# Generalized method of Feynman-Pines diagram technique in the theory of
energy spectrum of two-level quasiparticle renormalized due to multi-phonon
processes at cryogenic temperature

M.V. Tkach 111E-mail<EMAIL_ADDRESS>O.Yu. Pytiuk, O.M. Voitsekhivska,
Ju.O. Seti

(Received June 22, 2018, in final form July 31, 2018)

###### Abstract

Theory of the spectrum of localized two-level quasi-particle renormalized due
to interaction with polarization phonons at cryogenic temperature is developed
using the generalized method of Feynman-Pines diagram technique. Using the
procedure of partial summing of infinite ranges of the main diagrams, mass
operator is obtained as a compact branched chain fraction, which effectively
takes into account multi-phonon processes. It is shown that multi-phonon
processes and interlevel interaction of quasiparticle and phonons cardinally
change the renormalized spectrum of the system depending on the difference of
energies of two states, which either resonates with phonon energy or does not.
The spectrum of non-resonant system contains renormalized energies of the main
states and two similar infinite series of groups of phonon satellite levels.
The spectrum of a resonant system contains a renormalized ground state and
infinite series of satellite groups.

Key words: diagram technique, quasi-particle, mass operator, phonon, spectrum

PACS: 71.38.-k 72.10.Di 63.20.kk

###### Abstract

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áåçìåæíèõ ðÿäâ îñíîâíèõ äàãðàì îòðèìàíî ìàñîâèé îïåðàòîð ó êîìïàêòíîìó âèãëÿä
ðîçãàëóæåíîãî ëàíöþãîâîãî äðîáó, ÿêèé åôåêòèâíî âðàõîâó áàãàòîôîíîíí ïðîöåñè.
Ïîêàçàíî, ùî áàãàòîôîíîíí ïðîöåñè ìæðâíåâà âçàìîäÿ êâàçàñòèíêè ç ôîíîíàìè
êàðäèíàëüíî çìíþþòü ïåðåíîðìîâàíèé ñïåêòð ñèñòåìè ó çàëåæíîñò âä òîãî, è
ðçíèöÿ åíåðãé ìæ îáîìà ñòàíàìè íåâçàìîäþî¿ êâàçàñòèíêè ðåçîíó, è íå ðåçîíó ç
ôîíîííîþ åíåðãþ. Ñïåêòð íåðåçîíàíñíèõ ñèñòåì ìñòèòü ïåðåíîðìîâàí åíåðã¿
îñíîâíèõ ñòàíâ äâ ïîäáí ìæ ñîáîþ áåçìåæí ñåð¿ ãðóï ôîíîííèõ ñàòåëòíèõ ðâíâ.
Ñïåêòð ðåçîíàíñíèõ ñèñòåì ìñòèòü ïåðåíîðìîâàíèé îñíîâíèé ðâåíü áåçìåæíó ñåðþ
ñàòåëòíèõ ãðóï ìóëüòèàíòèêðîñèíãâ.

Ключов слова: äàãðàìíà òåõíêà, êâàçàñòèíêà, ìàñîâèé îïåðàòîð, ôîíîí, ñïåêòð

##  1 Introduction

The theory of energy spectra of quasiparticles (electrons, holes, impurity
centers, excitons, etc.) interacting with phonons attracts a permanent
attention due to the fact that phonons, being a dissipative sub-system at
arbitrary temperature, essentially effects the physical processes and
phenomena. During the starting period, the states of the systems, renormalized
due to phonons, with the energies close to the energies of uncoupling quasi-
particles have been intensively studied [1]. From the physical considerations
it was clear that in the vicinity of so-called threshold energies (i.e., the
order of quasi-particles energy plus one phonon energy) the complexes of free
or bound-to-phonon states could exist. However, the lack of experimental data
and mathematical problems of the theory did not contribute to the active
research of even weak excited states. When a sufficient number of experimental
papers on electron-phonon complexes, cyclotron resonance, impurity centers,
etc. [2, 3, 4, 5, 6, 7, 8, 9, 10] had already appeared, the theoretical
investigations of the properties of the spectra of quasi-particles with the
energies close to the radiation threshold of one phonon became much more
active. The main theoretical results obtained during this period are presented
in the review [11]. Here, the near-threshold phenomena are analyzed in detail
within the framework of one-phonon model and approximation of vertices,
defined by the diagrams of mass operator, which contains dangerous crossings
only over one-quasiparticle and one-phonon lines.

The approximate methods of quantum field theory which are applied even in a
“schematic approach” to calculate the vertices [11], revealed three types of
states for the impurity centers interacting with polarization phonons. The
polaron states with a small effective number of phonons ($\langle N\rangle$)
in it can be observed far from resonance, hybrid states ($\langle N\rangle\sim
1/2$), corresponding to the dielectric modes known from the experiment — in
the resonance and bound state of quasiparticle plus phonons ($\langle
N\rangle\sim 1$) — can be observed near the resonance.

Usually, studying the spectra of quasi-particles interacting with phonons in
the interval of energies that do not significantly exceed the threshold of
phonon radiation, it was enough to use a one-phonon approximation. The poorly
substantiated “schematic” approaches taking account of multi-phonon processes
were used only in certain cases. In the systems with intermediate and strong
coupling and in a wide interval of energies, the above mentioned processes
play a dominant role. This requires the application of powerful methods of
quantum field theory, such as diagonalyzing the Hamiltonians using the exact
and approximated unitary transformations, Feynman and Matsubara diagram
technique and so on [12, 13, 14, 15]. Within these methods, the problem of the
spectrum of one-level localized quasi-particle interacting with polarization
phonons at $T=0$ K was solved exactly. It was shown that this spectrum
contained the ground state shifted into the low-energy region at a quantity
proportional to the coupling constant and infinite number of excited states
with equidistant energies (of one-phonon energy order) in the high-energy
region.

