Sign problem in Monte Carlo simulations of frustrated quantum spin systems

Abstract

We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of "semi-frustrated" systems (Heisenberg models with ferromagnetic couplings Jz(r) < 0 along the z-axis and antiferromagnetic couplings Jxy(r) = −Jz(r) in the xyplane, 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(−βH) to all orders). The solution relies on 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.

Anisotropy-Induced Ordering in the QuantumJ1−J2Antiferromagnet

“…The study of the quantum J 1 -J XXZ 2 model at finite temperature by quantum Monte Carlo (QMC) techniques is generally precluded by the sign problem [8] due to the transverse frustrating coupling α ⊥ . However, by removing that term completely we obtain the sign-problem-free…”

Tunnel-diode resonator and nuclear magnetic resonance studies of low-dimensional magnetic and superconducting systems

“…The QMC method could not be used to establish the magnetic properties of this system below 70 K due to the well-known negative-sign problem for frustrated spin systems [81]. of the exchange constant for any given nearest-neighbor pair is selected using a broad probability distribution constrained so that the mean value equals -115 K. Our analysis allows us to estimate the magnitude of the exchange disorder in the compound.…”

Marshall-positiveSU(N)quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians

“…6 In practice this capability is restricted to models that do not suffer from the notorious sign-problem. 7,8 The absence of this problem is guaranteed in standard world line methods only if the Marshall sign rule 9 ( α|H|β < 0 for α = β) is satisfied in a convenient local basis. Surprisingly, at the current time, there is no systematic knowledge of the extent of Marshall-positive spin Hamiltonians.…”