2→3, there is
only one transition from curve 1 to curve 2 going along the straight line
h = (2µ− U)
√
Ω2
g2 − (2µ− U)2
+ 1 + g − Jη. (17)
Besides the reverse motions along which transitions between the curves ηi(h)
occur, at the condition Ω < J/2 the resulting curve η(h) can contain its own reverse
motion of one of the curves ηi(h) (figure 2).
The graph of the thermodynamic potential ω(h) is composed of portions of the
parallel curves ωi(h) connected with segments of tangent parabolas (obtained by
integration of relations (15–17)). The domains where the dependence η(h) is many-
valued correspond to self-intersections of the curve ω(h). Let (h0, ω0) be coordinates
819
Yu.I.Dublenych
Figure 2. Cascade of phase transitions. Thick lines correspond to thermodynam-
ically stable states, dotted lines show “not realized” parts of the curves η i(h).
Letters a, b, c, d, f denote different phases. a) T = 0, J = 1, µ = −0.106,
U = 0.22, g = 0.28, Ω = 0.47; b) T = 0, J = 1, µ = 0.11, U = 0.22, g = 0.28,
Ω = 0.47.
820
Phase transitions and phase separation. . .
Figure 3. The (µ, h) phase diagrams at zero temperature. Dashed line shows the
phase diagram at Ω → 0. Letters a, b, c, d, e, f denote different phases (see
figure 2). a) J = 1, U = 0.22, g = 0.28, Ω = 0.47; b) J = 1, U = 0.15, g = 0.28,
Ω = 0.47; c) J = 1, U = 0.10, g = 0.28, Ω = 0.55; d) J = 1, U = 0.22, g = 0.28,
Ω = 0.7.
821
Yu.I.Dublenych
of the self-intersection point. If ω0 is minimal of all ω(h0), then at the point h0
there is a first order phase transition, where the value of η(h) changes by jump.
The maximum possible number of reverse motions of the curve η(h) as well as of
phase transitions are three. It is significant that the phase transitions related to
jumps from one curve η(h) to another do not disappear on increasing Ω but only
move along the h axis. At these transitions, η as well as n has a jump. If a phase
transition takes place on a single curve ηi(h), then η changes from −
√
1
4
− Ω2
J2 to
+
√
1
4
− Ω2
J2 and n remains constant.
The four possible topologically different diagrams in the plane (µ, h) at zero
temperature are shown in figure 3. The diagrams (a) and (b) correspond to the
case Ω < J/2 and the diagrams (c) and (d) correspond to the case Ω > J/2. The
diagrams (b) and (c) are possible only when U < g. As one can see, two types
of phase transitions exist at zero temperature: the first ones (horizontal lines in
figure 3 (a), (b)) disappear when Ω becomes equal to J/2, the other ones persist
at any large value of Ω. The phase transitions of the second type are caused by
simultaneous effect of pseudospin-electron interaction and transverse field.
If T → 0 the electron concentration n can take the values 0, 1 or 2. Therefore, at
fixed n different from these values a phase separation occurs. In figure 4 the graph
of the phase separation corresponding to the diagram of figure 3 (b) is shown. As
one can see, at zero temperature the domain of phase separation encloses all values
of h and n. If h < h1 or h > h2, then in the case 0 < n < 1 the phases with n = 0
and with n = 1 coexist, and in the case 1 < n < 2 there are the phases with n = 1
and with n = 2. If h1 < h < h2 the separation into phases with n = 0 and with
n = 2 occurs. At non-zero temperature the phase separation domain is bounded.
4. Temperature-field phase diagrams
If temperature deviates from zero, the curve η(h) (see figure 1) smooths out. On
increasing temperature, the phase transitions disappear, first the transition 1, then
the transition 2 (figure 5).
The set of phase coexistence curves for the temperature-field phase diagram is
symmetric with respect to the straight line h = g if µ = U/2. The curves of the
phase diagrams µ1 = U/2 −∆µ and µ2 = U/2 + ∆µ are mutually symmetric with
respect to the same straight line h = g, therefore it is sufficient to consider µ from
the interval ]−∞, U/2].
In the case Ω > J/2, phase transitions exist only at −g/2 < µ < g/2 or −g/2 +
U < µ < g/2 + U .