The same problem at $T\neq 0$ K could not be solved. Only at an additional
condition, when the probability of quasiparticle location in an arbitrary
state was either zero or one, the energy spectrum of the system was obtained
[13]. It did not depend on temperature at all and contained a ground state
which was shifted into the low-energy region and equidistant levels of bound
states in high- and low-energy regions. For physical reasons, this result is a
strange artifact of the theoretical model with an additional condition.

For a long time the problem remained unsolved but recently there was proposed
an approach [16], which, using the modified method of Feynman-Pines diagram
technique without artificial additional conditions, allowed taking into
account not only virtual but also “real” multi-phonon processes. It is
revealed that the complete renormalized spectrum of the system depends on
temperature within the averaged phonon occupation numbers. Besides the ground
state and quasi-equidistant satellite states, it contains new non-equidistant
bound states of quasiparticle with many phonons. The idea of a correct method
of solution appeared at the base of the known new approaches, which were
proposed and used in the theory of polarons and high-temperature
superconductivity [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The
diagrammatic Monte Carlo method and stochastic optimization (DMC) [17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27] and bold diagrammatic Monte Carlo (BDMC) [28,
29] played an important role in investigation of high excited states and
revealed the reasons of some questionable results obtained for the electron-
phonon interaction (for example, in the concept of a relaxed excited state).
These methods are essentially based on a computer algorithm of the calculation
of high-order diagrams of mass operators of Matsubara Green’s functions.
Therefore, one should expect that in the theory of renormalized spectrum of
localized quasiparticles interacting with polarization phonons at $T\neq 0$ K,
there must exist the computer algorithm for constructing Feynman diagrams of
mass operator and their analytical expressions of such high orders that allow
a partial summing of its main continuous series. Such a computer program was
presented in paper [16].

As can be seen, the development of the theory of interaction of quasi-
particles with phonons was almost always performed on the basis of a single-
band model. However, recently there appeared an information which proves that
the physics of multi-band models that describe the interacting electron-phonon
systems is significantly different from single-band models, [29, 30].

The rapid development of experimental and theoretical nano-physics during the
last decade stimulated a scrupulous study of low-dimensional heterostructures,
consisting of spatially confined structures (zero-dimensional quantum dots,
one-dimensional quantum wires, two-dimensional quantum layers) embedded into
the external bulk medium. A strong space confinement causes the quantization
of quasiparticle (electrons, holes, excitons, impurities, etc.) spectra and
the appearance of new types of phonons — confined, half-space, interface, and
propagating ones [31, 32, 33, 34].

At first, scientists studied the properties of quasiparticles and phonons
separately and then they took into account their interaction. The theory of
quantum transitions between two electron bands under the effect of
electromagnetic field and considering the interaction with phonons was mostly
developed in one-phonon approximation without taking account of inter-level
(inter-band) interaction [35, 36].

Even the appearance of the unique devices, such as quantum cascade lasers
(QCL) and quantum cascade detectors (QCD) whose main operating elements are
the cascades of multi-layered quantum wells, where the relaxation of electron
energy occurs due to the multi-phonon processes, did not essentially stimulate
the development of the theory of electron-phonon interaction in multi-band
models. It was considered that an especially important relaxation of the
electron energy in QCD was effectively provided by the radiation of single
phonons. They accompanied the “skip” of electrons from the upper level of the
active region of the previous cascade (through the “phonon ladder” of the
extractor) to the lower level of the active region of the next cascade [37,
38]. Of course, this process plays an important role, but it is not the only
one, since, as it was recently shown in experimental studies [39, 40, 41], QCD
effectively operates with the “torn phonon ladders”, where there is not a
required number of equidistant electron levels to create a “complete phonon
ladder” in extractor wells.

One of the mechanisms that can complement the “torn phonon ladder” of the
extractor may be the quantum transitions through the high excited satellite
states of the electron-phonon system of a cascade. In order to find out
whether such a mechanism could ensure a successful operation of QCD with the
“torn phonon ladder”, one should theoretically study the quantum transitions
between the states of the system renormalized due to electron-phonon
interaction under the action of an electromagnetic field using the multi-band
model and taking into account the multi-phonon processes. As far as we know,
this problem is not solved yet. Moreover, due to the well-known mathematical
difficulties in quantum field theory, a consistent theory of the renormalized
energy spectrum of such a system has not yet been established.

In this study, we propose the generalized Feynman-Pines diagram technique,
which makes possible the correct accounting of multi-phonon processes in the
calculation of the renormalized energy spectrum of a two-level localized
quasiparticle interacting with dispersionless phonons at $T=0$ K. As will be
seen, the applied approach, firstly, will provide new interesting data on the
properties of the renormalized spectrum of the main and satellite states of
the system, and, secondly, it can be extended to the systems of multi-band
quasi-particles interacting with phonons.

