Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity

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.

• The paper introduces a brickwall model with Dirichlet boundary conditions to explore quantum chaos in BTZ geometry.
• It applies normal mode analysis to extract spectral statistics, confirming chaotic signatures via Wigner-Dyson distributions and a linear spectral form factor ramp.
• Krylov complexity is computed using Lanczos coefficients, revealing operator growth dynamics and information spread in quantum black holes.

Detailed Summary of "Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity" (2412.12301)

Introduction to Quantum Chaology and the Brickwall Model

The paper explores quantum chaotic features using the brickwall model within the BTZ geometry by introducing a Dirichlet wall outside the event horizon. This effective model captures key properties of quantum black holes and is inspired by the stringy fuzzball proposal. The investigation focuses on the dynamics of scalar and fermionic probe fields, calculating their normal mode spectra against Gaussian-distributed boundary conditions. This approach allows for analyzing spectral statistics, such as level spacing distributions, the spectral form factor, and Krylov state complexity, as indicators of quantum chaos.

Key Highlights:

• The model reveals features aligning with @@@@1@@@@ (RMT), showing Wigner-Dyson distributions and a linear ramp in the spectral form factor.
• Krylov complexity is analyzed, exhibiting characteristic peaks without classical interior geometry.
• The presence of level repulsion and spectral rigidity highlights the model's chaotic behavior.

Implementation of the Brickwall Model

To implement the concepts portrayed in the paper, consideration of computational requirements, spectral analysis, and treatment of boundary conditions is essential. Below are key aspects and steps:

Normal Mode Analysis

1. Geometry Setup: The BTZ black hole metric is extended by the introduction of a stretched horizon with a Dirichlet boundary condition, effectively modeling a quantum black hole.
2. Field Equations: Formulate the equations of motion for scalar and fermionic fields in the BTZ metric with adjustments for the Dirichlet wall.
3. Spectral Calculation: Solve the Klein-Gordon and Dirac equations using hypergeometric functions to obtain the normal mode spectra.

from scipy.special import hyp2f1 def find_normal_modes(freq, J, mass): # Hypergeometric solution structure part1 = hyp2f1((1 - mass - freq)/2, J, 1 - freq, 1) part2 = hyp2f1((1 + mass + freq)/2, J, 1 + freq, 1) # Calculate modes at boundary return boundary_condition(freq, part1, part2) def boundary_condition(freq, part1, part2): # Checks for normalizability at Dirichlet wall return (abs(part1) - abs(part2)) < TOLERANCE # TOLERANCE defined for precision frequencies = [0.02, 0.025, 0.03] # Example frequency range for analysis normal_modes = [find_normal_modes(freq, J=1, mass=0) for freq in frequencies]

Spectral Statistics and Chaos Diagnostics

• Level Spacing Distribution: Evaluate distributions by comparing normalized spacings across spectrum intervals, discerning transitions between chaotic (Wigner-Dyson, Poisson) distributions.
• Spectral Form Factor (SFF): Calculate SFF to observe the characteristic linear ramp, verifying chaotic behavior consistent with RMT predictions.
• Krylov Complexity: Compute using Lanczos coefficients for system evolution and visualize complexity growth through time to understand operator complexity.

Analysing Spectral Results

The following plots are standard in analyzing spectral diagnostic outputs:

• Level Spacing Distribution: Histograms fitted against theoretical RMT distributions corroborate the quantum chaotic behavior.
• Linear SFF Ramp: Log-log plots exhibiting linearity in SFF validate underlying chaotic dynamics.
• Krylov Complexity Peak: Monitoring complexity’s peak and temporal behavior reflects system's movement towards equilibrium.

Conclusion and Implications

The study of the brickwall model opens avenues for understanding quantum chaos in black hole physics. The approach effectively demonstrates that chaotic signatures can arise without invoking classical interior geometries. The analysis of Krylov complexity extends our understanding of information spreading in quantum systems, further aligning theoretical underpinnings between quantum gravity and chaotic systems. Future investigations might explore other boundary conditions and broader parameter spaces within similar geometric configurations to refine the general understanding of quantum chaos in gravitation.