Density of states techniques for lattice field theories using the functional fit approach (FFA)

Abstract

We discuss a variant of density of states (DoS) techniques for lattice field theories, the so-called "functional fit approach" (FFA). The DoS FFA is based on a density of states ρ(x) which is parameterized on small intervals of the argument x of ρ(x). On these intervals restricted Monte Carlo simulations with an additional Boltzmann factor exp(λ x) allow to determine ρ(x) very precisely by obtaining its parameters from fitting the Monte Carlo data to a known function of λ .We describe the method in detail and show its applicability in four different systems, three of which have a complex action problem: The SU(3) spin model with a chemical potential, U(1) lattice gauge theory, the Z 3 spin model with chemical potential, and 2-dimensional U(1) lattice gauge theory with a topological term. In all cases we compare to reference calculations, which partly were done in a dual formulation where the complex action problem is absent. In all four cases we find a very encouraging performance of the DoS FFA.

Approaches to the sign problem in lattice field theory

“…This theory has a real dual formulation, 34,35,41 and can be efficiently simulated with the flux algorithm. 20 It is therefore ideally suited for testing the LLR approach (for a simulation with the related density of states FFA (functional fit approach) see 92,99,100 ). We introduce…”

Developing and testing the density of states FFA method in the SU(3) spin model

“…Here we present an exploratory study of a variant of DoS techniques, the Density of States Functional Fit Approach (DoS FFA) [14,15,16]. Similar to the LLR approach we parameterize the density on small intervals and evaluate restricted vacuum expectation values.…”

“…With a one-parameter fit of the Monte Carlo data one then can precisely determine ρ(x) on the corresponding interval. The Dos FFA has been introduced and tested for the Z 3 spin system and was found to reproduce the results of a reference simulation with dual variables for a relatively wide range of parameters [14,15,16].…”

Density of states approach for lattice field theory with topological terms

“…In principle, density of states techniques -initially introduced in lattice field theory in [1,2] -are a possible approach and, based on various recent technological developments, were applied in a wide range of applications with complex action problems [3] - [23]. However, the case of treating the complex action problem emerging from a topological term is somewhat subtle, because in the usual formulation with periodic boundary conditions the topological charge becomes quantized to integers in the continuum limit such that the density will approach a superposition of Dirac deltas (see, e.g., [18]). If a geometrical or a fermionic definition is used the topological charge is quantized also at finite lattice constant, resulting in a superposition of Dirac deltas also at finite lattice spacing.…”