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arXiv:2005.00918 · Physical review. B./Physical review. B

Langevin simulations of the half-filled cubic Holstein model

Benjamin Cohen-Stead, Kipton M. Barros, Zi Yang Meng, Chuang Chen, Richard T. Scalettar, G. G. Batrouni

Published Oct 13, 2020• 32 citations
arXivHTMLPDF

Abstract

Over the past several years, reliable quantum Monte Carlo results for the charge density wave transition temperature ${T}_{\mathrm{cdw}}$ 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 ${T}_{\mathrm{cdw}}$ 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.

DOI
10.1103/physrevb.102.161108

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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.
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.
The effect of electron-phonon coupling on Dirac fermions was recently explored numerically on a honeycomb lattice, leading to precise quantitative values for the finite temperature and quantum critical points. In this paper, we use the unbiased determinant quantum Monte Carlo method to study the Holstein model on a half-filled staggered-flux square lattice and compare the results with the honeycomb lattice geometry, presenting results for a range of phonon frequencies $0.1\ensuremath{\leqslant}\ensuremath{\omega}\ensuremath{\leqslant}2.0$. We find that the interactions give rise to charge density wave order, but only above a finite coupling strength ${\ensuremath{\lambda}}_{\mathrm{crit}}$. The transition temperature is evaluated and presented in a ${T}_{c}\text{\ensuremath{-}}\ensuremath{\lambda}$ phase diagram. An accompanying mean-field theory calculation also predicts the existence of a quantum phase transition, but at a substantially smaller coupling strength.
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.
We have performed numerical studies of the Hubbard-Holstein model in two dimensions using determinant quantum Monte Carlo (DQMC). Here, we present details of the method, emphasizing the treatment of the lattice degrees of freedom, and then study the filling and behavior of the fermion sign as a function of model parameters. We find a region of parameter space with large Holstein coupling where the fermion sign recovers despite large values of the Hubbard interaction. This indicates that studies of correlated polarons at finite carrier concentrations are likely accessible to DQMC simulations. We then restrict ourselves to the half-filled model and examine the evolution of the antiferromagnetic structure factor, other metrics for antiferromagnetic and charge-density-wave order, and energetics of the electronic and lattice degrees of freedom as a function of electron-phonon coupling. From this we find further evidence for a competition between charge-density-wave and antiferromagnetic orders 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.
Despite being relevant to better understand the properties of honeycomblike systems, as graphene-based compounds, the electron-phonon interaction is commonly disregarded in theoretical approaches. That is, the effects of phonon fields on interacting Dirac electrons is an open issue, in particular when investigating long-range ordering. Thus, here we perform unbiased quantum Monte Carlo simulations to examine the Hubbard-Holstein model (HHM) in the half-filled honeycomb lattice. By performing careful finite-size scaling analysis, we identify semimetal-to-insulator quantum critical points, and determine the behavior of the antiferromagnetic and charge-density wave phase transitions. We have, therefore, established the ground state phase diagram of the HHM for intermediate interaction strength, determining its behavior for different phonon frequencies. Our findings provide quantitative and qualitative descriptions of the model at intermediate coupling strengths, and may shed light on the emergence of many-body properties in honeycomblike systems.