Langevin simulations of a long-range electron-phonon model

Abstract

We present a Quantum Monte Carlo (QMC) study, based on the Langevin equation, of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The bosonic part of the action helps control the variation of the field in imaginary time. As a consequence, the iterative conjugate gradient solution of the fermionic action, which depends on the boson coordinates, converges more rapidly than in the case of electron-electron interactions, such as the Hubbard Hamiltonian. Fourier Acceleration is shown to be a crucial ingredient in reducing the equilibration and autocorrelation times. After describing and benchmarking the method, we present results for the phase diagram focusing on the range of the electron-phonon interaction. We delineate the regions of charge density wave formation from those in which the fermion density is inhomogeneous, caused by phase separation. We show that the Langevin approach is more efficient than the Determinant QMC method for lattice sizes N 8 × 8 and that it therefore opens a potential path to problems including, for example, charge order in the 3D Holstein model.

One-dimensional Hubbard-Holstein model with finite-range electron-phonon coupling

“…The Langevin method we used is introduced and benchmarked in 36 where some additional results for U = 0 are also presented.…”

“…Detailed expressions for S Bose and matrix M are found in 36,42 . M is a large sparse matrix of dimension LL τ and the method is free of the sign problem as the determinant of M is squared.…”

Dynamical tuning of the chemical potential to achieve a target particle number in grand canonical Monte Carlo simulations

“…An analogous formalism is used for sampling the gluon field in lattice quantum chromodynamics (QCD), and we can borrow techniques from that community. In particular, Langevin 7 and hybrid Monte Carlo (HMC) sampling 26,27 have both proven effective for simulating electron-phonon models, 28,29 and make it possible to update the entire field x τ,i at a cost that scales near-linearly with system size. Here we employ HMC.…”

Benchmark study of an auxiliary-field quantum Monte Carlo technique for the Hubbard model with shifted-discrete Hubbard-Stratonovich transformations

“…The way of transforming a quartic interaction term into a quadratic one via the Hubbard-Stratonovich transformation (HST) [22] is not unique and affects the efficiency of simulations [23][24][25][26][27][28]. Recently the popularity of this technique is substantially increased, because it has been realized that, with continuous auxiliary fields, one can treat interaction terms beyond the on-site Hubbard interaction, up to the complete treatment of the long-range Coulomb interaction [29][30][31][32][33], or of the long-range electron-phonon interaction [34], and even both of them on the same footing [35], without being vexed by the sign problem in a certain parameter region on bipartitle lattices. Interestingly, such a parameter region coincides with the one where rigorous statements on the ground state of an extended Hubbard-Holstein model are available [36,37].…”