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 $\ensuremath{\beta}$, 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.