A subset solution to the sign problem in random matrix simulations

Abstract

We present a solution to the sign problem in dynamical random matrix simulations of a two-matrix model at nonzero chemical potential. The sign problem, caused by the complex fermion determinants, is solved by gathering the matrices into subsets, whose sums of determinants are real and positive even though their cardinality only grows linearly with the matrix size. A detailed proof of this positivity theorem is given for an arbitrary number of fermion flavors. We performed importance sampling Monte Carlo simulations to compute the chiral condensate and the quark number density for varying chemical potential and volume. The statistical errors on the results only show a mild dependence on the matrix size and chemical potential, which confirms the absence of sign problem in the subset method. This strongly contrasts with the exponential growth of the statistical error in standard reweighting methods, which was also analyzed quantitatively using the subset method. Finally, we show how the method elegantly resolves the Silver Blaze puzzle in the microscopic limit of the matrix model, where it is equivalent to QCD.

18 pages, 11 figures, as published in Phys. Rev. D; added references; in Sec. VB: added discussion of model satisfying the Silver Blaze for all N (proof in Appendix E)

References in corpus (12)

• Phase of the Fermion Determinant at Nonzero Chemical Potential
• The QCD Sign Problem for Small Chemical Potential
• Overlap Dirac operator at nonzero chemical potential and random matrix theory
• The QCD phase diagram according to the center group
• Efficient simulation of relativistic fermions via vertex models
• Equivalence of QCD in the epsilon-regime and chiral Random Matrix Theory with or without chemical potential
• Random matrix analysis of the QCD sign problem for general topology
• Chiral Condensate at Nonzero Chemical Potential in the Microscopic Limit of QCD
• Evading the sign problem in random matrix simulations
• Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential
• Baryon Number Dirac Spectrum in QCD
• The QCD sign problem and dynamical simulations of random matrices

Cited by in corpus (8)

• Complex Langevin Simulation of a Random Matrix Model at Nonzero Chemical Potential
• Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics
• Subset method for one-dimensional QCD
• New developments for dual methods in lattice field theory at non-zero density
• Subsets and the canonical partition functions
• Positivity of center subsets for QCD
• Analysis of the QCD Kondo phase using random matrices
• Sign problem and subsets in one-dimensional QCD