92-117 Lieb E.H., Loss M.

FLUXES, LAPLACIANS AND KASTELEYN'S THEOREM (91K, Plain TeX) Sep 23, 92

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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.

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