Zone determinant expansions for nuclear lattice simulations

We consider quantum simulations of nuclear matter on the lattice. In particular, we address the problem of calculating the contribution of nucleon/nucleon-hole loops at nonzero nucleon density. With the help of auxiliary boson fields, all nucleon interactions can be written in terms of one-body interactions in a fluctuating background. In the grand canonical ensemble, the contribution of nucleon/nucleon-hole loops to the partition function equals the determinant of the onebody interaction matrix. Since the determinant of the interaction matrix for a general boson field configuration is not positive, stochastic methods such as hybrid Monte Carlo [1–3] do not give the sign or phase of the determinant. Instead one must rely on much slower and more memory intensive algorithms based on LU factorization, which decomposes matrices in terms of a product of upper and lower triangular matrices. The number of required operations in LU factorization for an n3n matrix scales as n3. It has been shown in the literature that repeated calculations of matrix determinants with only localized changes can be streamlined in various ways [4,5]. However it is difficult to avoid the poor scaling inherent in the method. If V is the spatial volume and b is the inverse temperature measured in lattice units, a simulation that includes nucleon/nucleon-hole loops requires sVbd3 times more operations than the corresponding quenched simulation without loops. This slowdown should not be confused with the infamous fermion sign or phase problem [30] which becomes significant at temperatures Tø1 MeV. The computational bottleneck we are considering is due to the inefficiencies of the algorithm and persists at all temperatures. It is this numerical challenge which sets current limits on nuclear lattice simulations. In this paper we introduce a new approach to approximating nucleon matrix determinants. We begin with a review of the current status of nuclear matter simulations on the lattice and look to chiral effective theory to determine the relative importance of various interactions. We then introduce the concept of spatial zones and suggest a new expansion of the nucleon determinant in powers of the hopping parameter connecting neighboring zones. Rigorous bounds on the convergence of this expansion are given as well as an estimate of the required size of the spatial zones as a function of temperature. We apply the expansion to a realistic lattice simulation of the interactions of neutrons and neutral pions.