Energy transport in local random matrix Hamiltonians

Random matrix theory (RMT) yields valuable insights into universal features of quantum many-body chaotic systems. While traditional RMT focuses on all-to-all interactions, many interesting dynamical questions such as energy transport or quantum information spreading require a notion of spatially local interactions. We study energy transport in few-body and 1D nearest-neighbor chains of locally interacting Hamiltonian RMT terms. In the few-body case, we analytically compute two-point energy-density correlators in the limit of large local Hilbert space dimension. Also, in this limit we show that overlapping spin-spin like RMT interaction terms become freely independent despite their strict locality. In the 1D chain, we numerically study the energy transport at small local Hilbert space dimension and examine how the choice of local RMT ensemble affects the transport coefficients. We discuss further extensions through free probability theory.