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Physical review. B, Condensed matter

Polaron conductivity in the narrow-band Hubbard model

Yuhei Suzuki, Philip A. Pincus, Alan J. Heeger

Published Oct 1, 1991• 11 citations
Publisher

Abstract

We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}$(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-${\mathit{T}}_{\mathit{c}}$ superconductors is briefly discussed.

DOI
10.1103/physrevb.44.7127

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Polaronic approach to strongly correlated electron systems with strong electron-phonon interaction

I. A. Makarov, Elena I. Shneyder, P. A. Kozlov +1 more

The three-band $p\text{\ensuremath{-}}d$ model of strongly correlated electrons interacting with optical phonons via diagonal and off-diagonal electron-phonon interactions is considered within the cluster perturbation theory. In the beginning, the exact diagonalization of the Hamiltonian of a ${\mathrm{CuO}}_{4}$ cluster results in the construction of local polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ with hole numbers ${n}_{h}=0,1,2$ per unit cell. The intercluster hoppings and interactions are exactly written in terms of Hubbard operators ${X}_{f}^{pq}=\left|p\right\ensuremath{\rangle}\ensuremath{\langle}q|$ determined via the multielectron polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ at site $\mathbf{f}$. The Fermi-type single-electron quasiparticle dispersion and spectral weight are calculated for the undoped antiferromagnetic parent insulator like ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$. The quasiparticle dispersion of Hubbard polarons is determined by a hybridization of the Hubbard fermion subbands with local Franck-Condon resonances so the main polaronic effect of the quasiparticle band structure is a splitting of the Hubbard bands on the number of Hubbard polaron subbands. Increasing of the EPI constant results in an increase of splitting, decrease of the subband width, transfer of the spectral weight to high-energy multiphonon excitations, and subsequent localization of the charge carriers. Herewith, the effect of such renormalization for the conduction band and the valence one differs depending on the ratio of the diagonal and off-diagonal EPI. In the framework of the GTB method, the Franck-Condon broadening of the spectral function of polaronic excitations is also reproduced for strongly correlated systems with strong electron-phonon interaction.

