Addressing the sign-problem in Euclidean path integrals with radial basis function neural networks

TL;DR

The paper tackles the sign problem in finite-density Euclidean path integrals by introducing a radial basis function (RBF) expansion that expresses nonlinear interaction terms as Gaussian mixtures, enabling tractable computation of observables in a complex scalar field theory at finite chemical potential. By moving to momentum space and diagonalizing the resulting kernel, the partition function is approximated in a factorized form with complexity scaling as $O(K N^4)$, which is orders of magnitude more efficient than brute-force Gaussian sums. The authors determine a critical chemical potential $μ_c \approx 1.17 \pm 0.018$ for Bose condensation and demonstrate the silver blaze phenomenon (μ-independence of observables for $μ<μ_c$) on large lattices, with results aligning with complex Langevin dynamics. The approach shows promise for extension to fermionic and non-Abelian gauge theories at finite density, potentially impacting simulations of QCD-like systems.

Abstract

Solving interacting field theories at finite densities remains a numerically and conceptually challenging task, even with modern computational capabilities. In this paper, we propose a novel approach based on an expansion of the Euclidean path integrals using radial basis function neural networks, which allows the calculation of observables at finite densities and overcomes the sign problem in a numerically very efficient manner. The method is applied to an interacting complex scalar field theory at finite chemical potential in 3+1 dimensions, which exhibits both the sign problem and the silver blaze phenomenon, similar to QCD. The critical chemical potential at which phase transition occurs is estimated to be $μ_c=1.17 \pm 0.018$, and the silver blaze problem is accurately described below $μ_c$.

Figures (4)

• Figure 1: Dependence of the relative difference between $\ln Z_{U}$, and $\ln Z_{U=0}$ on the number of combinations $N_c$ included for different RBF parametrizations and chemical potentials.
• Figure 2: Normalized distribution of the relative error for uniformly distributed centers (dashed) and for a fixed center configuration (full) with $K=25$, $A=1$, and $\mu=1$.
• Figure 3: Vacuum expectation value of the fields determined from the effective potential. The critical chemical potential is $\mu_c = 1.17 \pm 0.018$.
• Figure 4: Particle number density for the full-, and for the phase quenched theories on a $10^4$ lattice.