Spectral correlations of individual quantum graphs

Abstract

We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the 'energy'-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric non-linear σ-model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries.

Relationship between scattering matrix and spectrum of quantum graphs

“…Persistent deviations from the predicted behavior were found in star graphs [24,4], and Tanner [40] proposed a precise condition on the graphs to follow the random matrix theory prediction. This question was then attacked analytically by various methods, with results reported, in particular, in [7,16,2]. For more information we refer the reader to a recent review [17].…”

“…Generic quantum graphs exhibit this universal behavior and represent the most likely system for which a full mathematically rigorous proof of this universality will first be found. Important steps in this direction have been taken in [16].…”

Experimental Study of Quantum Graphs with Simple Microwave Networks: Non-Universal Features

“…According to theory, the universal behavior is obtained only in the limit of infinitely intricate graphs with infinitely many bonds and nodes [21]. Fieldtheoretical results for spectral statistics in finite quantum graphs have largely focused on the size of these deviations, and criteria for their disappearance in the limit of large graphs [21,24]. Numerical work shows that many statistical properties of finite-size graphs are consistent with random matrix theory, but others, such as the second-order level velocity autocorrelation functions and the parametric curvature distribution, are not in agreement [25].…”

A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

“…The χ j and ψ j are column-vectors and the χ * j and ψ * j are row-vectors. In contrast to (2), there is no restriction on m and N in (3). The integration on the right-hand side in (3) is over the entire space of complex m × m matrices and dµ …”

“…Later on it was realized, again by Zirnbauer, see e.g [44,10], that actually all forms of the CFT are just manifestations of a very deep algebraic fact related to the so-called Howe duality [20]. Since their introduction, the Colour-Flavour Transformations proved to be a very useful tool, finding diverse applications in such areas of physics as lattice gauge theory [42,7,36], random network models [1,43], quantum chaos models [2,3], and the random matrix theory [41,16].…”