Abstract:
Recent study on doping effects in the heavy fermion superconductor CeCoIn$_5$ has shown that a small amount of doping induces unexpectedly large broadening of the second order transition into the high field and low temperature (HFLT) phase of this material. Motivated by this observation, effects of quenched disorder on the second order transition into a longitudinal Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state are examined. The observed large broadening of the transition is naturally explained as a consequence of softness of each FFLO nodal plane. The present results strongly support the scenario identifying the HFLT phase of CeCoIn$_5$ with a longitudinal FFLO vortex state.

Abstract:
A Zeeman magnetic field can induce a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in spin-singlet superconductors. Here we argue that there is a non-trivial solution for the FFLO vortex phase that exists near the upper critical field in which the wavefunction has only spatial line nodes that form intricate and unusual three-dimensional structures. These structures include a crisscrossing lattice of two sets of non-parallel line nodes. We show that these solutions arise from the decay of conventional Abrikosov vortices into pairs of fractional vortices. We propose that neutron scattering studies can observe these fractional vortex pairs through the observation of a lattice of 1/2 flux quanta vortices. We also consider related phases in non-centrosymmetric (NC) superconductors.

Abstract:
We demonstrate that the vortex state in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase may be very different depending on the field orientation relative to the crystalline axes. We calculate numerically the upper critical field near the tricritical point taking into account the modulation of the order parameter along the magnetic field as well as the higher Landau levels. For s-wave superconductors with the anisotropy described by an elliptical Fermi surface we propose a general scheme of the analysis of the angular dependence of upper critical field at all temperatures on the basis of the exact solution for the order parameter. Our results show that the transitions (with tilting magnetic field) between different types of mixed states may be a salient feature of the FFLO phase. Moreover we discuss the reasons for the first-order phase transition into the FFLO state in the case of CeCoIn5 compound.

Abstract:
We study the phase diagram of a superconducting ring threaded by an Aharonov-Bohm flux and an in-plane magnetic Zeeman field. The simultaneous presence of both the external flux and the in-plane magnetic field leads to the competition between the Fulde-Ferrell (FF) phase and the Larkin-Ovchinnikov (LO) phase. Using the Bogoliubov-de Gennes equation, we investigate the spacial profile of the order parameter. Both the FF phase and the LO phase are found to exist stably in this system. The phase boundary is determined by comparing the free energy. The distortion of the phase diagrams due to the mesoscopic effect is also studied.

Abstract:
The realizations of spin-orbit coupling in cold atoms lead to a burst of research activities in the searching of topological matters in ultracold atom systems. The very recent theoretical predictions show that topological Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids can be realized with proper spin-orbit coupling and Zeeman fields. In this work, a comprehensive understanding of the pairing symmetry, phase diagram and the edge modes in this new topological matter are presented. The momentum of the Cooper pairs plays the role of renormalizing the in-plane Zeeman field and chemical potential. The in-plane Zeeman field and finite momentum pairing induce asymmetry to the effective $p$-wave pairing, apart from a small fraction of higher orbital components. The phase diagram is composed by different phases, which are determined by the topology and band gap nature of the superfluids. Especially, the gapped and gapless topological FFLO phase have totally different finite size effect. These novel features show that the spin-orbit coupled cold atoms provides an important platform in realizing topological matters which may not be materialized with solids.

Abstract:
In three dimensional Fermi gases with spin imbalance, a competition exists between Cooper pairing with zero and with finite momentum. The latter gives rise to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase, which only exists in a restricted area of the phase diagram as a function of chemical potential imbalance and interaction strength. Applying an optical potential along one direction enhances the FFLO region in this phase diagram. In this paper, we construct the phase diagram as a function of polarization and interaction strength in order to study the competition between the FFLO phase and the spin balanced BCS phase. This allows to take into account the region of phase separation, and provides a more direct connection with experiment. Subsequently, we investigate the effects of the wavelength and the depth of the optical potential, which is applied along one direction, on the FFLO state. It is shown that the FFLO state can exist up to a higher level of spin imbalance if the wavelength of the optical potential becomes smaller. Our results give rise to an interesting effect: the maximal polarization at which the FFLO state can exist, decreases when the interaction strength exceeds a certain critical value. This counterintuitive phenomenon is discussed and the connection to the optical potential is explained.

Abstract:
The recent experimental support for the presence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in the CeCoIn5 directed the attention towards the mechanisms responsible for this type of superconductivity. We investigate the FFLO state in a model where on--site/inter--site pairing coexists with repulsive pair hopping interaction. The latter interaction is interesting in that it leads to pairing with nonzero momentum of the Cooper pairs even in the absence of the external magnetic field (the so-called eta-pairing). It turns out that depending on the strength of the pair hopping interaction the magnetic field can induce one of two types of the FFLO phase with different spatial modulations of the order parameter. It is argued that the properties of the FFLO phase may give information about the magnitude of the pair hopping interaction. We also show that eta-pairing and d-wave superconductivity may coexist in the FFLO state. It holds true also for superconductors which in the absence of magnetic field are of pure d-wave type.

Abstract:
We studied the phase diagram for a two-dimensional d-wave superconducting system under an in-plane magnetic field or an exchange field. According to the spatial configuration of the order parameter, we show that there exists quantum phase transitions in which the uniform phase transforms to the one-dimensional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, and then to two-dimensional FFLO state upon increasing the exchange field. The local density of states are calculated and suggested to be signatures to distinguish these phases.

Abstract:
Effect of the phase fluctuations of the order parameter on the stability of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states are examined in exactly two-dimensional (2D) type-II superconductors with cylindrically symmetric Fermi surface on the basis of a generalized Ginzburg-Landau theory. It is found that for the FFLO states with oscillations in a single direction, not only the long-range order but also quasi-long-range order (QLRO), which is characterized by a power law decay of the order parameter correlation function, is suppressed by the phase fluctuations at any finite temperatures. On the other hand, for the FFLO states with order parameter structures such as triangular and square lattices, it is shown that the QLRO is possible as the uniform BCS state. Systems with anisotropy in the Fermi surface and pairing are also discussed.

Abstract:
We consider a formation of the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase in a quasi-one-dimensional (Q1D) conductor in a magnetic field, parallel to its conducting chains, where we take into account both the paramagnetic spin-splitting and orbital destructive effects against superconductivity. We show that, due to a relative weakness of the orbital effects in a Q1D case, the LOFF phase appears in (TMTSF)$_2$ClO$_4$ superconductor for real values of its Q1D band parameters. We compare our theoretical calculations with the recent experimental data by Y. Maeno's group [S. Yonezawa et al., Phys. Rev. Lett. \textbf{100}, 117002 (2008)] and show that there is a good qualitative and quantitative agreement between the theory and experimental data.