Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling
The partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Frohlich Hamiltonian is performed using the Feynman-Pines diagram technique. The renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calcul...
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nasplib_isofts_kiev_ua-123456789-1552682025-02-09T14:12:30Z Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling Перенормована енергiя основного i першого збудженого стану полярона Фелiха зi слабким зв’язком Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. Pytiuk, O.Yu. The partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Frohlich Hamiltonian is performed using the Feynman-Pines diagram technique. The renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for the weak electron-phonon coupling at T=0 K. It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra. The ground state stays stationary and the excited one - decays at bigger coupling constant. Методом дiаграмної технiки Фейнмана-Пайнса виконано парцiальне пiдсумовування безмежного ряду дiаграм двофононного масового оператора полярона, що описується гамiльтонiаном Фрeлiха. Перенормованi спектральнi параметри основного та першого збудженого поляронного стану (фононного повторення) коректно розрахованi для електрон-фононної системи зi слабким зв’язком при T = 0 К. Показано, що сильнiша електрон-фононна взаємодiя змiщує енергiї обох станiв у низькоенергетичну область спектра. Основний стан залишається стацiонарним, а загасання збудженого — зростає при збiльшеннi константи зв’язку. 2015 Article Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling / M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska, O.Yu. Pytiuk // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33707: 1–12 . — Бібліогр.: 28 назв. — англ. 1607-324X DOI:10.5488/CMP.18.33707 arXiv:1510.06912 PACS: 71.38.-k, 63.20.kd, 63.20.dk, 72.10.Di https://nasplib.isofts.kiev.ua/handle/123456789/155268 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Frohlich Hamiltonian is performed using the Feynman-Pines diagram technique. The renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for the weak electron-phonon coupling at T=0 K. It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra. The ground state stays stationary and the excited one - decays at bigger coupling constant. |
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| author |
Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. Pytiuk, O.Yu. |
| spellingShingle |
Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. Pytiuk, O.Yu. Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling Condensed Matter Physics |
| author_facet |
Tkach, M.V. Seti, Ju.O. Voitsekhivska, O.M. Pytiuk, O.Yu. |
| author_sort |
Tkach, M.V. |
| title |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling |
| title_short |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling |
| title_full |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling |
| title_fullStr |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling |
| title_full_unstemmed |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling |
| title_sort |
renormalized energy of ground and first excited state of fröhlich polaron in the range of weak coupling |
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Інститут фізики конденсованих систем НАН України |
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2015 |
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https://nasplib.isofts.kiev.ua/handle/123456789/155268 |
| citation_txt |
Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling / M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska, O.Yu. Pytiuk // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33707: 1–12
. — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T16:41:29Z |
| last_indexed |
2025-11-26T16:41:29Z |
| _version_ |
1849871875407085568 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 3, 33707: 1–12
DOI: 10.5488/CMP.18.33707
http://www.icmp.lviv.ua/journal
Renormalized energy of ground and first excited
state of Fröhlich polaron in the range
of weak coupling
M.V. Tkach∗, Ju.O. Seti, O.M. Voitsekhivska, O.Yu. Pytiuk
Chernivtsi National University, 2 Kotsyubinsky St., 58012 Chernivtsi, Ukraine
Received June 10, 2015, in final form July 2, 2015
Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröh-
lich Hamiltonian is performed using the Feynman-Pines diagram technique. Renormalized spectral parameters
of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon
coupling at T = 0 K. It is shown that the stronger electron-phonon interaction shifts the energy of both states
into low-energy region of the spectra. The ground state stays stationary and the excited one decays at a bigger
coupling constant.
Key words: polaron, phonon, electron-phonon interaction, Green’s function, mass operator
PACS: 71.38.-k, 63.20.kd, 63.20.dk, 72.10.Di
1. Introduction
The concept of polaron, as electron interacting with the polarization vibrations of the crystal, intro-
duced by Landau in 1933 [1, 2] has attracted a permanent attention lately. For a long time, the theory
of polaronic phenomena was developed in the framework of different physical models using various
mathematical approaches [3, 4]. Following the Fröhlich’s introduction of the Hamiltonian of electron
interacting with nondispersive (optical) phonons of a dielectric medium via its polarization, in represen-
tation of second quantization [5], the methods of quantum field theory were used to solve the polaron
problems [6–8]. Studying the renormalized energy of ground state and the effective mass of structureless
Fröhlich polaron, three ranges for electron-phonon coupling were established. The mobility, impedance
and optical conductivity were investigated in detail for these ranges.
