Strong-Weak Coupling Self-Duality in the Two-Dimensional Quantum Phase Transition of $p+ip$ Superconducting Arrays

The 2D quantum phase transition that occurs in a square lattice of Josephson-coupled $p\pm ip$ superconductors is an example of how four-body interactions in $d=2$ reproduce nonperturbative effects caused by two-body interactions in $d=1$. The ordered phase has an unconventional “bond order” of the local $T$-breaking variable. This problem can be analyzed using an exact self-duality; this duality in classical notation is the 3D generalization of the Kramers-Wannier duality of the 2D Ising model, and there are similar exact dualities in dimensions $d\ge 3$. We discuss the excitation spectrum and experimental signatures of the ordered and disordered phases, and the relationship between our model and previously studied behavior of 2D boson models with four-boson interactions.

Figure 1

The classical anisotropic 3D problem that describes the quantum critical point of the model (2). The two types of interactions (shaded bonds) are plaquette interactions in the planes normal to $\widehat{\mathbf{z}}$, and bond interactions along $\widehat{\mathbf{z}}$.