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
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 diagram...
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irk-123456789-1574642019-06-21T01:29:37Z 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 Tkach, M.V. Pytiuk, O.Yu. Voitsekhivska, O.M. Seti, Ju.O. 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. Узагальненим методом дiаграмної технiки Фейнмана-Пайнса розвинена теорiя перенормованого взаємодiєю з поляризацiйними фононами спектра локалiзованої дворiвневої квазiчастинки при крiогеннiй температурi. Парцiальним пiдсумовуванням безмежних рядiв основних дiаграм отримано масовий оператор у компактному виглядi розгалуженого ланцюгового дробу, який ефективно враховує багатофононнi процеси. Показано, що багатофононнi процеси i мiжрiвнева взаємодiя квазiчастинки з фононами кардинально змiнюють перенормований спектр системи у залежностi вiд того, чи рiзниця енергiй мiж обома станами невзаємодiючої квазiчастинки резонує, чи не резонує з фононною енергiєю. Спектр нерезонансних систем мiстить перенормованi енергiї основних станiв i двi подiбнi мiж собою безмежнi серiї груп фононних сателiтних рiвнiв. Спектр резонансних систем мiстить перенормований основний рiвень i безмежну серiю сателiтних груп мультиантикросингiв. 2018 Article 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, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43703: 1–14. — Бібліогр.: 41 назв. — англ. 1607-324X PACS: 71.38.-k 72.10.Di 63.20.kk DOI:10.5488/CMP.21.43703 arXiv:1812.08570 http://dspace.nbuv.gov.ua/handle/123456789/157464 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
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. |
| format |
Article |
| author |
Tkach, M.V. Pytiuk, O.Yu. Voitsekhivska, O.M. Seti, Ju.O. |
| spellingShingle |
Tkach, M.V. Pytiuk, O.Yu. Voitsekhivska, O.M. Seti, Ju.O. 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 Condensed Matter Physics |
| author_facet |
Tkach, M.V. Pytiuk, O.Yu. Voitsekhivska, O.M. Seti, Ju.O. |
| author_sort |
Tkach, M.V. |
| title |
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 |
| title_short |
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 |
| title_full |
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 |
| title_fullStr |
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 |
| title_full_unstemmed |
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 |
| title_sort |
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 |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157464 |
| citation_txt |
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, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43703: 1–14. — Бібліогр.: 41 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-14T09:53:31Z |
| last_indexed |
2025-07-14T09:53:31Z |
| _version_ |
1837615611222425600 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 4, 43703: 1–14
DOI: 10.5488/CMP.21.43703
http://www.icmp.lviv.ua/journal
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 ∗, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
Chernivtsi National University, 2 Kotsyubinsky St., 58012 Chernivtsi, Ukraine
Received June 22, 2018, in final form July 31, 2018
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 tech-
nique. Using the procedure of partial summing of infinite ranges of the main diagrams, mass operator is ob-
tained 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 en-
ergies 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
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
evenweak excited states.When a sufficient number of experimental papers on electron-phonon complexes,
cyclotron resonance, impurity centers, etc. [2–10] 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 (〈N〉) in it can be
∗E-mail: m.tkach@chnu.edu.ua
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43703-1
https://doi.org/10.5488/CMP.21.43703
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
observed far from resonance, hybrid states (〈N〉 ∼ 1/2), corresponding to the dielectric modes known
from the experiment — in the resonance and bound state of quasiparticle plus phonons (〈N〉 ∼ 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 pro-
cesses 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 [12–15].
Within these methods, the problem of the spectrum of one-level localized quasi-particle interacting with
polarization phonons atT = 0 Kwas 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 , 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 [17–27]. The diagrammatic Monte
Carlo method and stochastic optimization (DMC) [17–27] 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 , 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 quantumwires, 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 [31–34].
