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Physical review. B, Condensed matter

Pairing and charge-density-wave correlations in the Holstein model at half-filling

F. Marsiglio

Published Aug 1, 1990• 109 citations
Publisher

Abstract

We study the two-dimensional Holstein model at half-filling for small systems using quantum Monte Carlo methods and through diagrammatic perturbation techniques in the Migdal-Eliashberg approximation. It is found that the latter approximation is very accurate as long as the phonon self-energy is taken into account in a self-consistent way. Vertex corrections are not required.

DOI
10.1103/physrevb.42.2416

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We study the two-dimensional Holstein model at half-filling for small systems using quantum Monte Carlo methods and through diagrammatic perturbation techniques in the Migdal-Eliashberg approximation. It is found that the latter approximation is very accurate as long as the phonon self-energy is taken into account in a self-consistent way. Vertex corrections are not required.
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.
Abstract Determining the range of validity of Migdal’s approximation for electron-phonon (e-ph) coupled systems is a long-standing problem. Many attempts to answer this question employ the Holstein Hamiltonian, where the electron density couples linearly to local lattice displacements. When these displacements are large, however, nonlinear corrections to the interaction must also be included, which can significantly alter the physical picture obtained from this model. Using determinant quantum Monte Carlo and the self-consistent Migdal approximation, we compared superconducting and charge-density-wave correlations in the Holstein model with and without second-order nonlinear interactions. We find a disagreement between the two cases, even for relatively small values of the e-ph coupling strength, and, importantly, that this can occur in the same parameter regions where Migdal’s approximation holds. Our results demonstrate that questions regarding the validity of Migdal’s approximation go hand in hand with questions of the validity of a linear e-ph interaction.
We use an unbiased, continuous-time quantum Monte Carlo method to address the possibility of a zero-temperature phase without charge-density-wave (CDW) order in the Holstein and, by extension, the Holstein-Hubbard model on the half-filled square lattice. In particular, we present results spanning the whole range of phonon frequencies, allowing us to use the well understood adiabatic and antiadiabatic limits as reference points. For all parameters considered, our data suggest that CDW correlations are stronger than pairing correlations even at very low temperatures. These findings are compatible with a CDW ground state that is also suggested by theoretical arguments.
Phase diagram of the two-dimensional Hubbard-Holstein model: enhancement of sss-wave pairing between charge and magnetic orders
We investigate the role of electron-electron and electron-phonon interactions in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model at half-filling. We study the competition and interplay between antiferromagnetism (AFM) and charge-density wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favoring (nonlocal) sss-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.
We study the electron and phonon spectral-weight functions and densities of states for the Holstein model obtained by analytically continuing numerical single-particle Green's functions. At half-filling we observe the appearance of a charge-density-wave gap in the electron spectral-weight function and a softening and broadening of the phonon spectrum in the density of states as a function of the inverse temperature. Away from half-filling we have a transition to a superconducting phase indicated by the formation of a gap in the electron density of states. Our results are consistent with previously studied correlation functions, susceptibilities, and self-energies obtained from quantum Monte Carlo calculations and diagrammatic techniques.