Stability of current-carrying states in hard-core bosons with long-range hopping on a square lattice

We investigate the stability of current-carrying states with quasi-momentum $K$ in the Bose-condensed phase of the hard-core Bose-Hubbard model on a square lattice, where particles transfer between two sites separated by distance $r$ with hopping amplitude decaying algebraically with $r$ as $\propto r^{-\alpha }$. Using a mean-field theory, we analyze the excitation spectrum and determine the critical quasi-momenta associated with Landau and dynamical instabilities. We find that the long-range hopping suppresses the critical quasi-momenta and makes them vanish at $\alpha =3$. Near $\alpha =3$, we show that the critical quasi-momentum $K_{\mathrm{c}}$ for the dynamical instability exhibits the scaling behavior $K_{\mathrm{c}}\propto {\mathrm{\Delta }}^{1+\mathrm{\Delta }}$ with $\mathrm{\Delta }=\alpha -3$, where the scaling exponent explicitly depends on $\mathrm{\Delta }$, as a consequence of the long-range nature of the hopping.