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We study the continuous time quantum walk of a single particle (initiallylocalized at a single site) on a one-dimensional spatial lattice with complexnearest neighbour and next-nearest neighbour hopping amplitudes. Complexcouplings lead to chiral propagation and a causal cone structure asymmetricabout the origin. We provide a hydrodynamic description for quantum walkdynamics in large space time limit. We find a global "quasi-stationary state"which can be described in terms of the local quasi-particle densitiessatisfying Euler type of hydrodynamic equation and is characterized by aninfinite set of conservation laws satisfied by scaled cumulative positionmoments. Further, we show that there is anomalous sub-diffusive scaling nearthe extremal fronts, which can be described by higher order hydrodynamicequations. The long time behaviour for any complex next-nearest neighbourhopping with a non-zero real component is similar to that of purely realhopping (apart from asymmetric distribution). There is a critical couplingstrength at which there is a Lifshitz transition where the topology of thecausal structure changes from a regime with one causal cone to a regime withtwo nested causal cones. On the other hand, for purely imaginary next-nearestneighbour hopping, there is a transition from one causal cone to a regime withtwo partially overlapping cones due to the existence of degenerate maximalfronts (moving with the same maximal velocity). The nature of the Lifshitztransition and the scaling behaviour (both) at the critical coupling strengthis different in the two cases.

Long time dynamics of a single particle extended quantum walk on a one  dimensional lattice with complex hoppings: a generalized hydrodynamic  description