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)whosemain operating elements are the cascades ofmulti-layered quantumwells,
43703-2
Generalized method of Feynman-Pines diagram technique
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 QCDwas 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 [39–41], 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 QCDwith 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 quasiparticleGreen’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 atT = 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]
Ĥ =
2∑
µ=1
Eµa+µaµ +
∑
®q
Ω
(
b+
®q
b
®q
+
1
2
)
+
∑
®q
2∑
µ1,µ2=1
ϕµ1µ2 a+µ1 aµ2
(
b
®q
+ b+
−®q
)
. (2.1)
Here, Eµ = E + dµ are the energies of uncoupling quasiparticle (further, dµ=1 = 0, dµ=2 = d),Ω is an
energy of polarization phonon, ϕµ1µ2 are the binding constants characterizing the intra-level (at µ1 = µ2)
and inter-level (at µ1 , µ2) interaction. All these quantities are assumed as known parameters of the
system. The operators aµ, a+µ and b ®q, b+
®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 [Gµµ(ω)] of quasiparticle. According to the general
theory [14] and taking into account the Hamiltonian (2.1), the matrix functions Gµµ′(ω) satisfy the
system of two equations.
Gµµ′(ω) = (ω − Eµ + iη)−1
[
δµµ′ +
2∑
µ1=1
Mµµ1(ω)Gµ1µ′(ω)
]
, µ, µ′ = 1, 2; (η→ +∞) , (2.2)
where Mµµ1(ω) is a matrix MO.
43703-3
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
The further calculations are performed using the dimensionless functions, variables and constants
gµµ′(ξ) = Gµµ′Ω; mµµ′(ξ) = Mµµ′Ω
−1; ξ = (ω − E)Ω−1;
ξµ = (ω − Eµ)Ω−1; αµ1µ2 = ϕµ1µ2Ω
−1; δ = d Ω−1.
(2.3)
Then, the exact solution of equation (2.2) for the dimensionless function of m-th level gµ(ξ) = gµµ(ξ)
is defined by Dyson equation (~ = 1)
gµ(ξ) =
[
ξµ −mµ(ξ)
]−1 (2.4)
through the complete MO mµ(ξ) of this level, which can be written as a sum of two components
mµ(ξ) =mµµ(ξ) +mg
µµ(ξ). (2.5)
Here, the component mµµ(ξ) is determined only by the diagonal matrix elements of the complete MO
mµµ′(ξ), while the component mg
µµ′(ξ) also contains non-diagonal components
[m12(ξ),m21(ξ)
]
of
the same MO, which appear due to the non-diagonal elements of gµµ′(ξ)matrix. Hence, the components
mg
11(ξ) and mg
22(ξ) are written as follows:
mg
11(ξ) =
m12(ξ)m21(ξ)
ξ2 −m22(ξ)
, mg
22(ξ) =
m12(ξ)m21(ξ)
ξ1 −m11(ξ)
. (2.6)
The matrix mµµ′(ξ) of the complete MO is defined by the rules of Feynman-Pines diagram tech-
nique [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),
MOmµµ′(ξ) is obtained in such diagrammatic formwhich 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.
43703-4
Generalized method of Feynman-Pines diagram technique
.
(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
(µ1, µ2, . . . , µN ), except the outer indices (µ, µ′), 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 µl.
For example, the first two components ofmµµ′(ξ) 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 mµµ′(ξ), 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
m[p]
µµ′(ξ). For example, the partial 2-phonon MO is
.
(3.1)
Then, the complete MO mµµ′(ξ) can be written as follows:
mµµ′(ξ) =m(1)
µµ′(ξ) +
∞∑
p=2
m[p]
µµ′(ξ) . (3.2)
In each diagram of the MO, the analytical contributions of its elements are summed up over all
µl 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, µ1, µ2, µ3, µ4, µ5, µ6, µ7) one should separate the multiplicative component with
certain equal indices (for example, µ1, µ2, µ3, µ4, µ5 = µ3, µ6 = µ2, µ7 = µ1), where the summing is
performed only over the indices µ1, µ2, µ3, µ4, and non-multiplicative component with the indices (µ1,
43703-5
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
µ2, µ3, µ4, µ̄5 = µ5 , µ3, µ̄6 = µ6 , µ2, µ̄7 = µ7 , µ1), 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:
m(2)
µµ′(ξ) =m(2,m)
µµ′ (ξ) +m(2,nm)
µµ′ (ξ)
=
2∑
µ1=1
αµµ1m(1)
µ1µ1(ξ − 1)αµ1µ′
(ξµ1 − 1)2
+
2∑
µ1=1
αµµ1
ξµ1 − 1
2∑
µ3,µ1
m(1)
µ1µ3(ξ − 1)αµ3µ′
ξµ3 − 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 m(1)
µµ′(ξ) belongs to the m-class of diagrams, from (3.2) we obtain:
mµµ′(ξ) =m[m]
µµ′(ξ) +m[nm]
µµ′ (ξ) =
∞∑
p=1
m[p,m]
µµ′ (ξ) +
∞∑
p=2
m[p, nm]
µµ′ (ξ). (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 (α11 > α22 � α12, that is
fulfilled for themajority of semiconductor structures), in the diagrams of any p-th order, the multiplicative
components
{m[p,m]
µµ′ (ξ)
}
are much bigger than non-multiplicative ones
{m[p, nm]
µµ′ (ξ)
}
. 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 m(m)
µµ′(ξ), 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 m(m)
µµ′(ξ) 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 MOm(m)
µµ′(ξ), which describe
one-, two- and three-phonon processes
43703-6
Generalized method of Feynman-Pines diagram technique
.
