The Bogolubov representation of the polaron model and its completely integrable RPA-approximation
The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at finite temperature is calculated analytically. The polaron free energy in the cons...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2010 |
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation / N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23703: 1-10. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859656582600589312 |
|---|---|
| author | Bogolubov (jr.), N.N. Prykarpatsky, Ya.A. Ghazaryan, A.A. |
| author_facet | Bogolubov (jr.), N.N. Prykarpatsky, Ya.A. Ghazaryan, A.A. |
| citation_txt | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation / N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23703: 1-10. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at finite temperature is calculated analytically. The polaron free energy in the constant magnetic field at finite temperature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable.
Досліджено модель полярона в іонному кристалі в представленні Боголюбова, використовуючи особливе наближення хаотичних фаз. Виведено та описано нову точно розв'язну наближену модель полярона. Аналітично одержано вільну енергію такої моделі за ненульової температури. Розглянуто вільну енергію полярона в постійному магнітному полі за ненульової температури. На базі унітарного перетворення Боголюбова для поляронного гамільтоніана одержано важливий новий результат: повна модель полярона є точно розв'язною.
|
| first_indexed | 2025-12-07T13:39:56Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 2, 23703: 1–10
http://www.icmp.lviv.ua/journal
The Bogolubov representation of the polaron model and
its completely integrable RPA-approximation
N.N. Bogolubov (jr.)1,2∗, Ya.A. Prykarpatsky3,4†, A.A. Ghazaryan5
1 Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russian Federation
2 Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
3 Institute of Mathematics, Pedagogical University of Cracow, Poland
4 Drohobych Ivan Franko State Pedagogical University, Lviv region, Ukraine
5 The Lomonosov Moscow State University, Department of Physics, Moscow, Russian Federation
Received November 13, 2009, in final form March 25, 2010
The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approxima-
tion. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at
finite temperature is calculated analytically. The polaron free energy in the constant magnetic field at finite tem-
perature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian
there is stated a very important new result: the full polaron model is exactly solvable.
Key words: polaron, polaron model, constant magnetic field, RPA-approximation
PACS: 71.38.+i
1. Introduction
The polaron concepts, which was first introduced by Landau [2], is one of the main pillars
which the theoretical analysis of materials with strong electron-phonon coupling rests on. In these
compounds, the coupling between the electron and the lattice leads to a lattice deformation whose
potential tends to bind the electron to the deformed region of the crystal. This process, which has
been called self-trapping because of the potential depending on the state of the electron, does not
destroy translational invariance, even if the lattice deformation is confined to a single lattice site
(small polaron) [1, 5, 7–9]. Quantum mechanical tunneling between different lattice sites restores
this symmetry and ensures that a self-trapped electron forms [10, 11] an itinerant polaronic quasi-
particle.
A polaron is a quasiparticle composed of a charge and its accompanying polarization field. A
slowly moving electron in a dielectric crystal, interacting with lattice ions through long-range forces
will be permanently surrounded by a region of lattice polarization and deformation caused by the
moving electron. Moving through the crystal, the electron carries the lattice distortion along with
it. Thus, one may speak of a cloud of phonons accompanying the electron.
The resulting lattice polarization acts as a potential wall that hinders the movements of the
charge and thus decreases its mobility. Polarons have a spin, though two close-by polarons are
spinless. The latter is called a bipolaron. In materials science and chemistry, a polaron is formed
when a charge within a molecular chain effects the local nuclear geometry, causing an attenuation
(or even reversal) of the nearby bond alternation amplitudes. This “excited state” possesses an
energy level between the lower and upper bands.
As it is well known [1, 13, 14], the quantum model of the polaron in the ion crystal of volume
∗E-mail: nikolai bogolubov@hotmail.com
†E-mail: yarpry@gmail.com
c© N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan 23703-1
http://www.icmp.lviv.ua/journal
N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan
Λ ⊂ E3 can be described [1] by means of the Hamiltonian operator
Ĥp =
p̂2
2m
⊗ 1 +
∑
(f)
1⊗
(
b+f bf + 1/2
)
~ωf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
ei〈f,r〉1 ⊗
(
bf + b+−f
)
, (1)
acting in the Hilbert space L2(Λ; C) ⊗ Φ(Λ; C), where Φ(Λ; C) is the corresponding Fock space
for the phonon quasi-particle states in the crystal, m is an effective electron mass, p̂ := ~
i ∇ is its
momentum operator, b+f and bf , f ∈ 2πΛ−1/3Z3, are, respectively, Bose-operators of creation and
annihilation of phonons with energy ~ωf ∈ R+, the coefficient Lf =
√
α|f |−1, α ∈ R+, is an
intensity parameter of the polaron bond in the crystal and 〈., .〉 is the ordinary scalar product in
the Euclidean space E3.
