A geometric way to ask how chaotic a quantum system is

Why look for another measure of chaos?

A many-body quantum system is described by a Hamiltonian, the mathematical object that specifies its energy and governs its evolution. Changing a field or interaction in that Hamiltonian can rotate the system’s energy eigenstates—the states with definite energy. How much those states shift depends both on the perturbation and on the system’s structure.

Existing indicators of quantum chaos capture different features: energy-level statistics, eigenstate entanglement, or the time evolution of observables. The paper asks whether some of these differences can be understood through one shared quantity, and whether it can help compare degrees of chaos. There is no universally accepted definition of quantum chaos, and even the phrase “maximally chaotic” can mean different things in different contexts.

Measuring response across allowed changes

The authors build a geometric susceptibility tensor from the system’s response to a set of allowed Hamiltonian deformations. These are chosen to respect the model’s physical constraints, such as locality and symmetries. Rather than measuring sensitivity to just one selected perturbation, the tensor describes responses across the accessible set of changes. Its trace gives an overall susceptibility; its principal directions show which combinations of changes produce stronger or weaker responses.

A change that merely rescales the whole Hamiltonian does not alter its eigenstates, so the analysis separates that direction from changes that alter the Hamiltonian’s composition. The construction also uses a regularization: a small frequency scale that limits the influence of extremely close energy levels. The authors interpret the susceptibility as especially sensitive to slow processes, because its underlying response weights low-frequency transitions strongly.

They test the framework numerically in a one-dimensional mixed-field Ising spin chain, with transverse and longitudinal fields and a nearest-neighbor interaction. Most analyses use eigenstates near the middle of the energy spectrum. They compare the susceptibility with eigenstate randomness and with spectral functions, which describe how strongly local operators connect states separated by different energies. They also examine spatially varying operators and some operators outside the original set of controls.

What the calculations show

In this model, the overall susceptibility develops a minimum in a region the authors identify as strongly chaotic. That region is close to one identified independently in earlier work using eigenstate entanglement statistics. The match is notable because the two approaches measure different things: one concerns responses to Hamiltonian changes, the other the structure of individual eigenstates.

Near the susceptibility minimum, the eigenstate-resolved responses are more similar to one another than they are near the model’s integrable limits. The principal response directions are also approximately alike in magnitude there. Meanwhile, the spectral response is broad and smooth, with less low-frequency structure than near the integrable regimes. The paper interprets this as evidence that local perturbations mix states broadly and that nonhydrodynamic slow behavior is suppressed.

Slow response does not disappear altogether. The authors find low-frequency features for operators that overlap with energy, a conserved quantity whose long-wavelength behavior is hydrodynamic. In their calculations, operators with weaker overlap with this slow sector generally show broader response. Thus, the proposed picture is not that a chaotic system has no slow dynamics, but that some slow behavior is imposed by conservation laws and locality.

How far does the conclusion reach?

The evidence supports a connection among susceptibility, eigenstate randomness, spectral broadening, and slow dynamics in the models studied. It does not prove that all these signs of chaos are universally governed by one geometry. The main analysis focuses on a spin chain and a particular set of local controls; choosing a different set of physically allowed changes could alter the measure. The authors describe broader application to other local many-body systems as a conjecture and identify additional symmetries, unusual hydrodynamics, and long-range interactions as open directions.

The numerical work also uses finite chains and a selected window of central-spectrum eigenstates. The paper reports checks of system-size and regularization dependence, while noting finite-size effects near integrable limits. Those checks strengthen the case within the studied setting, but they are not a substitute for tests across other models or experiments. The framework may help researchers search for strongly chaotic regimes in controllable quantum devices, but that use remains a proposed application, not an experimental demonstration reported here.

This article is based on the authors’ arXiv preprint, version 1, rather than an independently verified account of peer review. The paper’s claims should therefore be read as research findings and interpretations that may be revised or reassessed as the work is scrutinized further.