The rest of the paper is organized as follows: in section 2, the Hamiltonian
of the system is introduced and the rules of generalized Feynman-Pines diagram
technique are formulated for the matrix of mass operator (MO) of quasiparticle
Green’s function. In section 3, the principle of a sequential partial summing
of the MO diagrams and its representation in the form of a infinite branched
chain fraction is proposed. In section 4, this method is used for the
calculation of the renormalized spectrum of ground state and phonon complexes
of satellite states of the system. The obtained results and conclusions are
presented in section 5.

##  2 Hamiltonian of the system. Mass operator of quasiparticle Green’s
function at $T=0$ K

The system of localized two-level quasiparticle interacting with polarization
phonons is described by Hamiltonian of Fröhlich type, like in [5, 12, 13]

$\widehat{H}=\sum_{\mu=1}^{2}E_{\mu}a_{\mu}^{+}a_{\mu}+\sum_{\vec{q}}\Omega{\rm\;}\Big{(}b_{\vec{q}}^{+}b_{\vec{q}}+\frac{1}{2}\Big{)}+\sum_{\vec{q}}\sum_{\mu_{1},\mu_{2}=1}^{2}\varphi_{\mu_{1}\mu_{2}}a_{\mu_{1}}^{+}a_{\mu_{2}}\big{(}b_{\vec{q}}+b_{-\vec{q}}^{+}\big{)}.$
(2.1)

Here, $E_{\mu}=E+d_{\mu}$ are the energies of uncoupling quasiparticle
(further, $d_{\mu=1}=0$, $d_{\mu=2}=d$), $\Omega$ is an energy of polarization
phonon, $\varphi_{\mu_{1}\mu_{2}}$ are the binding constants characterizing
the intra-level (at $\mu_{1}=\mu_{2}$) and inter-level (at
$\mu_{1}\neq\mu_{2}$) interaction. All these quantities are assumed as known
parameters of the system. The operators $a_{\mu},\,a_{\mu}^{+}$ and
$b_{\vec{q}},\,b_{\vec{q}}^{+}$ satisfy Bose commutative relationships.

The energy spectrum of the system, renormalized due to the interaction, at
cryogenic temperature (formally, at $T=0$ K), is obtained using the method of
Feynman-Pines diagram technique [6, 12, 14] for the Fourier image of casual
Green’s function [$G_{\mu\mu}(\omega)$] of quasiparticle. According to the
general theory [14] and taking into account the Hamiltonian (2.1), the matrix
functions $G_{\mu\mu^{\prime}}(\omega)$ satisfy the system of two equations.

$G_{\mu\mu^{\prime}}(\omega)=(\omega-E_{\mu}+\rm
i\eta)^{-1}\bigg{[}\delta_{\mu\mu^{\prime}}+\sum_{\mu_{1}=1}^{2}M_{\mu\mu_{1}}(\omega)G_{\mu_{1}\mu^{\prime}}(\omega)\bigg{]},\qquad\mu,\mu^{\prime}=1,2;\qquad(\eta\to+\infty)\,,\vspace{-1mm}$
(2.2)

where $M_{\mu\mu_{1}}(\omega)$ is a matrix MO.

The further calculations are performed using the dimensionless functions,
variables and constants

$\begin{array}[]{l}{g_{\mu\mu^{\prime}}(\xi)=G_{\mu\mu^{\prime}}\Omega;\qquad{\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)=M_{\mu\mu^{\prime}}\Omega^{-1};\qquad\xi=(\omega-E)\Omega^{-1}};\\\
{\xi_{\mu}=(\omega-
E_{\mu})\Omega^{-1};\qquad\alpha_{\mu_{1}\mu_{2}}=\varphi_{\mu_{1}\mu_{2}}\Omega^{-1};\qquad\quad\delta=d\frac{}{}\Omega^{-1}.}\end{array}$
(2.3)

Then, the exact solution of equation (2.2) for the dimensionless function of
$m$-th level $g_{\mu}(\xi)=g_{\mu\mu}(\xi)$ is defined by Dyson equation
($\hbar=1$)

$g_{\mu}(\xi)=\big{[}\xi_{\mu}-{\Large{\textbf{m}}}_{\mu}(\xi)\big{]}^{-1}$
(2.4)

through the complete MO ${\Large{\textbf{m}}}_{\mu}(\xi)$ of this level, which
can be written as a sum of two components

${\Large{\textbf{m}}}_{\mu}(\xi)={\Large{\textbf{m}}}_{\mu\mu}(\xi)+{\Large{\textbf{m}}}_{\mu\mu}^{g}(\xi).$
(2.5)

Here, the component ${\Large{\textbf{m}}}_{\mu\mu}(\xi)$ is determined only by
the diagonal matrix elements of the complete MO
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$, while the component
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{g}(\xi)$ also contains non-diagonal
components
$\big{[}{\Large{\textbf{m}}}_{12}(\xi),{\Large{\textbf{m}}}_{21}(\xi)\big{]}$
of the same MO, which appear due to the non-diagonal elements of
$g_{\mu\mu^{\prime}}(\xi)$ matrix. Hence, the components
${\Large{\textbf{m}}}_{11}^{g}(\xi)$ and ${\Large{\textbf{m}}}_{22}^{g}(\xi)$
are written as follows:

${\Large{\textbf{m}}}_{11}^{g}(\xi)=\frac{{\Large{\textbf{m}}}_{12}(\xi){\Large{\textbf{m}}}_{21}(\xi)}{\xi_{2}-{\Large{\textbf{m}}}_{22}(\xi)}\,,\qquad{\Large{\textbf{m}}}_{22}^{g}(\xi)=\frac{{\Large{\textbf{m}}}_{12}(\xi){\Large{\textbf{m}}}_{21}(\xi)}{\xi_{1}-{\Large{\textbf{m}}}_{11}(\xi)}\,.$
(2.6)