As it was also shown in [1, 4], Hamiltonian (1) can be transformed by means of the unitary
transformation
U := exp
i
∑
(f)
〈f, r〉1 ⊗ b+f bf
(2)
into the following form:
Ĥp =
1
2m
p̂⊗ 1−1⊗
∑
(f)
~fb+f bf
2
+
∑
(f)
~ωf1⊗b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1⊗(bf +b+−f ), (3)
where it is clearly seen a quantum nature of the polaron structure, which does not depend on the
force of the interaction parameter Lf . This conclusion can be also made from the fact that model
(1) possesses a conservation law of the general electron-phonon momentum:
P̂ = p̂⊗ 1 + 1 ⊗
∑
(f)
~fb+f bf , (4)
that is [Ĥp, P̂ ] = 0. It is interesting to note, that the analytical studies of statistical properties of
the model of polaron in all of the works on this problem [1, 3, 6, 13, 21], except the works [4, 5],
were based on the Hamiltonian expression (1). But, as it was pointed out in [1], the statistical
properties of the model do not depend on the unitary-equivalent representation choice for operator
(1). In particular, expression (3) can be easily rewritten equivalently, making use of the normal
operator ordering, as
Ĥp =
1
2m
p̂⊗ 1− 1⊗
∑
(f)
~fb+f bf
2
+
∑
(f)
~ωf1⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)
1/21⊗ (bf + b+−f )
=
1
2m
p̂2 ⊗ 1 − 1
m
∑
(f,g)
〈p̂g, ~f〉 ⊗ b+g bgb
+
f bf +
1
2m
∑
(f,g)
〈~f, ~g〉1⊗ b+f bf b
+
g bg
+
∑
(f)
~ωf1⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1⊗ (bf + b+−f )
=
1
2m
p̂2 ⊗ 1 −
∑
(f)
1
2m
p̂2
f ⊗ b+f bf +
∑
(f)
[(p̂f − ~f)2 + ~ωf ] ⊗ b+f bf
+
1
2m
∑
(f,g)
〈~f, ~g〉1⊗ b+f b
+
g bgbf − 1
m
∑
(f,g)
〈p̂f , ~g〉 ⊗ b+f b
+
g bgbf
+
1
Λ1/2
∑
(f)
Lf (
~
2ωf
)1/21⊗ (bf + b+−f )
=
1
2m
p̂2 ⊗ 1 −
∑
(f)
1
2m
p̂2
f ⊗ b+f bf +
1
2m
∑
(f)
[(p̂f − ~f)2 + ~ωf1] ⊗ b+f bf
23703-2
RPA-approximation of the Bogolubov’s polaron model. Part 1
− 1
2m
∑
(f,g)
(〈p̂f , ~g〉 + 〈p̂g, ~f〉) ⊗ b+f b
+
g bgbf
+
1
2m
∑
(f,g)
〈~f, ~g〉1⊗ b+f b
+
g bgbf +
1
Λ1/2
∑
(f)
Lf (
~
2ωf
)1/21⊗ (bf + b+−f ), (5)
where we made use of the tensor operator representation p̂⊗ 1 =
∑
(f) p̂f ⊗ b+f bf =
∑
(f) p̂N̂
−1 ⊗
b+f bf , N̂ :=
∑
(f) b
+
f bf , in the Fock space Φ(Λ; C) and of the related commutation properties:
[p̂f , p̂g] = 0 = [p̂g, b
+
f bf ] for all f, g ∈ 2πΛ−1/3Z3. We need to mention here that this representation
of the Hamiltonian operator (3) possesses a natural physical interpretation as a polaron model
with collectively separated interaction potentials. This fact proves to have become very important
in our analysis that follows below.
