Localization in the kicked Ising chain

Determining the border between ergodic and localized behavior is of central interest for interacting many-body systems. We consider here the recently very popular spin-chain model that is periodically excited. A convenient description of such a many-body system is achieved by the dual operator that evolves the system in contrast to the time-evolution operator not in time but in particle direction. We identify in this paper the largest eigenvalue of a function based on the dual operator as a convenient tool to determine if the system shows ergodic or many-body localized features. By perturbation theory in the vicinity of the noninteracting system we explain analytically the eigenvalue structure and compare it with numerics in [P. Braun, D. Waltner, M. Akila, B. Gutkin, T. Guhr, Phys. Rev. E $\bf{101}$, 052201 (2020)] for small times. Furthermore we identify a quantity that allows based on extensive large-time numerical computations of the spectral form factor to distinguish between localized and ergodic system features and to determine the Thouless time, i.e. the transition time between these regimes in the thermodynamic limit.