The matrix ${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$ of the complete MO is
defined by the rules of Feynman-Pines diagram technique [14], which are
generalized for the case of multi-level systems. The energies in Hamiltonian
(2.1) are dispersionless, contrary to the classic Fröhlich one-band
Hamiltonian [5, 12, 14], where the energy of uncoupling quasiparticle is a
function of quasi-momentum and, consequently, its MO contains diagrams both
without and with the crossing phonon lines. Thus, the equivalent diagrams with
and without crossing phonon lines correspond to the identical analytical
expressions. Hence, in dimensionless variables (2.3), MO
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$ is obtained in such diagrammatic
form which contains all possible non-equivalent diagrams without crossing
phonon lines. The number of equivalent diagrams of this type is given by the
integers before the respective diagrams.

$\hskip 2.84526pt.$ (2.7)

All MO diagrams in an arbitrary $p$-th order over the power of pairs of
coupling constants, which are the same as the number of phonon lines in a
respective diagram, are definitely calculated using a computer program.
Further on it will become clear that in order to establish the exact rules of
partial summing of infinite ranges of diagrams, it is quite sufficient to take
into account all the first 64 diagrams to the seventh order inclusive.

The analytical expression for the arbitrary MO diagram is obtained as a sum
over all inner indices ($\mu_{1},\mu_{2},\dots,\mu_{N}$), except the outer
indices ($\mu,\mu^{\prime}$), of the products of all vertices and solid lines

(2.8)

where $p$ is a number of dashed (phonon) lines placed above the solid
(quasiparticle) line with index $\mu_{\textit{l}}$. For example, the first two
components of ${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$ have the following
diagrammatic and analytical forms

(2.9) (2.10)

##  3 Partial summing of MO diagrams

For partial summing of diagrams in the complete MO
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$, it is convenient to group them
into classes which are separated by brackets in expression (2.7). It is clear
that the $p$-th class of diagrams together with the factor $p!$ contains an
infinite number of only exactly those diagrams (together with numerical
factors), whose arbitrary blocks, in their turn, contain not more than p
dashed lines over any solid line. Such a class of diagrams is further referred
to as a partial $p$-phonon MO and is denoted as
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p]}(\xi)$. For example, the partial
2-phonon MO is

$\hskip 2.84526pt.$ (3.1)

Then, the complete MO ${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)$ can be
written as follows:

${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)={\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(1)}(\xi)+\sum_{p=2}^{\infty}{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p]}(\xi)\,.$
(3.2)

In each diagram of the MO, the analytical contributions of its elements are
summed up over all $\mu_{l}$ indices, since, in general form, none of them is
multiplicative. However, the analytical expressions show that each diagram can
be expressed as a sum of two components: multiplicative (m) and non-
multiplicative (nm) one. For this sake, in any diagram, in which summing is
performed over all inner indices (for example, $\mu_{1}$, $\mu_{2}$,
$\mu_{3}$, $\mu_{4}$, $\mu_{5}$, $\mu_{6}$, $\mu_{7}$) one should separate the
multiplicative component with certain equal indices (for example, $\mu_{1}$,
$\mu_{2}$, $\mu_{3}$, $\mu_{4}$, $\mu_{5}=\mu_{3}$, $\mu_{6}=\mu_{2}$,
$\mu_{7}=\mu_{1}$), where the summing is performed only over the indices
$\mu_{1}$, $\mu_{2}$, $\mu_{3}$, $\mu_{4}$, and non-multiplicative component
with the indices ($\mu_{1}$, $\mu_{2}$, $\mu_{3}$, $\mu_{4}$,
$\bar{\mu}_{5}=\mu_{5}\neq\mu_{3}$, $\bar{\mu}_{6}=\mu_{6}\neq\mu_{2}$,
$\bar{\mu}_{7}=\mu_{7}\neq\mu_{1}$), where summing is performed over all these
indices. In this approach, the diagrams of the second and third order over the
number of phonon lines can be expressed in the following form:

$\displaystyle{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(2)}(\xi)={\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(2,m)}(\xi)+{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(2,nm)}(\xi)\qquad\qquad\qquad$
$\displaystyle=\sum_{\mu_{1}=1}^{2}\frac{\alpha_{\mu\mu_{1}}{\Large{\textbf{m}}}_{\mu_{1}\mu_{1}}^{(1)}(\xi-1)\,\alpha_{\mu_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)^{2}}+\sum_{\mu_{1}=1}^{2}\,\frac{\,\alpha_{\mu\mu_{1}}}{\xi_{\mu_{1}}-1}\sum_{\mu_{3}\neq\mu_{1}}^{2}\frac{{\Large{\textbf{m}}}_{\mu_{1}\mu_{3}}^{(1)}(\xi-1)\alpha_{\mu_{3}\mu^{\prime}}}{\xi_{\mu_{3}}-1}\,,$
(3.3) (3.4)

Separating in complete MO (2.7) all diagrams of all orders at m- and nm-
classes and taking into account that one-phonon MO
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(1)}(\xi)$ belongs to the m-class of
diagrams, from (3.2) we obtain:

${\Large{\textbf{m}}}_{\mu\mu^{\prime}}(\xi)={\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[m]}(\xi)+{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[nm]}(\xi)=\sum_{p=1}^{\infty}{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p,m]}(\xi)+\sum_{p=2}^{\infty}{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p,\,nm]}(\xi).$
(3.5)

