We introduce an inhomogeneous bosonic mixture composed of two kinds of hardcore and semihardcore boson with different nilpotency conditions and demonstrate that in contrast with the standard hardcore Bose Hubbard model, our bosonic mixture with nearest and next nearest neighbor interactions on a square lattice develops the checkerboard supersolid phase characterized by the simultaneous superfluid and checkerboard solid orders. Our bosonic mixture is created from a twoorbital Bose-Hubbard model including two kinds of bosons: a single orbital boson and a two-orbital boson. By mapping the bosonic mixture to an anisotropic inhomogeneous spin model in the presence of a magnetic field, we study the ground state phase diagram of the model by means of cluster mean field theory and linear spin wave theory and show that various phases such as solid, superfluid, supersolid and Mott insulator appear in the phase diagram of the mixture. Competition between the interactions and magnetic field causes the mixture to undergo different kinds of first and second order phase transitions. By studying the behavior of the spin wave excitations we find the reasons of all first and second order phase transitions. We also obtain the temperature phase diagram of the system using cluster mean field theory. We show that the checkerboard supersolid phase persists at finite temperature comparable with the interaction energies of bosons.

We study the ground state phase diagram of a two dimensional mixed-spin system of coupled alternating spin-1 and 1/2 chains with a stripe supersolid phase. Utilizing different analytical and numerical approaches such as mean field approximation, cluster mean field theory and linear spin wave theory, we demonstrate that our system displays a rich ground state phase diagram including novel stripe supersolid, solids with different fillings and super-counterfluid phases, in addition to a stripe solid with half filling, superfluid and Mott insulating phases. In order to find a minimal mixed-spin model for stripe supersolidity, in the second part of the paper we consider two kinds of mixed-spin system of coupled alternating spin-1 and 1/2 chains with (i) anisotropic nearest neighbor interactions, (ii) anisotropic hoppings and study their ground state phase diagrams. We demonstrate that, for the systems with uniform hoppings, the repulsive intra-chains interactions are necessary for stripe supersolidity. In this case the minimal two dimensional mixed-spin model is a system of spin-1 and spin-1/2 XXZ chains, interacting via Ising Hamiltonian. In the case of anisotropic hoppings, a system of coupled Ising chains is the minimal model.

Recently, it has been shown that two dimensional frustrated mixedspin systems with anisotropic exchange interactions display supersolid phases in their ground state phase diagrams even in the absence of long-range interactions. In this paper, using cluster mean field theory, we investigate the effects of thermal fluctuations on the ground state phases of this kind of systems and show that various thermal solids and thermal insulators emerge around the ground state solid and Mott insulating phases. We also study the thermodynamic properties and magnetocaloric effect of these systems and demonstrate that at low temperatures, a large cooling rate is seen in the vicinity of the solid-supersolid, solid-superfluid and Mott insulator-superfluid critical points, with the large accumulation of the entropy and the minimums of the isentropes. Our results show the sign change of the magnetocaloric parameter inside the solids and the Mott insulator, which is a characteristic of ordered phases.

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