arXiv:1506.05368•Physical Review B•Oct 26, 2015•16 citations
We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity \ensuremathσpot{\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}\ensuremathσpot​(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is briefly discussed.
The thermodynamic effects of polaronic interactions in the Hubbard model are studied by high-temperature perturbation theory. Two electron-lattice interaction terms are added to the Hubbard Hamiltonian. The transfer term is treated as the perturbation, so the results are valid for narrow bandwidths and high temperatures. A one-particle Green's-function method is used to find an explicit expression for the secondorder change in the partition function, and a simple analytic expression is given for the weak-binding limit. The magnetic susceptibility and specific heat are computed numerically and given for several cases of polaron binding energies and molecular energy levels. The results are compared with the limit of no electron-lattice coupling and it is shown that they cannot be explained as a simple renormalization of the Hubbard parameters ttt and UUU.
Motivated by aspects of layered high-temperature superconductors and related quasi-one-dimensional materials, we consider polaron structure, dynamics, and coherence in certain extended Peierls–Hubbard models. We emphasize the qualitative importance of electron–lattice interactions even in the presence of dominant electron–electron correlations, the signatures of polaron structure and dynamics in energy-resolved pair-distribution structure functions, and the effect of disorder on polaron propagation and stability. Introduction The formation and dynamics of polarons (and bipolarons), despite a halfcentury of theoretical and experimental study, remain fascinating topics in many-body physics, combining as they do (often competing) aspects of coupled fields with distinct natural time scales (e.g., electron–phonon, spin–phonon, exciton–phonon), electron–electron interactions, lattice discreteness, nonadiabaticity, collective quantum tunneling, thermal fluctuations, competitions between disorder and polaronic localization, etc. Direct observation of polarons through real-space imaging of any of the coupled fields is rare even with the advent of STEM , AFM techniques, etc. Likewise, global measurements such as that of electronic band-structure are insensitive to polaron features. Thus, experiments and theoretical techniques have had to focus on indirect effects on microscopic probes such as transport coefficients, electronic absorption, vibrational spectroscopy, and so forth. Our purpose here is to briefly review three qualitative effects in polaron physics that have arisen in modeling two components of the lattice structure of the layered cuprate superconductors [1] – namely, extended multi-band Peierls–Hubbard models of (a) the active CuO 2 planes and (b) polarizable Cu–O clusters in the axial direction (polarizable interplanar medium).
We consider the form of the frequency dependent conductivity function for the Hubbard Model when there is one electron per atom. The conductivity is written in terms of the possible paths of an extra electron and hole in which the particle-hole pair avoid being simultaneously on the same lattice site. The absorptive part of the conductivity can be crudely described as a Gaussian line centered at energy U of width 2 (zt)y3 as inferred from a calculation of the moments. The line is somewhat more narrow for a random spin system than for the antiferromagnetic Néel state. We also consider the question of the anisotropy of the conductivity in layered spin structures and show that it is not expected to be large.
We study the frequency-dependent electrical conductivity of a structurally disordered single-band Hubbard model near half-filling and in the strong-correlation limit. The electronic transfer integral is assumed to have a quasi-exponential form as used in the impurity-band problem of doped semiconductors. The short-time expansion of the current correlation function is calculated diagrammatically to the second order in the inverse temperature β and up to the seventh order in electronic hopping. Based on this expansion an approximate current correlation function is constructed. By means of the Kubo formula, the configurational average of the electrical conductivity is then calculated to order β2 for various atomic densities. The validity of the high-temperature expansion is examined. We also study the effect of spin polarization on the absorption line shape and the dc resistivity at different temperatures.
The polaron band theory of Lee, Low and Pines is extended beyond the effective mass approximation to a general form of the electron–phonon interaction within two different self-consistent schemes. In the first one a variational Bloch state depending on the polaron crystalline momentum is used to study polaron features taking into account umklapp processes. In particular, the self-energy and the effective mass at the Γ point of the Brillouin zone are calculated as functions of the electron–phonon coupling constant α. It is found that a sudden transition of these quantities from a large band to a narrow band polaron behaviour occurs at a critical value αc. Such value increases indefinitely with the bare electron band width; moreover the transition becomes sharper. The theory recovers, as particular cases, a) the Lee-Low-Pines regime valid for small α and large electron band width Δ˜(Δ˜ ≫ ħω and b) the Holstein limit for large α and small band width (Δ˜ ≫ ħω). This occurs as result of the competing role of α and Δ˜. In the second scheme, the electron–phonon interaction is considered only in a single elementary cell and the self-trapped polaron state is calculated self-consistently. It is shown that, if only one electronic band is taken into account, such state becomes a Δ function with infinitely negative energy when α is larger than a critical value depending on the electron band width. To avoid this fact, electronic band states of higher energy must be considered. Finally, through the tight-binding method, the polaron band is set up. It is shown that at the Γ point of the Brillouin zone, the contribution of the higher electronic bands prevents the self-energy and the polaron mass to have a sharp increase from the large to the narrow band regime.
The three-band $p\text{\ensuremath{-}}d$ model of strongly correlated electrons interacting with optical phonons via diagonal and off-diagonal electron-phonon interactions is considered within the cluster perturbation theory. In the beginning, the exact diagonalization of the Hamiltonian of a CuO4{\mathrm{CuO}}_{4}CuO4​ cluster results in the construction of local polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ with hole numbers nh=0,1,2{n}_{h}=0,1,2nh​=0,1,2 per unit cell. The intercluster hoppings and interactions are exactly written in terms of Hubbard operators ${X}_{f}^{pq}=\left|p\right\ensuremath{\rangle}\ensuremath{\langle}q|$ determined via the multielectron polaronic eigenstates $\left|p\right\ensuremath{\rangle}$ at site f\mathbf{f}f. The Fermi-type single-electron quasiparticle dispersion and spectral weight are calculated for the undoped antiferromagnetic parent insulator like La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​. The quasiparticle dispersion of Hubbard polarons is determined by a hybridization of the Hubbard fermion subbands with local Franck-Condon resonances so the main polaronic effect of the quasiparticle band structure is a splitting of the Hubbard bands on the number of Hubbard polaron subbands. Increasing of the EPI constant results in an increase of splitting, decrease of the subband width, transfer of the spectral weight to high-energy multiphonon excitations, and subsequent localization of the charge carriers. Herewith, the effect of such renormalization for the conduction band and the valence one differs depending on the ratio of the diagonal and off-diagonal EPI. In the framework of the GTB method, the Franck-Condon broadening of the spectral function of polaronic excitations is also reproduced for strongly correlated systems with strong electron-phonon interaction.