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arXiv:2312.01491

Flat band projections: Sign problem mapping for frustrated spin systems

Maksim Ulybyshev, Anika Götz, Fakher F. Assaad

Published Dec 3, 2023• 0 citations
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Abstract

Projection of the Coulomb potential onto flat bands paves the way to design various interactions in the particle-hole and particle-particle channels. Here we pose the question if we can use this mapping to overcome the negative sign problem for the simplest possible frustrated spin system consisting a trimer of spins-1/2 coupled with an antiferromagnetic exchange interaction. While the answer is negative, we show that we can map the sign problem for frustrated spin systems onto a problem where we need to simulate particle-hole symmetric systems but with long-ranged Coulomb interactions prevailing over short-ranged ones. While the latter systems are currently not accessible to auxiliary field determinant quantum Monte Carlo, this mapping motivates algorithmic development that may overcome this issue.

DOI
10.48550/arxiv.2312.01491

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Projection of the Coulomb potential onto flat bands paves the way to design various interactions in the particle-hole and particle-particle channels. Here we pose the question if we can use this mapping to overcome the negative sign problem for the simplest possible frustrated spin system consisting a trimer of spins-1/2 coupled with an antiferromagnetic exchange interaction. While the answer is negative, we show that we can map the sign problem for frustrated spin systems onto a problem where we need to simulate particle-hole symmetric systems but with long-ranged Coulomb interactions prevailing over short-ranged ones. While the latter systems are currently not accessible to auxiliary field determinant quantum Monte Carlo, this mapping motivates algorithmic development that may overcome this issue.
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.
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 Jz(r)<0{J}_{z}(r)<0Jz​(r)<0 along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the xy\mathrm{xy}xy plane, for arbitrary distances r],r],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.
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.
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.
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.