Two types of quantum chaos: testing the limits of the Bohigas-Giannoni-Schmit conjecture

TL;DR

The authors separate eigenbasis chaos from spectral chaos and interrogate the BGS conjecture by constructing Poissonian Hamiltonian ensembles that share a fixed density of states but have uncorrelated spectra. They show that ETH and early-time chaos can persist without Random Matrix Universality, challenging a strong form of the conjecture, and they demonstrate that typicality depends on whether one fixes matrix entries or the density of states. The work then investigates the role of k-locality, showing Poisson ensembles are not k-local and that enforcing exact k-locality via a Metropolis-like procedure can yield Hamiltonians with Poisson spectra yet preserved DOS and early-time chaos, implying a nuanced link between locality and spectral universality. The discussion broadens to implications for quantum field theory, quantum gravity, and black hole microstate physics, proposing refined formulations of BGS and potential gravity-inspired models that decouple spectral chaos from early-time dynamics.

Abstract

There are two types of quantum chaos: eigenbasis chaos and spectral chaos. The first type controls the early-time physics, e.g. the thermal relaxation and the sensitivity of the system to initial conditions. It can be traced back to the Eigenstate Thermalization Hypothesis (ETH), a statistical hypothesis about the eigenvectors of the Hamiltonian. The second type concerns very late-time physics, e.g. the ramp of the Spectral Form Factor. It can be traced back to Random Matrix Universality (RMU), a statistical hypothesis about the eigenvalues of the Hamiltonian. The Bohigas-Giannoni-Schmit (BGS) conjecture asserts a direct relationship between the two types of chaos for quantum systems with a chaotic semiclassical limit. The BGS conjecture is challenged by the Poissonian Hamiltonian ensembles, which can be used to model any quantum system displaying RMU. In this paper, we start by analyzing further aspects of such ensembles. On general and numerical grounds, we argue that these ensembles can have chaotic semiclassical limits. We then study the Poissonian ensemble associated with the Sachdev-Ye-Kitaev (SYK) model. While the distribution of couplings peaks around the original SYK model, the Poissonian ensemble is not $k$-local. This suggests that the link between ETH and RMU requires of physical $k$-locality as an assumption. We test this hypothesis by modifying the couplings of the SYK Hamiltonian via the Metropolis algorithm, rewarding directions in the space of couplings that do not display RMU. The numerics converge to a $k$-local Hamiltonian with eigenbasis chaos but without spectral chaos. We finally comment on ways out and corollaries of our results.

Figures (6)

• Figure 1: The OTOC (\ref{['OTOCp']}) at temperatures $\beta=0,1,2,3$ (corresponding to orange, yellow, green, and blue respectively) for a single sample of an $N=22$ SYK Hamiltonian (darker colors) and its associated Poisson-ized version (lighter colors).
• Figure 2: The squared magnitudes of the $|I|$-fermion parts of the "Poissonized" Hamiltonian, averaged over $64$ instances of SYK hamiltonians.
• Figure 3: The norm of the non-local part of the "Poisson-ized" $N=10,14,18,22$ SYK Hamiltonian (blue dots are individual samples, their average at the blue x), and the difference from the original SYK Hamiltonian (orange dots are individual samples, their average at the orange $+$), all divided by the norm of the whole "Poisson-ized" Hamiltonian. The black line has slope $-1$ for reference.
• Figure 4: A histogram (probability density norm) of successive eigenvalue gaps of $64$ samples each of: Poisson/independently sampled eigenvalues (red line), normal $N=22$ SYK (black line), "Poisson-ized" $N=22$ SYK (blue histogram), and "Re-localized" $N=22$ SYK Hamiltonians.
• Figure 5: A histogram of the eigenvalues (left) and a histogram of the successive eigenvalue gap ratios (right), for an $N=18$ SYK model (orange) and for an $N=18$ model adjusted by the Metropolis algorithm (\ref{['metalg']}) (blue). The adjusted Hamiltonian is the same as the one used in Fig. \ref{['fig:metrocorrelation_comp']}.