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

arXiv:cond-mat/9802215 · Physical Review Letters

Phase Diagram of the One-Dimensional Holstein Model of Spinless Fermions

Robert J. Bursill, Ross H. McKenzie, Chris J. Hamer

Published Jun 22, 1998• 104 citations
arXivHTMLPDF

Abstract

The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.

DOI
10.1103/physrevlett.80.5607

Related papers

Quantum Monte Carlo study of the one-dimensional Holstein model of spinless fermions

Ross H. McKenzie, Chris J. Hamer, D. W. Murray

The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (${\mathit{g}}^{2}$\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9512046•Physical review. B, Condensed matter•Apr 15, 1996•62 citations

Luttinger parameters and momentum distribution function for the half-filled spinless fermion Holstein model: A DMRG approach

S. Ejima, H. Fehske

We reexamine the nature of the metallic phase of the one-dimensional half-filled Holstein model of spinless fermions. To this end, we determine the Tomonaga-Luttinger–liquid correlation parameter K ρ by large-scale density matrix renormalisation group (DMRG) calculations, exploiting i) the leading-order scaling relations between the ground-state energy and the single-particle excitation gap and ii) the static charge structure factor in the long-wavelength limit. While both approaches give almost identical results for intermediate-to-large phonon frequencies, we find contrasting behaviour in the adiabatic regime: i) K ρ >1 (attractive) vs. ii) K ρ < 1 (repulsive). The latter result for the correlation exponent is corroborated by data obtained for the momentum distribution function n ( k ), which puts the existence of an attractive metallic state in the spinless fermion Holstein model into question. We conclude that the scaling relation must be modified in the presence of electron-phonon interactions with noticeable retardation.

arXiv:0905.1214•Europhysics Letters (EPL)•Jul 1, 2009•36 citations

Density-Matrix Embedding Theory Study of the One-Dimensional Hubbard–Holstein Model

Teresa E. Reinhard, Uliana Mordovina, Claudius Hubig +5 more

We present a density-matrix embedding theory (DMET) study of the one-dimensional Hubbard-Holstein model, which is paradigmatic for the interplay of electron-electron and electron-phonon interactions. Analyzing the single-particle excitation gap, we find a direct Peierls insulator to Mott insulator phase transition in the adiabatic regime of slow phonons in contrast to a rather large intervening metallic phase in the anti-adiabatic regime of fast phonons. We benchmark the DMET results for both on-site energies and excitation gaps against density-matrix renormalization group (DMRG) results and find good agreement of the resulting phase boundaries. We also compare the full quantum treatment of phonons against the standard Born-Oppenheimer (BO) approximation. The BO approximation gives qualitatively similar results to DMET in the adiabatic regime but fails entirely in the anti-adiabatic regime, where BO predicts a sharp direct transition from Mott to Peierls insulator, whereas DMET correctly shows a large intervening metallic phase. This highlights the importance of quantum fluctuations in the phononic degrees of freedom for metallicity in the one-dimensional Hubbard-Holstein model.

arXiv:1811.00048•Journal of Chemical Theory and Computation•Feb 26, 2019•37 citations

Metallicity in the half-filled Holstein-Hubbard model

Holger Fehske, Georg Hager, Eric Jeckelmann

We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove —in the course of a careful finite-size scaling analysis— recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequencies the metallic region is even more extended than previously expected, and subdivided into ordinary Luttinger-liquid and bipolaronic-liquid phases.

arXiv:0808.1675•Europhysics Letters (EPL)•Nov 24, 2008•80 citations

Metal-insulator transition of spinless fermions coupled to dispersive optical bosons

Florian Lange, Holger Fehske

Including the previously ignored dispersion of phonons we revisit the metal-insulator transition problem in one-dimensional electron-phonon systems on the basis of a modified spinless fermion Holstein model. Using matrix-product-state techniques we determine the global ground-state phase diagram in the thermodynamic limit for the half-filled band case, and show that in particular the curvature of the bare phonon band has a significant effect, not only on the transport properties characterized by the conductance and the Luttinger liquid parameter, but also on the phase space structure of the model as a whole. While a downward curved (convex) dispersion of the phonons only shifts the Tomonaga-Luttinger-liquid to charge-density-wave quantum phase transition towards stronger EP coupling, an upward curved (concave) phonon band leads to a new phase-separated state which, in the case of strong dispersion, can even completely cover the charge-density wave. Such phase separation does not occur in the related Edwards fermion-boson model.