For the system of a two-level quasiparticle that weakly couples with the
phonons under the condition that the intra-level interaction is much bigger
than the inter-level interaction
($\alpha_{11}\geqslant\alpha_{22}\gg\alpha_{12}$, that is fulfilled for the
majority of semiconductor structures), in the diagrams of any $p$-th order,
the multiplicative components
$\big{\\{}{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p,m]}(\xi)\big{\\}}$ are
much bigger than non-multiplicative ones
$\big{\\{}{\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{[p,\,nm]}(\xi)\big{\\}}$.
Taking into account this condition, the non-multiplicative MO components are
not considered further. However, we should note that for the systems with an
arbitrary intralevel and interlevel interactions, non-multiplicative MO
components must be taken into account either as corrections, or by separating
the new types of multiplicative diagrams from them, followed by their partial
summing. The component ${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(m)}(\xi)$,
due to its multiplicative structure, can be exactly partially summed. Finally,
it assumes a form of an infinite branched chain fraction with simple typical
links.

A complete partial summing of all diagrams of
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(m)}(\xi)$ MO is performed within the
successive renormalization of the energy denominators of its components of a
lower order by the contributions of all the diagrams of the next higher order.
Thus, let us observe the diagrams of MO
${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(m)}(\xi)$, which describe one-, two-
and three-phonon processes

., (3.6)

Now, using the multiplicative structure and the rules of diagram technique, it
is possible to perform the first stage of the renormalization of one-phonon MO
by all only two-phonon processes

(3.7)

In order to perform the next (second) stage, we should note that, as it is
seen from formula (3.6), each two-phonon diagram, belonging to infinite
series, can be renormalized by the summing of the respective infinite series
of three-phonon diagrams. In particular, the first component of two-phonon
diagrams is reformed by the infinite series of three-phonon diagrams in the
following way:

(3.8)

The second component of two-phonon diagrams is reformed by the following
infinite series of three-phonon diagrams:

${\color[rgb]{1,1,1}\blacklozenge}.$ (3.9)

In the same way, all two- and three-phonon multiplicative diagrams are summed
and, taking into account the one-phonon MO, we obtain:

$\displaystyle=\sum_{\mu_{1}=1}^{2}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu^{\prime}}}{\xi_{\mu_{1}}-1-2\displaystyle\sum_{\mu_{2}=1}^{2}\frac{\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu_{1}}}{\xi_{\mu_{2}}-2-3\displaystyle\sum_{\mu_{3}=1}^{2}\frac{\alpha_{\mu_{2}\mu_{3}}\alpha_{\mu_{3}\mu_{2}}}{\xi_{\mu_{3}}-3}}}\,.\qquad\qquad$
(3.10)

Using the reforming of $p$-phonon multiplicative diagrams by $(p+1)$-phonon
diagrams with their further summing, we obtain the exact representation of
completely renormalized MO ${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(m)}(\xi)$
in the form of infinite branched chain fraction

${\Large{\textbf{m}}}_{\mu\mu^{\prime}}^{(m)}(\xi)=\sum_{\mu_{1}=1}^{2}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu^{\prime}}}{\xi_{\mu_{1}}-1-2\displaystyle\sum_{\mu_{2}=1}^{2}\frac{\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu_{1}}}{\xi_{\mu_{2}}-2-3\displaystyle\sum_{\mu_{3}=1}^{2}\frac{\alpha_{\mu_{2}\mu_{3}}\alpha_{\mu_{3}\mu_{2}}}{\xi_{\mu_{3}}-3-...-p\displaystyle\sum_{\mu_{p}=1}^{2}\frac{\alpha_{\mu_{p-1}\mu_{p}}\alpha_{\mu_{p}\mu_{p-1}}}{\xi_{\mu_{p}}-p-...}}}}\,.$
(3.11)

It is worth noting that the obvious advantage of the realized representation
of the MO lies in the fact that each chain has a typical compact structure
and, therefore, submits to a simple computer algorithm that provides the
maximum speed of numerical calculations. Besides, it is also important that
this representation completely solves the “problem of sign” in the diagram
technique because the sign-varying series are absent here.

##  4 Properties of the energies of main and satellite states of two-level
quasiparticle interacting with phonons

As far as the renormalized energy spectrum is defined by the poles of Fourier
images of $g_{\mu}(\xi)$ Green’s functions and for the system under study, the
imaginary term of MO is absent at $T=0$ K, then, according to (2.4), the poles
of the functions $g_{1}(\xi)$ and $g_{2}(\xi)$ are fixed by the equations

$\xi={\Large{\textbf{m}}}_{1}(\xi)\,,\qquad\qquad\xi=\delta+{\Large{\textbf{m}}}_{2}(\xi)\,.$
(4.1)

Their solutions give the same spectrum due to the presence of interaction
between the quasiparticle and phonons.

An example of functions ${\Large{\textbf{m}}}_{1}(\xi)$ and the formation of
the lower part of the spectrum for the system with the coupling constants
$\alpha_{11}=0.3;\alpha_{22}=0.2;\alpha_{12}=0.075$ in the interval of
energies $\xi\leqslant 3.5$ for the non-resonant (NR) case ($\delta$=0.5) and
for first resonant (R) case ($\delta$=1) are presented in figure 1. We should
note that here and further ${\Large{\textbf{m}}}_{\mu=1}(\xi)$ (2.5) contains
only the basic approximation where in the chain (3.11) for the
${\Large{\textbf{m}}}_{1}^{(m)}(\xi)$, not more than five-six upper links were
taken into account due to convergence.

