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
∝r−α. 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
α=3. Near
α=3, we show that the critical quasi-momentum
Kc for the dynamical instability exhibits the scaling behavior
Kc∝Δ1+Δ with
Δ=α−3, where the scaling exponent explicitly depends on
Δ, as a consequence of the long-range nature of the hopping.
Beliaev damping in a superfluid is the decay of a collective excitation into two lower-frequency collective excitations; it represents the only decay mode for a bosonic collective excitation in a superfluid at
T=0. The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final-state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS--Bose-Einstein condensate crossover: Here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum that is well established both theoretically and experimentally and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.
In the presence of dipolar interactions, the excitation spectrum of a Bose gas can acquire a local minimum.The corresponding quasiparticles are known as rotons.They are gapped and do not decay at zero temperature.Here we study the decay of rotons in one-dimensional Bose gases at low temperatures.It predominantly occurs due to the backscattering of thermal phonons on rotons.The resulting rate scales with the third power of temperature and is inversely proportional to the sixth power of the roton gap near the solidification phase transition.The hydrodynamic approach used here enables us to find the decay rate for quasiparticles at practically any momenta, with minimal assumptions on the exact form of the interparticle interactions.Our results are an essential prerequisite for the description of all the dissipative phenomena in dipolar gases and have direct experimental relevance.