arXiv:2407.14187•Scientific Reports•Aug 5, 2024•1 citation

One-dimensional Holstein model revisited

Sijia Zhao, Zhaoyu Han, Steven Allan Kivelson +1 more

We analyze the global ground-state (quantum) phase diagram of the one-dimensional spinful Holstein model at half filling as a function of the strength of the electron-phonon coupling (represented by the strength of the phonon-induced attraction, $U$) and the phonon frequency ${\ensuremath{\omega}}_{0}$. In addition to reanalyzing the various asymptotic regimes, we carry out density-matrix renormalization group simulations to correct previous inferences concerning the antiadiabatic (large ${\ensuremath{\omega}}_{0}$) and strong-coupling (large $U$) regimes. There are two distinct phases---a fully gapped commensurate charge-density-wave phase and a spin-gapped Luther-Emery phase with a gapless charge mode---separated by a phase boundary, with a shape that reflects different microscopic physics in the weak- and strong-coupling limits.

arXiv:2209.05498•Physical review. B./Physical review. B•Feb 21, 2023•3 citations
The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
We reexamine the nature of the metallic phase of the one-dimensional half-filled Holstein model of spinless fermions. To this end, we determine the Tomonaga-Luttinger–liquid correlation parameter K ρ by large-scale density matrix renormalisation group (DMRG) calculations, exploiting i) the leading-order scaling relations between the ground-state energy and the single-particle excitation gap and ii) the static charge structure factor in the long-wavelength limit. While both approaches give almost identical results for intermediate-to-large phonon frequencies, we find contrasting behaviour in the adiabatic regime: i) K ρ >1 (attractive) vs. ii) K ρ < 1 (repulsive). The latter result for the correlation exponent is corroborated by data obtained for the momentum distribution function n ( k ), which puts the existence of an attractive metallic state in the spinless fermion Holstein model into question. We conclude that the scaling relation must be modified in the presence of electron-phonon interactions with noticeable retardation.
We present a density-matrix embedding theory (DMET) study of the one-dimensional Hubbard-Holstein model, which is paradigmatic for the interplay of electron-electron and electron-phonon interactions. Analyzing the single-particle excitation gap, we find a direct Peierls insulator to Mott insulator phase transition in the adiabatic regime of slow phonons in contrast to a rather large intervening metallic phase in the anti-adiabatic regime of fast phonons. We benchmark the DMET results for both on-site energies and excitation gaps against density-matrix renormalization group (DMRG) results and find good agreement of the resulting phase boundaries. We also compare the full quantum treatment of phonons against the standard Born-Oppenheimer (BO) approximation. The BO approximation gives qualitatively similar results to DMET in the adiabatic regime but fails entirely in the anti-adiabatic regime, where BO predicts a sharp direct transition from Mott to Peierls insulator, whereas DMET correctly shows a large intervening metallic phase. This highlights the importance of quantum fluctuations in the phononic degrees of freedom for metallicity in the one-dimensional Hubbard-Holstein model.
We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove —in the course of a careful finite-size scaling analysis— recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequencies the metallic region is even more extended than previously expected, and subdivided into ordinary Luttinger-liquid and bipolaronic-liquid phases.
Including the previously ignored dispersion of phonons we revisit the metal-insulator transition problem in one-dimensional electron-phonon systems on the basis of a modified spinless fermion Holstein model. Using matrix-product-state techniques we determine the global ground-state phase diagram in the thermodynamic limit for the half-filled band case, and show that in particular the curvature of the bare phonon band has a significant effect, not only on the transport properties characterized by the conductance and the Luttinger liquid parameter, but also on the phase space structure of the model as a whole. While a downward curved (convex) dispersion of the phonons only shifts the Tomonaga-Luttinger-liquid to charge-density-wave quantum phase transition towards stronger EP coupling, an upward curved (concave) phonon band leads to a new phase-separated state which, in the case of strong dispersion, can even completely cover the charge-density wave. Such phase separation does not occur in the related Edwards fermion-boson model.
We analyze the global ground-state (quantum) phase diagram of the one-dimensional spinful Holstein model at half filling as a function of the strength of the electron-phonon coupling (represented by the strength of the phonon-induced attraction, UUU) and the phonon frequency ${\ensuremath{\omega}}_{0}$. In addition to reanalyzing the various asymptotic regimes, we carry out density-matrix renormalization group simulations to correct previous inferences concerning the antiadiabatic (large ${\ensuremath{\omega}}_{0}$) and strong-coupling (large UUU) regimes. There are two distinct phases---a fully gapped commensurate charge-density-wave phase and a spin-gapped Luther-Emery phase with a gapless charge mode---separated by a phase boundary, with a shape that reflects different microscopic physics in the weak- and strong-coupling limits.