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Journal of Physics Condensed Matter

Solution of the local polaron model with a boson continuum

Michele Cini, Massimiliano Cuozzo

Published Nov 11, 1996• 3 citations
Publisher

Abstract

We study the zero-temperature Green function for the local polaron model that describes the interaction of an electron in an empty band with a boson continuum; the coupling takes place at one site in the lattice. To this end, we propose a new exact approach which consists in two steps: first, the problem is reduced to the solution of an infinite linear algebraic system of recursion equations by the method of excitation amplitudes; then, a systematic expansion of the solution is generated by Feenberg's formalism. Previously, the problem had been solved exactly in the narrow-band limit; otherwise, the solution was achieved for one or a few boson modes. Here we consider a boson continuum and obtain an infinite expansion of the self-energy in terms of the dressed Green function. In this case, the recursive, continued-fraction technique is equivalent to self-consistent perturbation theory. The results are illustrated by selected numerical examples, and show that the satellites are much more sensitive to the boson dispersion than the main line.

DOI
10.1088/0953-8984/8/46/007

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We study the zero-temperature Green function for the local polaron model that describes the interaction of an electron in an empty band with a boson continuum; the coupling takes place at one site in the lattice. To this end, we propose a new exact approach which consists in two steps: first, the problem is reduced to the solution of an infinite linear algebraic system of recursion equations by the method of excitation amplitudes; then, a systematic expansion of the solution is generated by Feenberg's formalism. Previously, the problem had been solved exactly in the narrow-band limit; otherwise, the solution was achieved for one or a few boson modes. Here we consider a boson continuum and obtain an infinite expansion of the self-energy in terms of the dressed Green function. In this case, the recursive, continued-fraction technique is equivalent to self-consistent perturbation theory. The results are illustrated by selected numerical examples, and show that the satellites are much more sensitive to the boson dispersion than the main line.
The spectral function of a single electron coupled to Einstein phonons (Holstein model) is calculated exactly in the limit of large space dimensionality, for any coupling strength and phonon frequency. A self-consistent mapping onto an effective impurity model leads to the solution in terms of a continued-fraction expansion for the electron propagator at zero temperature. True self-trapping is obtained only for finite bandwidth, such as the case of a Bethe lattice in infinite dimensions. It is shown that at intermediate coupling multiphonon resonances develop, corresponding to polaron states. Only the lowest one has long lifetime, and the polaron mass is calculated.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
Numerical methods whereby exact solutions to the problem of a few particles interacting with one another and with several bosonic excitation branches are presented. The diagrammatic Monte Carlo method allows the exact calculation of the Matsubara Green function, and the stochastic optimization technique provides an approximation-free analytic continuation. In this review, results unobtainable by conventional methods are discussed, including the properties of excited states in the self-trapping phenomenon, the optical spectra of polarons in all coupling regimes, the validity range analysis of the Frenkel and Wannier approximations relevant to the exciton, and the peculiarities of photoemission spectra of a lattice-coupled hole in a Mott insulator.
We develop and compare several analytical approximations for the ground state and zero-temperature polaron dispersion in finite-width, nonparabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. In addition, we revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results. All methods are applied to the Holstein model on a tight-binding lattice and are benchmarked against numerically exact calculations, including diagrammatic Monte Carlo (both our calculations and preceding works) and exact diagonalization results. Furthermore, the analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.
We develop a bosonization formalism that captures nonperturbatively the interaction effects on the Q=0\mathbf{Q}=0Q=0 continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the Q=0\mathbf{Q}=0Q=0 neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single-particle Green's function. We apply this machinery to compute the interaction corrections to the optical conductivity of two-dimensional Dirac fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong-coupling regime.
We introduce a method, numerically exact in principle, for computing the momentum- and frequency-resolved single-particle Green's function of a polaron at finite temperature. The method, which we refer to as the finite-temperature Green's function cluster expansion, combines two ingredients: the generalized Green's function cluster expansion, a numerically exact extension of the momentum average family of methods that solves the polaron problem at zero temperature through a hierarchy of equations of motion for restricted phonon cloud configurations; and the thermofield double formalism, which maps the thermal trace onto a pure-state expectation value over a doubled Hilbert space. The resulting equations of motion have the same algebraic structure as those of the multi-boson zero-temperature theory, with the temperature entering through a Bogoliubov-type mixing angle that controls the coupling to a set of fictitious bath bosons. We implement the method in our open-source software package and benchmark it on the one-dimensional Holstein polaron, computing spectral functions, dispersions, lifetimes, and effective masses across coupling regimes and temperatures up to T/Ω∼1T/Ω\sim 1T/Ω∼1. Where finite-temperature density matrix renormalization group results are available, we find quantitative agreement at affordable computational cost. The method recovers momentum-resolved spectra directly in frequency space, with no time evolution or analytic continuation. We also discuss the practical costs of the approach. In particular, since the doubled phonon Hilbert space has a non-trivial configuration structure in which real and fictitious clouds compete, convergence in the corresponding cloud parameters requires care.