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arXiv:cond-mat/0001351 · Physical review. B, Condensed matter

Sign problem in Monte Carlo simulations of frustrated quantum spin systems

Patrik Henelius, Anders W. Sandvik

Published Jul 1, 2000• 134 citations
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Abstract

We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of ``semifrustrated'' systems [Heisenberg models with ferromagnetic couplings ${J}_{z}(r)<0$ along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the $\mathrm{xy}$ plane, for arbitrary distances $r],$ the sign problem present for algorithms operating in the z basis can be solved within a recent ``operator-loop'' formulation of the stochastic series expansion method [a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ to all orders]. The solution relies on the identification of operator loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models [Phys. Rev. Lett. 83, 3116 (1999)]. Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.

DOI
10.1103/physrevb.62.1102

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We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of semifrustrated'' systems [Heisenberg models with ferromagnetic couplings ${J}_{z}(r)<0$ along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the $\mathrm{xy}$ plane, for arbitrary distances $r],$ the sign problem present for algorithms operating in the z basis can be solved within a recent operator-loop'' formulation of the stochastic series expansion method [a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of exp(\ensuremath−\ensuremathβH)\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)exp(\ensuremath−\ensuremathβH) to all orders]. The solution relies on the identification of operator loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models [Phys. Rev. Lett. 83, 3116 (1999)]. Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.
Quantum Monte Carlo methods are sophisticated numerical techniques for simulating interacting quantum systems. In some cases, however, they suffer from the notorious ``sign problem'' and become too inefficient to be useful. A recent publication [J. Wojtkiewicz, Phys. Rev. B 75, 174421 (2007)] claims to have solved the sign problem for a certain class of frustrated quantum spin systems through the use of a bipartite valence bond basis. We show in this Comment that the apparent positivity of the path integral is due to a misconception about the resolution of the identity operator in this basis and that consequently the sign problem remains a severe obstacle for the simulation of frustrated quantum magnets.
We propose a mechanism for solving the "negative sign problem"-the inability to assign non-negative weights to quantum Monte Carlo configurations-for a toy model consisting of a frustrated triplet of spin-1/2 particles interacting antiferromagnetically. The introduced technique is based on the systematic grouping of the weights of the recently developed off-diagonal series expansion of the canonical partition function [Phys. Rev. E 96, 063309 (2017)PREHBM2470-004510.1103/PhysRevE.96.063309]. We show that, although the examined model is easily diagonalizable, the sign problem it encounters can nonetheless be very pronounced, and we offer a systematic mechanism to resolve it. We discuss the prospects of generalizing the suggested scheme and the steps required to extend it to more general and larger spin models.
The sign problem in quantum Monte Carlo calculations is analyzed using the meron-cluster solution. A meron is a loop that alters the sign of the configuration, and the concept of merons can be used to solve the sign problem for a limited class of models. Here we show that the method can be used to reduce the sign problem in a wider class of models. We investigate how the meron solution evolves between a point in parameter space where it eliminates the sign problem and a point where it does not affect the sign problem at all. In this intermediate regime, the merons can be used to reduce the sign problem. The average sign still decreases exponentially with system size and inverse temperature, but with a different prefactor. The sign exhibits the slowest decrease in the vicinity of points where the meron-cluster solution eliminates the sign problem. We have used stochastic series expansion quantum Monte Carlo combined with the concept of directed loops.
When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.
Building on a recent investigation of the Shastry-Sutherland model [S. Wessel et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to eliminate the Monte Carlo sign problem near the zero-temperature limit in frustrated quantum spin models. If the Hamiltonian of interest and the sign-problem-free Hamiltonian, obtained by making all off-diagonal elements negative in a given basis, have the same ground state and this state is a member of the computational basis, then the average sign returns to one as the temperature goes to zero. We illustrate this technique by studying the triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field above saturation, as well as the Heisenberg antiferromagnet on a modified Husimi cactus in the dimer basis. We also provide detailed Appendices on using linear programming techniques to automatically generate efficient directed loop updates in quantum Monte Carlo simulations.
Quantum Monte Carlo simulations provide one of the more powerful and versatile numerical approaches to condensed matter systems. However, their application to frustrated quantum spin models, in all relevant temperature regimes, is hamstrung by the infamous “sign problem.” Here we exploit the fact that the sign problem is basis-dependent. Recent studies have shown that passing to a dimer (two-site) basis eliminates the sign problem completely for a fully frustrated spin model on the two-leg ladder. We generalize this result to all partially frustrated two-leg spin-1/2 ladders, meaning those where the diagonal and leg couplings take any antiferromagnetic values. We find that, although the sign problem does reappear, it remains remarkably mild throughout the entire phase diagram. We explain this result and apply it to perform efficient quantum Monte Carlo simulations of frustrated ladders, obtaining accurate results for thermodynamic quantities such as the magnetic specific heat and susceptibility of ladders up to L = 200 L = 200 rungs (400 spins 1/2) and down to very low temperatures.