Mean-field calculation of the doping-iduced quantum spin Hall state to superconductor transiton

The mean-field theory and simulation result of the easy-plane t-λ model and the mean-field phase diagram are given in this paper． Two mean-field scenarios exist for doping the pure QSH state: 1) a first-order QSH-SSC transition occurs in the weak interaction region; 2) a continuous QSH to QSH-SSC co-existence transition followed by a first-order QSH-vanishing transition in the strong interaction region． The mean-field theory is found to capture features for symmetry-broken phases but give the wrong conclusion of QSH-SSC co-existence due to ignoring fluctuation． It is unable to characterize transitions with topological excitation．