Spectral determinant on quantum graphs

TL;DR

This paper derives new formulas for the spectral determinant of the Laplacian on finite quantum graphs, linking it to periodic orbits and enabling analysis of mesoscopic network properties.

Contribution

It introduces a path integral approach and diagrammatic method to express the spectral determinant in terms of periodic orbits, generalizing to magnetic fields.

Findings

Spectral determinant expressed as infinite product over periodic orbits.

Diagrammatic method simplifies computation of spectral determinants.

Application to mesoscopic networks' thermodynamic and transport properties.

Abstract

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.