Almost fifty years ago J. Devreese with colleagues [9, 10] were investigating the polaron complexes re-
lated to the excited states of electron-phonon system. In particular, it was established that in the regimes
of intermediate and strong coupling, the so-called relaxed excited states (RES) exist [9–13] in the region
of energies a little bit bigger than the renormalized energy of the ground polaron state plus the energy of
one phonon while the Franck-Condon (FC) excited states are located higher in the energy scale. The treat-
ment of the structure of the excited quasi-stationary states spectra is a complicated problem, constantly
attracting attention of theoretical community. The results obtained within different methods were com-
pared in original papers [14–16] and in reviews [17–19].
In the process of theoretical investigation of electron-phonon systems, the physical picture was stud-
ied in detail and a new mathematical approach [14–16] made it possible to avoid the contradictive re-
sults previously obtained in the approximations used in earlier papers. The exact diagrammatic quantum
Monte Carlo (DMC) method [14] was used in order to solve the problem of Fröhlich polaron RES and FC
states and analyze their dependence on the regime of coupling. In the cited papers it was proven that
the one-phonon approximation, used in the early papers of J. Devreese with colleagues, was not capable
of correctly describing the optical conductivity in the limit of strong coupling because, in particular, the
∗
E-mail: ktf@chnu.edu.ua
©M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska, O.Yu. Pytiuk, 2015 33707-1
http://dx.doi.org/10.5488/CMP.18.33707
http://www.icmp.lviv.ua/journal
M.V. Tkach et al.
energy density for RES and FC states was not correctly defined even at a very small coupling constant
(α= 0.05). Further, in reference [16], the main results of reference [14] were confirmed by DMC method
for a wide range of the coupling constant. These results correlated well with the ones obtained in the
memory function formalism (MFF) and strong coupling expansion (SCE) which assumed the FC principle.
We should mention that DMC method [14–17] made it possible to establish the properties of op-
tical conductivity in the region of excited states of Fröhlich polaron and was used to study the high-
temperature superconductivity. In particular, in reference [20] it was proven that electron-phonon inter-
action, together with magnetic sub-system plays a substantial role in the formation of high-temperature
superconductivity.
The search for mathematical approaches to the study of the excited states of Fröhlich polarons with
intermediate and strong coupling overshadows the solution of the same problems in the range of weak
electron-phonon coupling for 3D systems. However, with the appearance of low dimensional nano-systems
(quantum dots, quantum wires and quantum layers, characterized by weak coupling), where the differ-
ence between electron energies resonates with the energy of confined optical and interface phonons, the
attention payed to the excited polaron states has grown essentially [18, 21–27]. Now it is necessary to
study the renormalized spectrum of excited states of 3D polaron in the range of weak coupling.
From the papers [5–8, 22] we know that atα¿ 1 in one-phonon approximation for the mass operator,
the energy of polaron ground state (E ) at T = 0 K shifts into the low-energy region due to the electron-
phonon interaction. In the vicinity of E +Ω energies, where Ω is the polarization phonon energy, there
is observed a wide peak, related to the bound state of polaron with one phonon. This fact is clear and
coordinates with physical considerations. However, when the coupling constant increases, the energy of
excited state shifts into the opposite side of the spectrum, contradicting physical consideration because at
T = 0 K the virtual phonons are not capable of providing their energy to create a new quasi-particle. So,
it is clear that one-phonon approximation is not valid for an accurate calculation of mass operator (MO).
Thus, the further approximation was to take into account the two two-phonon diagrams, proportional
to α2
in the polaron Green’s function, besides the one-phonon, proportional to α. Herein, the magnitude
of renormalized energy of the ground state became more accurate (the red shift increased) but the peak
of the energy in the region of the bound state, being located in the left-hand part respectively to that in
one-phonon approximation, shifted into the high-energy region when α increased.
The abovementioned result brings us to the conclusion that the finite number of diagrams being taken
into account in polaron MO is insufficient to obtain the correct physical behavior of the excited polaron
state when the coupling constant varies in a wide range. It is evident that one has to perform a partial
summing of infinite number of diagrams of the respective order.
In this paper we study the renormalized energies of Fröhlich polaronwithweak electron-phonon cou-
pling in such approximation for the MO, which correctly takes into account partially summed one- and
two-phonon diagrams over all orders of the coupling constant. The result of this approach is that using
the Feynman-Pines diagram technique we obtained a physically correct conclusion: both the ground and
the first excited polaron states for a system with weak coupling shift into the low-energy region when α
increases.
2. The Fröhlich polaron at T = 0 K
We consider polaron as an electron interacting with polarization phonons described by Fröhlich
Hamiltonian
H =∑
~k
E~k a+
~k
a~k +
∑
~q
Ω~q
(
b+
~q b~q + 1
2
)
+ ∑
~k ,~q
ϕ (~q) a+
~k+~q a~k
(
b~q +b+
−~q
)
, (1)
where
E~k = E + ħ2k2
2m
, Ω~q =Ω (2)
are the energies of electron and optical phonons, respectively,
ϕ(q) = i
q
√√√√2πħΩα
V
√
2Ω
m
(3)
33707-2
Renormalized energy of ground and first excited state of Fröhlich polaron
— their binding function expressed within the coupling constant
α= e2
ħ
(
1
ε∞
− 1
ε0
) √
m
2Ω
=
(
1
ε∞
− 1
ε0
) √
m
m0
Ry
Ω
, (4)
withm0 — the mass of electron in vacuum and Ry= 13.6 eV.
