Lieb E.H., Loss M.
FLUXES, LAPLACIANS AND KASTELEYN'S THEOREM
(91K, Plain TeX)

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.