Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles

Abstract

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.

Supersymmetric SYK model and random matrix theory

“…If we arrange them in ascending order E n < E n+1 , we define, ∆E n = E n − E n−1 to be the level spacing, and we compute the ratio for the nearest neighbourhood spacing as r n = ∆E n /∆E n+1 . For matrices from the standard Dyson ensemble, the distribution of level spacing ratio satisfies the Wigner-Dyson statistics [80] (which is called the Wigner surmise)…”

Anderson localization in sigma models

“…In our view the last point makes P(r) more suitable for numerical studies, since increased statistics is directly added to the interval [0, 1], as opposed to the tail of the algebraic decay. For more details on P(r) and P(r) see [8].…”

Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

“…In contrast, extended states exhibit level repulsion and obey Wigner-Dyson statistics [48]. To distinguish between these distributions, it is convenient to use the ratio between the spacings of adjacent quasienergy levels [49][50][51]. Choosing the quasienergy zone to be between −π=T and π=T (i.e., choosing −i log e iεT ¼ εT for −π=T ≤ ε < π=T), we label quasienergies in ascending order.…”