Emergent fractional vortices and inter-layer Ising order in a bilayer of XY models.

2021
We study a bilayer system of coupled two-dimensional XY models by using tensor-network methods, showing that the inter-layer coupling can significantly alter the uncoupled topological Berezinskii-Kosterlizt-Thouless (BKT) transition. In the tensor-network representation, the partition function is mapped into a product of one-dimensional quantum transfer operator, whose eigen-equation can be accurately solved by an algorithm of matrix product states. The entanglement entropy of this one-dimensional quantum correspondence exhibits singularity, which can be used to determine various phase transitions. In the low temperature phase, an inter-layer Ising long-range order is developed,accompanying with vortex-antivortex bindings 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-long-range ordered phase, where vortex-antivortex bindings occur only in the layer with the larger intra-layer coupling but half-vortex pairs associated with topological strings emerge in the other layer. These results are supported by the correlation functions of the XY spins and nematic spins, and can be used to understand the novel properties in bilayer systems of superfluids, superconductors and quantum Hall systems.
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