The full momentum dependence of spectrum of a point-like impurity immersed in a dilute onedimensional Bose gas is calculated on the mean-field level. In particular we elaborate, to the finite-momentum Bose polaron, the path-integral approach whose semi-classical approximation leads to the conventional mean-field treatment of the problem while quantum corrections can be easily accounted by standard loop expansion techniques. The extracted low-energy parameters of impurity spectrum, namely, the binding energy and the effective mass of particle, are shown to be in qualitative agreement with the results of quantum Monte Carlo simulations.

The properties of a Bose polaron immersed in a dilute two-dimensional medium at finite temperatures are discussed. Assuming that the impurity is weakly coupled to bath particles, we have perturbatively calculated the polaron energy, effective mass, quasiparticle residue and damping rate. The parameters of the impurity spectrum are found to be well-defined in the whole temperature region whereas the pole structure of the impurity Green's function is visible only at absolute zero. At any finite temperatures the quasiparticle residue is logarithmically-divergent signalling of the branch-cut behavior of the polaron propagator.

We determined perturbatively the low-energy universal thermodynamics of dilute one-dimensional bosons with the three-body repulsive forces. The final results are presented for the limit of vanishing potential range in terms of three-particle scattering length. An analogue of Tan's energy theorem for considered system is derived in generic case without assuming weakness of the interparticle interaction. We also obtained an exact identity relating the three-body contact to the energy density.

We rigorously analyze the low-temperature properties of homogeneous three-dimensional twocomponent Bose mixture with dipole-dipole interaction. For such a system the effective hydrodynamic action that governs the behavior of low-energy excitations is derived. The infrared structure of the exact single-particle Green's functions is obtained in terms of macroscopic parameters, namely the inverse compressibility and the superfluid density matrices. Within one-loop approximation we calculate the anisotropic superfluid and condensate densities and give the beyond mean-field stability condition for the binary dipolar Bose gas. A brief variational derivation of the coupled equations that describe macroscopic hydrodynamics of the system in the external non-uniform potential at zero temperature is presented.

The thermodynamic and superfluid properties of the dilute twodimensional binary Bose mixture at low temperatures are discussed. We also considered the problem of the emergence of the long-range order in these systems. All calculations are performed by means of celebrated Popov's pathintegral approach for the Bose gas with a short-range interparticle potential.Keywords two-dimensional Bose mixtures · superfluid properties · offdiagonal long-range orderThe spatial dimensionality plays a crucial role in the behavior of interacting many-boson systems. Perhaps, the most exciting phase diagram is obtained in the two-dimensional case (for review, see [1,2]), where the ground-state Bose condensate state [3,4,5,6,7,8] is altered by the low-temperature Beresinskii-Kosterlitz-Thouless (BKT) phase with the characteristic power-law [9] decay of the one-particle density matrix. To describe these systems appropriately one needs to extend [10] the standard approach with the separated condensate and to use the phase-density formulation [11], renormalization-group [12,13,14,15, 16,17] or the effective field-theoretic [18,19] treatments. Particularly Popov's theory allows to find out the low-energy structure of one-particle Green's functions [20] and improved version of this approach [21,22] which takes into account phase fluctuations exactly is capable to explain [23] experiments with two-dimensional Bose gases. In contrast to the two-dimensional Bose systems