It is well known [28] that at T = 0 K, the renormalized electron-phonon spectrum is obtained from
the poles of Fourier image of polaron Green’s function which, in its turn, through the Dyson equation
G (~k,ω) =
{
ħω−E~k −M (~k,ω)
}−1
(5)
is related with MO expressed in a diagrammatic form. An analytical calculation of MO in two first or-
ders over the powers of the coupling constant α is performed. According to the rules of Feynman-Pines
diagram technique, MO of the first order over α is defined as
= M1(~k,ω) =∑
~q
|ϕ(~q)|2
ħω−E~k−~q −Ω+ iη
. (6)
Transiting from summing to integration over ~q and accounting for (2) and (3), we obtain
M1(~k,ω) = e2Ω
4π2
(
1
ε∞
− 1
ε0
)∫
d3~q
q2
[
ħω−E − ħ2
2m (~k −~q)2 −Ω+ iη
] . (7)
Further, it is convenient to introduce the dimensionless MOm= M/Ω, with dimensionless energy ξ
and quasi-momentum (~K and ~Q)
ξ= ħω−E
Ω
, ~K = ħp
2mΩ
~k , ~Q = ħp
2mΩ
~q . (8)
In these variables the expression (7) is rewritten as follows:
m1(~K ,ξ) = α
2π2
∞∫
−∞
d3~Q
Q2
[
ξ−1− (~K − ~Q)2 + iη
] . (9)
Integrating in the spherical coordinate system, an exact analytical expression is obtained
m1(K ,ξ) =−α
K
arctan
(
K√
1−ξ
)
, ξÉ 1,
i
2
ln
∣∣∣∣∣
√
ξ−1+K√
ξ−1−K
∣∣∣∣∣+ π
2
θ(K −
√
ξ−1), ξÊ 1.
(10)
Herein, at K = 0
Rem1(ξ) =−αθ(1−ξ)√
1−ξ
, Imm1(ξ) =−αθ(ξ−1)√
ξ−1
. (11)
The analysis of MOm1(K ,ξ) and of the peculiarities of polaron spectra will be performed further
on. Now we should note that as far as inm1(K ,ξ) the electron interacts only with one virtual phonon,
it is called a one-phonon MO. Diagram technique proves that contrary to the two-, three- and n-phonon
MO,m1(K ,ξ) is a unitary one, which does not contain an infinite number of terms. The other terms of
complete MO contain an infinite number of diagrams over all powers of coupling constant.
33707-3
M.V. Tkach et al.
Now we observe the MO of the second order over the power of the coupling constant, corresponding
to the diagram without an intersection of phonon lines. According to the rules of diagram technique, it is
written as follows:
M a
2 (~k,ω) = = ∑
~q1,~q2
ϕ2(~q1)ϕ2(~q2)(
ħω−E~k−~q1
−Ω+ iη
)2 (
ħω−E~k−~q1−~q2
−2Ω+ iη
) . (12)
Transiting here from summing to integration, accounting for (2) and (3) and using the dimensionless
parameters, we obtain
ma
2 (~K ,ξ ) = α2
4π4
∞∫
−∞
d3~Q1
Q2
1
[
ξ−1− (~K − ~Q1)2 + iη
]2
∞∫
−∞
d3~Q2
Q2
2
[
ξ−2− (~K − ~Q1 − ~Q2)2 + iη
] . (13)
We are going to study the renormalized energies of the bottom of the ground and excited states of po-
laron. Thus, here and further we put ~K = 0 in order to simplify the analytical calculations. As a result, the
integration in (13) is performed exactly and in the region ξÉ 1 the expression forma
2 (ξÉ 1) containing
only the real part is obtained in the following form:
ma
2 (ξÉ 1) =− α2
(1−ξ)2
ln
(
1+
√
1−ξ
2−ξ
)
−
√
1−ξ
2
(√
1−ξ+ √
2−ξ
)
. (14)
Continuing analytically this expression into the region 1 É ξ É 2, one can obtain both the real and the
imaginary parts, while in the region ξÊ 2, only the real part [22].
