Loop- and Grassmann Techniques for Lattice Systems

In physics, as well as in various other fields, rewriting techniques are a strong mathematical tool. Different formulations of one particular problem allow for several advantages. In this application we concentrate on techniques which lead to the so-called Grassmann- or Loop representations. For the Schwinger model there have been several studies in the strong coupling expansion, while for weak coupling rather little is known. One of the major problems is that the fermion sign problem limits the studies of the model from first principles with numerical methods. It has been shown for Wilson fermions that the sign problem gets mildened if one rewrites the action in terms of loops. However, in order to completely eliminate the sign problem we suggest to use staggered fermions instead, which indeed results in a representation free from any sign problem. We want to further investigate this model, rewrite observables in terms of loops, implement cluster algorithms and perform an extensive study for the weak coupling regime. The second sub-project will concentrate on the Ising anti-ferromagnet in two dimensions. While the analytic solution of the anti-ferromagnet on bipartite lattices is trivial, solutions for other lattice geometries are not known. Rewriting the model in terms of loops allows for an elegant solution. It consists of a series of mappings, which finally lead to Peierls contours that can be represented as a Gaussian Grassmann integral. However, for an anti- ferromagnet the structure of Peierls contours is quite different, such that the approach used for the square lattice, is not applicable. Nevertheless it is possible to perform a duality transformation, where the anti-ferromagnet on a square (triangular, honeycomb) lattice is mapped onto a ferromagnet on a square (honeycomb, triangular) lattice. We plan to construct a Grassmann representation for the dual model, such that we can solve the anti-ferromagnet on the non-bipartite triangular lattice in closed form. Our third sub-project considers surfaces, rather than loops. Also surfaces in three or more dimensions can be represented in terms of Grassmann integrals. To each surface of a particular configuration is assigned a weight, which however is a function of the couplings appearing in the Grassmann integral. The main goal is to systematically explore the space of couplings and the corresponding mapping functions onto the surface weights. We expect to find new and potentially powerful representations of fermionic models in terms of surfaces.