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


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][8][9].Quantum mechanical tunneling between different lattice sites restores this symmetry and ensures that a self-trapped electron forms [10,11] an itinerant polaronic quasiparticle.
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 Λ ⊂ E 3 can be described [1] by means of the Hamiltonian operator acting in the Hilbert space L 2 (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, p := i ∇ is its momentum operator, b + f and b f , f ∈ 2πΛ −1/3 Z 3 , are, respectively, Bose-operators of creation and annihilation of phonons with energy is an intensity parameter of the polaron bond in the crystal and ., . is the ordinary scalar product in the Euclidean space E 3 .
As it was also shown in [1,4], Hamiltonian (1) can be transformed by means of the unitary transformation into the following form: where it is clearly seen a quantum nature of the polaron structure, which does not depend on the force of the interaction parameter L f .This conclusion can be also made from the fact that model (1) possesses a conservation law of the general electron-phonon momentum: that is [ Ĥp , P ] = 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 where we made use of the tensor operator representation p 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 K 0 r 2 /2, where , was added into the initial Hamiltonian.Under these conditions, the resulting Hamiltonian persists to be translationalinvariant 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 V(1) being written in the normally ordered secondly quantized form, gives rise to the following effective twoparticle operator expression in an N -particle invariant Fock subspace [14,16,17,19]: acting in the Hilbert space L 2 (Λ; C) ⊗ L 2,sym (Λ N ; C), where we denoted by Π(y), y ∈ Λ, the respectively modified momentum operator of the crystal deformations and p(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 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 RPAapproximation 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: being a well defined operator expression bounded from below in the Hilbert space L 2 (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 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, −mω c x, 0) ∈ E 3 is directed along the axis Oy of the Euclidian space E 3 and ω c ∈ R + denotes the corresponding "cyclotronic" frequency.In this case Hamiltonian (5) has the form where by definition p(µ) f = (p fx , pfy + mω c x) , f = (f x , f y ) , pertaining to the quadratic structure of the phonon operators.

The RPA-approximated polaron model
To study the thermodynamics of the RP A-approximated polaron model ( 5), we need to calculate the statistical sum where β = 1/kT is the inverse temperature in the Boltzmann units.Then, taking into account that where the operators Here, by definition, we denoted the "phonon" part of the statistical sum as 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 transforms it to the canonical diagonal form Here we need to mention that the operator ωf : L 2 (Λ; C) → L 2 (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 ω f = ω −f for all vectors f ∈ 2πΛ −1/3 Z 3 .A similar symmetry condition also holds for the operator quantities ωf , that is ωf = ω+ −f for all f ∈ 2πΛ −1/3 Z 3 .Based on representation (13), it is easy to find the following expression for the phonon part of statistical sum (11): where we denoted by Ñf := b+ f bf , f ∈ 2πΛ −1/3 Z 3 , the shifted phonon density operators and ωf (p, , 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: where we used the known trick [1,13,14] of changing the discrete sum (k) (. . . ) by a corresponding integral: Thus, if we calculate the internal sums (f ) (. . . ) of expression (15) for the given values of phonon frequencies ω f and the interaction parameter L f for all discrete values f ∈ 2πΛ −1/3 Z 3 , then the quasi-gaussian integral E 3 (. . .)d 3 k can be calculated analytically or by means of approximate and asymptotic methods.In particular, assuming that L f = α 1/2 /|f |, and ω f = ω 0 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.

RPA-approximated polaron model in the static magnetic field
To study RPA-approximated polaron model in the static magnetic field we will use the Hamiltonian operator (8), which persists to be quadratic in the phonon operators.Taking into account that where the operators (p commute to each other.Put, by definition, for f = (f x , f y ) ∈ E 2 , the respectively split statistical sum can be successfully calculated, where Let us, first, find the statistical sum for which, using ( 19) and ( 21), we obtain Using now the fact that Hamiltonian ( 18) is quadratic in the phonon variables and making transformation (12), we obtain Thus, from ( 23) and ( 21), one obtains that where ωf (p 24) into (22), we obtain an expression for the statistical sum of RPA-approximated polaron model in the magnetic field: Here, by definition, we have put as an effective polaron Hamiltonian with the operator potential, defined by the expression exp Thus, the statistical sum (25) is determined completely by means of the spectrum of the one-particle two-dimensional self-adjoint problem where the eigenvalues ε n (β) ∈ R + , n ∈ Z + , compile the corresponding modified Landau's spectrum [12][13][14] and ψ n ∈ L ∞ (E 2 ; C), n ∈ Z + , are eigenfunctions of a suitable two-dimensional operator with periodic boundary conditions: for all (l x , l y ) ∈ Z 2 , (x, y) ∈ E 2 .Then from ( 25) and ( 27) it is easy to obtain that where the quantities of the Landau spectrum ε n (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 E 0 = − lim β→∞ ln Z p , being also a very interesting and important problem.

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) ∈ L 2 (Λ; C) ⊗ Φ(Λ; C) at a small fixed momentum p (µ) ∈ E 2 , if the interaction constant L f = √ α|f | −1 , f ∈ 2πΛ −1/3 Z 3 , and α ∈ R + is the corresponding dimensionless intensity parameter.Then the polaron energy could be defined as where E 0 (α) satisfies the eigenvalue equation Ĥ(µ) p |p (µ) = E(α; p (µ) )|p (µ)   under the conditions p(µ) |p (µ) = p (µ) |p (µ) for a small vector p (µ) ∈ E 2 .
In the case of a non-zero temperature T > 0 the polaron free energy is determined by means of the following expression: where, by definition, Z we can calculate expression(9) classically, having reduced it to the form Z (0) p = Sp(−βH (0) p ) = Sp (e) Z