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arXiv:2010.01141 · Physical review. B./Physical review. B

Mitigating the fermion sign problem by automatic differentiation

Zhou‐Quan Wan, Shi‐Xin Zhang, Hong Yao

Published Dec 20, 2022• 16 citations
arXivHTMLPDF

Abstract

As an intrinsically unbiased method, the quantum Monte Carlo (QMC) method is of unique importance in simulating interacting quantum systems. Although the QMC method often suffers from the notorious sign problem, the sign problem of quantum models may be mitigated by finding better choices of the simulation scheme. However, a general framework for identifying optimal QMC schemes has been lacking. Here, we propose a general framework using automatic differentiation to automatically search for the best QMC scheme within a given ansatz of the Hubbard-Stratonovich transformation, which we call ``automatic differentiable sign optimization'' (ADSO). We apply the ADSO framework to the honeycomb lattice Hubbard model with Rashba spin-orbit coupling and demonstrate that ADSO is remarkably effective in mitigating and even solving its sign problem. Specifically, ADSO finds a sign-free point in the model which was previously thought to be sign-problematic. For the sign-free model discovered by ADSO, its ground state is shown by sign-free QMC simulations to possess spiral magnetic ordering; we also obtained the critical exponents characterizing the magnetic quantum phase transition.

DOI
10.1103/physrevb.106.l241109

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As an intrinsically unbiased method, the quantum Monte Carlo (QMC) method is of unique importance in simulating interacting quantum systems. Although the QMC method often suffers from the notorious sign problem, the sign problem of quantum models may be mitigated by finding better choices of the simulation scheme. However, a general framework for identifying optimal QMC schemes has been lacking. Here, we propose a general framework using automatic differentiation to automatically search for the best QMC scheme within a given ansatz of the Hubbard-Stratonovich transformation, which we call ``automatic differentiable sign optimization'' (ADSO). We apply the ADSO framework to the honeycomb lattice Hubbard model with Rashba spin-orbit coupling and demonstrate that ADSO is remarkably effective in mitigating and even solving its sign problem. Specifically, ADSO finds a sign-free point in the model which was previously thought to be sign-problematic. For the sign-free model discovered by ADSO, its ground state is shown by sign-free QMC simulations to possess spiral magnetic ordering; we also obtained the critical exponents characterizing the magnetic quantum phase transition.
As an intrinsically-unbiased method, quantum Monte Carlo (QMC) is of unique importance in simulating interacting quantum systems. Unfortunately, QMC often suffers from the notorious sign problem. Although generically curing sign problem is shown to be hard (NP-hard), sign problem of a given quantum model may be mitigated (sometimes even cured) by finding better choices of simulation scheme. A universal framework in identifying optimal QMC schemes has been desired. Here, we propose a general framework using differentiation (AD) to automatically search for the best continuously-parameterized QMC scheme, which we call automatic differentiable sign mitigation (ADSM). We further apply the ADSM framework to the honeycomb lattice Hubbard model with Rashba spin-orbit coupling and demonstrate ADSM's effectiveness in mitigating its sign problem. For the model under study, ADSM leads a significant power-law acceleration in computation time (the computation time is reduced from MMM to the order of MνM^{\nu}Mν with ν≈2/3\nu\approx 2/3ν≈2/3).
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
As an intrinsically-unbiased approach, quantum Monte Carlo (QMC) is of vital importance in understanding correlated phases of matter. Unfortunately, it often suffers notorious sign problem when simulating interacting fermion models. Here, we show for the first time that there exist interacting fermion models whose sign problem becomes less severe for larger system sizes and eventually disappears in the thermodynamic limit, which we dub as "asymptotic sign free". We demonstrate asymptotically-free sign in determinant QMC for various interacting models. Moreover, based on renormalization-group-like ideas we propose a heuristic understanding of the feature of asymptotic sign free. We believe that asymptotic sign free behavior could shed new lights to deepening our understanding of sign problem. More importantly, it can provide a promising way to decipher intriguing physics in correlated models which were conventionally thought not accessible by QMC.
Quantum Monte Carlo simulations of quantum many-body systems are plagued by the Fermion sign problem. The computational complexity of simulating Fermions scales exponentially in the projection time β and system size. The sign problem is basis dependent and an improved basis, for fixed errors, leads to exponentially quicker simulations. We show how to use sign-free quantum Monte Carlo simulations to optimize over the choice of basis on large two-dimensional systems. We numerically illustrate these techniques decreasing the "badness" of the sign problem by optimizing over single-particle basis rotations on one- and two-dimensional Hubbard systems. We find a generic rotation which improves the average sign of the Hubbard model for a wide range of U and densities for L×4 systems. In one example improvement, the average sign (and hence simulation cost at fixed accuracy) for the 16×4 Hubbard model at U/t=4 and n=0.75 increases by exp[8.64(6)β]. For typical projection times of β⪆100, this accelerates such simulation by many orders of magnitude.
Quantum Monte Carlo (QMC) methods for the frustrated quantum spin systems occasionally suffer from the negative sign problem, which makes simulations exponentially harder for larger systems at lower temperatures and severely limits QMC's application across a wide range of spin systems. This problem is known to depend on the choice of representation basis. We propose a systematic approach for mitigating the sign problem independent of the given Hamiltonian or lattice structure. We first introduce the concept of negativity to characterize the severity of the negative sign problem. We then demonstrate the existence of a locally defined quantity, the L1 adaptive loss function, which effectively approximates negativity, especially in frustration-free systems. Using the proposed loss function, we demonstrate that optimizing the representation basis can mitigate the negative sign. This is evidenced by several frustration-free models and other important quantum spin systems. Furthermore, we compare the effectiveness of unitary transformations against the standard orthogonal transformation and reveal that unitary transformations can effectively mitigate the sign problem in certain cases.
The sign problem limits determinant quantum Monte Carlo studies of strongly correlated fermion systems. In the spin-channel Hubbard-Stratonovich decoupling, spin correlations are exactly related to auxiliary-field correlations. This relation implies that an antiferromagnetic transition reorganizes auxiliary-field configurations and thereby changes the statistical structure of the resulting fermion matrices. We use the adjacent gap ratio of low-lying singular values of the space-time fermion matrix to probe interaction-driven transitions in two half-filled honeycomb-lattice Hubbard models. In the sign-free honeycomb Hubbard model, the statistic tracks the established transition from a Dirac semimetal to an antiferromagnetic Mott insulator. In the complex-weight Haldane-Hubbard model, the transition-sensitive feature remains visible in the phase-quenched reference ensemble and occurs near previous estimates of the transition. Moreover, phase reweighting only weakly modifies the gap ratio over the regimes investigated, despite the rapid suppression of the average phase. These results establish singular-value statistics as a sign-problem-resilient probe of interaction-driven transitions in determinant quantum Monte Carlo.