Quantum valley Hall effects and spin-valley locking in topological Kane-Mele circuit networks

The Kane-Mele model of time-reversal symmetry, which simultaneously contains spin and valley degrees of freedom, is simulated in a topological circuit network. By varying capacitances and inductances, we are able to demonstrate the quantum valley Hall states, in addition to the quantum spin Hall states, as well as the topological phase transition between them. We also study the rich topological interface states between nontrivial phases with different spin and valley Chern numbers via tuning capacitances and inductances so as to tune the couplings and gauge fields. Finally, we demonstrate the spin-valley locking in the Kane-Mele circuit network. The topological circuit network provides us with a platform to simultaneously control spin and valley degrees of freedom so that a spin splitter and a valley splitter can be realized.