In a series of papers on the polaron theory [1, 6] the oscillator approximation of the Hamiltonian
(1) was analyzed. This approximation leads to the so-called “linearized” model of the polaron, when
only the first member of the expansion exp(i〈r, f〉) ' 1 + i〈r, f〉 in (1) was taken into account
and the quadratic compensating part K0r
2/2, where K2
0 = 1
3Λ
∑
(f) L
2
ff
2ω−2
f , was added into the
initial Hamiltonian. Under these conditions, the resulting Hamiltonian persists to be translational-
invariant which makes it possible to analyze its thermodynamic properties as N → ∞,Λ → ∞.
Concerning the Hamiltonian representation (5) one can make the following very important
observation: the operator term V̂(1)
p := 1
2m
∑
(f,g)(〈p̂f , ~g〉+ 〈p̂g, ~f〉+ 〈~f, ~g〉1)⊗b+f b+g bgbf , being
written in the normally ordered secondly quantized form, gives rise to the following effective two-
particle operator expression in an N -particle invariant Fock subspace [14, 16, 17, 19]:
V̂(1)
N =
1
2m
N
∑
k 6=j=1
(
1
N
〈
p̂(yj) ⊗ Π̂(yk) + p̂(yk) ⊗ Π̂(yj)〉 + 〈Π̂(yk), Π̂(yj)
〉
)
, (6)
acting in the Hilbert space L2(Λ; C) ⊗ L2,sym(ΛN ; C), where we denoted by Π̂(y), y ∈ Λ, the re-
spectively modified momentum operator of the crystal deformations and p̂(y), y ∈ Λ, the uniformly
distributed polaron momentum.
Taking into account the well-known [14, 17, 18] random phase approximation (RPA) for the
two-particle phonon excitations in crystal, one can obtain that expression (6) transforms into zero,
since
N
∑
k 6=j=1
〈Π̂(yk), Π̂(yj)〉 = 0 =
N
∑
k 6=j=1
1
N (〈p̂(yj)⊗Π̂(yk) +p̂(yk)⊗Π̂(yj)〉) weakly in L2,sym(ΛN ; C)
owing to the stability of the crystal deformations, generated by the polaron interaction. Since
we are interested in the statistical properties of our polaron model, the above discussed RPA-
approximation well fits to this aim, because the corresponding statistical sum is calculated as the
average value of statistical operator over all Hamiltonian (5) eigenstates. Thus, we can consider a
polaron model Hamiltonian within the RPA-approximation in the following reduced form:
Ĥ(0)
p =
1
2m
p̂2 ⊗ (1− N̂
−1
) +
1
2m
∑
(f)
(p̂f − ~f)2 ⊗ b+f bf
+
∑
(f)
~ωf1⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1⊗ (bf + b+−f ), (7)
being a well defined operator expression bounded from below in the Hilbert space L2(Λ; C) ⊗
Φ(Λ; C).
It can be shown using the standard considerations, devised in [6, 20] jointly with the Bogolubov
inequality [1], that the corresponding statistical sums for the Hamiltonian operators (7) and (3)
in the thermodynamical limit are asymptotically equal as the intensity parameter α → ∞. The
analytical details of this analysis will be presented in the Part 2 of this work. Thereby, we believe
that our model (7), describing the polaron properties within the RPA-approximation, is more
appropriate in studying its thermodynamics, remaining a priori translation-invariant. We will
23703-3
N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan
study this model herein below in detail. Moreover, we will investigate this polaron model in the
external magnetic field B = rotA, where, by definition, the vector potential A = (0,−mωcx, 0)ᵀ ∈
E3 is directed along the axis Oy of the Euclidian space E3 and ωc ∈ R+ denotes the corresponding
“cyclotronic” frequency. In this case Hamiltonian (5) has the form
Ĥ(µ)
p =
1
2m
(p̂(µ)2 + p̂2
z) ⊗ (1 − N̂
−1
) −
∑
(f)
1
2m
(p̂
(µ)2
f + p̂2
fz
) ⊗ b+f bf
+
1
2m
∑
(f̄)
(p̂
(µ)
f − ~f̄)2 ⊗ b+f bf +
1
2m
∑
(fz)
(p̂
fz
− ~fz)
2 ⊗ b+f bf
+
∑
(f)
~ωf1⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1⊗ (bf + b+−f ), (8)
where by definition p̂
(µ)
f = (p̂fx
, p̂fy
+mωcx)
ᵀ, f̄ = (fx, fy)ᵀ, pertaining to the quadratic structure
of the phonon operators.
