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arXiv:1906.04243 · Physical review. D/Physical review. D.

Applying the tempered Lefschetz thimble method to the Hubbard model away from half filling

Masafumi Fukuma, Nobuyuki Matsumoto, Naoya Umeda

Published Dec 17, 2019• 41 citations
arXivHTMLPDF

Abstract

The tempered Lefschetz thimble method is a parallel-tempering algorithm toward solving the numerical sign problem. It uses the flow time of the gradient flow as a tempering parameter and is expected to tame both the sign and multimodal problems simultaneously. In this paper, we further develop the algorithm so that the expectation values can be estimated precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate that this algorithm works well, we apply it to the quantum Monte Carlo simulation of the Hubbard model away from half filling on a two-dimensional lattice of small size and show that the numerical results agree nicely with exact values.

DOI
10.1103/physrevd.100.114510

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The tempered Lefschetz thimble method is a parallel-tempering algorithm toward solving the numerical sign problem. It uses the flow time of the gradient flow as a tempering parameter and is expected to tame both the sign and multimodal problems simultaneously. In this paper, we further develop the algorithm so that the expectation values can be estimated precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate that this algorithm works well, we apply it to the quantum Monte Carlo simulation of the Hubbard model away from half filling on a two-dimensional lattice of small size and show that the numerical results agree nicely with exact values.
The tempered Lefschetz thimble method (TLTM) is a parallel-tempering algorithm towards solving the numerical sign problem. It tames both the sign and ergodicity problems simultaneously by tempering the system with the flow time of continuous deformations of the integration region. In this article, after reviewing the basics of the TLTM, we explain a new algorithm within the TLTM that enables us to estimate the expectation values precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate the effectiveness of the algorithm, we apply the TLTM to the quantum Monte Carlo simulation of the Hubbard model away from half filling on a two-dimensional lattice of small size, and show that the obtained numerical results agree nicely with exact values.
We study the sign problem in the Hubbard model on the hexagonal lattice away from half-filling using the Lefschetz thimbles method. We identify the saddle points, reduce their amount, and perform quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour in complex space. Finally, the results are compared against exact diagonalization (ED). We show that the sign problem can be substantially weakened, even in the regime with low temperature and large chemical potential, where standard QMC techniques exhibit an exponential decay of the average sign.
The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is an efficient algorithm for addressing the numerical sign problem at moderate computational cost. It mitigates the sign problem while avoiding the ergodicity issues inherent in approaches based on Lefschetz thimbles. In this study, we apply WV-HMC to the two-dimensional Hubbard model doped away from half filling, which is known to suffer from a severe sign problem. We compute the number density and the energy density on lattices of size 6×66 \times 66×6 and 8×88 \times 88×8 at temperature T/t=1/6.4≃0.156T/t = 1/6.4 \simeq 0.156T/t=1/6.4≃0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0, using Trotter number Nt=20N_t = 20Nt​=20 (Trotter step ε=0.32ε= 0.32ε=0.32). Our results demonstrate that WV-HMC remains effective even in parameter regimes where standard (non-thimble) determinant quantum Monte Carlo methods fail. In this work, fermion matrix inversions are performed using direct solvers, leading to a computational cost of O(N3)O(N^3)O(N3), where NNN denotes the number of degrees of freedom and is proportional to the spacetime lattice volume. An alternative algorithm employing pseudofermions and iterative solvers, which reduces the cost to O(N2)O(N^2)O(N2) at the expense of careful parameter tuning, will be discussed in a separate publication.
The concept of Lefschetz thimble decomposition is one of the most promising possible modifications of Quantum Monte Carlo (QMC) algorithms aimed at alleviating the sign problem which appears in many interesting physical situations, e.g. in the Hubbard model away from half filling. In this approach one utilizes the fact that the integral over real variables with an integrand containing a complex fluctuating phase is equivalent to the sum of integrals over special manifolds in complex space ("Lefschetz thimbles"), each of them having a fixed complex phase factor. Thus, the sign problem can be reduced if the resulting sum contains terms with only a few different phases. We explore the complexity of the sign problem for the few-site Hubbard model on square lattice combining a semi-analytical study of saddle points and thimbles in small lattices with several steps in Euclidean time with results of test QMC calculations. We check different variants of conventional Hubbard-Stratonovich transformation based on Gaussian integrals. On the basis of our analysis we reveal a regime with minimal number of relevant thimbles in the vicinity of half filling. In this regime we found only two relevant thimbles for the few-site lattices studied in the paper. There is also indirect evidence of the existence of this regime in more realistic systems with large number of Euclidean time slices. In addition, we derive a new non-Gaussian representation of the interaction term, where the number of relevant Lefschetz is also reduced in comparison with conventional Gaussian Hubbard-Stratonovich transformation.
Recently, a method based on stochastic integration on the surfaces of steepest descent of the action was introduced to tackle the sign problem in quantum-field theories. We show how this method can be used in many-body theories to perform fully nonperturbative calculations of quantum corrections about mean-field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from other state-of-the-art calculations.
Recently, a new method, based on stochastic integration on the surfaces of steepest descent of the action, was introduced to tackle the sign problem in quantum field theories. We show how this method can be used in many body theories to perform fully non-perturbative calculations of quantum corrections about mean field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from the recent state of the art cluster dynamical mean field theory calculations.