Simulation of the multiphase configuration and phase transitions with quantum walks utilizing a step-dependent coin

Abstract

Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed matter physics. Here, we utilize a 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) ±1 which are similar to the ones observed in topological insulators and polyacetylene. We confirm that the 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 (multiphase 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.

Toward simulation of topological phenomenas with one-, two- and three-dimensional quantum walks

“…In our previous studies in Refs. [36,37], we showed that simple-and split-step quantum walks with protocols…”

“…9 for two-dimensional quantum walk with only PHS, the same could be done for the protocol given in Eq. (36).…”

“…Based on these concepts, we have plotted Figs. 10 and 11 for the protocol with only PHS (28) and protocol without any of three symmetries (36), respectively.…”

“…The protocol without the three symmetries (36) can simulate A family of the topological phases in which the topological invariant is an integer, Z. Since the TRS is absent, we use the doubling procedure and find the following protocol in which τ y is a Pauli matrix associating to the two additional flavors A and B, U d is given in (36). This protocol (44) possess only TRS with T 2 = −1.…”

“…In fact, all types of the topological phases in one and two dimensions are simulable by one-and two-dimensional quantum walks [28][29][30][31][32]. The simulated topological phenomenas with the quantum walks can be used to extract topological invariants [33], suppress the limitations on strongly-driven systems, check the robustness of the edge states [13] and investigation of topological phase transitions [34][35][36]. So far, the simulation of the topological phases and boundary states in three dimensions with quantum walks have not been addressed in literature.…”