2. The RPA-approximated polaron model
To study the thermodynamics of the RPA-approximated polaron model (5), we need to calcu-
late the statistical sum
Z(0)
p = Sp exp(−βĤ(0)
p ), (9)
where β = 1/kT is the inverse temperature in the Boltzmann units. Then, taking into account that
Ĥ(0)
p =
1
2m
p̂2 ⊗ (1 − N̂
−1
) +
∑
(f)
~ω̂f ⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1 ⊗ (bf + b+−f ),
where the operators
ω̂f :=
[
1
2m
(p̂f/~ − f)2 + ωf
]
, n̂f := b+f bf
commute to each other, that is [ω̂f , n̂f ] = 0, we can calculate expression (9) classically, having
reduced it to the form
Z(0)
p = Sp(−βH(0)
p ) = Sp(e)Z
(0)
ph (p̂). (10)
Here, by definition, we denoted the “phonon” part of the statistical sum as
Z
(0)
ph (p̂) = Sp(ph) exp(−βĤ(0)
p ). (11)
Since the operator Ĥ(0)
ph is a quadratic form with respect to the phonon operators, it is easy to
calculate that the change of variables
bf = b̃f − cf (p̂, N̂f ), b+f = b̃+f − cf (p̂, N̂f ), cf (p̂, N̂f ) =
1
Λ1/2
∑
(f)
Lf
ω̂f
(
~
2ωf
)1/2
, (12)
transforms it to the canonical diagonal form
Ĥ(0)
ph =
1
2m
p̂2 ⊗ (1− N̂
−1
)+
∑
(f)
~ω̂f ⊗ b̃+f b̃f +
1
Λ
∑
(f)
L2
f
2ω̂fωf
⊗ 1. (13)
Here we need to mention that the operator ω̂f : L2(Λ; C) → L2(Λ; C) is a strongly positive defined
expression. We also notice here, as it was done in [1], that the phonon frequencies satisfy the
symmetry condition ωf = ω−f for all vectors f ∈ 2πΛ−1/3Z3. A similar symmetry condition
also holds for the operator quantities ω̂f , that is ω̂f = ω̂+
−f for all f ∈ 2πΛ−1/3Z3. Based on
23703-4
RPA-approximation of the Bogolubov’s polaron model. Part 1
representation (13), it is easy to find the following expression for the phonon part of statistical
sum (11):
Z
(0)
ph (p̂) = exp(− β
2m
p̂2)Sp(ph)
∏
(f)
∑
(Ñf )
exp
[
βp̂2
2m[N̂f + c2f (p̂, Ñf )]
− β~Ñf ω̄f (p̂, Ñf )
−β
L2
f
2Λω̄f (p̂, Ñf )ωf
]
= exp(− β
2m
p̂2) exp
∑
(f)
∑
(Nf∈Z+)
exp
[
βp̂2
2m[Nf + c2f (p̂, Nf )]
−β~Nf ω̄f (p̂, Nf ) − β
L2
f
2Λω̄f (p̂, Nf )ωf
]
:= exp(− β
2m
p̂2) exp
−
∑
(f)
Ff (p̂;β)
, (14)
where we denoted by Ñf := b̃+f b̃f , f ∈ 2πΛ−1/3Z3, the shifted phonon density operators and
ω̄f (p̂, Nf ) := {( 1
2m (p̂/[(Nf + c2f (p̂, Nf )~] − f)2 + ωf}, f ∈ 2πΛ−1/3Z3, the related shifted phonon
frequencies. Substituting now (14) in (10), we obtain, as a result, an analytical expression for the
complete statistical sum of our approximated polaron model:
Z(0)
p = Sp(e) exp
− β
2m
p̂2 −
∑
(f)
Ff (p̂;β)
=
Λ
(2π)3
∫
E3
d3k exp
− β
2m
k2 −
∑
(f)
Ff (k;β)
, (15)
where we used the known trick [1, 13, 14] of changing the discrete sum
∑
(k)(. . . ) by a corresponding
integral:
∑
(k)
(. . . ) =
Λ
(2π)3
∫
E3
(. . . )d3k. (16)
Thus, if we calculate the internal sums
∑
(f)(. . . ) of expression (15) for the given values of phonon
frequencies ωf and the interaction parameter Lf for all discrete values f ∈ 2πΛ−1/3Z3, then the
quasi-gaussian integral
∫
E3(. . . )d
3k can be calculated analytically or by means of approximate
and asymptotic methods. In particular, assuming that Lf = α1/2/|f |, and ωf = ω0 for all f ∈
2πΛ−1/3
Z
3, from (15) and (16) it is easy to obtain an explicit expression for the statistical sum of
the polaron model (5), which is not going to be dealt with in this paper.
