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arXiv:0906.1233 · Europhysics Letters (EPL)

Holstein polaron in the presence of disorder

Mona Berciu, Andrei S. Mishchenko, Naoto Nagaosa

Published Feb 1, 2010• 31 citations
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Abstract

Non-local, inhomogeneous and retarded response similar to that observed in experiments is studied theoretically by introducing the Inhomogeneous Momentum Average (IMA) approximation for single-polaron problems with disorder in the on-site potential and/or spatial variations of the electron-phonon couplings and/or phonon frequencies. We show that the electron-phonon coupling gives rise to an additional inhomogeneous, strongly retarded potential. This potential describes essential physics ignored by "instanteneous" approximations. The accuracy of IMA is demonstrated by comparison with single-impurity results from the approximation-free Diagrammatic Monte Carlo (DMC) method. Its simplicity allows for easy study of many problems that were previously unaccessible.

DOI
10.1209/0295-5075/89/37007

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•Jan 1, 2009•1 citation
Non-local, inhomogeneous and retarded response similar to that observed in experiments is studied theoretically by introducing the Inhomogeneous Momentum Average (IMA) approximation for single-polaron problems with disorder in the on-site potential and/or spatial variations of the electron-phonon couplings and/or phonon frequencies. We show that the electron-phonon coupling gives rise to an additional inhomogeneous, strongly retarded potential. This potential describes essential physics ignored by "instanteneous" approximations. The accuracy of IMA is demonstrated by comparison with single-impurity results from the approximation-free Diagrammatic Monte Carlo (DMC) method. Its simplicity allows for easy study of many problems that were previously unaccessible.
We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling is responsible for a strong renormalization of the impurity potential, resulting in an effective potential with significant retardation effects, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our approximation is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency, and contrast and highlight the difference between these cases.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.
The Hubbard model augmented by the Holstein interaction of localized electrons and optical dispersionless phonons is reduced (with the help of a canonical transformation) to a model, in which there occur jumps of polarons and scattering on “impurity” scatterers, which are accompanied by jumps and scatterings of an unlimited number of phonons. This circumstance and the fact that there is a strong on-site interaction of electrons inherent in the Hubbard model, lead to the need for the formulation of a new theory of the phenomenon and the use of a diagrammatic technique. A theory of perturbations in disorder for electrons moving in the random field of “impurity” scatterers has been developed. A collective mode of the phonon cloud surrounding a polaron has been found and the influence of the processes of scattering of this mode on the energy spectrum of both localized and delocalized polarons has been established. The motion of the delocalized polarons related to their scattering is described in the approximation of chain diagrams, in which the manyparticle irreducible Green’s functions of the polaron are ignored. The electron-energy spectrum obtained consists of four subbands.
Using the momentum average (MA) approximation, we find an analytical expression for the disorder-averaged Green's function of a Holstein polaron in a three-dimensional simple cubic lattice with random on-site energies. The on-site disorder is assumed to be weak compared to the kinetic energy of the polaron, and is treated perturbationally. Within this scheme, the states at the bottom of the polaron band are found to have an infinite lifetime, signaling a failure of perturbation theory at these energies. The higher-energy polaron states have a finite lifetime. We study this lifetime and the disorder-induced energy shift of these eigenstates for various strengths of disorder and electron-phonon coupling. We compare our findings to the predictions of Fermi's golden rule and the average TTT-matrix method, and find a significant quantitative discrepancy at strong electron-phonon coupling, where the polaron lifetime is much shorter than Fermi's golden rule prediction. We attribute this to the renormalization of the on-site potential by the electron-phonon coupling.
We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.
In this work we present a highly efficient and accurate analytical approximation for the Green's function of a polaron: the momentum average (MA) approximation. It is obtained by summing all of the self-energy diagrams, but with each diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling, and is accurate everywhere in the parameter space. The approximation is first used to investigate the Holstein model. A detailed analysis of its accuracy is provided through diagrammatics and spectral weight sum rules. It is shown that the resulting Green's function satisfies exactly the first six spectral weight sum rules, and all higher order sum rules are satisfied with great accuracy. Comparison with numerical data also confirms this accuracy. We then show how to improve the MA approximation by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the zeroth-order MA approximation. The quantitative agreement with numerical data is also improved. Next, we generalize the MA approximation to study the properties of models with momentum-dependent electron-phonon coupling, and then show that further improvements can be obtained based on variational considerations, using the 1D breathing-mode Hamiltonian as a specific example. For example, by using this variational MA, we obtain ground state energies within at most 0.3% error of the numerical data. Finally, we study the effects of a nearby surface on the spectral weight of a Holstein polaron. The broken translational symmetry is accounted for without any additional approximations, and the resulting inhomogeneous MA approximation continues to be accurate for all coupling strengths. We show that the surface changes properties significantly, with bulk values being recovered only very far away from it. We find that the electron-phonon coupling gives rise to an additional surface potential which is responsible for binding surface states even when they are not normally expected. These results demonstrate that interpretation in terms of bulk properties of spectroscopic data sensitive only to a few surface layers is not straightforward.