The MO of the second order over the power of the coupling constant, corresponding to the diagram
with the intersection of phonon lines has the following form:
M b
2 (~k,ω) =
= ∑
~q1,~q2
ϕ2(~q1)ϕ2(~q2)(
ħω−E~k−~q1
−Ω+ iη
)2 (
ħω−E~k−~q1−~q2
−2Ω+ iη
)(
ħω−E~k−~q2
−Ω+ iη
) . (15)
In dimensionless variables it is rewritten as follows:
mb
2 (~K ,ξ) = α2
4π4
∞∫
−∞
d3~Q1
Q2
1
[
ξ−1− (~K − ~Q1)2 + iη
]
×
∞∫
−∞
d3~Q2
Q2
2
[
ξ−1− (~K − ~Q2)2 + iη
][
ξ−2− (~K − ~Q1 − ~Q2)2 + iη
] . (16)
At ~K = 0, integration in (16) is performed exactly. As a result, in the region ξ É 1 the expression for
mb
2 (ξÉ 1) containing only the real part is obtained
mb
2 (ξÉ 1) =− α2
(1−ξ)2 ln
[ √
2−ξ+ √
1−ξ
ξ
√
2−ξ+ (2−ξ)
√
1−ξ
]
. (17)
Continuing analytically this expression into the region 1 É ξ É 2, one can obtain both the real and the
imaginary parts while in the region ξÊ 2, only the real part [22].
33707-4
Renormalized energy of ground and first excited state of Fröhlich polaron
Finally, the functionsma
2 (ξÉ 1) andmb
2 (ξÉ 1) and their analytical continuations completely define
the MO of the second order over the power of the coupling constant
m(2)
2 (ξ) =ma
2 (ξ)+mb
2 (ξ) (18)
as a complex function of dimensionless energy ξ in the whole range of its variation.
The analytical expressions prove thatm1(ξ) has a discontinuity at ξ= 1whilem(2)
2 (ξ)— at ξ= 1 and
ξ= 2.
-1 1
-3
-1
1
Imm
1
( )
Imm
(2)
2
( )
Imm
(2)
2
( )
Rem
1
( )
Rem
(2)
2
( )
Rem
(2)
2
( )
Rem( ), Imm( )
0 0
-2
2
Figure 1. Dependences ofm1,m(2)
2 andm2
2Σ terms on
ξ at α= 0.5.
In figure 1 typical dependences of MO
m(2)
2Σ(ξ) =m1(ξ)+m(2)
2 (ξ) and its terms on
ξ calculated at α = 0.5 are presented. Here,
one can see the properties of MO terms and
how they influence the formation of renor-
malized energy of the ground polaron state
(ξ0).
At ξ É 1 the real parts of all terms
[Rem(2)
2Σ(ξ)] are negative and regularly de-
cay tending to −∞ at ξ→ 1 from the left. In
this region, Imm(2)
2Σ(ξ) = 0, thus, the renor-
malized energy of polaron ground state is
obtained from the expression ξ−m(2)
2Σ(ξ) =
0. It is clear from figure 1 that two-phonon
m(2)
2 (ξ) being taken into account in addition
to the one-phononm1(ξ), makes ξ0 magni-
tude more precise by increasing its absolute
value.
At ξ É 1, Rem1(ξ) = 0 and Imm1(ξ) < 0,
decaying over the absolute magnitude from
−∞ at ξ = 1 to zero at ξ→ ∞. In the range
1 É ξ É 2, Rem(2)
2 (ξ) varies from ∞ to −∞
and Imm(2)
2 (ξ) varies from −∞ to the finite
negative value. At 2 É ξ Imm(2)
2 (ξ) = 0 and
Rem(2)
2 (ξ) varies from −∞ at ξ= 2 to zero at
ξ→∞.
These properties of MO terms determine the features of polaron spectrum manifested through the
dependence of spectral density ρ on dimensionless energy ξ
ρ(ξ) =−2Img (ξ) =− 2Imm(ξ)
[ξ−Rem(ξ)]2 + [Imm(ξ)]2
. (19)
In figure 2, the function ρ(ξ) is presented at α = 0.25, 0.5, 0.75 calculated within MO m1(ξ) and
m(2)
2Σ(ξ). It is clear that independently of the approximated MO, besides δ-peak corresponding to the
renormalized energy of ground polaron state, the asymmetric peak of one-phonon repetition with big
width (γ) is observed because polaron in this state has a small lifetime. At bigger α, the renormalized
energy (ξ0) shifts into the negative region while the position of the maximum of one-phonon repetition
(ξ1) shifts into the region of higher energies with the increasing width (γ) of this peak.
Finally, we should note that one- and two-phononMO, proportional toα andα2
respectively, cause the
renormalization of polaron ground state energy in such a way that it shifts into the region of smaller en-
ergies when α increases, according to the physical considerations. As for the excited state, where polaron
is bound with one phonon, it is manifested as asymmetric peak in function ρ(ξ). When α increases, its
maximum shifts into the region of higher energies, being incorrect from the physical point of view. In the
next section we show that in order to correctly calculate the spectral parameters of the excited states, one
should perform a partial summing of infinite ranges of MO diagrams instead of a finite number thereof.