3. RPA-approximated polaron model in the static magnetic field
To study RPA-approximated polaron model in the static magnetic field we will use the Hamil-
tonian operator (8), which persists to be quadratic in the phonon operators. Taking into account
that
Ĥ(µ)
p =
1
2m
(p̂(µ)2 + p̂2
z) ⊗ (1− N̂
−1
) +
1
2m
∑
(f̄)
(p̂
(µ)
f − ~f̄)2 ⊗ b+f bf
+
1
2m
∑
(fz)
(p̂fz
− ~fz)
2 ⊗ b+f bf +
∑
(f)
~ωf1⊗ b+f bf
+
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1⊗ (bf + b+−f ), (17)
where the operators
(p̂
(µ)
f − ~f̄)2, (p̂fz
− ~fz)
2, n̂f = b+f bf (18)
commute to each other. Put, by definition,
ω̂f := ωf +
1
2m
(p̂
(µ)
f /~ − f̄)2 +
1
2m
(p̂fz
/~ − fz)
2, p̄
(µ)
f = (p̂fx
, p̂fy
+mωcx)
ᵀ,
23703-5
N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan
for f̄ = (fx, fy)
ᵀ ∈ E2, the respectively split statistical sum
Z(µ)
p = Sp exp(−βĤ(µ)
p ) = Sp(e)
[
Sp(ph) exp(−βĤ(µ)
ph )
]
(19)
can be successfully calculated, where
Ĥ(µ)
ph =
1
2m
(p̂(µ)2 + p̂2
z)⊗
(
1− N̂
−1
)
+
∑
(f̄)
~ω̂f ⊗ b+f bf +
1
Λ1/2
∑
(f)
Lf
(
~
2ωf
)1/2
1 ⊗ (bf + b+−f ). (20)
Let us, first, find the statistical sum
Z
(µ)
ph = Sp(ph) exp
(
−βĤ(µ)
ph
)
, (21)
for which, using (19) and (21), we obtain
Z(µ)
p = Sp(e)Z
(µ)
ph . (22)
Using now the fact that Hamiltonian (18) is quadratic in the phonon variables and making trans-
formation (12), we obtain
Ĥ(µ)
ph =
1
2m
(p̂(µ)2 + p̂2
z) ⊗
(
1− N̂
−1
)
+
∑
(f)
~ω̂f b̃
+
f b̃f +
1
Λ
∑
(f)
L2
f
2ω̂fωf
. (23)
Thus, from (23) and (21), one obtains that
Z(µ)
p = exp
[
− β
2m
(p̂(µ)2 + p̂2
z)
]
Sp(ph)
∏
(f)
∑
(N̂f )
exp
[
β(p̂(µ)2 + p̂2
z)
2m[Ñf + c2f (p̂(µ), kz ; Ñf )]
−β~N̂f ω̄f (p̂(µ), p̂z; N̂f ) −
βL2
f
2Λω̄f (p̂(µ), p̂z; N̂f )ωf
]
= exp
[
− β
2m
(p̂(µ)2 + p̂2
z)
]
exp
∑
(f)
∑
(Nf∈Z+)
exp
[
β(p̂(µ)2 + p̂2
z)
2m[Ñf + c2f (p̂(µ), kz ; Ñf )]
−β~Nf ω̄f (p̂(µ), p̂z;Nf ) +
βL2
f
2Λω̄f (p̂(µ), p̂z;Nf )ωf
]
:= exp
− β
2m
(p̂(µ)2 + p̂2
z) −
∑
(f)
Ff (p̂(µ), p̂z;β)
, (24)
where ω̄f (p̂(µ), kz ;β) = ωf + 1
2m (p̂(µ)/[~(Nf + c2f )] − f̄)2 + 1
2m (kz − fz)
2. Substituting (24) into
(22), we obtain an expression for the statistical sum of RPA-approximated polaron model in the
magnetic field:
Z(µ)
p = Sp(e) exp
− β
2m
(p̂(µ)2 + p̂2
z) −
∑
(f)
Ff (p̂(µ), p̂z;β)
= Sp(e) exp
(
−βH̄(µ)
e
)
. (25)
Here, by definition, we have put
H̄(µ)
e (β) :=
1
2m
p̂(µ)2 + V̄ (µ)
e (p̂(µ);β) (26)
23703-6
RPA-approximation of the Bogolubov’s polaron model. Part 1
as an effective polaron Hamiltonian with the operator potential, defined by the expression
exp[−βV̄ (µ)
e (p̂(µ);β)] =
Λ1/3
2π
∫
R
dkz exp
−
∑
(f)
Ff (p̂(µ), kz ;β)
. (27)
Thus, the statistical sum (25) is determined completely by means of the spectrum of the one-particle
two-dimensional self-adjoint problem
H̄(µ)
e (β)ψn = εn(β)ψn , (28)
where the eigenvalues εn(β) ∈ R+, n ∈ Z+, compile the corresponding modified Landau’s spectrum
[12–14] and ψn ∈ L∞(E2; C), n ∈ Z+, are eigenfunctions of a suitable two-dimensional operator
with periodic boundary conditions:
ψn(x, y) = ψn
(
x+ lxΛ−1/3, y + lyΛ
−1/3
)
for all (lx, ly)ᵀ ∈ Z2, (x, y)ᵀ ∈ E2. Then from (25) and (27) it is easy to obtain that
Z(µ)
p =
∑
n∈Z+
exp[−βεn(β)], (29)
