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 the Physical Society of Japan

The Hubbard–Holstein Model with Anharmonic Phonons in One Dimension

Ashok Chatterjee, Yasutami Takada

Published Apr 15, 2004• 18 citations
Publisher

Abstract

Effects of cubic and quartic anharmonic phonons on the polaronic properties are investigated in the one-dimensional Hubbard–Holstein model at half filling, using the variable-displacement Lang–Firsov canonical transformation and the exact solution of Lieb and Wu. Although the quartic anharmonicity always reduces the effect of the Coulomb repulsion U , the cubic one mainly responsible for the thermal expansion of lattice brings about the effect of strong asymmetric nature in the sign of the electron–phonon coupling constant g ; it enhances the effect of U for positive g , while it suppresses for negative g . As a result, the overall features of the anharmonic system are basically similar to those in the harmonic one for positive g , but even in the range of the realistic magnitudes for those anharmonicities, they may become qualitatively so different for negative g as to provide a first-order phase transition.

DOI
10.1143/jpsj.73.964

Related papers

On the Anharmonic Polaronic Model

Jerzy Konior

Anharmonic polaronic model is considered.The model consists of the electronic subsystem, described within the extended Hubbard model, coupled to anharmonic local phonons.By the canonical Holstein-Lang-Firsov transformation the electron-phonon interaction term is ruled out and, then, the effective electronic Hamiltonian is obtained by averaging over the ground state of the phononic subsystem.It is shown that anharmonicity introduces two main additional factors as compared with the harmonic case.First, the band narrowing factor is less rapidly decaying function of the electron-phonon interaction strength.Second, anharmonicity introduces further renormaliza - tion of the on-and intersite interaction between fermions.

Acta Physica Polonica A•Nov 1, 1992•0 citations

Metallicity in a Holstein-Hubbard Chain at Half Filling with Gaussian Anharmonicity

Ch. Uma Lavanya, I. V. Sankar, Ashok Chatterjee

The Holstein-Hubbard model with Gaussian phonon anharmonicity is studied in one-dimension at half filling using a variational method based on a series of canonical transformations. A fairly accurate phonon state is chosen to average the transformed Holstein-Hubbard Hamiltonian to obtain an effective Hubbard model which is then solved using the exact Bethe - ansatz following Lieb and Wu to obtain the ground state energy, the average lattice displacement and the renormalized parameters. The Mott-Hubbard criterion, local spin moment and the von Neumann entropy (which is a measure of quantum entanglement) are calculated to determine the ground state phase diagram which shows that the width of the metallic phase flanked by the SDW and CDW phases increases with increasing anharmonicity at low and moderate values of anharmonicity but eventually saturates when the anharmonicity becomes substantially large.

Scientific Reports•Jun 13, 2017•13 citations

Quantum Monte Carlo study of an anharmonic Holstein model

G. Paleari, F. Hébert, Benjamin Cohen-Stead +3 more

We study the effects of anharmonicity on the physics of the Holstein model, which describes the coupling of itinerant fermions and localized quantum phonons, by introducing a quartic term in the phonon potential energy. We find that the presence of this anharmonic term reduces the extent of the charge density wave (CDW) phase at half-filling as well as the transition temperature to this phase. Doping away from half-filling, we observe a first-order phase transition between the CDW and a homogeneous phase which is also present in the harmonic model. In addition, we study the evolution of the superconducting susceptibility in the doped region and show that anharmonicity can enhance the superconducting response.

arXiv:2101.08285•Physical review. B./Physical review. B•May 10, 2021•13 citations

Holstein model in infinite dimensions

James K. Freericks, Mark Jarrell, Douglas J. Scalapino

Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.

Physical review. B, Condensed matter•Sep 1, 1993•155 citations

Phase diagram of the one-dimensional Hubbard-Holstein model at half and quarter filling

Rahul P. Hardikar, Rudolf Torsten Clay

The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling, the Holstein electron-phonon coupling promotes on-site pairs of electrons and a Peierls charge-density wave, while the Hubbard on-site Coulomb repulsion $U$ promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) as the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions, the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-$U$ Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger $U$, the two transitions merge at a tricritical point into a single first-order Mott/Peierls transition. In addition, we show that an intermediate phase also occurs in the quarter-filled model.

