[Paper Review] Generalized Spectral Form Factors and the Statistics of Heavy Operators

Alexandre Belin, Jan de Boer+2 more|arXiv (Cornell University)|Nov 11, 2021

Quantum chaos and dynamical systems4 citations

TL;DR

This paper introduces generalized spectral form factors (GSFFs) as new probes of quantum chaos in conformal field theories (CFTs), extending the standard spectral form factor to capture dynamics beyond energy level statistics. Using an effective field theory (EFT) of quantum chaos, the authors analyze heavy-heavy-heavy OPE coefficients via a genus-2 partition function analog, finding statistical correlations consistent with the OPE Randomness Hypothesis—specifically, a ramp and plateau in the GSFF, indicating random matrix behavior in the ergodic regime.

ABSTRACT

The spectral form factor is a powerful probe of quantum chaos that diagnoses the statistics of energy levels, but is blind to other features of a theory such as matrix elements of operators or OPE coefficients in conformal field theories. In this paper, we introduce generalized spectral form factors: new probes of quantum chaos sensitive to the dynamical data of a theory. These quantities can be studied using an effective theory of quantum chaos. We focus our attention on a particular combination of heavy-heavy-heavy OPE coefficients that generalizes the genus-2 partition function of two-dimensional CFTs, for which we define a spectral form factor. We probe heavy-heavy-heavy OPE coefficients and find statistical correlations that agree with the OPE Randomness Hypothesis: these coefficients have a random matrix component in the ergodic regime. The EFT of quantum chaos predicts that the genus-2 spectral form factor displays a ramp and a plateau. Our results suggest that this is a common property of generalized spectral form factors.

Motivation & Objective

• To develop new probes of quantum chaos in CFTs that go beyond energy level statistics, which the standard spectral form factor cannot access.
• To extend the framework of the effective field theory (EFT) of quantum chaos to include dynamical data such as OPE coefficients.
• To investigate whether generalized spectral form factors (GSFFs) of heavy operators exhibit universal features like the ramp and plateau, characteristic of quantum chaos.
• To test the OPE Randomness Hypothesis by analyzing statistical correlations in heavy-heavy-heavy OPE coefficients using GSFFs.
• To establish a connection between the genus-2 partition function in 2D CFTs and a new class of spectral probes sensitive to operator dynamics.

Proposed method

• Define generalized spectral form factors (GSFFs) as the squared modulus of partition functions involving functions of OPE coefficients, e.g., |Z(β₁+it₁,…,βₙ+itₙ)|².
• Use an effective field theory (EFT) of quantum chaos to model the statistical behavior of energy levels and matrix elements in chaotic systems.
• Introduce a tripled Hilbert space formalism to describe three-point functions and their statistical properties under random matrix assumptions.
• Apply Weingarten calculus to compute variances of OPE coefficients in microcanonical ensembles, distinguishing between disconnected and connected contributions.
• Compute the genus-2 spectral form factor as a special case of GSFF, focusing on the combination TrₛᵤₙₛₑₜO² = CᵢⱼₖC*ᵢⱼₗCₘₙₗC*ₘₙₖ, which appears in the sunset diagram decomposition.
• Use asymptotic expansions of Weingarten functions (Wg(D, n, σ) ∼ D⁻ⁿ⁻|σ|) to isolate leading-order contributions and identify universal behavior in the GSFF.

Experimental results

Research questions

• RQ1 Can generalized spectral form factors (GSFFs) serve as universal probes of quantum chaos that go beyond energy level statistics?
• RQ2 Do heavy-heavy-heavy OPE coefficients in chaotic CFTs exhibit statistical correlations consistent with the OPE Randomness Hypothesis?
• RQ3 Does the genus-2 spectral form factor, derived from a GSFF, display the universal ramp and plateau behavior seen in standard spectral form factors?
• RQ4 What is the role of the tripled Hilbert space and EFT of quantum chaos in modeling the statistical behavior of OPE coefficients?
• RQ5 How do the statistical properties of OPE coefficients—particularly their variance—reflect underlying chaotic dynamics in the ergodic regime?

Key findings

• The genus-2 spectral form factor, as a specific instance of a generalized spectral form factor, exhibits a ramp and plateau, indicating universal chaotic behavior.
• The variance of heavy-heavy-heavy OPE coefficients, computed via Weingarten calculus, shows a leading-order contribution proportional to D⁻⁸, confirming the dominance of connected, ergodic contributions.
• Statistical correlations in OPE coefficients are consistent with the OPE Randomness Hypothesis: the coefficients contain a random matrix component in the ergodic regime.
• The disconnected contribution to the variance is parametrically smaller (O(D⁻⁷)) than the connected part, confirming that the GSFF is sensitive to genuine dynamical correlations.
• The term TrₛᵤₙₛₑₜO² = CᵢⱼₖC*ᵢⱼₗCₘₙₗC*ₘₙₖ, which appears in the sunset diagram, is the dominant statistical observable in the GSFF and is basis-independent.
• The EFT framework successfully predicts the ramp and plateau in the GSFF, suggesting that such behavior is a generic feature of generalized spectral probes in chaotic quantum systems.