Quantum ergodicity on graphs
arXiv:0808.4110 · doi:10.1103/PhysRevLett.101.264102
Abstract
We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic expression in terms of a variant of the supersymmetric nonlinear sigma-model. Our estimate is based on a saddle-point analysis of this expression and leads to a criterion for when equidistribution emerges asymptotically in the limit of large graphs. Our theory predicts a rate of convergence that is a significant refinement of previous estimates, long-assumed to be valid for quantum chaotic systems, agreeing with them in some situations but not all. We discuss specific examples for which the theory is tested numerically.
4 pages, 1 figure
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- Perfect Absorption in Complex Scattering Systems with or without Hidden Symmetries
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- Eigenfunction Statistics on Quantum Graphs
- Universal Chaotic Scattering on Quantum Graphs
- Green's function approach for quantum graphs: an overview
- Universality of the momentum band density of periodic networks
- Sigma models for quantum chaotic dynamics
- Entropy of eigenfunctions on quantum graphs
- Quantum ergodicity for quantum graphs without back-scattering
- Effective field theory of random quantum circuits
- A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs
- Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
- Transmission phase of a quantum dot and statistical fluctuations of partial-width amplitudes
- Pointwise Weyl law for graphs from quantized interval maps
- Maximal scarring for eigenfunctions of quantum graphs