Ordered statistics of energy levels

Ordered (extreme value) statistics of energy levels for 1D disordered quantum mechanics

Dirac equation with a random mass is known to support a critical extended state at zero energy (note that Dirac operator is directly related to the supersymmetric Schrödinger Hamiltonian). We analyse the average density of states for 1D Dirac operator in a bounded domain. Due to delocalisation, the DoS is sensitive to the boundaries for E → 0. This provides a possible definition of the localisation length, much larger than the inverse localisation length, that usually provides a definition of localisation in 1D problems. We argue that Lyapunov exponent is not able to capture the essential physics of low energy localisation near the delocalisation point (E=0).

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