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

Abstract

It was known that by a duality transformation, interacting bosons at filling factor f = p/q hopping on a lattice can be mapped to interacting vortices hopping on the dual lattice subject to a fluctuating dual " magnetic field" whose average strength through a dual plaquette is equal to the boson density f = p/q. So the kinetic term of the vortices is the same as the Hofstadter problem of electrons moving in a lattice in the presence of f = p/q flux per plaquette. Motivated by this mapping, we study the Hofstadter bands of vortices hopping in the presence of magnetic flux f = p/q per plaquette on 5 most common bipartite and frustrated lattices namely square, honeycomb, triangular, dice and Kagome lattices. We count the total number of bands, determine the number of minima and their locations in the lowest band. We also numerically calculate the bandwidths of the lowest Hofstadter bands in these lattices that directly measure the mobility of the dual vortices. The less mobil the dual vortices are, the more likely in a superfluid state the bosons are. We find that except the Kagome lattice at odd q, they all satisfy the exponential decay law W = Ae −cq even at the smallest q. At given q, the bandwidth W decreases in the order of Triangle, Square and Honeycomb lattice. This indicates that the domain of the superfluid state of the original bosons increases in the order of the corresponding direct lattices: Honeycome, Square and Triangular. When q = 2, we find that the the lowest Hofstadter band is completely flat for both Kagome and dice lattices. There is a gap on Kagome lattice, but no gap on dice lattice. This indicates that the boson ground state at half filling with nearest neighbor hopping on Kagome lattice is always a superfluid state. The superfluid state remains stable slightly away from the half filling. Our results show that the behaviours of bosons at or near half filling on Kagome lattice are quite distinct from those in square, honeycomb and triangular lattices studied previously.

Quantum magnetism of spinor bosons in optical lattices with synthetic non-Abelian gauge fields

“…In Fig.7, we contrast the gauge field configurations in the U(1) basis with that quantum spin Hall effect realized in recent experiments [12][13][14]. [12][13][14][41][42][43][44]. Both are translational invariant along the y direction, so only one row is shown.…”

“…where the x is the x− coordinate [41][42][43][44] of the site i. For irrational β, this Hamiltonian completely breaks the lattice translational symmetry.…”

“…For irrational β, this Hamiltonian completely breaks the lattice translational symmetry. For a rational 2β = p/q, it contains q sites per unit cell (RBZ is 1/q of the original BZ, for details, see [41][42][43][44]). However, the U(1) basis Fig.7a only breaks the lattice into A and B sublattices for any ( irrational ) value β.…”

SU(2)-Invariant Continuum Theory for an Unconventional Phase Transition in a Three-Dimensional Classical Dimer Model

“…The dual of kagomé is the dice lattice [11,12], and so the gauge field A is defined on the links of a lattice of stacked dice planes.…”

Quantum phases, supersolids and quantum phase transitions of interacting bosons in frustrated lattices

“…In the hard core limit U = ∞, Eqn.1 can be mapped to a XXZ quantum spin model 4,7 . Using a 1/S spin wave expansion on the resulting quantum spin model, the authors in 1 found that a SS state is more robust in a triangular lattice with only t and V 1 terms in Eqn.1 and a SS state with √ 3 × √ 3 pattern is stable even at half filling f = 1/2.…”