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EDN: IWKRAA УДК 538.915
Lang-Firsov Transformation in the Generalized Tight-binding Method
Alexander V. Dudarev*
Reshetnev Siberian State University of Science and Technology
Krasnoyarsk, Russian Federation
Elena I. Shneyder^
Kirensky Institute of Physics, Federal Research Center KSC SB RAS Krasnoyarsk, Russian Federation
Received 10.08.2023, received in revised form 25.09.2023, accepted 30.10.2023 Abstract. The effects of strong electron-phonon interaction in a realistic model of a system with strong Coulomb correlations are analyzed using the Lang and Firsov transformation. It is shown that polaronic and bipolaronic transformations, widely discussed in the literature and associated with a smooth or sharp transitions in the properties of charge carriers when the strength of the electron-phonon coupling changes, determine the switching of the system between the regimes of correlated carriers, polarons or bipolarons. These transitions are controlled by the local electron-phonon interaction of the Holstein type. At the same time, the non local electron-lattice contribution associated with the modulation of the hopping integral plays a major role in the crossover of the polaron and bipolaron regimes in the limit of strong electron correlations.
Keywords: Lang-Firsov transformation, polaron and bipolaron crossover, pd-model.
Citation: A.V. Dudarev, E.I. Shneyder, Lang-Firsov Transformation in the Generalized Tight-binding Method, J. Sib. Fed. Univ. Math. Phys., 2023, 16(6), 780-785. EDN: IWKRAA.
1. Lang-Firsov transformation
Let us consider the classical canonical transformation for systems with the strong electron-phonon coupling of the Holstein type. It was proposed by Lang and Firsov [1] for systematic consideration of the perturbation theory of the mobility of small-radius polarons. They managed to record a significant part of the short-ranged electron-phonon interaction in such a way that, during the transformation of the Hamiltonian, they could collect infinite series for the transformed operators. The authors [1] simplified the Hamiltonian of the system by limiting the contribution of the electron-phonon interaction to the local electron variables:
H ealam Tga^+gam ( rfqbq + 1 j Ku^lam (Um,q^ + U*m^bq) , (1)
m m,q q m,q
where ajn (am) are operators of creation (annihilation) of an electron at a site m, bq (bq) are operators of creation (annihilation) of a phonon with quasi momentum q; e is the ground state
* aleksdudarev2013@gmail.com
tshneyder@iph.krasn.ru https://orcid.org/0000-0001-6285-2258 © Siberian Federal University. All rights reserved
electron energy; the parameter Tg characterizes the hopping energy from site g to m and vice versa; Umq is the electron-phonon interaction (EPI) parameter and is the phonon mode frequency.
According to the original paper, the canonical transformation H = eSHe-S of the Hamilto-nian (1) with the unitary operator S in the form:
S ^ ] alnam (bqUm,q — bqUm,q) , m,q
leads to a new Hamiltonian:
H = Y1 a'mam (e - A) + Y^ ^bq + 2 j + Tg<m,g aL+g
m q ^ ' m,g
which describes the energy of polarons, lattice oscillators, and their residual interaction. Here A = q \Um,q\2 is the polaron shift of the atomic energy of the ground states of charge car-
riers and <1 m g = exp
is the renormalization of the hopping
Sq (bqAm,m+g,q bqAm,m+g,q)
integral, it depends on the strength of the electron-phonon coupling: Am m+g q — Um q — Um+g q. In the transformed Hamiltonian, the contribution characterizing the interaction of electrons at different sites through the exchange of virtual phonons is discarded, since it is less than the Coulomb correlations of electrons not taken into account in the original model.
The modified Lang and Firsov transformation, which includes the nonlocal electron-phonon interaction, is also found in the literature. In the general case, analysis of the obtained renor-malizations is complicated, however, there are some results that generalize the Holstein theory of polarons. For example, within the framework of the Holstein-Peierls model, the authors of [2, 3] obtain an expression describing the temperature narrowing of the polaron bandwidth and demonstrate that nonlocal electron-phonon coupling plays an important role in organic molecular crystals.
m
2. Extended pd-model
Realistic models of systems with strong electron-phonon interaction contain more numbers of contributions compared to the Hamiltonian (1). In addition to the nonlocal electron-lattice interaction, these can be multiband effects, Coulomb correlations, and much more. Let us consider an extended pd-model combining contributions essential for describing the nature of high-temperature superconductors based on copper oxides. The Hamiltonian of the model has the form:
H = Hpd + He-ph + Hph, (2)
here Hpd is a well-known pd model describing the hole carriers in the orbitals of oxygen p and copperd; it takes into account both the Coulomb interaction of holes in the same and different orbitals and the strong overlap of the orbitals [4].
The term He-ph describes the interaction of electrons with lattice vibrations. We consider only one optical fully symmetric mode, the dispersion of which can be neglected. Deviations of atoms from their equilibrium positions when bonds are stretching lead to modulations of the single-site energy of charge carriers in copper orbitals (local Holstein-type EPI) and of the hopping integral between the copper and oxygen orbitals (non-local Peierls-type EPI). The Hph part corresponds to the energy of free phonons.
