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arXiv:1906.02726

Taming the sign problem of the finite density Hubbard model via Lefschetz thimbles

Maksim Ulybyshev, Christopher Winterowd, Savvas Zafeiropoulos

Published Jun 6, 2019• 12 citations
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Abstract

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.

DOI
10.48550/arxiv.1906.02726

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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.
In this paper, we propose a framework for studying the properties of the Lefschetz thimbles decomposition for lattice fermion models approaching the thermodynamic limit. The proposed set of algorithms includes the Schur complement solver and the exact computation of the derivatives of the fermion determinant. It allows us to solve the gradient flow equations taking into account the fermion determinant exactly, with high performance. Being able to do so, we can find both real and complex saddle points and describe the structure of the Lefschetz thimbles decomposition for large enough lattices which allows us to extrapolate our results to the thermodynamic limit. We describe the algorithms for a general lattice fermion model, with emphasis on two widely used types of lattice discretizations for relativistic fermions (staggered and Wilson fermions), as well as on interacting tight-binding models for condensed matter systems. As an example, we apply these algorithms to the Hubbard model on a hexagonal lattice. Several technical improvements allow us to deal with lattice volumes as large as $12\ifmmode\times\else\texttimes\fi{}12$ with ${N}_{\ensuremath{\tau}}=256$ steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low-temperature, and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) transformation were studied, and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we provide evidence for the existence of an optimal regime for the hexagonal lattice Hubbard model, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the gradient flow to deform the integration contour into the complex plane. These calculations were made on small volumes (Ns=8{N}_{s}=8Ns​=8 sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit an exponential decay of the average sign. The results are compared versus exact diagonalization, and we demonstrate the importance of choosing an optimal form for the HS transformation for the Hubbard model to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS (Blankenbecler, Scalapino, and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.
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.
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 Lefschetz-thimble approach to path integrals is applied to a one-site model of electrons, i.e., the one-site Hubbard model. Since the one-site Hubbard model shows a non-analytic behavior at the zero temperature and its path integral expression has the sign problem, this toy model is a good testing ground for an idea or a technique to attack the sign problem. Semiclassical analysis using complex saddle points unveils the significance of interference among multiple Lefschetz thimbles to reproduce the non-analytic behavior by using the path integral. If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities. Analyzing such singular behaviors semiclassically, we propose a criterion to identify the necessary number of Lefschetz thimbles. We argue that this interference of multiple saddle points is a key issue to understand the sign problem of the finite-density quantum chromodynamics.
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.
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.