Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity

TL;DR

This paper explores the quantum chaotic properties of the brickwall model in BTZ black hole geometry, analyzing spectral statistics and Krylov complexity to identify chaos signatures without relying on classical interior geometries.

Contribution

It demonstrates that the brickwall model exhibits quantum chaos features such as Wigner-Dyson statistics and spectral form factor ramp, using boundary conditions on the stretched horizon.

Findings

Spectral statistics match random matrix theory ensembles.

Krylov complexity shows a peak indicative of chaos.

Spectral rigidity alone can produce Krylov complexity peaks.

Abstract

We investigate quantum chaotic features of the brickwall model, which is obtained by introducing a stretched horizon - a Dirichlet wall placed outside the event horizon - within the BTZ geometry. This simple yet effective model has been shown to capture key properties of quantum black holes and is motivated by the stringy fuzzball proposal. We analyze the dynamics of both scalar and fermionic probe fields, deriving their normal mode spectra with Gaussian-distributed boundary conditions on the stretched horizon. By interpreting these normal modes as energy eigenvalues, we examine spectral statistics, including level spacing distributions, the spectral form factor, and Krylov state complexity as diagnostics for quantum chaos. Our results show that the brickwall model exhibits features consistent with random matrix theory across various ensembles as the standard deviation of the Gaussian distribution is varied. Specifically, we observe Wigner-Dyson distributions, a linear ramp in the spectral form factor, and a characteristic peak in Krylov complexity, all without the need for a classical interior geometry. We also demonstrate that non-vanishing spectral rigidity alone is sufficient to produce a peak in Krylov complexity, without requiring Wigner-Dyson level repulsion. Finally, we identify signatures of integrability at extreme values of the Dirichlet boundary condition parameter.