Continuous-time quantum walks on nonorientable surfaces: analytical solutions for Mobius strips and Klein bottles

In this paper, we present an extensive analytical study of continuous-time quantum walks (CTQWs) occurring on two-dimensional lattices under various boundary conditions, focusing on the M x N lattice wrapped on Mobius strips and Klein bottles, which are featured by the twisted boundary conditions. We find that the eigenvalue spectra and transport efficiency (both quantum mechanical and classical) for the two structures show comparable results with those for other well-known two-dimensional lattices, including rectangles, cylinders and tori. We also demonstrate that the behaviors of CTQWs depend on the initial node of the excitation and the network size, both on short and long timescales. In addition, we discover the asymmetric behaviors of limiting probabilities for Mobius strips and Klein bottles, which are quite different from each other and are also compared to those discovered in other two-dimensional networks. Our work provides a comprehensive understanding of recent results about CTQWs on two-dimensional lattices, and sheds light on quantum dynamics on lattices, especially those with different boundary conditions.