Dynamical multicritical front behaviour in an integrable quantum walk model

We consider an integrable quantum walk model on a one-dimensional lattice with finite range hopping. Starting from a spatially inhomogeneous initial state, we show the emergence of a generalized hydrodynamic description in the large space-time limit by studying the time dependence of various observables of interest like the cumulative probability distribution, cumulative currents, full counting statistics, cumulants, entanglement entropy, etc, using a combination of the analytic method of stationary phase approximation and exact numerics. We show the existence of a global ”quasi-stationary state” which can be described in terms of the local density of quasi-particle excitations satisfying Euler type of hydrodynamic equations. The global quasi-stationary state is characterized by an infinite set of conservation laws. Furthermore, we show that there is anomalous scaling behaviour in the vicinity of the extremal fronts which can be described in terms of higher-order hydrodynamic equations. Thus, we find that while the observables show global scaling behaviour in the bulk, they exhibit anomalous subdiffusive scaling behaviour with multicritical exponents near extremal fronts. We point out interesting connections of the model with multicritical random matrix models and fermions in nonharmonic traps. We also connect the study to that of domain wall dynamics in spin chain systems with finite-range spin exchange interactions.