State-Dependent Mobility Edge in Kinetically Constrained Models

In this work, we show that the kinetically constrained quantum East model lies between a quantum scarred and a many-body localized system featuring an unconventional type of mobility edge in the spectrum. We name this scenario the state-dependent mobility edge: while the system does not exhibit a sharp separation in energy between thermal and nonthermal eigenstates, the abundance of nonthermal eigenstates results in slow entanglement growth for many initial states, such as product states, below a finite energy density. We characterize the state-dependent mobility edge by looking at the complexity of classically simulating dynamics using tensor networks for system sizes well beyond those accessible via exact diagonalization. Focusing on initial product states, we observe a qualitative change in the dynamics of the bond dimension needed as a function of their energy density. Specifically, the bond dimension typically grows polynomially in time up to a certain energy density, where we locate the state-dependent mobility edge, enabling simulations for long times. Above this energy density, the bond dimension typically grows exponentially, making the simulation practically unfeasible beyond short times, as generally expected in interacting theories. We correlate the polynomial growth of the bond dimension to the presence of many nonthermal eigenstates around that energy density, a subset of which we compute via tensor networks. The outreach of our findings encompasses quantum sampling problems and the efficient simulation of quantum circuits beyond Clifford families.