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

Journal of Physics A Mathematical and General

Crossover between large and small polarons in many-electron system with on-site repulsion

Hongxing Zheng

Published Aug 20, 2003• 0 citations
Publisher

Abstract

An analytical approach has been developed for the electron–phonon coupling in the Holstein–Hubbard model. The Hubbard U is treated approximately by the slave-boson approach. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ω p /2 Dt is small, ω p /2 Dt ≪ 1, our result is similar to that of the Migdal–Eliashberg theory. When ω p /2 Dt > 1, our result approaches that of the Lang–Firsov transformation and small polaron theory. We have calculated the effective mass of electrons at the Fermi surface to show that the new perturbation approach works well in the intermediate region where ω p /2 Dt ∼ 1.

DOI
10.1088/0305-4470/36/35/325

Related papers

Intermediate region between large and small polarons in many-electron systems

H. Zheng, M. Avignon

An analytical approach has been developed for a many-electron Holstein model in one, two, and three dimensions. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}$ $(2\mathrm{Dt}$ is the half bandwidth in D dimensions) is small, ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\ll}1,$ our result is similar to that of the Migdal-Eliashberg theory. When ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}>1,$ our result approaches that of the Lang-Firsov transformation and small polaron theory. We have calculated the Green's function, the effective mass of electrons at the Fermi surface, the number of electrons ${n}_{\mathbf{k}}$ in a momentum state $\mathbf{k},$ and the jump of ${n}_{\mathbf{k}}$ at the Fermi surface to show that the perturbation approach works well in the intermediate region in which ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\sim}1.$

Physical review. B, Condensed matter•Jul 10, 2003•4 citations

A CONVERGENT PERTURBATION METHOD FOR THE HOLSTEIN MODEL AND ITS APPLICATION TO THE DOUBLE EXCHANGE SYSTEM

A. N. Das, Jayita Chatterjee, M. MITRA

We look for an analytical method which can describe the Holstein model over the entire region of electron–phonon (e–ph) coupling strength. Considering a toy model of two sites the perturbation expansions are developed for the Holstein Hamiltonian using different standard phonon bases. It is found that the modified Lang Firsov phonon basis shows satisfactory convergence for almost the entire range of e–ph coupling from the antiadiabatic limit to the intermediate range of hopping. This method is used to study the magnetic transition and the small to large polaron crossover in a double exchange model with e–ph interaction. Implications of the results in the context of manganites are discussed.

International Journal of Modern Physics B•Aug 10, 2001•1 citation

TWO-SITE POLARON PROBLEM: A PERTURBATION APPROACH WITH VARIATIONAL BASIS STATES

A. N. Das, JAYITA CHATTERJEE

A convergent perturbation method using modified Lang Firsov transformation is developed for a two-site single-polaron system. The method is applicable for the entire range of the electron–phonon coupling strength from the antiadiabatic limit to the intermediate region of hopping. The single-electron energies, oscillator wavefunctions and correlation functions, calculated using this method, are in good agreement with the exact results.

arXiv:cond-mat/9905373•International Journal of Modern Physics B•Dec 30, 1999•14 citations

Effective electron-phonon coupling and polaronic transition in the presence of strong correlation

Paolo Barone, Roberto Raimondi, Massimo Capone +1 more

We study the Hubbard-Holstein model using slave-boson mean-field and a variational Lang-Firsov transformation. We identify weak and strong $e$-ph coupling solutions, whose stability depends both on the bare $e$-ph coupling and on the correlation strength. At mean field level the evolution from weak to strong electron-phonon coupling occurs via a first-order polaronic transition if the adiabatic parameter is below a critical value. In the strongly correlated regime and in the adiabatic limit, the region in which the weak-coupling solution is stable is sizeably enlarged with respect to the weakly correlated system and the Mott metal-insulator transition is found to be robust with respect to $e$-ph interaction.

arXiv:cond-mat/0509090•Physical Review B•Feb 28, 2006•15 citations

Numerical study of the crossover from free electrons to small polarons

Zhou Li

The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.

