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Received July 12, 2017, in final form August 5, 2017
Abstract

The theory of renormalized energy spectrum of localized quasi-particle interacting with polarization phonons at finite temperature is developed within the Feynman-Pines diagram technique. The created computer program effectively takes into account multi-phonon processes, exactly defining all diagrams of mass operator together with their analytical expressions in arbitrary order over the coupling constant. Now it is possible to separate the pole and non-pole mass operator terms and perform a partial summing of their main terms. The renormalized spectrum of the system is obtained within the solution of dispersion equation in the vicinity of the main state where the high- and low-energy complexes of bound states are observed. The properties of the spectrum are analyzed depending on the coupling constant and the temperature.

\keywords

quasi-particles, phonon, Green’s function, mass operator

\pacs

71.38.-k, 63.20.kd, 63.20.dk, 72.10.Di

Abstract

На основi дiаграмної технiки Феймана-Пайнса запропонована теорiя перенормування енергетичного спектру локалiзованої квазiчастинки, яка взаємодiє з поляризацiйними фононами при скiнченнiй температурi. Розроблена комп’ютерна програма, яка ефективно враховує багатофононнi процеси, точно визначаючи усi дiаграми масового оператора i їх аналiтичнi вирази у довiльному порядку за константою зв’язку. Це дозволило розмежувати неполюснi й полюснi доданки масового оператора та виконати парцiальне пiдсумовування їх головних складових. Розв’язанням дисперсiйного рiвняння отримано перенормований спектр системи в околi основного стану, де спостерiгаються високо- та низькоенергетичнi комплекси зв’язаних станiв. Проаналiзовано властивостi спектру в залежностi вiд константи зв’язку й температури. \keywordsквазiчастинка, фонон, функцiя Грiна, масовий оператор

\issue

201720443706 \doinumber10.5488/CMP.20.43706 Energy spectrum of localized quasiparticles]Energy spectrum of localized quasiparticles renormalized by multi-phonon processes at finite temperature M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti]M.V. Tkach111E-mail: ktf@chnu.edu.ua , O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti \addressChernivtsi National University, 2 Kotsyubinsky St., 58012 Chernivtsi, Ukraine

1 Introduction

Obviously, there are many reasons that despite of the long period of its successful development, the theory of the interaction of quasi-particles (electrons, excitons and so on) with quantized fields (phonons, photons) remains in the center of physicists attention. One of them is that the new phenomena permanently discovered in physics (superfluidity, superconductivity, high-temperature superconductivity and so on) in the classic 3d and 2d structures [1, 2, 3] and, further [4, 22] in low-dimensional heterostructures (in particular, nano-structures) demanded a deep understanding of the specific interaction between quasi-particles proper or with different fields and urged the development of mathematical methods in theoretical physics.

A fundamental progress in the theory of interaction of quasi-particles-quantized fields in condensed materials has been achieved late in the twentieth century when the methods of quantum field theory were developed [6, 7], in particular, the Feynman diagram technique in the method of thermodynamic Green’s functions [2, 8, 9]. However, this universal and powerful mathematical method is not devoid of the difficulties at the stage of its direct application to the specific problems where the perturbation method or the variational one is not applicable. For instance, the difficulties associated with the peculiar problem of sign [10] appear when the excited hybrid states of the system [11] are studied and it is necessary to take into account the multi-phonon processes. In any representation, except the Matsubara one, the expanded Green’s function in the representation of interaction contains complex terms with the sign that changes in general case. Hence, the mathematical structure of mass operator (MO) ranges becomes complicated. Therefore, in order to reliably define the spectrum of a quasi-particle interacting with phonons in a wide range of energies, it is not enough to account for the finite number of diagrams, but it is necessary to perform a partial summing of an infinite number of diagrams, if not all of them, then at least the main ones.

The long period of a lack of reliable and accurate data on the main and excited states in macroscopic structures has stimulated a search for new approaches to the mathematical methods of their research. This, in particular, contributed to the creation of diagrammatic Monte Carlo method, which together with the method of stochastic optimization made it possible to calculate the Green’s functions of quasi-particles in different models [10, 13, 12, 14, 15, 16, 17, 18, 19, 20, 21] almost exactly and to get reliable data where they were incomplete or very controversial [10], like in the concept of the relaxed excited states. This method does not require any explicit ‘‘circumcision’’ of the orders of the ranges of Green’s functions [10] and satisfies the central limit theorem, since, using a sufficient reserve of memory of modern computers, one can always achieve an insignificantly small effect of the system fluctuations and obtain an almost exact result.

