Eigenfunction statistics on quantum graphs

Abstract

We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric σ model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder.

The Green function for simplest quantum graphs

“…Nonrelativistic particle dynamics in networks described by the Schrödinger equation on graphs has been studied in the context of quantum chaos in the Refs. [1][2][3]. Despite the progress made in the study Schrödinger equation on graphs, the Green function based approach to particle transport still remains undeveloped.…”

“…Formally, it coincides with the Schrödinger equation, which was extensively studied in the context of quantum chaos theory [1][2][3][4][5][6][7][8][9][10]. Linear waves in optical waveguide networks provide experimental realization of quantum graphs [11].…”

“…In most cases, such transport can be modeled by evolution equations on so-called metric graphs. The latter is called quantum graphs for nanometer scales and has become a rapidly developing subject over the last three decades (see, e.g., reviews [1][2][3] and references therein). In this case, one deals with the Schrödinger equation for which the boundary conditions on graph vertices graphs are imposed [1].…”

Time-dependent quantum graph

“…However, the concept of studying a system confined to a graph dates back to Pauling [4], who suggested the use of such systems for modeling free electron motion in organic molecules. Over the last two decades, quantum graphs have found numerous applications in modeling different discrete structures and networks in nanoscale and mesoscopic physics (e.g., see reviews [1][2][3] and references therein).…”

“…Usually, such systems are modeled by so-called quantum graphs. These systems have attracted much attention in physics [1][2][3] and mathematics [5][6][7] over the past two decades.…”