, (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:
�.
(3.9)
In the same way, all two- and three-phonon multiplicative diagrams are summed and, taking into account
43703-7
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
the one-phonon MO, we obtain:
=
2∑
µ1=1
αµµ1αµ1µ′
ξµ1 − 1 − 2
2∑
µ2=1
αµ1µ2αµ2µ1
ξµ2 − 2 − 3
2∑
µ3=1
αµ2µ3αµ3µ2
ξµ3 − 3
. (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 m(m)
µµ′(ξ) in the
form of infinite branched chain fraction
m(m)
µµ′(ξ) =
2∑
µ1=1
αµµ1αµ1µ′
ξµ1 − 1 − 2
2∑
µ2=1
αµ1µ2αµ2µ1
ξµ2 − 2 − 3
2∑
µ3=1
αµ2µ3αµ3µ2
ξµ3 − 3 − ... − p
2∑
µp=1
αµp−1µpαµpµp−1
ξµp − 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 gµ(ξ)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 g1(ξ) and g2(ξ) are fixed by the equations
ξ =m1(ξ) , ξ = δ +m2(ξ) . (4.1)
Their solutions give the same spectrum due to the presence of interaction between the quasiparticle and
phonons.
An example of functions m1(ξ) and the formation of the lower part of the spectrum for the system
with the coupling constants α11 = 0.3;α22 = 0.2;α12 = 0.075 in the interval of energies ξ 6 3.5 for the
non-resonant (NR) case (δ=0.5) and for first resonant (R) case (δ=1) are presented in figure 1. We should
note that here and furthermµ=1(ξ) (2.5) contains only the basic approximation where in the chain (3.11)
for the m(m)
1 (ξ), not more than five-six upper links were taken into account due to convergence.
Figure 1 shows the formation of an energy spectrum of the main (ε1, ε2) and the complexes (ε3...ε12)
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 (δ) 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
43703-8
Generalized method of Feynman-Pines diagram technique
0 1 2 3
0
1
2
3
m
1
(
)
m
1
(
)
a
0 2
2.89 2.90
0
1
2
3
0 1 2 3
0
1
2
3
m
1
(
)
m
1
(
)
b
0 2 3
2.914 2.916
0
1
2
3
Figure 1. Mass operator m(m)
1 as function of x and renormalized energy spectrum of two main states
(ε1, ε2) and complexes of bound-to-phonons satellite states (ε3...ε12) in NR-case at δ=0.5 (a) and in
R-case at δ=1 (b) at α11 = 0.3;α22 = 0.2;α12 = 0.075.
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.
43703-9
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
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 6 ξ 6 3.5)
depending on the energy difference (δ) between two levels of the quasiparticle uncoupling with phonons.
The typical example of a spectrum εn(δ) is presented in figure 2 for the system with coupling constants
α11 = 0.3, α22 = 0.2, α12 = 0 [figure 2 (a)] and α12 = 0.075 [figure 2 (b)]. These figures reveal the main
properties of a renormalized energy spectrum.
