We use a combination of quantum Monte Carlo (QMC) methods and analytic continuation techniques to study the formation of charge-density-wave (CDW) gaps at half-filling in the two-dimensional Holstein model. From QMC data for the dressed unequal-time Green's function G(\ensuremath{\tau},p) we calculate the real frequency spectral function A(\ensuremath{\omega},p) for different temperatures T, electron-phonon coupling constants g, phonon frequencies ${\mathrm{\ensuremath{\omega}}}_{0}$, and spatial lattice sizes N. We also compute the spectral functions by approximate diagrammatic methods and find these results to be in good qualitative agreement with the QMC results. We discuss attempts to get quantitative agreement for a range of T and ${\mathrm{\ensuremath{\omega}}}_{0}$ by using a single renormalized vertex g\ifmmode\bar\else\textasciimacron\fi{} in place of g, an approach that proved useful in previous studies of the Hubbard model. Finally, comparisons are made with previous estimates of the CDW transition temperature based on measurements of static correlation functions and susceptibilities.
We examine the behavior of the electron and phonon self-energies obtained from quantum Monte Carlo simulations of the two-dimensional Holstein model as a function of Matsubara frequency and momentum. For the half-filled system, there is a finite-temperature phase transition to an insulating charge-density-wave state, and the electron self-energy develops a distinctive divergence. The renormalized phonon frequency softens strongly with temperature at the commensurate wave vector. Away from half-filling, the electron self-energy behaves in a manner consistent with that of a Fermi liquid.
By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping
t, the on-site electron-electron interaction
U, the phonon energy ${\ensuremath{\omega}}_{0}$, and the electron-phonon coupling
g. At half filling, the ground state is an antiferromagnetic insulator for $U\ensuremath{\gtrsim}2{g}^{2}/{\ensuremath{\omega}}_{0}$, while it is a charge-density-wave (or bipolaronic) insulator for $U\ensuremath{\lesssim}2{g}^{2}/{\ensuremath{\omega}}_{0}$. In addition to these phases, we find a superconducting phase that intrudes between them. For ${\ensuremath{\omega}}_{0}/t=1$, superconductivity emerges when both
U/t and $2{g}^{2}/t{\ensuremath{\omega}}_{0}$ are small; then, by increasing the value of the phonon energy ${\ensuremath{\omega}}_{0}$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.
We use a recently developed extension of the weak-coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model with phonon frequency ${\ensuremath{\omega}}_{0}=0.1t$. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant
2kf charge fluctuations. Emphasis is placed on the temperature dependence of the single-particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low-temperature Luttinger liquid phase we observe features that compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap that matches half the spin gap, whereas at temperatures above which this quasiparticle gap closes characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to
L=20 for two-particle properties and on cluster dynamical mean-field theory calculations with clusters up to 12 sites for the single-particle spectral function.
Single- and two-particle excitation spectra of the one-dimensional, half-filled Holstein-Hubbard model are calculated using the continuous-time quantum Monte Carlo method. In the metallic phase, the results are consistent with a Luther-Emery liquid that has gapped spin and single-particle excitations but a gapless charge mode. However, given the initially exponential dependence of the spin gap on the backscattering matrix element, the numerical excitation spectra appear gapless in the weak-coupling regime, and therefore resemble those of a Luttinger liquid. The Mott phase has the expected charge gap and gapless spin excitations. The Peierls state shows a charge, spin, and single-particle gap, a soft phonon mode, backfolded shadow bands, and soliton excitations. Arguments and numerical evidence for the existence of a nonzero spin gap throughout the metallic phase are provided in terms of equal-time spin and charge correlation functions.