Transport and entanglement growth in long-range random Clifford circuits*

Conservation laws and hydrodynamic transport can constrain entanglement dynamics in isolated quantum systems, manifest in a slowdown of higher Renyi entropies. Here, we introduce a class of long-range random Clifford circuits with U(1) symmetry, which act as minimal models for more generic quantum systems and provide an ideal framework to explore this phenomenon. Depending on the exponent α controlling the probability ∼r^-α of gates spanning a distance r, transport in such circuits varies from diffusive to superdiffusive with a corresponding dynamical transport exponent z. We unveil that  the different hydrodynamic regimes reflect themselves in the asymptotic entanglement growth according to S(t) ∼ t^1/z, where the value of z depends on α. We explain this finding in terms of the inhibited operator spreading in U(1)-symmetric Clifford circuits, where the emerging light cones are intimately related to the transport behavior and are significantly narrower compared to circuits without conservation law. For sufficiently small α, we show that the presence of hydrodynamic modes becomes irrelevant such that S(t) behaves similarly in circuits with and without conservation law.

Based on: J. Richter, O. Lunt, and A. Pal, arXiv:2205.06309