33707-5
M.V. Tkach et al.
-1.0 -0.5 0.5 1.0 1.5 2.0
0.2
0.4
0.6
0.8
1.0
= 0.75
= 0.5
= 0.25
1
( ),
(2)
2
( )
0 1
0
Figure 2. Dependences of spectral density ρ on energy ξ at different values of the coupling constant α
within different approximations for MO. ρ1 — thin curve, ρ
(2)
2Σ — thick curve.
3. Ground and first excited polaron states renormalized due to one- and
two-phonon processes
Taking into account the infinite number of diagrams inMO describing one- and two-phonon processes
over all powers of the coupling constant but containing no diagram of three- or more phonons, brings
us to the physically correct behavior of the renormalized energy both of the ground and the first excited
state.We refer to suchMO as the two-phononMO [M2(~k,ω)]. In diagrammatic representation it is written,
according to the rules of Feynman-Pines diagram technique, as follows:
M2(~k,ω) =
. (20)
Performing a complete partial summing of this range of diagrams, we obtain
M2(~k,ω) =∑
~q1
∣∣ϕ(q1)
∣∣2
ε̃~k−~q1
+ ∑
~q1~q2
∣∣ϕ(q1)
∣∣2 ∣∣ϕ(q2)
∣∣2
ε̃~k−~q1
ε~k−~q1−~q2
ε̃~k−~q2
+ ∑
~q1~q2~q3
∣∣ϕ(q1)
∣∣2 ∣∣ϕ(q2)
∣∣2 ∣∣ϕ(q3)
∣∣2
ε̃~k−~q1
ε~k−~q1−~q2
ε̃~k−~q2
ε~k−~q2−~q3
ε̃~k−~q3
+ ... , (21)
33707-6
Renormalized energy of ground and first excited state of Fröhlich polaron
where
ε̃~k−~q1
= ε~k−~q1
−∑
~q2
∣∣ϕ(q2)
∣∣2
ε~k−~q1−~q2
=ħω−E~k−~q −Ω−∑
~q2
∣∣ϕ(q2)
∣∣2
ħω−E~k−~q1−~q2
−2Ω
(22)
is the energetic denominator renormalized due to two-phonon processes.
Transiting here from summing to integration, integrating over the angular variables and using the
dimensionless parameters (8), we obtain an exact analytical expression for a dimensionless two-phonon
MO
m2(~K ,ξ) =∑
n
m(n)
2 (~K ,ξ). (23)
At K = 0, its terms are as follows:
m(1)
2 (ξ) = 2α
π
∞∫
0
dQ1
ξ−Q2
1 −1+αA(ξ,Q1)+ iη
, (24)
m(2)
2 (ξ) = α2
π2
∞∫
0
∞∫
0
dQ1
Q1
dQ2
Q2
ln
[
2−ξ+ (Q1 −Q2)2
2−ξ+ (Q1 +Q2)2
]
[
ξ−Q2
1 −1+αA(ξ,Q1)+ iη
][
ξ−Q2
2 −1+αA(ξ,Q2)+ iη
] , (25)
m(n)
2 (ξ) = 4
( α
2π
)n
∞∫
0
(n)
...
∞∫
0
Q2
Q1
n∏
s=2
Q−2
s ln
[
2−ξ+ (Qs −Qs−1)2
2−ξ+ (Qs +Qs−1)2
]
n∏
s=1
[
ξ−Q2
s −1+αA(ξ,Qs )+ iη
] dQ1 . . .dQn , (n = 2, 3, . . . ,∞), (26)
where
A(ξ,Q) =
1
Q
arctan
(
Q√
2−ξ
)
, ξÉ 2,
π
Q
θ(Q −
√
ξ−2)+ i
2Q
ln
∣∣∣∣∣Q + √
ξ−2
Q − √
ξ−2
∣∣∣∣∣ , ξÊ 2.
(27)
The integrals in formulae (24)–(26) are calculated within numerical methods. They are typical and
contain smooth (without peculiarities) functions of real variables [numerator in formula (26)] multiplied
by generalized functions
∏n
s=1[ξ−Q2
s −1+αA(ξ,Qs )+ iη]−1
. The presence of the latter causes a different
integration, depending on the range of ξ. There are three specific ranges.
At ξÊ 2, all factors
ξ−Q2
s −1+ πα
Qs
θ (Qs −
√
ξ−2)+ iα
2Qs
ln
∣∣∣∣∣Qs +
√
ξ−2
Qs −
√
ξ−2
∣∣∣∣∣
are complex functions, thus, they are all integrals and, hence,m(n)
2 (ξÊ 2) is a complex function contain-
ing Rem(n)
2 (ξÊ 2) and Imm(n)
2 (ξÊ 2) parts.
