Supersolidity refers to the state of bosonic matter which exhibits both crystalline order associated with localized bosons and superfluid order which is associated with delocalized bosons. Such a state of matter is truly bizarre for systems such as He4 in the continuum. Understanding the elastic properties of such a state leads to some interesting questions - for instance, does the shear modulus of such a system vanish or is it finite? A much simpler problem, first recognized by Michael Fisher and coworkers, is to consider bosons moving on a lattice - in such a system, crystallization refers to breaking of lattice translational symmetry, and the supersolid phase of bosons can be viewed, semiclassically, as a particular ordered state of magnets with the boson occupation on a lattice site (n=0,1) playing the role of a spin-1/2 degree of freedom.

Motivated by exploring the interplay of geometric frustration and strong interactions between bosons we were led to study a toy model of bosons on the triangular lattice. We have used variational wave functions and quantum Monte Carlo numerics to show that the ground state of the spin-1/2 XXZ model (which is equivalent to a boson Hamiltonian) on the triangular lattice exhibits such supersolid order over a wide swath of its phase diagram. In addition, we have explored the effect of superflow in the superfluid phase of this model when the system is close to becoming a supersolid. Remarkably, we find that a supercurrent in this regime can induce supersolid order even when the quiescent state is a uniform superfluid with no crystallinity.