Phase coherence of pairs of Cooper pairs as quasi-long-range order of half-vortex pairs in a two-dimensional bilayer system.

It is well established that the phase coherence of Cooper pairs in two-dimensional superconductivity is described by the formation of vortex-antivortex pairs with the quasi-long-range order (quasi-LRO) in the order-parameter phase field, corresponding to the Berezinskii-Kosterlizt-Thouless (BKT) transition of a two-dimensional XY model. Here we study the phase coherence of Cooper pairs in a bilayer system with two-coupled XY models, showing that the second-order Josephson coupling can induce an exotic superconducting phase. By using tensor-network methods, we map the partition function into a product of one-dimensional quantum transfer operator, whose eigen-equation can be solved by an algorithm of matrix product states. The entanglement entropy of the quantum correspondence exhibits singularity, which can be used to accurately determine various phase transitions. Below the BKT phase transition, an inter-layer Ising long-range order is established, accompanying with quasi-LRO of vortex-antivortex pairs in both intra-layers and inter-layers. For two identical coupled layers, the Ising transition coincides with the BKT transition at a multi-critical point. For two inequivalent coupled layers, however, there emerges an intermediate quasi-LRO phase: the vortex-antivortex bindings occur in the layer with the larger intra-layer 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.