Universality and Critical Exponents of the Fermion Sign Problem

Abstract

Initial characterizations of the fermion sign problem focused on its evolution with spatial lattice size L and inverse temperature β, emphasizing the implications of the exponential nature of the decay of the average sign S 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 S with density ρ, 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 a new analysis of the sign problem which demonstrates that the spin-resolved sign Sσ already possesses signatures of universal behavior traditionally associated with order parameters, even in the absence of symmetry protection that makes S = 1. When appropriately scaled, Sσ 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.

Hamming distance and the onset of quantum criticality

“…It has been recently demonstrated that the average sign of the weights extracted over the course of the importance sampling in quantum Monte Carlo simulations of various fermionic models can be used to both qualitatively [5] and quantitatively [6] identify the loci of quantum critical points. Here, we show that this analysis does not in principle hold in the case of the Hubbard model in the triangular lattice, both on its U(1) or SU(2) formulations, at least using the 'standard' Hubbard-Stratonovich transformation described in the Appendix A.…”

“…While recent studies have in fact suggested that the average sign of weights in the latter already pinpoints the regimes of strong quantum fluctuations [4][5][6][7], here we focus on other statistical properties that also aid in locating quantum phase transitions. In particular, we investigate a specific class of QMC methods for ddimensional fermionic systems, referred to as auxiliary field QMC [2,8,9], which provides a framework to stochastically average observables by sampling a fictitious field in d + 1-dimensions, introduced in a path integral formulation of the partition function.…”

How universal is the relation between sign problem and phase transition

“…Or more strictly speaking, those positive examples where the average sign can probe the critical value can be excep- tional cases. On the other hand, in previous studies, the phase transition can be related to the sign value itself or its derivative in different circumstances without an explicit criterion [32][33][34][35][36][37][38]. This non-conformity has also been discussed in the present work.…”

Energy response and spatial alignment of the perturbed electron gas

“…Therefore, the formation of permutation cycles [43], which are exclusively responsible for a non-unity sign in standard PIMC, is suppressed, which explains the observed trend. While the average sign by itself does not constitute a physical observable, it can nevertheless give insights into the physical behaviour of a given system in some situations [146,147]. In the top panel of Fig.…”