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arXiv:2609.38740

Sign problem and criticality in world-line quantum Monte Carlo methods

Rubem Mondaini, Richard T. Scalettar

Published Sep 30, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2609.38740

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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.
Sign problems in fermion quantum Monte Carlo (QMC) simulation appears to be an extremely hard problem. Traditional lore passing around for years tells people that when there is a sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space. We, however, analytically show this is not always the case and manage to find physical bounds for the average sign. Our understanding is based on a direct connection between the sign bounds and a well-defined partition function of the reference system and could distinguish when the bounds have the usual exponential scaling and when they are bestowed on an algebraic scaling at the low-temperature limit. We analytically explain such algebraic sign problems found in flat-band moir\'e lattice models at the low-temperature limit. At finite temperature, a domain-size argument based on sign bounds also explains the connection between sign behavior and finite-temperature phase transition. Sign bounds, as a well-defined observable, may have the ability to ease or even make use of 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.
Initial characterizations of the fermion sign problem focused on its evolution with spatial lattice size LLL and inverse temperature $\ensuremath{\beta}$, emphasizing the implications of the exponential nature of the decay of the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ for the complexity of its solution and associated limitations of quantum Monte Carlo studies of strongly correlated materials. Early interest was also on the evolution of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ with density $\ensuremath{\rho}$, either because commensurate filling is often associated with special symmetries for which the sign problem is absent, or because particular fillings are often primary targets, e.g., those densities, which maximize superconducting transition temperature (the top of the ``dome'' of cuprate systems). Here we describe an analysis of the sign problem, which demonstrates that the spin-resolved sign $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ already possesses signatures of universal behavior traditionally associated with order parameters, even in the absence of symmetry protection that makes $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}=1$. When appropriately scaled, $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ exhibits universal crossings and data collapse. Moreover, we show these behaviors occur in the vicinity of quantum critical points of three well-understood models, exhibiting either second-order or Kosterlitz-Thouless phase transitions. Our results pave the way for using the average sign as a minimal correlator that can potentially describe quantum criticality in a variety of fermionic many-body problems.
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.
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.
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.