Modern computers significantly expanded the scope of applying the Feynman diagram technique in the method of Green’s functions for the problems of quasi-particles-phonons interaction, especially in the cases where it is necessary to consider the multi-phonon processes [3, 11, 10]. Recently, using a modified Feynman-Pines diagram technique, computer calculations of the leading classes of MO diagrams were performed [22]. Herein, a physically correct picture was established, which was missing earlier, for the ground and excited states of Frohlich polaron in the regime of a weak electron-phonon coupling at  K.

A renormalized spectrum of a localized quasiparticle weakly   interacting   with   phonons  at  K was obtained in [23] using the Feynman-Pines diagram technique in the general model with the Frohlich Hamiltonian. It was shown that the new complexes of bound states existed in the system besides the ground and bound states which were known earlier [1, 22] from the simplified model with the additional condition for the operators of quasi-particles. The energy spectrum was calculated using the MO, which contained all diagrams till the third order. Of course, this was not enough to identify the leading diagrams for their partial summing. However, it was impossible to hope for the analytical calculation of high-order diagrams without a computer, since their number in the -th order was proportional to .

In this paper, we use a new computer program for analytical calculation of MO in such a high order over the coupling constant, which is limited by the computer resource only. Although we take into account all diagrams till the tens order, using a personal computer, this is enough for a partial summing of the main MO diagrams. As a result, we confirm the existence of previously known and new states of a quasiparticle interacting with phonons and analyze the temperature dependence of the spectrum.

2 Hamiltonian of the system. Multi-phonon processes and structure of complete MO at  K

Localized dispersionless quasiparticles (excitons, impurities and so on) interacting with dispersionless polarization phonons are described by Frohlich Hamiltonian, like in [23]

(1)

where and are the energies of quasi-particles and phonons, respectively, which can be arbitrary in the typical ranges ( meV,  meV) for the solid states and nano-structures. It turns out that the form of binding function is irrelevant because the energy of quasi-particle-phonon interaction is uniquely characterized by a constant independent of , also arbitrary in the natural range, since the problem becomes zero-dimensional. The operators of second quantization of quasi-particles () and phonons () satisfy the Bose commutative relationships.

The renormalized spectrum of quasi-particles interacting with phonons at an arbitrary temperature (), like in [23], is obtained from the poles of Fourier image of quasi-particle Green’s function , which through the Dyson equation

(2)

is related with MO . It is calculated using the Feynman-Pines diagram technique when the concentration of localized quasi-particles is small. Introducing the dimensionless variables and values

(3)

the Dyson equation (2) is rewritten as

(4)

As far as the problem becomes ‘‘zero-dimensional’’, the complete MO is defined by the infinite sum of indexed diagrams (figure 1). The simple rules for these diagrams presented in [23], allow their definite programming and numerical calculation of all topologically unequal diagrams and their analytical expressions till such a high order over the powers of a coupling constant (), which is limited by a computer resource only (appendix A). We should note that in this paper all diagrams and their analytical expressions for the MO terms till the tenth order are obtained. Although in figure 1, all first terms of MO, including the third order, are presented for illustration and for further understanding.

Figure 1: Diagrammatic representation of MO terms in the first three orders of powers.

The rules of conformity between indexed diagrams and analytical expressions are simple:

(5)

where number is the sum of phonon lines with the arrows directed to the right (with the sign ‘‘’’) and with the arrows directed to the left (with the sign ‘‘’’) over the quasi-particle line (figure 1), is an average phonon occupation number. Thus, the analytical expression for the arbitrary diagram with indices is written as a product of contributions of all its lines and tops. For example, the analytical expressions are as follows:

 . (6)

A detailed analysis of the structure of complete MO, figure 1, till high orders over the powers of , , (1+) shows that it can be written in an analytical form which contains three types of terms. The terms with the symbol ‘‘’’: describe the processes of quasi-particle scattering accompanied by the prevailing creation of phonons, the terms with the symbol ‘‘’’: describe the processes of quasi-particle scattering accompanied by the prevailing annihilation of phonons and . Hence, the infinite ranges of complete MO terms can be written as

(7)

In this formula, the upper and the lower terms of the complete MO are defined by the respective infinite ranges of the terms , to which the diagrams without crossing phonon lines with arrays directed only to the right () and only to the left () correspond. These diagrams describe the unmixed (u) processes of quasi-particles scattering accompanied either only by the creation () or by annihilation () of phonons, respectively. After the partial summing, an exact analytical representation for these terms in the form of infinite chain fractions was obtained [23].