where the quantities of the Landau spectrum εn(β) ∈ R, n ∈ Z+, can be found by means of
appropriate quantum-mechanical methods, being already another problem to deal with. The energy
of the ground state for the reduced polaron model at zero temperature (as β → ∞) can be calculated
[1, 13], as the limit E0 = − lim
β→∞
lnZ
(µ)
p , being also a very interesting and important problem.
4. The polaron mass
It is important to mention that the description of our polaron system by means of the Bogolubov
canonical transformation (2) gives rise to a direct possibility of calculating the polaron mass in
magnetic field within our RPA-approximation both at zero and non-zero temperatures. Namely,
based on the considerations of [24], one can consider at zero temperature the least energy state
|p(µ)〉 ∈ L2(Λ; C)⊗Φ(Λ; C) at a small fixed momentum p(µ) ∈ E2, if the interaction constant Lf =√
α|f |−1, f ∈ 2πΛ−1/3Z3, and α ∈ R+ is the corresponding dimensionless intensity parameter.
Then the polaron energy could be defined as
E(α; p(µ)) = E0(α) +
|p(µ)|2
2m∗(α)
+ o
(
|p(µ)|2
)
, (30)
where E0(α) satisfies the eigenvalue equation
Ĥ(µ)
p |p(µ)〉 = E(α; p(µ))|p(µ)〉
under the conditions p̂(µ)|p(µ)〉 = p(µ)|p(µ)〉 for a small vector p(µ) ∈ E2.
In the case of a non-zero temperature T > 0 the polaron free energy is determined by means
of the following expression:
F (α;β) = −β−1 ln
(
Z(µ)
p /Z
(can)
ph
)
, (31)
where, by definition, Z
(can)
ph = Sp(ph) exp(−βĤ(can)
ph ) is the pure statistical sum of the phonon field,
and Ĥ(can)
ph =
∑
(f) 1⊗ (b+f bf + 1/2)~ωf , β = 1/kT . Making use of the result (31) obtained above
one can calculate the average polaron energy as
U(α;β) =
∂
∂β
[βF (α;β)] , (32)
23703-7
N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan
and the proper polaron energy as
E(α;β) = U(α;β) − 3/(2β).
At zero temperature one has:
E(α) = lim
β→∞
E(α;β).
5. Conclusion
The above described one-particle RPA-approximated polaron model in ion crystal can be ap-
plied [14], in particular, for analyzing the results of de Haas-van Alfven type experiments. We
would also like to mention a polaron gas model, having many interesting applications, which can
be similarly investigated by means of powerful methods of the many-particle quantum theory
[1, 13, 14, 22], in particular, by means of the methods of quantum current algebra [15] and its rep-
resentations, as it was recently demonstrated in [20]. Concerning our polaron model and the used
RPA-approximation, the following is important to mention: the Hamiltonian operator resulting
from the canonical Bogolubov transformation (2) was presented in the canonical normal ordered
form as a sum of an exactly solvable operator part and of a part responsible for the manyparticle
correlation interaction. The latter appeared to be of the exactly RPA-approximated form, owing
to which we could neglect it. This approximation can be used whether the crystal temperature is
high enough or the intensity parameter α ∈ R+ is not too small, being α > 5, 8, when a transition
of the polaron from a non-localized state to a self-localized state is realized [23, 24].