If µ → −∞ (or g = 0) the phase coexistence line is a vertical segment of length
1/β, where β = 1
Ω
ln J+2Ω
J−2Ω
. On increasing µ from −∞ to −g/2, this line bends more
and more to the right. As soon as µ becomes greater than −g/2, one more line
appears in the phase diagram. On increasing µ, it deforms, elongates, moves to the
right along the h axis, and joins the first curve (figure 6).
One can notice that the phase transitions corresponding at T → 0 to jumps from
822
Phase transitions and phase separation. . .
Figure 4. Phase separation at zero temperature corresponding to figure 3 (c).
Figure 5. The order parameter η as a function of the field h at non zero temper-
ature. Θ = 0.003, J = 1, µ = −0.01, U = 0.1, g = 0.28, Ω = 0.55. Thick lines
correspond to thermodynamically stable states.
823
Yu.I.Dublenych
Figure 6. The (Θ, h) phase diagram. J = 1, µ = −0.106, U = 0.22, g = 0.28,
Ω = 0.47. Dotted line shows the phase coexistence line for Ω = 0.
curve 2 to curve 3 or from curve 3 to curve 1 disappear well sooner on increasing
temperature than those corresponding at T → 0 to jumps from curve 3 to curve 1
or occurring on a single curve.
On increasing Ω, the critical temperature of each transition is lowered down. As
soon as Ω becomes equal to J/2, transitions from curve 2 to curve 1 or those on a
single curve disappear but transitions from curve 2 to curve 3 or from curve 3 to
curve 1 remain at any large value of Ω.
At fixed h a phase transition temperature can either increase or decrease on
increasing Ω (until ∆η becomes zero). It depends on the value of µ.
5. Results
Hence the transverse field Ω not only reduces the critical temperature for a first
order phase transition but at certain values of µ and sufficiently low temperature
gives rise to one or two additional phase transitions which contrary to the case g = 0
persist at any large value of Ω.
At zero temperature, instability with respect to phase separation exists not in a
bounded domain as in the case Ω = 0 but at any value of asymmetry parameter h.
824
Phase transitions and phase separation. . .
6. Acknowledgement
The author would like to thank professor I.V.Stasyuk for many helpful sugges-
tions and discussions.
References
1. Kurtz S.K., Hardy J.R., Flocken J.W. // Ferroelectrics, 1988, vol. 87, p. 29.
2. Plakida N.M. // Physica Scripta, 1989, vol. T29, p. 77.
3. Bussman-Holder A., Simon A., Butner H. // Phys. Rev. B, 1989, vol. 39, p. 207.
4. Müller V., Hucho C., Maurer D. // Ferroelectrics, 1992, vol. 130, p. 45–76.
5. Müller K.A. // Z. Phys. B., 1990, vol. 80, p. 193; Phase Transitions, 1998 (Special
issue).
6. Stasyuk I.V., Shvaika A.M. // Acta Physica Polonica A, 1993, vol. 84, p. 293.
7. Stasyuk I.V., Havrylyuk Yu. // Cond. Matt. Phys., 1999, vol. 2, p. 487.
8. Stasyuk I.V., Dublenych Yu.I. Phase transitions and phase separation in a pseudospin-
electron model with direct interaction of antiferroelectric type. Preprint of the Institute
for Condensed Matter Physics, ICMP-99-07U, Lviv, 1999, 26 p.
Фазові переходи та розділення фаз у
псевдоспін-електронній моделі з прямою
взаємодією псевдоспінів та поперечним полем
Ю.І.Дубленич
Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
Отримано 14 вересня 2000 р.
В наближенні середнього поля досліджено рівноважні стани псевдо-
спін-електронної моделі з прямою взаємодією псевдоспінів та з по-
перечним полем за відсутности перенесення електронів. У режимі
µ = const виявлено фазові переходи першого роду (зі стрибкоподіб-
ною зміною середнього значення псевдоспіна та середньої елек-
тронної концентрації) двох типів: перші у випадку T → 0 зникають зі
збільшенням поперечного поля, інші залишаються, яким би великим
не було це поле. Фазові переходи другого типу зумовлені одночас-
ним впливом псевдоспін-електронної взаємодії та поперечного по-
ля. Показано також, що в режимі n = const за нульової температури
відбувається розшарування фаз в усій області значень поздовжнього
поля h .
Ключові слова: псевдоспін-електронна модель, фазові переходи,
розділення фаз
PACS: 74.65.+n, 71.45.Gm
825
826
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