# Felix Fritzsch: Boundary chaos

Spatiotemporal correlation functions provide the key diagnostic tool for studying spatially extended complex quantum many-body systems. In ergodic systems scrambling causes initially local observables to spread uniformly over the whole available Hilbert space and causes exponential suppression of correlation functions with the spatial size of the system. In this talk, we present a perturbed free quantum circuit model, in which ergodicity is induced by a unitary impurity placed on the system's boundary. We refer to this setting as boundary chaos. It allows for computing the asymptotic scaling of correlations with system size.

This is achieved by mapping dynamical correlation functions of local operators in a system of linear size L at time t to a partition function with complex weights defined on a two-dimensional lattice of smaller size t/L × L with a helix topology. We evaluate this partition function in terms of suitable transfer matrices. As this drastically reduces the complexity of the computation of correlation functions, we are able to treat system sizes far beyond what is accessible by exact diagonalization. By studying the spectra of transfer matrices numerically and combining our findings with analytical arguments we determine the asymptotic scaling of correlation functions with system size.

For impurities that remain unitary under partial transpose, we demonstrate that correlation functions between local operators at the system’s boundary at times proportional to system size L are generically exponentially suppressed with L. In contrast, for generic unitary impurities or generic locations of the operators correlations show persistent revivals with a period given by the system size.

Moreover we justify the notion of boundary chaos by demonstrating that spectral fluctuations follow predictions from random matrix theory: We compute the spectral form factor exactly in the limit of large local Hilbert space dimension, which agrees with random matrix results after possible non-universal initial behavior. For small local Hilbert space dimension we support our claim by extensive numerical investigations.