As it is well known, the temperature dependence of the effective polaron mass is of great
interest for physical applications, in particular, with its relation to the interpretation of experiments
of cyclotronic resonance in polar crystals [4–6]. Measurements fulfilled in the crystals CdTe and
AgBr in weak magnetic fields at small cyclotronic frequencies demonstrate clearly enough that the
corresponding cyclotronic polaron mass grows with temperature at its low values. Thereby, one
can expect that the calculation of the magnetic polaron mass based on our results within the RPA-
approximation can explain this effect and determine the bounds of the applied method. Moreover,
one can expect that our results will confirm the prediction of the existence [13, 21, 22] of the first
kind phase transition of a polaron state from its self-localized state to a free state under the action
of a strong enough magnetic field. We plan to discuss these intriguing problems in the forthcoming
paper.
6. Supplement
The Bogolubov unitary transformed Hamiltonian operator (5) can be formally written down as
Ĥp := Ĥ(0)
p + V̂(1)
p , (33)
where the operator
V̂(1)
p :=
1
2m
∑
(f,g)
(〈p̂f , ~g〉 + 〈p̂g , ~f〉 + 〈~f, ~g〉1) ⊗ b+f b
+
g bgbf , (34)
is respective for the potential energy of crystal deformations caused by a polaron motion. The
following proposition is crucial for our further analysis of statistical properties of the polaron
model in the Bogolubov representation (33).
Proposition 6.1 The RPA-approximated polaron operator Ĥ(0)
p : L2(Λ; C) ⊗ Φ(Λ; C) and the
crystal deformation energy operator V̂(1)
p : L2(Λ; C)⊗Φ(Λ; C) are commuting to each other, that is
[Ĥ(0)
p , V̂(1)
p ] = 0 (35)
identically.
23703-8
RPA-approximation of the Bogolubov’s polaron model. Part 1
As a corollary from Proposition 6.1, the operators Ĥ(0)
p , V̂(1)
p : L2(Λ; C) ⊗ Φ(Λ; C), being self-
adjoint, possess a common set of eigenstates, being thereby suitably degenerate. The latter can be
used for more exact calculations of the full statistical sum
Zp = Sp exp
[
−β ˆ(H
(0)
p + V̂(1)
p )
]
, (36)
if we take into account the stability condition imposed on the ion crystal deformations. Namely,
we can assume that the following compatible selection conditions
〈(k;α(k))|V̂(1)
p |(k;α(k))〉 = 0, Ĥ(0)
p |(k;α(k))〉 = α(k)|(k;α(k))〉, (37)
hold for all physically permitted polaron eigenstates |(k;α(k))〉 ∈ L2(Λ; C) ⊗Φ(Λ; C) of the RPA-
approximated polaron operator Ĥ(0)
p : L2(Λ; C)⊗Φ(Λ; C). Then the corresponding statistical sum
(36) reduces, owing to constraints (37), to
Zp,red = Sp(red)
(
exp(−βĤ(0)
p ) exp(−βV̂(1)
p )
)
= Sp(red)
(
exp(−βĤ(0)
p )
)
, (38)
where the operation Sp(red)(. . . ) means the trace taken over the above selected states |(k;α(k))〉 ∈
L2(Λ; C) ⊗ Φ(Λ; C).
This problem, being important for suitable applications of the polaron model under regard, is
planned to be treated in detail in a work under preparation.
Acknowledgements
The authors are greatful to their colleagues from the ICTP Center in Trieste, Italy, for valu-
able discussions of the related physical aspects concerning the polaron model approximations.
Authors cordially appreciate the Referee, whose remarks and suggestions proved so instrumen-
tal when preparing the manuscript, Prof. Z. Popowicz (Wroclaw University, Poland) and Prof.
Denis L. Blackmore (NJIT, Newark NJ, USA) for very useful comments and discussions.
References
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6. Brosens F., Devreese J.T., Phys. Stat. Sol., 1988, 145(b), 517.
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13. Feynman R.P., Statistical Mechanics. W.A. Benjamin, 1972.
