Fluxes, Laplacians and Kasteleyn's Theorem

TL;DR

This paper investigates flux phase problems on planar graphs, extending to the Falicov-Kimball model, and provides a new proof of Kasteleyn's theorem by analyzing determinants of weighted adjacency matrices.

Contribution

It offers a partial solution to the flux phase problem, extends it to include potentials, and completely solves the determinant maximization problem for planar bipartite graphs, providing a new proof of Kasteleyn's theorem.

Findings

Partial solution to flux phase problem on planar graphs.

Complete solution for determinant maximization in bipartite planar graphs.

New, more transparent proof of Kasteleyn's theorem.

Abstract

The following problem, which stems from the ``flux phase'' problem in condensed matter physics, is analyzed and extended here: One is given a planar graph (or lattice) with prescribed vertices, edges and a weight $\vert t_{xy}\vert$ on each edge $(x,y)$. The flux phase problem (which we partially solve) is to find the real phase function on the edges, $\theta(x,y)$, so that the matrix $T:=\{\vert t_{xy}\vert {\rm exp}[i\theta(x,y)]\}$ minimizes the sum of the negative eigenvalues of $-T$. One extension of this problem which is also partially solved is the analogous question for the Falicov-Kimball model. There one replaces the matrix $-T$ by $-T+V$, where $V$ is a diagonal matrix representing a potential. Another extension of this problem, which we solve completely for planar, bipartite graphs, is to maximize $\vert {\rm det}\ T \vert$. Our analysis of this determinant problem is closely connected with Kasteleyn's 1961 theorem (for arbitrary planar graphs) and, indeed, yields an alternate, and we believe more transparent proof of it. {}.