Simulation of multi phase configuration and phase transitions with step-dependent-coined quantum walk

Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed matter physics. Here, we utilize step dependent coin in quantum walks and investigate what topological phases we can simulate with it, their topological invariants, bound states and possibility of phase transitions. These quantum walks simulate non-trivial phases characterized by topological invariants (winding number) $\pm 1$ which are similar to the ones observed in topological insulators and polyacetylene. We confirm that number of phases and their corresponding bound states increase step dependently. In contrast, the size of topological phase and distance between two bound states are decreasing functions of steps resulting into formation of multiple phases as quantum walks proceed (multi phase configuration). We show that in the bound states, the winding number and group velocity are ill-defined, and the second moment of the probability density distribution in position space undergoes an abrupt change. Therefore, there are phase transitions taking place over the bound states and between two topological phases with different winding numbers.