Field Theory of Borromean Super-counterfluids

We introduce a class of dynamical field theories for $N$-component "Borromean" ($N\geq 3$) super-counterfluid order, naturally formulated in terms of inter-species bosonic fields $\psi_{\alpha\beta}$. Their condensation breaks the normal-state [U(1)]$^N$ symmetry down to its diagonal U(1) subgroup, thereby encoding the arrest of the net superflow. This approach broadens our understanding of dynamical properties of super-counterfluids, at low energies capturing its universal properties, phase transition, counterflow vortices, and many of its other properties. Such super-counterfluid strikingly exhibits $N$ distinct flavors of energetically stable elementary vortex solutions, despite $\mathbb{Z}^{N-1}$ homotopy group of its $N\! -\! 1$ independent Goldstone modes, with $N\! -\! 1$ topologically distinct elementary vortex types, obeying modular arithmetic. The model leads to Borromean hydrodynamics as a low-energy theory, reveals counteflow AC Josephson effect, and generically predicts a first-order character of the phase transitions into Borromean super-counterfluid state in dimensions greater than two.