Spectral determinant on quantum graphs

Details Spectral determinant on quantum graphs Spectral determinant on quantum graphs

1999-11-12

arxiv.org/abs/cond-mat/9911183v1

Annals Phys. 284 (2000) 10-51

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

33 pages, submitted to Ann. Phys

Scientific paper

10.1006/aphy.2000.6056

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and of bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian either in terms of a V x V vertex matrix or a 2B x 2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed.

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