Figure 1: Mass operator ${\Large{\textbf{m}}}_{1}^{(m)}$ as function of x and
renormalized energy spectrum of two main states
($\varepsilon_{1},\varepsilon_{2}$) and complexes of bound-to-phonons
satellite states ($\varepsilon_{3}...\varepsilon_{12}$) in NR-case at
$\delta$=0.5 (a) and in R-case at $\delta$=1 (b) at
$\alpha_{11}=0.3;\alpha_{22}=0.2;\alpha_{12}=0.075$.

Figure 1 shows the formation of an energy spectrum of the main
($\varepsilon_{1},\varepsilon_{2}$) and the complexes
($\varepsilon_{3}...\varepsilon_{12}$) of satellite states, which appear due
to the interaction between a two-level quasiparticle and phonons, and how it
is changed depending on either the difference of energies ($\delta$) between
two levels of the non-interacting quasiparticle is multiple (resonant) or non-
multiple (non-resonant) with respect to the phonon energy. It is clear that in
the NR-case each of the main states creates its “own” group of satellite
states due to the complexes of quasiparticle states bound to the respective
number of phonons. The energy distance between the neighbour satellite groups
of each of the main states are of the order of one phonon energy. In figure 1
(a), the first main level and its satellites are shown with the circles while
the second main level and its satellites are shown with the triangles.

In R-case, one can see a superposition of the second main state and all its
satellites with all satellite states of the first main state. Hence, in the
vicinity of the energies multiple to the phonon energy, an infinite number of
groups of the respective number of satellite levels with multi-anti-crossings
(MACs) is observed. In figure 1 (b), three such groups are marked with ovals.

In order to uniquely classify the renormalized energy levels of the system
with respect to their formation and evolution, the spectrum was calculated in
a rather wide energy interval ($0\leqslant\xi\leqslant 3.5$) depending on the
energy difference ($\delta$) between two levels of the quasiparticle
uncoupling with phonons. The typical example of a spectrum
$\varepsilon_{n}(\delta)$ is presented in figure 2 for the system with
coupling constants $\alpha_{11}=0.3,\alpha_{22}=0.2,\alpha_{12}=0$ [figure 2
(a)] and $\alpha_{12}=0.075$ [figure 2 (b)]. These figures reveal the main
properties of a renormalized energy spectrum.

The complicated non-linear dependence $\varepsilon_{n}(\delta)$ is
qualitatively clear from physical considerations because it arises precisely
due to the inter-level interaction between a quasiparticle and phonons
($\alpha_{12}\neq 0$). This leads not only to the shift of both main levels
and their phonon satellites but also to the creation of two infinite series
[in figure 2 (b) their number is limited by the energy interval in which the
calculation was performed] of satellite groups with a different number of
levels in NR- and R-regions.

Really, if the inter-level interaction is absent ($\alpha_{12}=0$), an
idealized picture of the renormalized spectrum of the same system shows,
figure 2 (a), that near each of the two main levels
($\widetilde{e}_{10}=-\alpha_{11}^{2}$,
$\widetilde{e}_{20}=\delta-\alpha_{22}^{2}$) renormalized only by its “own”
intra-level interaction with phonons, there is observed its “own” infinite
series of equidistant satellite levels
($\widetilde{e}_{1n}=n-\alpha_{11}^{2},$
$\widetilde{e}_{2n}=n+\delta-\alpha_{22}^{2};n=1,2,\dots,\infty$). If $\delta$
increases, the behaviour of the spectrum is as follows: the whole series of
the first main level and its equidistant satellite levels, shifted into the
low-energy region by the quantity $\alpha_{11}^{2}$, remains unchanged while
the whole series, the second main level and its equidistant satellite levels,
also shifted into the low-energy region but by the quantity $\alpha_{22}^{2}$,
linearly moves into the high-energy region over $\delta$. Consequently, in the
vicinity of resonant energies ($\delta=1,2,3,\dots$) in the points with
coordinates
($\widetilde{D}_{n}=n-\Delta=n+\alpha_{22}^{2}-\alpha_{11}^{2},\tilde{\varepsilon}_{n}=\tilde{e}_{1n}=n-\alpha_{11}^{2}$),
the crossings of all satellite levels between each other and that of the
second main level with all satellites of the first main level are observed.

Analyzing the realistic renormalized spectrum at $\alpha_{12}\neq 0$, figure 2
(b), it is now clear how to classify all its levels. As one can see, the
realistic picture, contrary to the idealized one ($\alpha_{12}=0$), has a
fundamental difference: due to an inter-level interaction between the
quasiparticle and phonons, the satellite levels of both series undergo such a
splitting that none of them crosses independently of $\delta$. As a result,
the energies ($\varepsilon_{n=1,2,3,\dots}$) are single-value functions of
$\delta$. Density of levels, that is, their number per unit of energy,
essentially depend on the interval of $\delta$: either it is resonant
($\delta_{p=1,2,...}\approx 1,2,3,\dots$) or non-resonant ($0<\delta<1$,
$1<\delta<2,\dots$).

