Spectral determinant on quantum graphs

Eric Akkermans 1, 2, 3, Alain Comtet 3, Jean Desbois 3, Gilles Montambaux 2, Christophe Texier 3, 4

Annals of Physics 284 (2000) 10-51

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.

  • 1. Department of Physics (Technion),
    Technion-Israel Institute of Technology
  • 2. Laboratoire de Physique des Solides (LPS),
    CNRS : UMR8502 – Université Paris XI – Paris Sud
  • 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI – Paris Sud
  • 4. Département de Physique Théorique,
    University of Geneva
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