[challenge]: What is a Good Probe for Quantum Chaos in a Exact Degenerate Eigenstate Subspace?

Released by

Junkai Wang

Contact email

WangTheoPhys@outlook.com

Method

Exact Diagonalization

Challenge issue

Quantum chaos is connected to many deep topics: ETH, scrambling, quantum black holes, quantum gravity, and dirty superconductors (I was told by Mr.Xuanzhe and Prof.Zihong today about this concept). The puzzle of quantum chaos has many facets. The issue is not just that we lack a precise understanding and complete definition of quantum chaos (much like how the definition of quantum integrability remains unclear). It also stems from the fact that quantum chaos has several completely different operational definitions.

A common approach is to use random matrix theory ensembles, such as the Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE), or to use the Wigner semicircle law to study energy level repulsion. However, these spectral diagnostics fail when applied to a subspace composed of completely degenerate energy levels. Since all energy eigenvalues are identical in this case, it becomes extremely difficult to establish a reliable criterion for quantum chaos.

How should we understand this issue?

This question is highly relevant in high-energy physics:

1. In string theory, the BPS states of a black hole protected by supersymmetry form a subspace that is strictly degenerate. Yet, black holes are widely believed to be maximally chaotic.
2. This also applies to certain maximally chaotic quantum many-body systems, such as the Sachdev-Ye-Kitaev (SYK) model or four-dimensional supersymmetric Yang-Mills theory.
3. It has physical significance in condensed matter physics as well, such as in fractional Chern insulators.

Given this degeneracy, how do we characterize and diagnose quantum chaos within a completely degenerate subspace? What are the appropriate observables and diagnostic probes to use?

The issue names no files, tests, or code entry points. Start by reviewing the challenge statement and repository context, then establish the scope and acceptance criteria for a probe of chaos in an exactly degenerate subspace; the work is done when an agreed diagnostic approach and its observables are documented or implemented.