(8)

where

(9)

The rest terms of MO (7)

(10)

describe the mixed (m) processes of quasi-particle scattering accompanied by an equal (e) number of the created and annihilated phonons

(11)

by a prevailing number of the created phonons rather than annihilated ones

(12)

and by a prevailing number of annihilated phonons rather than the created ones

(13)

The functions and are polynomials, diagrammatic and analytical forms of which are numerically calculated within the computer using the diagram technique. Till the sixth order over the coupling constant (), the expressions for these functions are presented in appendix B, though in this paper we took into account all diagrams till the tenth order.

Finally, the complete MO (7) now has the form

(14)

where

(15)

Contrary to the unmixed terms (9), the partial summing in the mixed ones (11)–(2) is, in general, impossible. However, using the newly revealed analytical structure, this becomes possible for the arbitrary energies, highlighting in these MO terms the pole and non-pole factors and expanding them in Taylor series.

3 Partial summing in mixed MO using the non-pole factors expanded in Taylor series

For a system having a weak quasi-particle-phonon interaction () at the temperature when , we observe the range of energies () where, according to physical considerations, one should expect the renormalized energies of the main state and its nearest satellite bound-with-phonon states. In order to obtain them, both in high- and low-energy regions, we perform a partial summing in mixed MO (11)–(2).

First of all, we observe in the range , selecting pole and non-pole factors in it. Formula (11) proves that MO terms MO in this energy range can be submitted as a product

(16)

where one can see a multiplier independent of , pole multiplier and non-pole fractional-rational functions

(17)
(18)

which in the vicinity of are expanded into the Taylor series

(19)

where are the known numerical coefficients.

Substituting (17), (18) in (14) we see that now MO can be written as a sum of pole and non-pole terms

(20)

Similar analytics gives

(21)
(22)

where

(23)

Substituting formulae (3)–(22) in (11)–(2) we select pole and non-pole terms in the mixed MO and , in the range (). Taking into account formula (15), they are obtained in the range (). Thus, according to (10), a complete mixed MO is written as the sum of pole and non-pole term

(24)

The expression for obtained in the form of separated pole and non-pole terms gives an opportunity to perform a partial summing of their infinite ranges. It is clear that since the maximal order () of functions is limited by the maximal order of the accounted diagrams over , then in the complete one should take into account not more than (in our case ) mixed renormalized MOs []. The sums over quickly converge because -th terms of this MO contain small multipliers and . Avoiding sophisticated analytics, considering the symmetric relationship (15) and a similar structure of at all values, we demonstrate the method of partial summing by the example of MO in the range ().

In table 1 we present explicit analytical expressions for terms. Solid lines separate pole and non-pole terms. It is clear that the pole terms of can be written as a sum of and terms obtained within the partial summing of the pole terms and , only

(25)

where

(26)
     0  1    2    3    4   5   6

 0
 0
 0     0
 0     0
 0     0   0
 0     0   0
 0     0   0
 0     0   0

 0     0   0
Table 1: The terms of MO in the range ).

The main idea of partial summing of the main terms of pole MO is as follows: using the -th column, table 1, we study the analytical regularities of terms at small values and obtain an exact analytical expression for this MO at an arbitrary . Then, after the analytical summing of this infinite range, we obtain an exact analytical equation for renormalized MO , which is not limited by a finite number of accounted diagrams of the respective type. For example, at using table 1, we obtain

(27)

The renormalized of a higher order over are obtained in the same way. Still relatively not complicated MO at and have the form

(28)
(29)

By analogy, similar tables can be obtained at , from which the analytical expressions for are further calculated. For instance, we present table 2 at . We should note that if the values of and