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Importance Sampling and the Sign Problem

Sabyasachi Tarat, Bo Xiao, Rubem Mondaini, Richard T. Scalettar

Published Aug 3, 2021• 0 citations
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Abstract

Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field $x$ introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices $M_\sigma(x)$, where $\sigma$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\langle {\cal S} \rangle$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\langle {\cal S} \rangle$. Does $\langle {\cal S} \rangle \rightarrow 0$ because {\it randomly} chosen field configurations have ${\rm det}\big(M(x)\big) < 0$, or does the `sign problem' arise as a result of {\it importance sampling} in the simulations? In the latter case, the process of weighting the configurations with $|{\rm det}\big(M(x)\big)|$ might steer the simulation to a region of configuration space of $x$ where positive and negative determinants are equally likely, even though randomly chosen $x$ would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of $\langle {\cal S} \rangle$ in quantum simulations.

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Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field xxx introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices Mσ(x)M_\sigma(x)Mσ​(x), where σ\sigmaσ labels the species, and hence is typically not positive definite. Indeed, the average sign ⟨S⟩\langle {\cal S} \rangle⟨S⟩ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of ⟨S⟩\langle {\cal S} \rangle⟨S⟩. Does ⟨S⟩→0\langle {\cal S} \rangle \rightarrow 0⟨S⟩→0 because {\it randomly} chosen field configurations have det(M(x))<0{\rm det}\big(M(x)\big) < 0det(M(x))<0, or does the `sign problem' arise as a result of {\it importance sampling} in the simulations? In the latter case, the process of weighting the configurations with ∣det(M(x))∣|{\rm det}\big(M(x)\big)|∣det(M(x))∣ might steer the simulation to a region of configuration space of xxx where positive and negative determinants are equally likely, even though randomly chosen xxx would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of ⟨S⟩\langle {\cal S} \rangle⟨S⟩ in quantum simulations.
Auxiliary field quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field hhh introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices ${M}_{\ensuremath{\sigma}}(h)$, where $\ensuremath{\sigma}$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$. Does $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}\ensuremath{\rightarrow}0$ because randomly chosen field configurations have det[M(h)]<0\mathrm{det}[M(h)]<0det[M(h)]<0, or does the sign problem arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with ∣det[M(h)]∣|\mathrm{det}[M(h)]|∣det[M(h)]∣ might steer the simulation to a region of configuration space of hhh where positive and negative determinants are equally likely, even though randomly chosen hhh would preferentially have determinants with a single dominant sign. In this paper, we address the relative importance of these two mechanisms for the vanishing of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ in quantum simulations.
Quantum simulations are a powerful tool for exploring strongly correlated many-body phenomena. Yet, their reach is limited by the fermion sign problem, which causes configuration weights to become negative, compromising statistical sampling. In auxiliary-field Quantum Monte Carlo calculations of the doped Hubbard model, neglecting the sign S{\cal S}S of the weight leads to qualitatively wrong results -- most notably, an apparent suppression rather than enhancement of ddd-wave pairing at low temperature. Here we approach the problem from a different perspective: instead of identifying negative-weight paths, we examine the statistics of measured observables in a sign-resolved manner. By analyzing histograms of key quantities (kinetic energy, antiferromagnetic structure factor, and pair susceptibilities) for configurations with S=±1{\cal S}=\pm1S=±1, we derive an exact relation linking the bias from ignoring the sign to the difference between sign-resolved means, ΔμΔμΔμ, and the average sign, ⟨S⟩\langle {\cal S}\rangle⟨S⟩. Our framework provides a precise diagnostic of the origin of measurement bias in Quantum Monte Carlo and clarifies why observables such as the ddd-wave susceptibility are especially sensitive to the sign problem.
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.
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.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.