We study Andreev-Bashkin drag interactions between the superflow of different components in the superfluid phase of a three-component Bose-Hubbard model without component-mixing interactions, relevant for cold atom systems subjected to an optical lattice. By means of directed worm algorithm path integral Monte Carlo simulations we produce the phase diagram of the superfluid drag density as a function of the nearest-neighbor hopping amplitude and the chemical potential in the case of component-symmetric parameters. In both one and two dimensions, we find an expected region of negative drag close to the tip of the first Mott lobe and a generalization of the two-component supercounterfluid phase in parts of this region. In addition, we also find a more trivial positive drag with strongest presence below the Mott lobe. Further, we investigate the dependence of the drag on the details of the intercomponent interactions strengths in the weakly-coupled regime. In accordance with mean-field results obtained by Stian Hartman, we find that, unlike in the two-component case, the drag is no longer independent of the sign of the intercomponent interaction strengths. In particular, the drag between component a and b can either be strengthened or weakened by increasing interaction strength between component a and c, depending on the sign of the interaction strength between b and c.
In multicomponent superfluids and superconductors, co- and counterflows of components have, in general, different properties. A. F. Andreev and E. P. Bashkin [Sov. Phys. JETP 42, 164 (1975)] discussed, in the context of
He3/He4 superfluid mixtures, that interparticle interactions produce a dissipationless drag. The drag can be understood as a superflow of one component induced by phase gradients of the other component. Importantly, the drag can be both positive (entrainment) and negative (counterflow). The effect is known to have crucial importance for many properties of diverse physical systems ranging from the dynamics of neutron stars and rotational responses of Bose mixtures of ultracold atoms to magnetic responses of multicomponent superconductors. Although substantial literature exists that includes the drag interaction phenomenologically, only a few regimes are covered by quantitative studies of the microscopic origin of the drag and its dependence on microscopic parameters. Here we study the microscopic origin and strength of the drag interaction in a quantum system of two-component bosons on a lattice with short-range interaction. By performing quantum Monte Carlo simulations of a two-component Bose-Hubbard model we obtain dependencies of the drag strength on the boson-boson interactions and properties of the optical lattice. Of particular interest are the strongly correlated regimes where the ratio of coflow and counterflow superfluid stiffnesses can diverge, corresponding to the case of saturated drag.
The superfluid drag coefficient of a weakly interacting three-component Bose-Einstein condensate is computed on a square optical lattice deep in the superfluid phase, starting from a Bose-Hubbard model with component-conserving, on-site interactions and nearest-neighbor hopping. At the mean-field level, Rayleigh-Schr\"odinger perturbation theory is employed to provide an analytic expression for the drag density. In addition, the Hamiltonian is diagonalized numerically to compute the drag within mean-field theory at both zero and finite temperatures to all orders in intercomponent interactions. Moreover, path integral Monte Carlo simulations, providing results beyond mean-field theory, have been performed to support the mean-field results. In the two-component case the drag increases monotonically with the magnitude of the intercomponent interaction ${\ensuremath{\gamma}}_{AB}$ between the two components
A and
B. The increase is independent of the sign of the intercomponent interaction. This no longer holds when an additional third component
C is included. Instead of increasing monotonically, the drag can either be strengthened or weakened depending on the details of the interaction strengths, for weak and moderately strong interactions. The general picture is that the drag coefficient between component
A and
B is a nonmonotonic function of the intercomponent interaction strength ${\ensuremath{\gamma}}_{AC}$ between
A and a third component
C. For weak ${\ensuremath{\gamma}}_{AC}$ compared to the direct interaction ${\ensuremath{\gamma}}_{AB}$ between
A and
B, the drag coefficient between
A and
B can decrease, contrary to what one naively would expect. When ${\ensuremath{\gamma}}_{AC}$ is strong compared to ${\ensuremath{\gamma}}_{AB}$, the drag between
A and
B increases with increasing ${\ensuremath{\gamma}}_{AC}$, as one would naively expect. We attribute the subtle reduction of ${\ensuremath{\rho}}_{d,AB}$ with increasing ${\ensuremath{\gamma}}_{AC}$, which has no counterpart in the two-component case, to a renormalization of the intercomponent scattering vertex ${\ensuremath{\gamma}}_{AB}$ via intermediate excited states of the third condensate
C. We briefly comment on how this generalizes to systems with more than three components.
Motivated by previous work on drag effects in superfluid Bose-Einstein condensates, the Bose-Hubbard model on the square lattice is considered using mean-field theory, and a method for computing the superfluid drag density that does not rely on Galilean invariance is employed to re-derive the drag in the weakly interacting two-component Bose-Einstein condensate. The same method is then used on the Bose-Hubbard model generalized to three-components. An exact analytic expression is not possible because of the complexity of the resulting diagonalization problem. However, Rayleigh-Schrödinger perturbation theory is used to fourth order, which yields an analytic perturbative expression for the zero temperature drag, in addition to exact numerical diagonalization to find the exact behaviour at zero and finite temperatures within mean-field theory. It is found that the presence of a third component can alter the drag in a non-trivial manner: The drag between the two initial boson components can be completely mediated by the third, and it can be both enhanced and diminished depending on the inter-component interaction strengths and signs. An attempt is also made at finding the drag in a weakly interacting two-component Bose-Einstein condensate with spin-orbit coupling, but complications arise, which are discussed. Finally, a phase diagram for the two-component condensate with interactions and spin-orbit coupling is constructed, describing the spin-imbalance, degeneracy, and distribution of spin at zero temperature. A qualitative discussion of the finite temperature behaviour suggests that the inhomogeneous weakly interacting and spin-orbit coupled superfluid Bose-Einstein condensate may in some regions in parameter space favor spin-imbalance as the temperature is increased.