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Physical review. B, Condensed matter

Charge-density-wave-gap formation in the two-dimensional Holstein model at half-filling

Parhat Niyaz, James E. Gubernatis, Richard T. Scalettar, Ching Yin Fong

Published Dec 1, 1993• 36 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.16011

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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 \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​, 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 \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ 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 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.
Over the past several years, reliable quantum Monte Carlo results for the charge density wave transition temperature Tcdw{T}_{\mathrm{cdw}}Tcdw​ of the half-filled two-dimensional Holstein model in square and honeycomb lattices have become available for the first time. Exploiting the further development of numerical methodology, here we present results in three dimensions, which are made possible through the use of Langevin evolution of the quantum phonon degrees of freedom. In addition to determining Tcdw{T}_{\mathrm{cdw}}Tcdw​ from the scaling of the charge correlations, we also examine the nature of charge order at general wave vectors for different temperatures, couplings, and phonon frequencies, and the behavior of the spectral function and specific heat.
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.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw{T}_{\mathrm{cdw}}Tcdw​ in two dimensions. Quantum Monte Carlo calculations have determined Tcdw{T}_{\mathrm{cdw}}Tcdw​ for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2d=2d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc{T}_{\mathrm{sc}}Tsc​ for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where ttt is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher TTT. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
We use determinant quantum Monte Carlo to study the half-filled `bond-Holstein' model on a square lattice. We find that the model exhibits a charge-density-wave (CDW) phase transition with a critical temperature TcdwT_\mathrm{cdw}Tcdw​ considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor ScdwS_{\rm cdw}Scdw​, we obtain TcdwT_\mathrm{cdw}Tcdw​ to greater than one percent accuracy. At the same time, local observables also show clear signatures consistent with the transition temperatures inferred from our scaling analysis. We attribute the enhanced CDW tendencies to a phonon-mediated nearest-neighbor electron repulsion that is directly proportional to the dimensionless electron-phonon coupling λλλ in the atomic (t→0t\rightarrow 0t→0) limit. This behavior contrasts with the site-Holstein case, where the same limit yields only an on-site attraction. We supplement our analysis with results from several unsupervised machine learning methods, which not only confirm our estimates of TcdwT_\mathrm{cdw}Tcdw​ but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.
We study the dynamical properties of the two-dimensional square lattice Holstein Hamiltonian using numerically exact determinant quantum Monte Carlo simulations. In particular, we report systematic calculations of the model's single-particle spectral function and optical conductivity over a range of phonon energies, electron-phonon coupling strengths, carrier concentrations, and temperatures. In doing so, we map the evolution of the system from a dressed metallic phase to a (bi)polaron liquid/insulator to a charge-density-wave insulator as the carrier concentration is tuned from dilute values to half-filling. We corroborate these results using several equal-time correlation measurements and related proxy quantities reconstructed from time-displaced correlation measurements. This paper is also accompanied by an extensive open data set, covering over 16,640 unique parameter values.