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arXiv:1501.02832 · Physical Review B

Geometry dependence of the sign problem in quantum Monte Carlo simulations

Vladimir Iglovikov, Ehsan Khatami, Richard T. Scalettar

Published Jul 9, 2015• 106 citations
arXivHTMLPDF

Abstract

The sign problem is the fundamental limitation to quantum Monte Carlo simulations of the statistical mechanics of interacting fermions. Determinant quantum Monte Carlo (DQMC) is one of the leading methods to study lattice fermions, such as the Hubbard Hamiltonian, which describe strongly correlated phenomena including magnetism, metal-insulator transitions, and possibly exotic superconductivity. Here, we provide a comprehensive dataset on the geometry dependence of the DQMC sign problem for different densities, interaction strengths, temperatures, and spatial lattice sizes. We supplement these data with several observations concerning general trends in the data, including the dependence on spatial volume and how this can be probed by examining decoupled clusters, the scaling of the sign in the vicinity of a particle-hole symmetric point, and the correlation between the total sign and the signs for the individual spin species.

DOI
10.1103/physrevb.92.045110

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The sign problem is the fundamental limitation to quantum Monte Carlo simulations of the statistical mechanics of interacting fermions. Determinant quantum Monte Carlo (DQMC) is one of the leading methods to study lattice fermions, such as the Hubbard Hamiltonian, which describe strongly correlated phenomena including magnetism, metal-insulator transitions, and possibly exotic superconductivity. Here, we provide a comprehensive dataset on the geometry dependence of the DQMC sign problem for different densities, interaction strengths, temperatures, and spatial lattice sizes. We supplement these data with several observations concerning general trends in the data, including the dependence on spatial volume and how this can be probed by examining decoupled clusters, the scaling of the sign in the vicinity of a particle-hole symmetric point, and the correlation between the total sign and the signs for the individual spin species.
Determinant quantum Monte Carlo (QMC) simulations of fermion models normally face the sign problem, which has a strong dependence of the interaction strength, electron filling, temperature, lattice geometry and lattice size. It is useful to set up the data set of such dependence, to make it clear which parameter region is accessible or not. To supplement the previous study, here we report the behavior of the average sign in the weak interaction region, as functions of lattice geometry, lattice size, electron filling and temperature. Furthermore, we find that in the weak interaction region, for several interesting lattice geometries (except for the kagome lattice) the average sign gets improved when the lattice sizes increase, which is in contrast to the general understanding of the lattice size dependence of the average sign. This finding points out that when the sign problem is present in the weak interaction region, the simulation of a larger lattice size can be better under control.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
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.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
Determinant quantum Monte Carlo (DQMC) is used to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters---a two-dimensional square lattice consisting of elemental $2\ifmmode\times\else\texttimes\fi{}2$ plaquettes with hopping ttt and on-site repulsion UUU coupled by an interplaquette hopping ${t}^{\ensuremath{'}}\ensuremath{\le}t$. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half filling by the inhomogeneous hopping or not. For U/t=4U/t=4U/t=4, DQMC indicates an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.4$ at which the pairing vertex is most attractive. The optimal ${t}^{\ensuremath{'}}/t$ increases with U/tU/tU/t. We then contrast our results for this plaquette model with a hamiltonian which instead involves a regular pattern of site energies whose large site energy limit is the three-band CuO2{\mathrm{CuO}}_{2}CuO2​ model; we show that there the inhomogeneity rapidly, and monotonically, suppresses pairing.
Discussions of the sign problem usually focus on its role as the primary limitation of the applicability of quantum Monte Carlo methods for reliably solving quantum many-body models. Its behavior as a function of spatial lattice size, doping, temperature, and interaction strengths has been carefully characterized within the determinant quantum Monte Carlo algorithm. Here, we consider the much less well-explored question of the temperature- and size-dependence of the sign problem in the world-line quantum Monte Carlo method. We review how, in this algorithm, even sampling the partition function of free fermions yields a sign problem, which we connect to non-analytic behavior in the corresponding reference model, hard-core bosons on a lattice. The latter exhibits a finite-temperature Kosterlitz-Thouless phase transition in 2D, and we show how this non-analytic behavior is imprinted on the average sign of the weights for the corresponding free fermion 2D tight-binding model. Our results thus show how phase-transition information can remain encoded in the sign problem even after direct sign sampling becomes impractical.