When ξÉ 2, the character of the integrals depends on whether there exists any solution of the equa-
tion
ξ−Q2 −1+ α
Q
arctan
Q√
2−ξ
= 0 (28)
at fixed ξ. AtQ = 0, this equation is rewritten as follows:
ξ+ α√
2−ξ
= 1, (29)
which has an exact real solution
ξ̄= 2− 4
3
cos2
[
1
3
arccos
(
3
p
3
2
α
)]
, αÉ 2
3
p
3
,
ch
2
[
1
3
arcch
(
3
p
3
2
α
)]
, αÊ 2
3
p
3
,
(30)
33707-7
M.V. Tkach et al.
producing two regions for ξÉ 2: ξÉ ξ and ξÉ ξÉ 2.
At ξ É ξ, the equation (28) does not have any solution, thus the functions in respective integrals for
m(n)
2 (ξ) have no poles and, hence, the MOm(n)
2 (ξ) are real functions.
In the region ξÉ ξÉ 2, the equation (28) has the solution in the pointQ0 at fixed ξ. Thus, the integrals
inm(n)
2 (ξ) contain this specific point and are calculated using Dirac identity
∞∫
0
Φ(Q)
f (Q)+ iη
dQ =P
∞∫
0
Φ(Q)
f (Q)
dQ − i
π
| f ′(Q)|Q=Q0
∞∫
0
Φ(Q)δ(Q −Q0)dQ, (31)
where Φ(Q) is the regular function and f (Q =Q0) = 0.
At ξ= 2, the equation (28) has an exact real solution
Q0(ξ= 2,α) = 2p
3
cos
[
1
3
arccos
(
3
p
3πα
4
)]
, αÉ 4
3
p
3π
,
ch
[
1
3
arcch
(
3
p
3πα
4
)]
, αÊ 4
3
p
3π
,
(32)
thus, the functionQ0(ξ,α) smoothly increases from 0 toQ0(2,α) in the range ξÉ ξÉ 2, figure 3.
The presented analysis proves thatm(1)
2 (ξ) is given by an analytical expression
m(1)
2 (ξ) = 2α
π
∞∫
0
dQ1
ξ−Q2
1 −1+ α
Q1
arctan
Q1√
2−ξ
, ξÉ ξ̄,
P
∞∫
0
dQ1
ξ−Q2
1 −1+ α
Q1
arctan
Q1√
2−ξ
− iπQ0∣∣∣∣∣3Q0 +1−ξ− α
√
2−ξ
2−ξ+Q2
0
∣∣∣∣∣
, ξ̄É ξÉ 2,
∞∫
0
[
ξ−Q2
1 −1+ πα
Q1
θ(Q1 −
√
ξ−2)− iα
2Q1
ln
∣∣∣∣∣Q1 +
√
ξ−2
Q1 −
√
ξ−2
∣∣∣∣∣
]
dQ1
[
ξ−Q2
1 −1+ πα
Q1
θ(Q1 −
√
ξ−2)
]2
+ α2
4Q2
1
[
ln
∣∣∣∣∣Q1 +
√
ξ−2
Q1 −
√
ξ−2
∣∣∣∣∣
]2 , 2 É ξ.
(33)
All integrals inm(n)
2 (ξ) are calculated in a similar way.
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
=0.2
=0.4
=0.6
Q
0
Figure 3.Q0 as function of ξ at α= 0.2, 0.4, 0.6.
33707-8
Renormalized energy of ground and first excited state of Fröhlich polaron
-0.5 0.5 1.0 1.5 2.0
-1.5
-1.0
-0.5
0.5
1.0
0
0.2
0.4
=0.6
0.2
0.4
=0.6
R
e
m
(1
)
2
,
Im
m
(1
)
2
Rem
(1)
2
Imm
(1)
2
-0.5 0.5 1.0 1.5 2.0
-1.5
-1.0
-0.5
0.5
1.0
0
0.4=0.6 0.2
0.20.4=0.6
R
e
m
(2
)
2
,
Im
m
(2
)
2
Rem
(2)
2
Imm
(2)
2
-0.5 0.5 1.0 1.5 2.0
-1.5
-1.0
-0.5
0.5
1.0
0
0.4
0.2
=0.6
R
e
m
(3
)
2
,
Im
m
(3
)
2
Rem
(3)
2
Imm
(3)
2
Figure 4.MOm(n)
2 as a function of ξ at α= 0.2, 0.4, 0.6.
The first three terms of MOm(n=1,2,3)
2 (ξ)
at α = 0.2, 0.4, 0.6 are shown in figure 4.
It is clear that at ξ É ξ, Imm(n)
2 (ξ) =
0 and Rem(n)
2 (ξ) < 0 independently of
the coupling constant α for all n, herein
lim
ξ→ξ
m(n)
2 (ξ) → −∞. The absolute magni-
tude |Rem(n)
2 (ξ)| increases for a bigger cou-
pling constant α. In the range ξ É ξ É 2,
the real and imaginary parts ofm(1)
2 (ξ) and
m(3)
2 (ξ) are negative while those ofm(2)
2 (ξ)
andm(4)
2 (ξ) are positive.
