Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Weak-Coupling Theory of the Polaron Energy-Momentum Relation

George Whitfield, Robert D. Puff

Published Jul 5, 1965• 96 citations
Publisher

Abstract

We present arguments to show that the energy-momentum relation for a polaron is quadratic for small $k$, but bends over and becomes horizontal when the energy approaches the optical-phonon energy above the ground state. We present a calculation for zero temperature based on a thermodynamic Green's-function theory which is especially designed to give a reliable polaron energy-momentum relation.

DOI
10.1103/physrev.139.a338

Related papers

Polaron Energy Spectrum

David M. Larsen

The energy spectrum of a polaron---an electron interacting with the longitudinal-optical phonons of a polar crystal---is studied with particular attention given to the region where the polaron excitation energy closely approaches the energy of a longitudinal-optical phonon. In this region, existing theories of the polaron spectrum are inadequate; in particular, the usual Rayleigh-Schr\"odinger perturbation theory is shown to be inconsistent. A self-consistent weak-coupling theory is developed, and a variational theory of the polaron spectrum which, for small coupling, reduces to this weak-coupling theory is presented. Whitfield and Puff, and Schultz have conjectured that the polaron energy $E\mathrm{E}p\mathrm{R}$, bends over and becomes horizontal as the polaron momentum $p$ approaches from below the value at which the polaron excitation energy, $E(p)\ensuremath{-}E(0)$, becomes equal to the optical-phonon energy. Using the weak-coupling method of the present paper, this conjecture has been verified to lowest order and next higher order in the coupling constant.

Physical Review•Apr 15, 1966•83 citations

Green's function theory of the polaron energy‐momentum relation

R. v. Baltz

Abstract The energy E(p) of a polaron is studied as a function of temperature and momentum p with particular attention focused on the region where the excitation energy E(p) – E(0) closely approaches the energy of a longitudinal optical phonon. It is shown that in this region a quasi‐particle description is not adequate, e.g., the horizontal tangent in E(p) does not give rise to a singularity in the density of electronic states. The shape of the spectral weight function indicates that even for small coupling strength there is an internal excited state of the polaron.

physica status solidi (b)•Mar 1, 1972•7 citations

Energy-Momentum Relation for Weakly Coupled Optical Polarons

Tianwei Wu, K. Y. Yang

We have investigated the energy-momentum relationship for optical polarons by taking into consideration the phonon-cross-exchange diagrams. Our result is similar to that of the improved Tamm-Dancoff method, except that the whole curve shifts downward. Therefore, the ground-state energy is lower than the upper bound obtained by the variational method.

Physical Review Letters•Aug 7, 1972•7 citations

A Green's Function Approach to Intermediate‐ and Strong‐Coupling Polaron Theory in the Presence of External Fields

M. Porsch

Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.

physica status solidi (b)•Jan 1, 1970•12 citations

Dynamical mean-field theory of the small polaron

Sergio Ciuchi, FERDINANDO DE PASQUALE, S. Fratini +1 more

A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.

arXiv:cond-mat/9703118•Physical review. B, Condensed matter•Aug 15, 1997•188 citations

Polaron inndimensions

Bo E. Sernelius

The energy dispersion for arbitrary momentum of a polaron in n dimensions is calculated by the Rayleigh-Schr\"odinger perturbation theory and by a modified Brillouin-Wigner perturbation theory. For the energy and effective mass of the polaron ground state, both methods give results that agree with the results of V. V. Paranjape and P. V. Panjat [Phys. Rev. B 35, 2942 (1987)] to linear order in the polaron coupling constant \ensuremath{\alpha}. The self-energy shifts are progressively weakened with increasing dimension. In particular, this weakening occurs for the structure in the energy dispersion near the momentum value at which the polaron starts decaying via the emission of phonons.

Physical review. B, Condensed matter•Dec 15, 1987•13 citations
We present arguments to show that the energy-momentum relation for a polaron is quadratic for small kkk, but bends over and becomes horizontal when the energy approaches the optical-phonon energy above the ground state. We present a calculation for zero temperature based on a thermodynamic Green's-function theory which is especially designed to give a reliable polaron energy-momentum relation.
The energy spectrum of a polaron---an electron interacting with the longitudinal-optical phonons of a polar crystal---is studied with particular attention given to the region where the polaron excitation energy closely approaches the energy of a longitudinal-optical phonon. In this region, existing theories of the polaron spectrum are inadequate; in particular, the usual Rayleigh-Schr\"odinger perturbation theory is shown to be inconsistent. A self-consistent weak-coupling theory is developed, and a variational theory of the polaron spectrum which, for small coupling, reduces to this weak-coupling theory is presented. Whitfield and Puff, and Schultz have conjectured that the polaron energy EEpRE\mathrm{E}p\mathrm{R}EEpR, bends over and becomes horizontal as the polaron momentum ppp approaches from below the value at which the polaron excitation energy, $E(p)\ensuremath{-}E(0)$, becomes equal to the optical-phonon energy. Using the weak-coupling method of the present paper, this conjecture has been verified to lowest order and next higher order in the coupling constant.
Abstract The energy E(p) of a polaron is studied as a function of temperature and momentum p with particular attention focused on the region where the excitation energy E(p) – E(0) closely approaches the energy of a longitudinal optical phonon. It is shown that in this region a quasi‐particle description is not adequate, e.g., the horizontal tangent in E(p) does not give rise to a singularity in the density of electronic states. The shape of the spectral weight function indicates that even for small coupling strength there is an internal excited state of the polaron.
Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
The energy dispersion for arbitrary momentum of a polaron in n dimensions is calculated by the Rayleigh-Schr\"odinger perturbation theory and by a modified Brillouin-Wigner perturbation theory. For the energy and effective mass of the polaron ground state, both methods give results that agree with the results of V. V. Paranjape and P. V. Panjat [Phys. Rev. B 35, 2942 (1987)] to linear order in the polaron coupling constant \ensuremath{\alpha}. The self-energy shifts are progressively weakened with increasing dimension. In particular, this weakening occurs for the structure in the energy dispersion near the momentum value at which the polaron starts decaying via the emission of phonons.