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arXiv:cond-mat/0510510

New solution for the polaron problem

И. Д. Феранчук, L I Komarov

Published Oct 19, 2005• 2 citations
arXivHTMLPDF

Abstract

New variational ansatz for the large-radius Fröhlich polaron is considered. The corresponding operator estimation for the energy of polaron proves to be very similar to the result found by Feynman on the basis of the variational principle for the functional integral of the system. It allows us to make clear the problem of ``phase transition'' and to anylize the structure of the localized state of the polaron in the intermediate coupling regime. It is shown that the analogous state exists also for the model of particle-field interaction with the divergent perturbation theory.

DOI
10.48550/arxiv.cond-mat/0510510

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Abstract New results for the free polaron ground state energy are presented, based on a non‐perturbative Fock approximation mean field theory. From variational calculations a new energy upper bound for all couplings is obtained, whose newness lies in its cylindrical symmetry, thu sbreaking the spherical symmetry of the Fröhlich Hamiltonian. For α (the electron–phonon coupling constant)≦3 a translational invariant state is found, for 3 ≦ α ⪅ 3.8 a two‐dimensional localized state while for α ⪆ 3.8, a three‐dimensional cylindrical localized state is obtained. By linking the variational approach to a dynamical approximation of the Fröhlich Hamiltonian, the possibility is suggested of second‐order phase transitions for the structure of the polaron, in the framework of the Fock approximation.

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•Oct 4, 1999•0 citations
New variational ansatz for the large-radius Fröhlich polaron is considered. The corresponding operator estimation for the energy of polaron proves to be very similar to the result found by Feynman on the basis of the variational principle for the functional integral of the system. It allows us to make clear the problem of ``phase transition'' and to anylize the structure of the localized state of the polaron in the intermediate coupling regime. It is shown that the analogous state exists also for the model of particle-field interaction with the divergent perturbation theory.
Abstract New results for the free polaron ground state energy are presented, based on a non‐perturbative Fock approximation mean field theory. From variational calculations a new energy upper bound for all couplings is obtained, whose newness lies in its cylindrical symmetry, thu sbreaking the spherical symmetry of the Fröhlich Hamiltonian. For α (the electron–phonon coupling constant)≦3 a translational invariant state is found, for 3 ≦ α ⪅ 3.8 a two‐dimensional localized state while for α ⪆ 3.8, a three‐dimensional cylindrical localized state is obtained. By linking the variational approach to a dynamical approximation of the Fröhlich Hamiltonian, the possibility is suggested of second‐order phase transitions for the structure of the polaron, in the framework of the Fock approximation.
The Fröhlich polaron model describes a ubiquitous class of problems concerned with understanding properties of a single mobile particle interacting with a bosonic reservoir. Originally introduced in the context of electrons interacting with phonons in crystals, this model found applications in such diverse areas as strongly correlated electron systems, quantum information, and high energy physics. In the last few years this model has been applied to describe impurity atoms immersed in Bose-Einstein condensates of ultracold atoms. The tunability of microscopic parameters in ensembles of ultracold atoms and the rich experimental toolbox of atomic physics should allow to test many theoretical predictions and give us new insights into equilibrium and dynamical properties of polarons. In these lecture notes we provide an overview of common theoretical approaches that have been used to study BEC polarons, including Rayleigh-Schrödinger and Green's function perturbation theories, self-consistent Born approximation, mean-field approach, Feynman's variational path integral approach, Monte Carlo simulations, renormalization group calculations, and Gaussian variational ansatz. We focus on the renormalization group approach and provide details of analysis that have not been presented in earlier publications. We show that this method helps to resolve striking discrepancy in polaron energies obtained using mean-field approximation and Monte Carlo simulations. We also discuss applications of this method to the calculation of the effective mass of BEC polarons. As one experimentally relevant example of a non-equililbrium problem we consider Bloch oscillations of Bose polarons and demonstrate that one should find considerable deviations from the commonly accepted phenomenological Esaki-Tsu model. We review which parameter regimes of Bose polarons can be achieved in various atomic mixtures.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.
A detailed study of the Frohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green function, within a novel approach. We present the best up to date results for the binding energy, and for the first time make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals non-trivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.
A detailed study of the Frohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green function, within a novel approach. We present the best up to date results for the binding energy, and for the first time make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals non-trivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.