Figure 2 (b) shows that in all NR-intervals of $\delta$, the energy levels of
all satellite states except the first level, are splitting and grouping in two
series with respect to their “own” main levels. Like their “own” main levels,
the respective satellite levels of the first series weakly depend on $\delta$
while those of the second series are proportional to $\delta$ in each NR-
interval. In the narrow near-resonant intervals of $\delta$, due to the
superposition of corresponding states, the satellite levels of both series are
mixed with each other and with the second main level in such a way that, as a
result, the dense groups of MACs are formed.

In order to determine the number of split levels in both series in NR-
intervals and in the groups of MACs in R-intervals, it is necessary to use two
numerical indices $(p_{\delta},p_{e}=1,2,3,\dots)$, which, in the units of
phonon energy, fix $p_{\delta}$-th group of resonant levels and $p_{e}$-th
group of phonon satellites. Then, for a further understanding of the physical
origin and properties of energy spectrum, in the first NR-interval it is
convenient to introduce the energies of the first $[e_{1p_{e}(k)}]$ and second
$[e_{2p_{e}(k)}]$ series of $p_{e}$-th region of satellite states, where index
k numerates the split levels of this group. In order to characterize the group
of levels in certain MAC, we introduce the point (center)
$C_{p_{\delta}p_{e}}$ with coordinates $D_{p_{\delta}}$, $E_{p_{e}}$, which is
located in the middle of its minimal width of splitting
$\Delta_{p_{\delta}p_{e}}=\min[e_{\max n}(\delta)-e_{\min n}(\delta)]$ in the
vicinity of the point ($p_{\delta},p_{e}$). Figure 2 (b) and the insert proves
that this parameter well defines the width of ($p_{\delta}$,$p_{e}$)-th MAC
along the both axes ($e$ and $\delta$). Therefore, it is used as a
characteristic size of this MAC, because beyond it, the energy levels of both
series of satellites and the second main state are the linear functions of
$\delta$. The proposed choice of centers and widths of MACs is based on the
fact that at $\alpha_{12}\to 0$ their widths tend to zero
($\Delta_{p_{\delta}p_{e}}\to 0$), and the points $C_{p_{\delta}p_{e}}$ are
transformed into the points of crossings $\widetilde{C}_{p_{\delta},p_{e}}$ in
figure 2 (a), respectively, as it should be from physical considerations. When
analyzing the calculated parameters of MACs, it is more convenient not to use
large values of the coordinates of their centers ($D_{p_{\delta}p_{e}}$,
$E_{p_{\delta}p_{e}}$) but the respective small deviations:
$\Delta_{p_{\delta}p_{e}}^{\delta}=D_{p_{\delta}p_{e}}-p_{\delta}$ and
$\Delta_{p_{\delta}p_{e}}^{e}=E_{p_{\delta}p_{e}}-p_{e}$. An example of MAC
parameters ($p_{\delta}=2$, $p_{e}=3$) is shown in the insert in figure 2 (b)
and the parameters of the respective crossing are shown in the insert in
figure 2 (a).

Figure 2: (Colour online) Renormalized energy levels of the main states and
complexes of satellite states of the system as functions of $\delta$ at
$\alpha_{11}=0.3,\alpha_{22}=0.2$, $\alpha_{12}=0$ (a) and $\alpha_{12}=0.075$
(b).

The analysis of the results of calculations shows [figure 2 (b)] that the
renormalized energy of the second main state is almost a linear function of
$\delta$ in all NR-intervals between “diagonal” MACs
($p_{\delta}=p_{e}=p=1,2,3,\dots$). It separates the plane of energies
($\delta,e$) into two regions: inter-level ($0\leqslant e\leqslant\delta$) and
above-level ($\delta\leqslant e\leqslant\infty$). Herein, the infinite number
of satellite groups of the first series with $N_{e}=2^{p_{e}-1}$ levels is
present in $p_{e}$-th group. In the above-level region, besides this series,
one can see the same infinite number of groups of satellite levels of the
second series with the same number of levels in $p_{e}$-th group. The
arbitrary “non-diagonal” MAC $(p_{\delta},p_{e}>p_{\delta})$ contains
$N_{p_{\delta},p_{e}}=2^{p_{e}-(p_{\delta}+1)}+2^{p_{e}-1}$ levels while the
“diagonal” one $(p_{\delta},p_{e}=p_{\delta}=p=1,2,\dots)$ contains
$N_{p}=1+2^{p-1}$ levels.

The examples of typical dependences of MAC parameters on the coupling
constants describing the intra- and inter-level interaction between the
quasiparticle and phonons are presented in table 1. This table, like figure 2
(b), shows that depending on $\alpha_{12}$, the centers (coordinates of points
$C_{p_{\delta}p_{e}}$) of different MACs can be located in the plane
($\delta-e$) in different positions with respect to the points
$\widetilde{C}_{p_{\delta},p_{e}}$ in figure 2 (a). However, regardless of
this, an increase of $\alpha_{12}$ leads to the widening of all MACs
($\Delta_{p_{\delta}p_{5}}$). The other dependences are clearly seen in the
table 1.

