Topological transition between gapless phases in quantum walks

Topological gapless phases of matter have been a recent interest among theoretical and experimental condensed matter physicists. Fermionic chains with extended nearest neighbor couplings have been observed to show unique topological transition at the multicritical points between distinct gapless phases. In this work, we show that such topological gapless phases and the transition between them can be simulated in a quantum walk. We consider a three-step discrete-time quantum walk and identify various critical or gapless phases and multicriticalities from the topological phase diagram along with their distinguished energy dispersions. We reconstruct the scaling theory based on the curvature function to study transition between gapless phases in the quantum walk. We show the interesting features observed in fermionic chains, such as diverging, sign flipping and swapping properties of curvature function, can be simulated in the quantum walk. Moreover, the renormalization group flow and Wannier state correlation functions also identify transition at the multicritical points between gapless phases. We observe the scaling law and overlapping of critical and fixed point properties at the multicritical points of the fermionic chains can also be observed in the quantum walk. Furthermore, we categorize the topological transitions at various multicritical points using the group velocity of the energy eigenstates. Finally, the topological characters of various gapless phases are captured using winding number which allows one to distinguish various gapless phases and also show the transitions at the multicritical points.