arXiv:cond-mat/0612273•Physical Review B•Jun 6, 2007•73 citations

Anharmonicity in One-Dimensional Electron–Phonon System

Jize Zhao, Kazuo Ueda

We investigate the effect of anharmonicity on the one-dimensional half-filled Holstein model by using the determinant quantum Monte Carlo method. By calculating the order parameters we find that with and without anharmonicity there is always an transition from a disorder phase to a dimerized phase. Moreover, in the dimerized phase a lattice dimerization and a charge density wave coexist. The anharmonicity represented by the quartic term suppresses the dimerization as well as the charge density wave, while a double-well potential favors the dimerization. In addition, by calculating the correlation exponents we show that the disorder phase is metallic with gapless charge excitations and gapful spin excitations while in the dimerized phase both excitations are gapful.

arXiv:1002.0315•Journal of the Physical Society of Japan•Jun 25, 2010•3 citations
Effects of cubic and quartic anharmonic phonons on the polaronic properties are investigated in the one-dimensional Hubbard–Holstein model at half filling, using the variable-displacement Lang–Firsov canonical transformation and the exact solution of Lieb and Wu. Although the quartic anharmonicity always reduces the effect of the Coulomb repulsion U , the cubic one mainly responsible for the thermal expansion of lattice brings about the effect of strong asymmetric nature in the sign of the electron–phonon coupling constant g ; it enhances the effect of U for positive g , while it suppresses for negative g . As a result, the overall features of the anharmonic system are basically similar to those in the harmonic one for positive g , but even in the range of the realistic magnitudes for those anharmonicities, they may become qualitatively so different for negative g as to provide a first-order phase transition.
Anharmonic polaronic model is considered.The model consists of the electronic subsystem, described within the extended Hubbard model, coupled to anharmonic local phonons.By the canonical Holstein-Lang-Firsov transformation the electron-phonon interaction term is ruled out and, then, the effective electronic Hamiltonian is obtained by averaging over the ground state of the phononic subsystem.It is shown that anharmonicity introduces two main additional factors as compared with the harmonic case.First, the band narrowing factor is less rapidly decaying function of the electron-phonon interaction strength.Second, anharmonicity introduces further renormaliza - tion of the on-and intersite interaction between fermions.
The Holstein-Hubbard model with Gaussian phonon anharmonicity is studied in one-dimension at half filling using a variational method based on a series of canonical transformations. A fairly accurate phonon state is chosen to average the transformed Holstein-Hubbard Hamiltonian to obtain an effective Hubbard model which is then solved using the exact Bethe - ansatz following Lieb and Wu to obtain the ground state energy, the average lattice displacement and the renormalized parameters. The Mott-Hubbard criterion, local spin moment and the von Neumann entropy (which is a measure of quantum entanglement) are calculated to determine the ground state phase diagram which shows that the width of the metallic phase flanked by the SDW and CDW phases increases with increasing anharmonicity at low and moderate values of anharmonicity but eventually saturates when the anharmonicity becomes substantially large.
We study the effects of anharmonicity on the physics of the Holstein model, which describes the coupling of itinerant fermions and localized quantum phonons, by introducing a quartic term in the phonon potential energy. We find that the presence of this anharmonic term reduces the extent of the charge density wave (CDW) phase at half-filling as well as the transition temperature to this phase. Doping away from half-filling, we observe a first-order phase transition between the CDW and a homogeneous phase which is also present in the harmonic model. In addition, we study the evolution of the superconducting susceptibility in the doped region and show that anharmonicity can enhance the superconducting response.
Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.
The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling, the Holstein electron-phonon coupling promotes on-site pairs of electrons and a Peierls charge-density wave, while the Hubbard on-site Coulomb repulsion UUU promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) as the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions, the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-UUU Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger UUU, the two transitions merge at a tricritical point into a single first-order Mott/Peierls transition. In addition, we show that an intermediate phase also occurs in the quarter-filled model.
We investigate the effect of anharmonicity on the one-dimensional half-filled Holstein model by using the determinant quantum Monte Carlo method. By calculating the order parameters we find that with and without anharmonicity there is always an transition from a disorder phase to a dimerized phase. Moreover, in the dimerized phase a lattice dimerization and a charge density wave coexist. The anharmonicity represented by the quartic term suppresses the dimerization as well as the charge density wave, while a double-well potential favors the dimerization. In addition, by calculating the correlation exponents we show that the disorder phase is metallic with gapless charge excitations and gapful spin excitations while in the dimerized phase both excitations are gapful.