Boson-vortex duality in compressible spin-orbit coupled BECs

Using a (1+2)-dimensional boson-vortex duality between non-linear electrodynamics and a two-component compressible Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, we obtain generalised versions of the hydrodynamic continuity and Euler equations where the phase defect and non-defect degrees of freedom enter separately. We obtain the generalised Magnus force on vortices under SO coupling, and associate the linear confinement of vortices due to SO coupling with instanton fluctuations of the dual theory

Necessary and sufficient conditions for flat bands in MM-dimensional NN-band lattices with complex-valued nearest-neighbour hopping

We formulate the necessary and sufficient conditions for the existence of dispersionless energy eigenvalues (so-called `flat bands') and their associated compact localized eigenstates in $M$-dimensional tight-binding lattices with $N$ sites per unit cell and complex-amplitude nearest-neighbour tunneling between the lattice sites. The degrees of freedom $M$ can be traded for longer-range complex hopping in lattices with reduced dimensionality. We show the conditions explicitly for $(M = 1, N\leq4)$, $(M = 2, N = 2,3)$, and $(M = 3, N = 2,3)$, and outline their systematic construction for arbitrary $N$, $M$. If and only if the conditions are satisfied, then the system has one or more flat bands. By way of an example, we obtain new classes of flat band lattice geometries by solving the conditions for the lattice parameters in special cases.Comment: 7 pages, 4 figure

Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlev\'e transcendent. In each case the potential can be chosen to be time-independent.Comment: 8 pages, 7 figures. Version 2: stability analysis of the dark solutio