The mobility of dual vortices in honeycomb, square, triangular, Kagome and dicelattices

Abstract

It was known that by a duality transformation, interacting bosons at filling factorf = p/q hopping on a lattice can be mapped to interacting vortices hopping onthe dual lattice subject to a fluctuating dual ‘magnetic field’ whoseaverage strength through a dual plaquette is equal to the boson densityf = p/q. So the kinetic term of the vortices is the same as the Hofstadterproblem of electrons moving in a lattice in the presence off = p/q flux per plaquette. Motivated by this mapping, we study theHofstadter bands of vortices hopping in the presence of magnetic fluxf = p/q per plaquette on five most common bipartite and frustrated lattices namelysquare, honeycomb, triangular, dice and Kagome lattices. We count the totalnumber of bands, and determine the number of minima and their locations inthe lowest band. We also numerically calculate the bandwidths of the lowestHofstadter bands in these lattices that directly measure the mobility of the dualvortices. The less mobile the dual vortices are, the more likely are the bosonsto be in a superfluid state. We find that apart from the Kagome lattice at oddq, they all satisfy theexponential decay law W = Ae−cq even at the smallest q. At given q, thebandwidth W decreases in the order of triangle, square and honeycomb lattice. This indicates thatthe domain of the superfluid state of the original bosons increases in the orderof the corresponding direct lattices: honeycomb, square and triangular. Whenq = 2, we find that the lowest Hofstadter band is completely flat for both Kagome and dicelattices. There is a gap on the Kagome lattice, but no gap on the dice lattice. Thisindicates that the boson ground state at half filling with nearest neighbour hopping onKagome lattice is always a superfluid state. The superfluid state remains stable slightlyaway from the half filling. Our results show that the behaviours of bosons at or near halffilling on Kagome lattices are quite distinct from those in square, honeycomb andtriangular lattices studied previously.