Chaos in Glassy Systems from a TAP Perspective

TL;DR

This paper investigates how external parameter changes induce level crossing among TAP solutions in mean-field spin-glass models with 1RSB, revealing strong chaos and providing an analytical framework for complexity computation.

Contribution

It introduces a generalized complexity approach to analyze TAP solutions under external perturbations in 1RSB models, linking level crossing to chaos phenomena.

Findings

Finite parameter variations cause level crossing between TAP states with different free-energies.

The induced linear response is self-averaging and matches disorder-averaged anomalies.

Analytical recipe validated on spherical multi-p-spin models under temperature changes.

Abstract

We discuss level crossing of the free-energy of TAP solutions under variations of external parameters such as magnetic field or temperature in mean-field spin-glass models that exhibit one-step Replica-Symmetry-Breaking (1RSB). We study the problem through a generalized complexity that describes the density of TAP solutions at a given value of the free-energy and a given value of the extensive quantity conjugate to the external parameter. We show that variations of the external parameter by any finite amount can induce level crossing between groups of TAP states whose free-energies are extensively different. In models with 1RSB, this means strong chaos with respect to the perturbation. The linear-response induced by extensive level crossing is self-averaging and its value matches precisely with the disorder-average of the non self-averaging anomaly computed from the 2nd moment of thermal fluctuations between low-lying, almost degenerate TAP states. We present an analytical recipe to compute the generalized complexity and test the scenario on the spherical multi-$p$ spin models under variation of temperature.