Let us show the complete form for each part of the Hamiltonian (2):
HPd = E [£anta,o- + 1 U*nL,a n^a,a) + E ^PP + HC) +
+ E PPdtPd +H-C0 + E VPdniuo-K,a' ■ (3)
g,l,a g,l,aa'
Here dg,a (dg,a) and pgia (pgl,a) are operators of creation (annihilation) of a hole with spin a on copper and oxygen orbitals in positions specified by vectors g and g; respectively. The index l enumerates oxygen atoms in the unit cell at the site g; naaa is the hole number operator, where ga = g for a = d, and ga = g; for a = p; ea is the local energy of a hole in the p or d orbital; tpp and tpd are hopping integrals, Ua, Vpd are Coulomb repulsion parameters. The factors Ppp and Ppd are equal to 1 (-1) if in the region of orbital overlap their phase signs coincide (differ). In the general case, the Hamiltonian (3) describes compounds of transition metal with an incompletely filled d orbitals.
The energy of free phonons in the Hamiltonian (2) has the form:
Hph = E huofffq + 1) = E flfg + 1) , (4)
q ^ ' g ^ '
where fq is the phonon creates operator with a quasi momentum q and fg is its Fourier image, w0 is the frequency of lattice oscillations.
A significant short-ranged part of the electron-phonon interaction can be written as [6]:
He-ph = E Md f + fg) dg,adg +E MPdPpd f + fg) (dg,aPgl, a + H.C.) , (5)
g a g, l , a
where Md and Mpd are the electron-phonon interaction parameters for Holstein and Peierls type contributions, respectively.
3. Lang-Firsov transformation for intercluster part
Assuming that the strong coupling approximation is valid for the system (2) in the limit of strong electron correlations, we analyze the effects of electron-phonon interaction using the Lang and Firsov transformation. Following the ideology of the generalized tight-binding method, we divide the system into clusters and represent the Hamiltonian (2) as the sum of the intracluster contribution Hc and the intercluster interaction Hcc:
H = Hc + Hcc■ (6)
Such splitting is possible if we go from the operators pgia (Pgl,a) to the new ones, bg,a (bg,a), centered on copper orbitals and orthogonal in the nearest cells. A similar transformation for the displacement operators [5] translates the operators fg (fg) into the new Ag (Ag) [6]. As a result, the intracluster part of the Hamiltonian (2) is given by:
Hc = Y. ( Spng,a + 1 Upn§,ang,a) + E (t00 (dg,abg,a + H.c) + E Vbdng,a+
E (£P 4,a + 1 Up nla ngg,a) + E (too (dg,a bg,a + Hc) + E Vbdnga ng,a )
g,a,P ' g,a \ a' J
+ E (Modo¥>Xa + Mbdvf [dg,abg,a + H.c.]) + £ ^0 (AgAg + ^ ■
g,a g
(7)
where 3 = d,b is the orbital index in the new representation. Changes in the parameters of the original Hamiltonian (2) are result from the transition to the basis of Wannier functions.
Omitting the insignificant details of the polaronic version of the generalized tight-binding method [6], we only indicate that M0do = Md^oo, M0do = —2Mpd^0o and the coefficient ^oo ~ 0.96.
Let us carry out the canonical transformation of the Hamiltonian Hc, limiting ourselves to local contributions of the Holstein type EPI, that is, we put Mpd = 0. Then unitary operator is:
* = E S <4 -
g,*
and the Hamiltonian (7) takes the form:
Hc = £[£ - A) ng,* + £5ng,*] + £ g,* g, 2*
XMdof
Ud - 2nSt ng*ng* + Ubng*ng*
+
+ £ V5d<,*ng* + £ t5do (dg*bg,* + H.c.) + £ huo UgAg + M . (8)
or /T/T' g * g \ '
The residual polaron-lattice interaction results from the renormalization of the orbital overlap
Md
integral, associated with the transformation dg,* = dg,* exp
rd )2
hw o
In the polaron
LJ0 -
2
shift of the atomic level A = v 00— {ng a) we have explicitly preserved the mean of the particle
(Mi
hw o
number operator. We assume that {ng a) is not necessarily close to 1 even at small doping of the system due to the redistribution of charges carries between the orbitals b and d.
The equation (8) correlates with the well-known paper [7] of A. Aleksandrov and co-authors. The differences are associated with the formulation of the Hamiltonian (2), which takes into account, in addition to Coulomb correlations, multiband effects, and the structure of the unit cell. In addition, for the estimates below we use the realistic parameters of the Hamiltonian (2), determined from ab initio calculations, and compare the results with the conclusions obtained [8,9] by the exact diagonalization of the cluster Hc.
