Ultrametricity Between States at Different Temperatures in Spin-Glasses

TL;DR

This paper demonstrates that in multi-$p$-spin spherical spin-glass models, equilibrium states at different temperatures are correlated and follow ultrametric relations, indicating no chaos in temperature and a consistent Parisi tree structure.

Contribution

It proves the existence of temperature correlations and ultrametricity in these models, showing the Parisi tree remains essentially unchanged across temperatures.

Findings

Equilibrium states at different temperatures are correlated.

Ultrametric relations hold between states at different temperatures.

The Parisi tree structure is consistent across temperatures.

Abstract

We prove the existence of correlations between the equilibrium states at different temperatures of the multi-$p$-spin spherical spin-glass models with continuous replica symmetry breaking: there is no chaos in temperature in these models. Furthermore, the overlaps satisfy ultrametric relations. As a consequence the Parisi tree is essentially the same at all temperatures with lower branches developing when lowering the temperature. We conjecture that the reference free energies of the clusters are also fixed at all temperatures as in the generalized random-energy model.