Superspin Renormalization and Slow Relaxation in Random Spin Systems

We develop an excited-state real-space renormalization group (RSRG-X) formalism to describe the slow relaxation of conserved densities in randomly interacting spin-1/2 systems. Our formalism is suitable for systems with U(1) and Z_2 symmetries, and we apply it to systems of randomly positioned spins with long-range dipolar XX+YY interactions in one- and two-dimensions, as relevant to a variety of quantum simulation platforms. Our RSRG-X involves effective Hamiltonians of "superspins": two-level collective degrees of freedom constructed from (anti)aligned microscopic spins. Conserved densities can then be understood as relaxing via coherent collective spin-flips. Besides this physical picture, our formalism leads to an algorithm capable of simulating the dynamics up to an otherwise inaccessible combination of large system size and late time. Focusing on disorder-averaged infinite-temperature autocorrelations, we demonstrate agreement between our algorithm and exact diagonalization at low but nonzero frequencies.