•Jan 1, 2012•0 citations

Coherent-state ansatz for the Holstein polaron in one and two dimensions

Connor M. Walsh, Igor Boettcher, F. Marsiglio

The Holstein model often serves as an archetype for electron-phonon interactions and polaron formation in solids. However, precise descriptions of the Holstein polaron are difficult when the phonon frequency is small and the electron-phonon coupling is strong, due to the presence of many phonons in the ground-state. We present a semi-analytical approximation that consists of a variational ansatz with clouds of phonons surrounding the electron in the form of coherent states. This becomes particularly simple and exact in the Lang-Firsov limit. We determine the domain of validity away from this limit, and further explore the improvement achieved with a removal of the requirement that the phonon clouds form coherent states. Both approximations work extremely well for the ground-state energy at strong coupling, and both work surprisingly well also at weak-coupling. For the effective mass, however, both approximations have difficulty, particularly in the crossover region. Moreover, the coherent-state ansatz (CSA) retains an unphysical jump in the crossover region. Nonetheless, the CSA provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy.

arXiv:2603.07290•Journal of Physics Condensed Matter•Aug 24, 2026•0 citations
An analytical approach has been developed for the electron–phonon coupling in the Holstein–Hubbard model. The Hubbard U is treated approximately by the slave-boson approach. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ω p /2 Dt is small, ω p /2 Dt ≪ 1, our result is similar to that of the Migdal–Eliashberg theory. When ω p /2 Dt > 1, our result approaches that of the Lang–Firsov transformation and small polaron theory. We have calculated the effective mass of electrons at the Fermi surface to show that the new perturbation approach works well in the intermediate region where ω p /2 Dt ∼ 1.
An analytical approach has been developed for a many-electron Holstein model in one, two, and three dimensions. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}$ (2Dt(2\mathrm{Dt}(2Dt is the half bandwidth in D dimensions) is small, ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\ll}1,$ our result is similar to that of the Migdal-Eliashberg theory. When ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}>1,$ our result approaches that of the Lang-Firsov transformation and small polaron theory. We have calculated the Green's function, the effective mass of electrons at the Fermi surface, the number of electrons nk{n}_{\mathbf{k}}nk​ in a momentum state k,\mathbf{k},k, and the jump of nk{n}_{\mathbf{k}}nk​ at the Fermi surface to show that the perturbation approach works well in the intermediate region in which ${\ensuremath{\omega}}_{p}/2\mathrm{Dt}\ensuremath{\sim}1.$
We look for an analytical method which can describe the Holstein model over the entire region of electron–phonon (e–ph) coupling strength. Considering a toy model of two sites the perturbation expansions are developed for the Holstein Hamiltonian using different standard phonon bases. It is found that the modified Lang Firsov phonon basis shows satisfactory convergence for almost the entire range of e–ph coupling from the antiadiabatic limit to the intermediate range of hopping. This method is used to study the magnetic transition and the small to large polaron crossover in a double exchange model with e–ph interaction. Implications of the results in the context of manganites are discussed.
We study the Hubbard-Holstein model using slave-boson mean-field and a variational Lang-Firsov transformation. We identify weak and strong eee-ph coupling solutions, whose stability depends both on the bare eee-ph coupling and on the correlation strength. At mean field level the evolution from weak to strong electron-phonon coupling occurs via a first-order polaronic transition if the adiabatic parameter is below a critical value. In the strongly correlated regime and in the adiabatic limit, the region in which the weak-coupling solution is stable is sizeably enlarged with respect to the weakly correlated system and the Mott metal-insulator transition is found to be robust with respect to eee-ph interaction.
The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.
The Holstein model often serves as an archetype for electron-phonon interactions and polaron formation in solids. However, precise descriptions of the Holstein polaron are difficult when the phonon frequency is small and the electron-phonon coupling is strong, due to the presence of many phonons in the ground-state. We present a semi-analytical approximation that consists of a variational ansatz with clouds of phonons surrounding the electron in the form of coherent states. This becomes particularly simple and exact in the Lang-Firsov limit. We determine the domain of validity away from this limit, and further explore the improvement achieved with a removal of the requirement that the phonon clouds form coherent states. Both approximations work extremely well for the ground-state energy at strong coupling, and both work surprisingly well also at weak-coupling. For the effective mass, however, both approximations have difficulty, particularly in the crossover region. Moreover, the coherent-state ansatz (CSA) retains an unphysical jump in the crossover region. Nonetheless, the CSA provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy.