Fluxes, Laplacians, and Kasteleyn’s theorem

The genesis of this paper was an attempt to understand a problem in condensed matter physics related to questions about electron correlations, superconductivity, and electron-magnetic field interactions. The basic idea, which was proposed a few years ago, is that a magnetic field can lower the energy of electrons when the electron density is not small. Certain very specific and very interesting mathematical conjectures about eigenvalues of the Laplacian were made, and the present paper contains a proof of some of them. Furthermore, those conjectures lead to additional natural conjectures about determinants of Laplacians which we both present and prove here. It is not clear whether these determinantal theorems have physical applications but they might, conceivably in the context of quantum field theory. Some, but not all, of the results given here were announced earlier in [LE].

[1]  E. Lieb,et al.  An itinerant electron model with crystalline or magnetic long range order , 1986 .

[2]  J. C. Kimball,et al.  Simple Model for Semiconductor-Metal Transitions: SmB6and Transition-Metal Oxides , 1969 .

[3]  Zee,et al.  Chiral spin states and superconductivity. , 1989, Physical review. B, Condensed matter.

[4]  U. Brandt,et al.  Ground state properties of a spinless Falicov-Kimball model; additional features , 1987 .

[5]  P. G. Harper.  The General Motion of Conduction Electrons in a Uniform Magnetic Field, with Application to the Diamagnetism of Metals , 1955 .

[6]  D. Hofstadter.  Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields , 1976 .

[7]  P. G. Harper,et al.  Single Band Motion of Conduction Electrons in a Uniform Magnetic Field , 1955 .

[8]  M. Fisher,et al.  Dimer problem in statistical mechanics-an exact result , 1961 .

[9]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .

[10]  P. Wiegmann.  Towards a gauge theory of strongly correlated electronic systems , 1988 .

[11]  Rokhsar.  Solitons in chiral-spin liquids. , 1990, Physical review letters.

[12]  U. Brandt,et al.  Exact results for the distribution of thef-level ground state occupation in the spinless Falicov-Kimball model , 1986 .

[13]  R. Rammal,et al.  An algebraic semi-classical approach to Bloch electrons in a magnetic field , 1990 .

[14]  R. Rammal,et al.  Ground state of the Fermi gas on 2D lattices with a magnetic field , 1990 .

[15]  Lederer,et al.  Theory of electronic diamagnetism in two-dimensional lattices. , 1989, Physical review letters.

[16]  Kotliar,et al.  Resonating valence bonds and d-wave superconductivity. , 1988, Physical review. B, Condensed matter.

[17]  I. Affleck,et al.  Large-n limit of the Heisenberg-Hubbard model: Implications for high-Tc superconductors. , 1988, Physical review. B, Condensed matter.

[18]  Carlos Tomei,et al.  On the determinant of the adjacency matrix for a planar sublattice , 1983, J. Comb. Theory, Ser. B.