Phase Coherence of Pairs of Cooper Pairs as Quasi-Long-Range Order of Half-Vortex Pairs in a Two-Dimensional Bilayer System.

Abstract

It is known that the loss of phase coherence of Cooper pairs in two-dimensional superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D XY model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analog, various phase transitions can be accurately determined. Below the BKT phase transition, an interlayer Ising long-range order is established at T_{Ising}, and the phase coherence of both intralayers and interlayers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multicritical point. For two inequivalent layers, however, there emerges an intermediate quasi-long-range order phase (T_{Ising}<T<T_{BKT}), where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. So our study provides a promising way to realize the charge-4e superconductivity in a bilayer system.