Deconvolving the components of the sign problem

Abstract

Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field $h$ introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices $M_{\sigma }(h)$, where $\sigma$ labels the species, and hence is typically not positive definite. Indeed, the average sign $⟨\mathcal{S}⟩$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $⟨\mathcal{S}⟩$. Does $⟨\mathcal{S}⟩\to 0$ because {\it randomly} chosen field configurations have $\det (M(h))<0$, or does the `sign problem' arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with $|\det (M(h))|$ might steer the simulation to a region of configuration space of $h$ where positive and negative determinants are equally likely, even though randomly chosen $h$ would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of $⟨\mathcal{S}⟩$ in quantum simulations.

14 pages, 17 figures, Title changed, several small modifications made in the text

References in corpus (11)

• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Quantum spin-liquid emerging in two-dimensional correlated Dirac fermions
• Absence of a Spin Liquid Phase in the Hubbard Model on the Honeycomb Lattice
• Accelerating Dynamical Fermion Computations using the Rational Hybrid Monte Carlo (RHMC) Algorithm with Multiple Pseudofermion Fields
• Fermionic quantum criticality in honeycomb and $\pi$-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo
• Geometry Dependence of the Sign Problem
• Quantum Monte Carlo Study of an Interaction-Driven Band Insulator to Metal Transition
• Nested Cluster Algorithm for Frustrated Quantum Antiferromagnets
• Auxiliary Field Quantum Monte Carlo for Multiband Hubbard Models: Controlling the Sign and Phase Problems to Capture Hund's Physics
• Marshall-positive SU($N$) quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians
• Is there a superconducting phase in the half-filled ionic Hubbard model ?

Cited by in corpus (10)

• Temperature Dependence of Spin and Charge Orders in the Doped Two-Dimensional Hubbard Model
• Fast and scalable quantum Monte Carlo simulations of electron-phonon models
• Fermion sign bounds theory in quantum Monte Carlo simulation
• Sign Problem in Quantum Monte Carlo Simulation
• An integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulation
• Observation of emergent scaling of spin-charge correlations at the onset of the pseudogap
• Attractive Su-Schrieffer-Heeger-Hubbard Model on a Square Lattice Away from Half-Filling
• Investigating Berezinskii-Kosterlitz-Thouless phase transitions in Kagome spin ice by quantifying Monte Carlo process: Distribution of Hamming distances
• SO(4) multicriticality of two-dimensional Dirac fermions
• Defining a universal sign to strictly probe a phase transition