Chaos in Glassy Systems from a TAP Perspective

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.