Table 1: Parameters of multi-anti-crossings ($\Delta_{p_{\delta}p_{e}}$,$\Delta_{p_{\delta}p_{e}}^{\delta}$,$\Delta_{p_{\delta}p_{e}}^{e}$) as functions of inter-level coupling constant ($\alpha_{12}$) at $\alpha_{11}=0.3$, $\alpha_{22}=0.2$. $\alpha_{11}=0.3\quad\alpha_{22}=0.2$ | $p_{\delta}p_{e}$ | 1, 1 | 1, 2 | 1, 3 | 2, 2 | 2, 3 | 3, 3
---|---|---|---|---|---|---|---
| $\Delta_{p_{\delta}p_{e}}^{\delta}$ |  $-$0.0485 | 0.015 |  $-$0.05 | 0.03 |  $-$0.075 |  $-$0.035
| $\Delta_{p_{\delta}p_{e}}^{e}$ |  $-$0.0882 |  $-$0.0525 |  $-$0.1284 | 0.0099 |  $-$0.0993 |  $-$0.071
$\alpha_{12}=0.1$ | $\Delta_{p_{\delta}p_{e}}$ | 0.1611 | 0.2726 | 0.3399 | 0.1981 | 0.1122 | 0.0481
| $\Delta_{p_{\delta}p_{e}}^{\delta}$ |  $-$0.043 | 0.061 |  $-$0.025 | 0.0525 |  $-$0.055 |  $-$0.02
| $\Delta_{p_{\delta}p_{e}}^{e}$ |  $-$0.0829 |  $-$0.0210 |  $-$0.1054 | 0.0217 |  $-$0.0841 |  $-$0.0576
$\alpha_{12}=0.075$ | $\Delta_{p_{\delta}p_{e}}$ | 0.1212 | 0.2283 | 0.26960 | 0.1840 | 0.0773 | 0.0279
| $\Delta_{p_{\delta}p_{e}}^{\delta}$ |  $-$0.0395 | 0.102 |  $-$0.015 | 0.065 |  $-$0.035 |  $-$0.01
| $\Delta_{p_{\delta}p_{e}}^{e}$ |  $-$0.0793 | 0.0049 |  $-$0.0934 | 0.0334 |  $-$0.0694 |  $-$0.0482
$\alpha_{12}=0.05$ | $\Delta_{p_{\delta}p_{e}}$ | 0.0810 | 0.1938 | 0.1968 | 0.1732 | 0.0483 | 0.0130

##  5 Main results and conclusions

For a two-level localized quasiparticle interacting with polarization phonons,
the Feynman-Pines diagram technique is generalized in order to calculate the
renormalized spectrum of the system at $T=0$ K. A consistent and adequate
method of partial summing of MO diagrams is proposed, which, avoiding the
known “problem of a sign” in quantum field theory, effectively takes into
account multi-phonon processes and presents the MO matrix in the form of a
branched infinite chain fraction with the links of the same type. It provides
a high speed computer calculation of renormalized spectrum.

Using Dyson equation for the system with weak quasiparticle-phonon
interaction, the renormalized spectrum is obtained almost in the whole high-
energy region, which contains the main levels and their infinite groups of
bound-to-phonons satellite levels. It is shown that depending on whether the
energy difference ($\delta$) between two levels of the uncoupling
quasiparticle is resonant (multiple) to the phonon energy or non-resonant to
it, the spectrum of satellite states is almost independent of $\delta$ in its
interval. However, in the region of energies bigger than the energy of the
second main state, the renormalized spectrum essentially depends on $\delta$.
This is quite clear from physical considerations because at $T=0$ K, the
multi-phonon processes can occur only with the creation of phonons, and,
therefore, the bound-to-phonon states (satellites) can be formed only in the
region of energies bigger than the energy of the main renormalized state.
Consequently, in a two-level model, the quasiparticle in the second main state
cannot create bound states with phonons in the energy interval between the two
main renormalized levels. However, the inter-level interaction of both main
states due to phonons causes the splitting of the $p$-th phonon satellite of
the first main level (in a one-level model) into the group of $2^{p-1}$
satellite levels in a two-level model.

In the region of energy equal to and bigger than the energy of the second main
level, the physical situation is different. Here, the both main states in the
NR-intervals of energy create their “own” groups of satellite levels, and in
near-R- and R-intervals they create MACs, which are groups of the resonant
energy levels formed by the superposition of the second main state and
satellite (bound) states of the first main state and superpositions of all
satellite states created by both main states.

In general, the revealed properties of the spectrum of a two-level
quasiparticle renormalized due to phonons, and, in particular, the “ladder” of
the equidistant groups of split satellite levels present in this spectrum, can
at least highlight the physical mechanism of functioning of QCDs with cascades
operating on the basis of extractors with “torn ladders”, formed only by the
electron levels of the system, in the near-infrared range. Indeed, the
cascades of experimental QCD operating in middle and far infrared ranges
differ from that in the near-infrared range by the fact that the sizes of
quantum wells of extractors (for middle and far ranges) are selected so that
they create a complete “phonon ladder” of equidistant (in one phonon) electron
levels. In QCD operating in the near-infrared range, only the upper part of
the “phonon ladder” can be created by selecting the sizes of deep quantum
wells, while the lower part remains torn. Consequently, the QCD would not
operate because the energy relaxation is impossible since the electron levels
of the lower part of the “phonon ladder” are absent. However, it operates.
Taking into account the obtained results, it seems that one of the mechanisms
of relaxation of the electron energy may be not only the electron levels of
the extractor but also the groups of phonon satellites located in the energy
region between the two operating levels of the cascade active region.

Of course, in order to develop a consistent theory of electron-satellite
mechanism of QCD operating within the torn “phonon ladder”, the approach
proposed in this paper should be generalized in such a way that it is capable
of effectively taking into account the multi-phonon processes for the systems
of multi-band quasiparticles interacting with polarization phonons of
different modes. Such a generalization seems possible.

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