The stability of topological edge states in non-linear quantum walks: 
Bifurcations unique to Floquet systems, revealed from non-unitary time-evolution operators

Ken Mochizuki (Hokkaido U.), Norio Kawakami, and Hideaki Obuse

Quantum walk, one kind of systems with discrete time-periodicity (Floquet systems), can possess non-trivial topological phases. Recently, quantum walks with non-linear effects have been proposed theoretically. Taking these features into account, we study the stability of topologically protected edge states in non-linear quantum walks. There are two approaches for the linear stability analysis of the edge states, one is by using a non-Hermitian Hamiltonian in the continuum limit of the quantum walk and the other one is by using a non-unitary time-evolution operator. While the former approach was considered in the previous work[1], we employ the latter approach[2], taking the discrete time-periodicity into account more rigorously. As a result, we find bifurcations where edge states change from stable attractors to unstable repellers as increasing the strength of non-linearity. The bifurcations are unique to Floquet non-linear systems, since they originate from the discrete time-periodicity of quantum walks. We analytically derive bifurcation points and show that analytical results agree well with numerical results.

[1] Y. Gerasimenko, B. Tarasinski, and C.W. J. Beenakker, Phys. Rev. A 93, 022329 (2016).
[2] K. Mochizuki, N. Kawakami, and H. Obuse, arXiv:1907.08464