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 $\beta$, emphasizing the implications of the exponential nature of the decay of the average sign $⟨\mathcal{S}⟩$ 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 $⟨\mathcal{S}⟩$ with density $\rho$, 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 a new analysis of the sign problem, which demonstrates that the {\it spin-resolved} sign $⟨{\mathcal{S}}_{\sigma }⟩$ already possesses signatures of universal behavior traditionally associated with order parameters, even in the absence of symmetry protection that makes $⟨\mathcal{S}⟩=1$. When appropriately scaled, $⟨{\mathcal{S}}_{\sigma }⟩$ 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.

15+6 pages, 10+6 figures; updated version

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• Energy response and spatial alignment of the perturbed electron gas
• Disorder in interacting quasi-one-dimensional systems: flat and dispersive bands
• Defining a universal sign to strictly probe a phase transition
• Quantum Transport Theory of Strongly Correlated Matter
• Weyl semimetallic, Néel, spiral, and vortex states in the Rashba-Hubbard model
• The extended Hubbard model on a honeycomb lattice
• Finite temperatures and flat bands: the Hubbard model on three-dimensional Lieb lattices
• Real-space topology and charge order in the Haldane-Holstein Model
• Hubbard Model on the Honeycomb Lattice with an Indefinite Long-Range Interaction
• Probing the Critical Behavior of a Sign-Problematic Model with Monte Carlo Simulations
• Spin-orbit coupled periodic Anderson model: Kondo-Dirac semimetal and orbital-selective antiferromagnetic semimetal