Evaluating Grassmann Integrals

I discuss a simple numerical algorithm for the direct evaluation of multiple Grassmann integrals. The approach is exact, suffers no Fermion sign problems, and allows arbitrarily complicated interactions. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems of order a thousand Grassmann variables can be evaluated on a workstation. The technique is illustrated with a spinless fermion hopping along a one dimensional chain. 02.70.Fj, 11.15.Ha, 11.15.Tk Typeset using REVTEX 1 In path integral formulations of quantum field theory, fermions are treated via integrals over anti-commuting Grassmann variables [1]. This gives an elegant framework for the formal establishment of Feynman perturbation theory. For non-perturbative approaches, such as Monte Carlo studies with a discrete lattice regulator, these variables are more problematic. Essentially all such approaches formally integrate the fermionic fields in terms of determinants depending only on the bosonic degrees of freedom. Further manipulations give rise to the algorithms which dominate current lattice gauge simulations. Frequently, however, this approach has serious shortcomings. In particular, when a background fermion density is desired, as for baryon rich regions of heavy ion scattering, these determinants are not positive, making Monte Carlo evaluations tedious on any but the smallest systems [2]. This problem also appears in studies of many electron systems, particularly when doped away from half filling. In this note I explore an alternative possibility of directly evaluating the fermionic integrals, doing the necessary combinatorics on a computer. This is inevitably a rather tedious task, with the expected effort growing exponentially with volume. Nevertheless, in the presence of the sign problem, all other known algorithms are also exponential. My main result is that this growth can be controlled to a transverse section of the system. I illustrate the technique with low dimensional systems involving of order a thousand Grassmann variables. I begin with a set of n anti-commuting Grassmann variables {ψi}, satisfying [ψi, ψj ]+ = ψiψj + ψjψi = 0. Integration is uniquely determined up to an overall normalization by requiring linearity and “translation” invariance ∫ dψf(ψ + ψ) = ∫ dψf(ψ). For a single variable, I normalize things so that