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arXiv:2412.13263

Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model

Weber, Manuel

Published Dec 17, 2024• 0 citations
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Abstract

The Hubbard-Holstein model is one of the central models that describe the competition between electron-electron and electron-phonon interactions. In one dimension and at half-filling, the interplay between an electronic spin-density wave and a phonon-driven charge-density wave is considered to stabilize an intermediate Luther-Emery liquid. Here we show that, once the extended metallic regime disappears, a narrow bond-order-wave phase emerges. We use an exact directed-loop quantum Monte Carlo method for retarded interactions to simulate system sizes of several hundred sites, necessary to observe the weak signatures of this novel phase. Our results suggest a second-order quantum phase transition between the two charge orders that only turns first-order at strong coupling. Our findings are reminiscent of the extended Hubbard model, but they are driven by competing frequency dependencies of the same dynamically-screened Hubbard interaction.

DOI
10.48550/arxiv.2412.13263

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The Hubbard-Holstein model is one of the central models that describe the competition between electron-electron and electron-phonon interactions. In one dimension and at half-filling, the interplay between an electronic spin-density wave and a phonon-driven charge-density wave is considered to stabilize an intermediate Luther-Emery liquid. Here we show that, once the extended metallic regime disappears, a narrow bond-order-wave phase emerges. We use an exact directed-loop quantum Monte Carlo method for retarded interactions to simulate system sizes of several hundred sites, necessary to observe the weak signatures of this novel phase. Our results suggest a second-order quantum phase transition between the two charge orders that only turns first-order at strong coupling. Our findings are reminiscent of the extended Hubbard model, but they are driven by competing frequency dependencies of the same dynamically-screened Hubbard interaction.
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 present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent K(rho) >1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency.
We use a stochastic series-expansion quantum Monte Carlo method to study the phase diagram of the one-dimensional extended Hubbard model at half-filling for small to intermediate values of the on-site U and nearest-neighbor V repulsions. We confirm the existence of a novel, long-range-ordered bond-order-wave (BOW) phase recently predicted by Nakamura [J. Phys. Soc. Jpn. 68, 3123 (1999)] in a small region of the parameter space between the familiar charge-density-wave (CDW) state for $V\ensuremath{\gtrsim}U/2$ and the state with dominant spin-density-wave (SDW) fluctuations for $V\ensuremath{\lesssim}U/2.$ We discuss the nature of the transitions among these states and evaluate some of the critical exponents. Further, we determine accurately the position of the multicritical point, ${(U}_{m}{,V}_{m})=(4.7\ifmmode\pm\else\textpm\fi{}0.1,2.51\ifmmode\pm\else\textpm\fi{}0.04)$ (in energy units where the hopping integral is normalized to unity), above which the two continuous SDW-BOW-CDW transitions are replaced by one discontinuous (first-order) direct SDW-CDW transition. We also discuss the evolution of the CDW and BOW states upon hole doping. We find that in both cases the ground state is a Luther-Emery liquid, i.e., the spin gap remains but the charge gap existing at half-filling is immediately closed upon doping. The charge and bond-order correlations decay with distance r as ${r}^{\ensuremath{-}{K}_{\ensuremath{\rho}}},$ where ${K}_{\ensuremath{\rho}}$ is approximately 0.50.50.5 for the parameters we have considered. We also discuss advantages of using parallel tempering (or exchange Monte Carlo)---an extended ensemble method that we here combine with quantum Monte Carlo---in studies of quantum phase transitions.
The ground-state phase diagram of the half filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron $(e\text{\ensuremath{-}}e)$ repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also $e\text{\ensuremath{-}}ph$ couplings. Our method [Tsai et al., Phys. Rev. B 72, 054531 (2005); Philos. Mag. B 86, 2631 (2006)] treats $e\text{\ensuremath{-}}e$ and $e\text{\ensuremath{-}}ph$ interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the SDW and CDW states. We study the effects of retardation, which are particularly important near the transition, and find that umklapp processes at finite frequencies drive the CDW instability close to the transition.
We study the interplay between the electron-phonon (e(e(e-ph) and on-site electron-electron (e-e) interactions in a three-orbital Hubbard-Holstein model on an extended one-dimensional lattice using determinant quantum Monte Carlo. For weak e-e and eee-ph interactions, we observe a competition between an orbital-selective Mott phase (OSMP) and a (multicomponent) charge-density-wave (CDW) insulating phase, with an intermediate metallic phase located between them. For large e-e and eee-ph couplings, the OSMP and CDW phases persist, while the metallic phase develops short-range orbital correlations and becomes insulating when both the e-e and eee-ph interactions are large but comparable. Many of our conclusions are in line with those drawn from a prior dynamical mean-field theory study of the two-orbital Hubbard-Holstein model [Phys. Rev. B 95, 121112(R) (2017)] in infinite dimension, suggesting that the competition between the eee-ph and e-e interactions in multiorbital Hubbard-Holstein models leads to rich physics, regardless of the dimension of the system.
We present a determinant quantum Monte Carlo study of the competition between instantaneous on-site Coulomb repulsion and retarded phonon-mediated attraction between electrons, as described by the two-dimensional Hubbard-Holstein model. At half filling, we find a strong competition between antiferromagnetism (AFM) and charge-density-wave (CDW) order. We demonstrate that a simple picture of AFM-CDW competition that incorporates the phonon-mediated attraction into an effective-U Hubbard model requires significant refinement. Specifically, retardation effects slow the onset of charge order so that CDW order remains absent even when the effective U is negative. This delay opens a window where neither AFM nor CDW order is well established and where there are signatures of a possible metallic phase.