We study the phase diagram of repulsively interacting spin-1 bosons in optical lattices at unit filling, showing that an externally induced quadratic Zeeman effect may lead to a rich physics characterized by various phases and phase transitions. We find that the main properties of the system may be described by an effective field model, which provides the precise location of the phase boundaries for any dimension, in excellent agreement with our numerical calculations for one-dimensional (1D) systems. Thus, our work provides a quantitative guide for the experimental analysis of various types of field-induced quantum phase transitions in spin-1 lattice bosons. These transitions, which are precluded in spin-12 systems, may be realized by using an externally modified quadratic Zeeman coupling, similar to recent experiments with spinor condensates in the continuum.

Spin-orbit coupling can lead to exotic states of matter and unexpected behavior of the system properties. In this paper, we investigate the influence of spin-orbit coupling induced by proximity effects on a monolayer of superconductor (with $s$-wave or $d$-wave pairing) placed on an insulating bulk. We show that the critical temperatures $T_c$ of the superconducting states can be tuned by the spin-orbit coupling both in the case of on-site and intersite pairing. Moreover, we discuss a possibility of changing the location of the maximal $T_c$ from the half filling into the underdoped or overdoped regimes.

We study the superconducting properties of population-imbalanced ultracold Fermi mixtures in one-dimensional optical lattices that can be effectively described by the spin-imbalanced attractive Hubbard model in the presence of a Zeeman magnetic field. We use the mean-field theory approach to obtain the ground-state phase diagrams including some unconventional superconducting phases such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, and the $\eta$ phase (an extremal case of the FFLO phase), both for the case of a fixed chemical potential and for a fixed number of particles. It allows us to determine optimal regimes for the FFLO phase as well as $\eta$-pairing stability. We also investigate the evolution from the weak coupling (BCS-like limit) to the strong coupling limit of tightly bound local pairs (BEC) with increasing attraction, at $T=0$. Finally, the obtained results show that in spite of the occurrence of the Lifshitz transition induced by an external magnetic field, the superconducting state can still exist in the system, at higher magnetic field values.

We study the influence of the quadratic Zeeman effect on the Mott-insulator phases of hard-core 1D spin-$3/2$ fermions. We show that, contrary to spinor bosons, the quadratic Zeeman coupling preserves an $\mathrm{SU}(2)\otimes \mathrm{SU}(2)$ symmetry, leading for large-enough quadratic Zeeman coupling to an isotropic pseudo-spin-$1/2$ Heisenberg antiferromagnet. Decreasing the quadratic Zeeman coupling, this phase undergoes, depending on the scattering lengths, either a Kosterlitz-Thouless transition into a gapped dimerized phase or a commensurate-incommensurate transition into a gapless spin liquid. This rich phase diagram can be observed experimentally in four-component fermions in optical lattices under similar entropy constraints to those needed for Néel order in spin-$1/2$ gases.

The paper is devoted to a study of superconducting properties of population-imbalanced fermionic mixtures in quasi-one-dimensional optical lattices. The system can be effectively described by the attractive Hubbard model with the Zeeman magnetic field term. We investigated the ground-state phase diagram of the model as a function of the chemical potential and the magnetic field. The ground state of the system exhibits the conventional BCS-type superconductivity as well as the unconventional so-called Fulde-Ferrell-Larkin-Ovchinnikov state, in which the total momentum of Cooper pairs is non-zero. We determine the orders of transitions as well as the behavior of order parameters with a change of the model parameters.