The complicated non-linear dependence εn(δ) is qualitatively clear from physical considerations
because it arises precisely due to the inter-level interaction between a quasiparticle and phonons (α12 , 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 (α12 = 0), an idealized picture of the renormalized
spectrum of the same system shows, figure 2 (a), that near each of the two main levels (ẽ10 = −α
2
11,
ẽ20 = δ − α
2
22) renormalized only by its “own” intra-level interaction with phonons, there is observed its
“own” infinite series of equidistant satellite levels (ẽ1n = n − α2
11, ẽ2n = n + δ − α2
22; n = 1, 2, . . . ,∞).
If δ 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 α2
11, 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 α2
22, linearly moves into the high-energy region over δ. Consequently,
in the vicinity of resonant energies (δ = 1, 2, 3, . . . ) in the points with coordinates (D̃n = n − ∆ =
n + α2
22 − α
2
11, ε̃n = ẽ1n = n − α2
11), 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 α12 , 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 (α12 = 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 δ. As
a result, the energies (εn=1,2,3,...) are single-value functions of δ. Density of levels, that is, their number
per unit of energy, essentially depend on the interval of δ: either it is resonant (δp=1,2,... ≈ 1, 2, 3, . . . ) or
non-resonant (0 < δ < 1, 1 < δ < 2, . . . ).
Figure 2 (b) shows that in all NR-intervals of δ, 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 δ while those of the second
series are proportional to δ in each NR-interval. In the narrow near-resonant intervals of δ, 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 (pδ, pe = 1, 2, 3, . . . ), which, in the
units of phonon energy, fix pδ-th group of resonant levels and pe-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 [e1pe (k)] and second [e2pe (k)] series of pe-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) Cpδ pe with coordinates Dpδ , Epe , which is located
in the middle of its minimal width of splitting ∆pδ pe = min[emax n(δ) − emin n(δ)] in the vicinity of the
point (pδ, pe). Figure 2 (b) and the insert proves that this parameter well defines the width of (pδ ,pe)-th
MAC along the both axes (e and δ). 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 δ. The proposed choice of centers and widths of MACs is based on the fact that at α12 → 0 their widths
tend to zero (∆pδ pe → 0), and the points Cpδ pe are transformed into the points of crossings C̃pδ,pe 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
(Dpδ pe , Epδ pe ) but the respective small deviations: ∆δpδ pe
= Dpδ pe − pδ and ∆epδ pe
= Epδ pe − pe. An
example of MAC parameters (pδ = 2, pe = 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).
43703-10
Generalized method of Feynman-Pines diagram technique
0 1 2 3
0
1
2
3
3
2,95
2,90
2,85
1,90 1,95
p p p
p
e
=
p
e
=
p
e
=
23e
~
3D
~
( )
( )
( )
23E
~
e
13C
~
23e
~
22e
~
21e
~
20e
~
2D
~
212
~
e
~
22e
~
21e
~
313
~
e
~
111
~
e
~
10e
~
e
a
1D
~
23C
~
33C
~
22C
~
12C
~
11C
~
( )
2
20e
~
p
e=3
e~
23
23
~
23D
~
23C
~
p
=2
,
0 1 2 3
0
1
2
3p
e
p
e
p
e
p p
3
2,95
2,90
2,85
1,90 1,95
e
p
C
33C
23C
13
C
22
C
12
e e
e
12(2)
e
12(1)
e
12(2)
e
11(1)
e
21
e
11
e
20
e
b
C
11
e
10
2
23
p
e
p
23
C
23
e
23
D
23
E
23
Figure 2. (Colour online) Renormalized energy levels of the main states and complexes of satellite states
of the system as functions of δ at α11 = 0.3, α22 = 0.2, α12 = 0 (a) and α12 = 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 δ in all NR-intervals between “diagonal” MACs
(pδ = pe = p = 1, 2, 3, . . . ). It separates the plane of energies (δ, e) into two regions: inter-level
(0 6 e 6 δ) and above-level (δ 6 e 6 ∞). Herein, the infinite number of satellite groups of the first series
with Ne = 2pe−1 levels is present in pe-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 pe-th group. The arbitrary “non-diagonal” MAC (pδ, pe > pδ) contains Npδ,pe = 2pe−(pδ+1) + 2pe−1
levels while the “diagonal” one (pδ, pe = pδ = p = 1, 2, . . . ) contains Np = 1 + 2p−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 α12, the centers (coordinates of points Cpδ pe ) of different
MACs can be located in the plane (δ − e) in different positions with respect to the points C̃pδ,pe in
figure 2 (a). However, regardless of this, an increase of α12 leads to the widening of all MACs (∆pδ p5).
