Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

Abstract

We describe an approach for many-body calculations with a finite-temperature, grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo (AFQMC) with a self-consistent constraint to control the sign problem. The usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the sign problem with most physical Hamiltonians, as is well known. Building on earlier ideas to constrain the paths in auxiliary-field space [Phys. Rev. Lett. 83, 2777(1999] and incorporating recent developments in zero-temperature, canonical-ensemble methods, we discuss how a self-consistent constraint can be introduced in the finite-temperature, grand-canonical-ensemble framework. This together with several other algorithmic improvements discussed here leads to a more accurate, more efficient, and numerically more stable approach for finite-temperature calculations. We carry out a systematic benchmark study in the two-dimensional repulsive Hubbard model at 1/8 doping. Temperatures as low as T = 1/80 (in units of hopping) are reached. The finite-temperature method is exact at very high temperatures, and approaches the result of the zero-temperature constrained-path AFQMC as temperature is lowered. The benchmark shows that systematically accurate results are obtained for thermodynamic properties.

Quantitative functional renormalization group description of the two-dimensional Hubbard model

“…In this work, we choose ∆τ t = 0.02 which has been tested to safely reach the ∆τ → 0 limit. In this work, we have also implemented our most recent improvements [71,72] of the dQMC algorithm. For the computation of dynamical quantities, we first measure the imaginary-time correlation functions, and then obtain the imaginary-frequency observables via Fourier transformation.…”

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

“…In classical MC simulations, for example, moves that modify the particle number can be useful for reducing decorrelation times or for studying coexistence between phases. [1][2][3][4][5] Similarly, the starting point of many [6][7][8][9][10][11][12] (but not all [13][14][15][16] ) finite temperature quantum Monte Carlo (QMC) simulations is the grand canonical partition function Z = Tr exp[−β(H + µN )], where β is the inverse temperature, H is the Hamiltonian, and N is the number operator. The trace above runs over all quantum wavefunctions, not just those constrained to a fixed particle number.…”

A Phaseless Auxiliary-Field Quantum Monte Carlo Perspective on the Uniform Electron Gas at Finite Temperatures: Issues, Observations, and Benchmark Study

“…[43][44][45][46][47] In addition, the bias introduced by the phaseless approximation is typically much smaller than the fixed node-error in DMC 48 which in principle should serve as an rough upper bound to the bias in RPIMC. Thus, it is important to assess the quality of ph-FT-AFQMC for realistic systems as to date the applications have largely focussed on model systems 49,50 or have not enforced the constraint [51][52][53] which is not a practical approach as the system size increases. Compared to FT-CCSD, ph-FT-AFQMC maintains favorable cubic scaling O(M 3 ) for each statistical sample, 54,55 which makes it better-suited for large-scale warm dense matter simulations.…”