Carrying out a similar procedure, we consider the renormalizations of the Hamiltonian Hc due to the contribution of the Peierls-type EPI; so, we set Md = 0. Then we define the unitary operator in a form which is similar to the nonlocal contribution of the EPI, and, using the Baker-Campbell-Hausdorff formula, we find a new representation of the Hamiltonian operators. Avoiding cumbersome expressions, we note that the transformation of a pair of hole creation (annihilation) operators in the orbitals of copper and oxygen, dg a (dg,a) and bg a (bg,a), formally
M bd
is a rotation in two-dimensional Minkowski space with angle 6 = 000 (Ag + Ag). Changes to
hw o g
phonon operators have the form:
Ag = Ag - £ M? (dg,*'bg,*' + H.c.). (9)
In the transformed Hamiltonian, equation (9) gives renormalizations proportional to the value
MM 2
000 and describing (i) the change in the interorbital Coulomb interaction, (ii) spin fluctuations
hw 0
in the b and d orbitals without charge transfer, as well as (iii) effective interorbital electron-phonon interaction associated with charge transfer without spin-flip and (iv) effective interorbital spin-phonon interaction, corresponding to charge transfer with a change of spin. These processes are caused by the exchange of virtual phonons between carriers in different orbitals. Note also that process (i) contributes both ed and eb energies to the polaron shift.
4. Analysis of the effects of local and non-local EPI
(M )2
We introduce dimensionless electron-phonon coupling constants Xd = -—— and Xbd =
nw0 W
(M )2
= ——WW' characterizing contributions of Holstein (local) and Peierls (non-local) types. Here,
W is the width of the bare bond, which is not renormalized by EPI and Coulomb effects. For ease of comparison, all electronic structures parameters of the pd-model (3) have been chosen in accordance with the paper [9], in which the effects of EPI were investigated for the cell Hc and the total 2 Hamiltonians in the framework of polaronic version of the generalized tight binding method: £d = 0, ep = 1.5, tpp = 0.86, tpd = 1.36, Ud = 9, Up = 4, Vpd = 1.5, W = 2.15, all in eV.
The first case is Xbd = 0. Let us consider the single-particle states, the atomic energy ed of
(Md )2
which decreases under strong local electron-phonon interaction by the value A = —--(ng a).
nw o 5'
If the polaron shift A is larger than the width of the charge carrier band W , then the formation of an energetically more favorable band of polaron states begins near the Fermi level of the system.
The critical EPI strength is Xd1 = 2 d—r For given parameters, this occurs for the electron-
M2 0 \ng,a )
phonon coupling strength X"1 > 1.1 if (ng¡a) ~ 1; at lower values of (ng,a), due to the strong hybridization of the orbitals p and d, the value of Xd1 increases. In the electronic structure of the extended pd-model (2), a narrow polaron band appears [8] in certain directions of the Brillouin zone.
Now let us consider two-particle states and estimate the value of the second critical parameter of the local EPI, at which the Coulomb interaction at the copper sites effectively vanishes or
becomes negative. The equation (8) gives the value of X"f = UdUr. So, for Ud = 9 we obtain
2M20 W
Xd2 ~ 2.2. The screening of the Coulomb potential by electron-phonon interaction forms a tendency to the formation of bipolarons (here, local bipolarons). The process is accompanied by a significant transfer of charge carries from the p to the d orbital at Xd > X"f [8].
In the limit of strong electron correation we get that X"1 < X"2. Obviously, a decrease in the Coulomb interaction Ud leads to a decrease in Xd2. In the Holstein model, (Xbd = 0) the crossover of the critical values marking the polaronic Xd1 and bipolaronic X"f regimes take place
at Ud < ———-2W. It corresponds to the case of weak or intermediate Coulomb correlations.
(ng,a )
However, in a more realistic approach that takes into account the contribution of nonlocal EPI, such a crossover occurs in the limit of strong electron correlations for (Xbd = 0) [9]. The replacement of the polaron and bipolaron regimes is accompanied by changes in the electronic structure associated with the redistribution of the density of charge carriers in favor of the p orbital and with the 'relocation" of the chemical potential into the band of the two-particle states for hole doped system. This proves the essential role of the nonlocal EPI in the formation of bipolarons in a correlated system and emphasizes the need to take its effects into account in such problems as, for example, the search for light bipolarons.
References
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Преобразование Ланга-Фирсова в обобщенном методе сильной связи
Александр В. Дударев
Сибирский государственный университет науки и технологий им. ак. М. Ф.Решетнева
Красноярск, Российская Федерация
Елена И. Шнейдер
Институт физики им. Л. В. Киренского СО РАН Красноярск, Российская Федерация
Аннотация. В настоящей работе с помощью преобразования Ланга и Фирсова анализируются эффекты сильного электрон-фононного взаимодействия в реалистичной модели системы с сильными кулоновскими корреляциями. Показано, что широко обсуждаемые в литературе поляронные и биполяронные трансформации, связанные с плавным или резким изменением свойств носителей заряда при изменении силы электрон-фононной связи, определяют переключение системы между режимами коррелированных носителей, поляронов или биполяронов. При этом существенную роль в кроссовере поляронного и биполяронного режимов в пределе сильных электронных корреляций играет нелокальный электрон-фононный вклад, связанный с модуляцией интеграла перескока между различными орбиталями.
Ключевые слова: преобразование Ланга и Фирсова, кроссовер поляронов и биполяронов, р^модель.