The other dependences are clearly seen in the table 1.
43703-11
M.V. Tkach, O.Yu. Pytiuk, O.M. Voitsekhivska, Ju.O. Seti
Table 1. Parameters of multi-anti-crossings (∆pδ pe ,∆δpδ pe ,∆
e
pδ pe ) as functions of inter-level coupling
constant (α12) at α11 = 0.3, α22 = 0.2.
α11 = 0.3 α22 = 0.2 pδpe 1, 1 1, 2 1, 3 2, 2 2, 3 3, 3
∆δpδ pe
−0.0485 0.015 −0.05 0.03 −0.075 −0.035
∆epδ pe
−0.0882 −0.0525 −0.1284 0.0099 −0.0993 −0.071
α12 = 0.1
∆pδ pe 0.1611 0.2726 0.3399 0.1981 0.1122 0.0481
∆δpδ pe
−0.043 0.061 −0.025 0.0525 −0.055 −0.02
∆epδ pe
−0.0829 −0.0210 −0.1054 0.0217 −0.0841 −0.0576
α12 = 0.075
∆pδ pe 0.1212 0.2283 0.26960 0.1840 0.0773 0.0279
∆δpδ pe
−0.0395 0.102 −0.015 0.065 −0.035 −0.01
∆epδ pe
−0.0793 0.0049 −0.0934 0.0334 −0.0694 −0.0482
α12 = 0.05
∆pδ pe 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 (δ) 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 δ in its interval.
However, in the region of energies bigger than the energy of the second main state, the renormalized
spectrum essentially depends on δ. 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 2p−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 quantumwells, while the lower part remains
torn. Consequently, the QCD would not operate because the energy relaxation is impossible since the
43703-12
Generalized method of Feynman-Pines diagram technique
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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Узагальнений метод дiаграмної технiки Фейнмана-Пайнса у
теорiї енергетичного спектра дворiвневої квазiчастинки
перенормованого багатофононними процесами при
крiогеннiй температурi
М.В.Ткач, О.Ю.Питюк, О.М.Войцехiвська,Ю.О.Сетi
Чернiвецький нацiональний унiверситет iм.Ю.Федьковича,
вул. Коцюбинського, 2, 58012 Чернiвцi, Україна
Узагальненим методом дiаграмної технiки Фейнмана-Пайнса розвинена теорiя перенормованого взає-
модiєю з поляризацiйними фононами спектра локалiзованої дворiвневої квазiчастинки при крiогеннiй
температурi. Парцiальним пiдсумовуванням безмежних рядiв основних дiаграм отримано масовий опе-
ратор у компактному виглядi розгалуженого ланцюгового дробу, який ефективно враховує багатофонон-
нi процеси. Показано, що багатофононнi процеси i мiжрiвнева взаємодiя квазiчастинки з фононами кар-
динально змiнюють перенормований спектр системи у залежностi вiд того, чи рiзниця енергiй мiж обома
станами невзаємодiючої квазiчастинки резонує, чи не резонує з фононною енергiєю. Спектр нерезонан-
сних систем мiстить перенормованi енергiї основних станiв i двi подiбнi мiж собою безмежнi серiї груп
фононних сателiтних рiвнiв. Спектр резонансних систем мiстить перенормований основний рiвень i без-
межну серiю сателiтних груп мультиантикросингiв.
Ключовi слова: дiаграмна технiка, квазiчастинка, масовий оператор, фонон, спектр
43703-14
https://doi.org/10.1088/0268-1242/15/4/315
https://doi.org/10.1016/j.spmi.2018.01.021
https://doi.org/10.1016/j.spmi.2015.03.023
https://doi.org/10.12693/APhysPolA.128.343
https://doi.org/10.1109/JQE.2009.2017929
https://doi.org/10.1063/1.4772501
https://doi.org/10.1063/1.4775374
https://doi.org/10.1088/0268-1242/28/7/074022
Introduction
Hamiltonian of the system. Mass operator of quasiparticle Green's function at T=0 K
Partial summing of MO diagrams
Properties of the energies of main and satellite states of two-level quasiparticle interacting with phonons
Main results and conclusions
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