Nonequilibrium quantum order at infinite temperature: Spatiotemporal correlations and their generating functions

Localization-protected quantum order extends the idea of symmetry breaking and order in ground states to individual eigenstates at arbitrary energy. Examples include many-body localized static and pi spin glasses in Floquet systems. Such order is inherently dynamical and difficult to detect as the order parameter typically varies randomly between different eigenstates, requiring specific superpositions of eigenstates to be targeted by the initial state. We show that two-time correlators overcome this, reflecting the presence or absence of eigenstate order even in fully mixed, infinite temperature states. We show how spatiotemporal correlators are generated by the recently introduced dynamical potentials, demonstrating this explicitly using an Ising spin glass and a Floquet pi spin glass and focusing on features mirroring those of equilibrium statistical mechanics such as bimodal potentials in the symmetry-broken phase