In figure 5, the functions of real and
imaginary parts of MOm(n=1,2,3)
2Σ (ξ) (a) and
densities of energies ρ(n=1,2,3)
2Σ (ξ) (b) are pre-
sented at ξ É 2. The region ξ > 2 is not ob-
served because here for correct results one
has to take into account the MO renormal-
ized due to the three-phonon processes. Fig-
ure 5 (a) proves that when ξÉ ξ, Imm(n)
2Σ (ξÉ
ξ) = 0 and, thus, there is no decay of the
renormalized ground state. The real part of
MO Rem(n)
2Σ (ξ É ξ) is negative and its abso-
lute value increases at bigger α. Both fea-
tures bring us to the fact that the energy den-
sity has a δ-like shape ρ(n)
2Σ (ξ) = δ(ξ− ξ(n)
0Σ )
with the peak at the renormalized ground
state energy.
In the range ξÉ ξÉ 2, the energy density
ρ(n)
2Σ (ξ) in all approximations has the shape
of asymmetric quasi-Lorentz peak, arising
due to the first excited state of polaron, in-
terpreted as a bound state of electron with
one phonon. The maximum of ρ(n)
2Σ in the en-
ergy scale fixes the energy of this bound state
and the width (γ(n)
2Σ ) of the peak at the half
of its height defines the decay of this quasi-
stationary state or its lifetime (τ(n)
2Σ =×/γ(n)
2Σ ).
We should note that in spectroscopy, ac-
cording to the physical characteristics of
renormalized ground and the first excited
states, the respective peaks of the function
ρ2(ξ) are referred to as phononless and one-
phonon repetitions. The function ρ(n)
2Σ (ξ) and
its spectral parameters (ξ(n)
0Σ , ξ
(n)
1Σ , γ
(n)
1Σ ) are
shown in figure 5 (b) for different α and MO
approximation (m(n)
2Σ ).
According to the physical considerations
and the behavior of optical conductivity at
small α revealed in paper [14], the energies of both states shift into the low-energy region when the
coupling constant increases. Herein, the ground state is a stationary one (not decaying) and the decay of
the excited quasi-stationary state increases.
The spectral parameters also depend on the approximation of MOm(n)
2Σ . Figure 5 (b) and table 1 prove
33707-9
M.V. Tkach et al.
-0
.5
0
.5
1
.0
1
.5
2
.0
-1
.5
-1
.0
-0
.5
0
.5
1
.0
(a
)
0
.2
0
.4
=
0
.6
0
.2
0
.4
=
0
.6
Rem
(1)
2, Imm
(1)
2
R
e
m
(1
)
2
I
m
m
(1
)
2
(1
)
0
0
-0
.5
0
.5
1
.0
1
.5
2
.0
-1
.5
-1
.0
-0
.5
0
.5
1
.0
0
.4
0
0
.2
=
0
.6
(2
)
0
0
.4
=
0
.6
0
.2
0
.2
0
.4
=
0
.6
Rem
(2)
2, Imm
(2)
2
R
e
m
(2
)
2
I
m
m
(2
)
2
-0
.5
0
.5
1
.0
1
.5
2
.0
-1
.5
-1
.0
-0
.5
0
.5
1
.0
0
.4
0
.2
=
0
.6
0
0
.2
0
.4
=
0
.6
0
.4
0
.2
=
0
.6
Rem
(3)
2, Imm
(3)
2
R
e
m
(3
)
2
I
m
m
(3
)
2
(3
)
0
-0
.5
0
.0
0
.5
1
.0
1
.5
2
.0
0
.5
1
.0
1
.5
2
.0
2
.5
3
.0
3
.5
=0.6
=0.4
=0.2
(1
)
1
(1
)
0
(b
)
0
.2
0
.4
=
0
.6
(1
)
2
(1
)
1
-0
.5
0
.0
0
.5
1
.0
1
.5
0
.5
1
.0
1
.5
2
.0
2
.5
3
.0
3
.5
=0.6
=0.4
=0.2
(2
)
1
(2
)
0
(2
)
2
0
.4
=
0
.6
0
.2
(2
)
1
-0
.5
0
.0
0
.5
1
.0
1
.5
2
.0
0
.5
1
.0
1
.5
2
.0
2
.5
3
.0
3
.5
=0.6
=0.4
=0.2
(3
)
1
(3
)
0
(3
)
2
0
.4
0
.2
=
0
.6
(3
)
1
Figure 5.MOm(n)
2Σ (a) and energy density ρ
(n)
2Σ (b) as functions of ξ at n = 1, 2, 3 and α= 0.2, 0.4, 0.6.
