Statistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model

TL;DR

This paper analyzes the spectral form factor in the self-dual kicked Ising model, showing it matches Random Matrix Theory predictions with unique symmetry properties and enhanced fluctuations due to an anti-unitary symmetry.

Contribution

It provides an exact lower bound for moments of the spectral form factor and identifies the specific random matrix ensemble describing the model's spectral statistics.

Findings

Spectral form factor distribution matches RMT predictions at large times.

The relevant ensemble is a restricted symmetric space, not the circular orthogonal ensemble.

Enhanced fluctuations are due to an additional anti-unitary symmetry.

Abstract

We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated. We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory if one identifies the appropriate ensemble of random matrices. We find that this ensemble is not the circular orthogonal one - composed of symmetric random unitary matrices and associated with time-reversal-invariant evolution operators - but is an ensemble of random matrices on a more restricted symmetric space (depending on the parity of the number of sites this space is either ${Sp(N)/U(N)}$ or ${O(2N)/{O(N)\!\times\!O(N)}}$). Even if the latter ensembles yield the same averaged spectral form factor as the circular orthogonal ensemble they show substantially enhanced fluctuations. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.