14. Ishihara A., Statistical Physics. Academic Press, New York, 1971.
15. Bogolubov N.N. (Jr.), Prykarpatsky A.K., Phys. Part. Nuclei, 1986, 17, No. 4, 789–827.
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Systems. Naukova Dumka Publ., Kyiv, 1987 (in Russian).
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tion. World Scientific, 2009.
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Ukraine, Institute of Mathematics Publ., Kyiv, 1995 (in Russian).
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N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan
20. Doria M.M., Menikoff R., Sharp D.H., Phys. Rev. A, 1988, 37, No. 7, 2605–2607.
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Представлення Боголюбова для поляронної моделi та її
повнiстю iнтегроване наближення хаотичних фаз
М.М. Боголюбов (мол.)1,2, Я.А. Прикарпатський3,4, А.А. Казарян5
1 Математичний iнститут iм. Стєклова Росiйської академiї наук, Москва, Росiйська Федерацiя
2 Мiжнародний центр теоретичної фiзики iм. Абдус Салама, Трiєст, Iталiя
3 Математичний iнститут, Кракiвський педагогiчний унiверситет, Польща
4 Дрогобицький педагогiчний унiверситет iм. Iвана Франка, Львiвська область, Україна
5 Московський державний унiверситет iм. Ломоносова, фiзичний факультет, Москва, Росiйська
Федерацiя
Дослiджується модель полярона в iонному кристалi в представленнi Боголюбова, використовуючи
особливе наближення хаотичних фаз. Виводиться та детально описується нова точно розв’язна на-
ближена модель полярона. Аналiтично отримано вiльну енергiю такої моделi при ненульовiй темпе-
ратурi. Також обговорюється вiльна енергiя полярона в постiйному магнiтному полi при ненульовiй
температурi. На основi унiтарного перетворення Боголюбова для поляронного гамiльтонiана отри-
мано дуже важливий новий результат: повна модель полярона є точно розв’язною.
Ключовi слова: полярон, модель полярона, постiйне магнiтне поле, наближення хаотичних фаз
23703-10
Introduction
The RPA-approximated polaron model
RPA-approximated polaron model in the static magnetic field
The polaron mass
Conclusion
Supplement
|
| id | nasplib_isofts_kiev_ua-123456789-32100 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:39:56Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bogolubov (jr.), N.N. Prykarpatsky, Ya.A. Ghazaryan, A.A. 2012-04-08T16:29:49Z 2012-04-08T16:29:49Z 2010 The Bogolubov representation of the polaron model and its completely integrable RPA-approximation / N.N. Bogolubov (jr.), Ya.A. Prykarpatsky, A.A. Ghazaryan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23703: 1-10. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 71.38.+i https://nasplib.isofts.kiev.ua/handle/123456789/32100 The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at finite temperature is calculated analytically. The polaron free energy in the constant magnetic field at finite temperature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable. Досліджено модель полярона в іонному кристалі в представленні Боголюбова, використовуючи особливе наближення хаотичних фаз. Виведено та описано нову точно розв'язну наближену модель полярона. Аналітично одержано вільну енергію такої моделі за ненульової температури. Розглянуто вільну енергію полярона в постійному магнітному полі за ненульової температури. На базі унітарного перетворення Боголюбова для поляронного гамільтоніана одержано важливий новий результат: повна модель полярона є точно розв'язною. The authors are greatful to their colleagues from the ICTP Center in Trieste, Italy, for valuable discussions of the related physical aspects concerning the polaron model approximations. Authors cordially appreciate the Referee, whose remarks and suggestions proved so instrumental when preparing the manuscript, Prof. Z. Popowicz (Wroclaw University, Poland) and Prof. Denis L. Blackmore (NJIT, Newark NJ, USA) for very useful comments and discussions. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The Bogolubov representation of the polaron model and its completely integrable RPA-approximation Представлення Боголюбова для поляронної моделі та її повністю інтегроване наближення хаотичних фаз Article published earlier |
| spellingShingle | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation Bogolubov (jr.), N.N. Prykarpatsky, Ya.A. Ghazaryan, A.A. |
| title | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation |
| title_alt | Представлення Боголюбова для поляронної моделі та її повністю інтегроване наближення хаотичних фаз |
| title_full | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation |
| title_fullStr | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation |
| title_full_unstemmed | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation |
| title_short | The Bogolubov representation of the polaron model and its completely integrable RPA-approximation |
| title_sort | bogolubov representation of the polaron model and its completely integrable rpa-approximation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32100 |
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