We discuss a new strategy for treating the complex action problem of lattice field theories with a $theta$-term based on density of states (DoS) methods. The key ingredient is to use open boundary conditions where the topological charge is not quanti
zed to integers and the density of states is sufficiently well behaved such that it can be computed precisely with recently developed DoS techniques. After a general discussion of the approach and the role of the boundary conditions, we analyze the method for 2-d U(1) lattice gauge theory with a $theta$-term, a model that can be solved in closed form. We show that in the continuum limit periodic and open boundary conditions describe the same physics and derive the DoS, demonstrating that only for open boundary conditions the density is sufficiently well behaved for a numerical evaluation. We conclude our proof of principle analysis with a small test simulation where we numerically compute the density and compare it with the analytical result.
We explore new representations for lattice gauge theories with fermions, where the space-time lattice is divided into dynamically fluctuating regions, inside which different types of degrees of freedom are used in the path integral. The first kind of
regions is a union of so-called bags, in which the dynamics is described by the free propagation of composite degrees of freedom of the original fermions. In the second region, called complementary domain, configurations of the remaining interacting degrees of freedom are used to describe the dynamics. We work out the bag representation for the gauge groups SU(2) and SU(3) and address the nature of the strong coupling effective degrees of freedom, which are fermions for SU(3) and bosons for SU(2). We discuss first steps towards a numerical simulation of the bag representations.
We study $phi^4$ lattice field theory at finite chemical potential $mu$ in two and four dimensions, using a worldline representation that overcomes the complex action problem. We compute the particle number at very low temperature as a function of $m
u$ and determine the first three condensation thresholds, where the system condenses 1, 2 and 3 particles. The corresponding critical values of the chemical potential can be related to the 1-, 2- and 3-particle energies of the system, and we check this relation with a direct spectroscopy determination of the $n$-particle energies from $2n$-point functions. We analyze the thresholds as a function of the spatial size of the system and use the known finite volume results for the $n$-particle energies to relate the thresholds to scattering data. For four dimensions we determine the scattering length from the 2-particle threshold, while in two dimensions the full scattering phase shift can be determined. In both cases the scattering data computed from the 2-particle threshold already allow one to determine the 3-particle energy. In both, two and four dimensions we find very good agreement of this prediction with direct determinations of the 3-particle energy from either the thresholds or the 6-point functions. The results show that low temperature condensation is indeed governed by scattering data.