33707-10
Renormalized energy of ground and first excited state of Fröhlich polaron
Table 1. Dependences of the spectral parameters in the first three orders on the magnitude of α.
α 0.2 0.4 0.6
ξ(1)
0 –0.194 –0.376 –0.548
ξ(1)
1 0.856 0.780 0.728
ξ(1)
1 −ξ(1)
0 1.050 1.154 1.276
γ(1)
1 0.247 0.494 0.694
ξ(2)
0Σ –0.199 –0.395 –0.583
ξ(2)
1Σ 0.854 0.767 0.705
ξ(2)
1Σ−ξ(2)
0Σ 1.053 1.162 1.288
γ(2)
1Σ 0.077 0.172 0.273
ξ(3)
0Σ –0.20 –0.396 –0.585
ξ(3)
1Σ 0.84 0.74 0.69
ξ(3)
1Σ−ξ(3)
0Σ 1.04 1.136 1.275
γ(3)
1Σ 0.118 0.374 0.594
that the energies of the ground and the excited states (ξ(n)
0Σ , ξ
(n)
1Σ ) for bigger n are uniformly defined more
precisely, while the decay γ(n)
2Σ is defined varying between minimal values at odd n and maximal values
at even ones. From the table 1 it is clear that at smallα, the difference between the energies of the ground
and excited states correlates with the magnitude of the phonon energy. At a bigger α it increases, which
is probably not a physical property but the result of insufficiency of two-phonon approximation in MO
used for the calculation of renormalized energies.
4. Conclusions
Using the Feynman-Pines diagram technique, the exact analytical calculation of MO for the Fourier
image of polaron Green’s function is performed in the first and the second order over the electron-phonon
coupling constant. It is shown that though such an approximation makes the renormalized energy of
polaron ground state more precise, but even at a weak coupling (α < 1) the energy and decay of the
first excited state is evaluated very roughly. Firstly, the peak of the energy density of the first phonon
repetition in the energy scale is located higher than the energy E +Ω, however, according to the physical
considerations it should be lower because it is produced by the interaction between electron and virtual
phonon (T = 0 K). Secondly, the increasing coupling constant causes the shift of the peak of one-phonon
repetition into the high-energy region, which is not correct either.
A partial summing of all the infinite range of MO diagrams that do not contain three- and more
phonon energies is performed. The two-phonon MO and the energy density are calculated and their
properties are analyzed for the renormalized ground and first excited polaron states. It is shown, for the
first time, that according to the physical considerations, the stronger electron-phonon interaction, i.e.,
an increasing coupling constant, shifts the energy of the ground and first excited polaron states into the
low-energy region. Herein, the ground state stays stationary and the decay of the excited state increases.
At small α, the difference between the energies of the ground and the excited polaron states correlates
with the magnitude of the phonon energy.
The developed approach of partial summing of diagrams containing three-, four- and n-phonon pro-
cesses in polaron MO principally makes it possible to obtain more accurate renormalized energies and
decays both of the ground state and multi-phonon repetitions. However, an increasing number of rep-
etitions and the exactness of their spectral parameters brings us to complicated analytical and numeric
calculations.
33707-11
M.V. Tkach et al.
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Перенормована енергiя основного i першого збудженого
стану полярона Фелiха зi слабким зв’язком
М.В. Ткач,Ю.О. Сетi, О.М. Войцехiвська, О.Ю. Питюк
Чернiвецький нацiональний унiверситет iм.Ю. Федьковича,
вул. Коцюбинського, 2, 58012 Чернiвцi, Україна
Методом дiаграмної технiки Фейнмана-Пайнса виконано парцiальне пiдсумовування безмежного ряду
дiаграм двофононного масового оператора полярона, що описується гамiльтонiаном Фрeлiха. Перенор-
мованi спектральнi параметри основного та першого збудженого поляронного стану (фононного повто-
рення) коректно розрахованi для електрон-фононної системи зi слабким зв’язком при T = 0 К. Показа-
но, що сильнiша електрон-фононна взаємодiя змiщує енергiї обох станiв у низькоенергетичну область
спектра. Основний стан залишається стацiонарним, а загасання збудженого — зростає при збiльшеннi
константи зв’язку.
Ключовi слова: полярон, фонон, електрон-фононна взаємодiя, функцiя Грiна, масовий оператор
33707-12
http://dx.doi.org/10.1103/PhysRev.90.297
http://dx.doi.org/10.1080/00018735400101213
http://dx.doi.org/10.1103/PhysRev.97.660
http://dx.doi.org/10.1103/PhysRev.139.A338
http://dx.doi.org/10.1103/PhysRev.137.A1902
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Introduction
The Fröhlich polaron at T=0 K
Ground and first excited polaron states renormalized due to one- and two-phonon processes
Conclusions
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