Abstract:
There seems to be a one to one correspondence between the phases of atomic and molecular matter (AMOM) and vortex matter (VM) in superconductors. Crystals, liquids and glasses have been experimentally observed in both AMOM and VM. However, quasi-crystals also exist in AMOM, thus a new phase of vortex matter is proposed here: the vortex quasi-crystal. It is argued that vortex quasi-crystals are stabilized due to boundary and surface energy effects for samples of special shapes and sizes, and that a phase transition between a vortex crystal and a vortex quasi-crystal occurs as a function of magnetic field and temperature as the sample size is reduced.

Abstract:
Assuming that the order parameter corresponds to an equal spin triplet pairing symmetry state, we calculate the effect of phase fluctuations in quasi-one-dimensional superconductors at high magnetic fields applied along the y (b') axis. We show that phase fluctuations can destroy the theoretically predicted triplet reentrant superconducting state, and that they are responsible for melting the magnetic field induced Josephson vortex lattice above a magnetic field dependent melting temperature Tm.

Abstract:
We discuss the time-of-flight expansion of dilute p-wave Fermi condensates on the BEC side of Feshbach resonances, as a way to extract information about the order parameter symmetry for superfluidity. We show that the cloud profile is in general sensitive to the interaction strength between fermions, the magnitude and direction of external magnetic fields, and to the angular momentum projection of the order parameter. In particular, due to the anisotropic nature of p-wave interactions we show that the time-of-flight expansion of a p-wave superfluid is anisotropic even if the superfluid is confined to a completely isotropic trap, unlike the case of Bose or s-wave Fermi condensates, which under the same circumstances expand isotropically. Furthermore, we demonstrate that expanding p-wave superfluids released from axially symmetric traps experience anisotropy inversions, where the aspect ratio between the axial and radial directions change during expansion, while the axial symmetry can also be lost reflecting the spatial anisotropy of the underlying interaction.

Abstract:
The ground state phase diagram of Fermi-Fermi mixtures in optical lattices is analyzed as a function of interaction strength, population imbalance, filling fraction and tunneling parameters. It is shown that population imbalanced Fermi-Fermi mixtures reduce to strongly interacting Bose-Fermi mixtures in the molecular limit, in sharp contrast to homogeneous or harmonically trapped systems where the resulting Bose-Fermi mixture is weakly interacting. Furthermore, insulating phases are found in optical lattices of Fermi-Fermi mixtures in addition to the standard phase-separated or coexisting superfluid/excess fermion phases found in homogeneous systems. The insulating states can be a molecular Bose-Mott insulator (BMI), a Fermi-Pauli insulator (FPI), a phase-separated BMI/FPI mixture or a Bose-Fermi checkerboard (BFC).

Abstract:
The ground state phase diagram of fermion mixtures in optical lattices is analyzed as a function of interaction strength, fermion filling factor and tunneling parameters. In addition to standard superfluid, phase-separated or coexisting superfluid/excess-fermion phases found in homogeneous or harmonically trapped systems, fermions in optical lattices have several insulating phases, including a molecular Bose-Mott insulator (BMI), a Fermi-Pauli (band) insulator (FPI), a phase-separated BMI/FPI mixture or a Bose-Fermi checkerboard (BFC). The molecular BMI phase is the fermion mixture counterpart of the atomic BMI found in atomic Bose systems, the BFC or BMI/FPI phases exist in Bose-Fermi mixtures, and lastly the FPI phase is particular to the Fermi nature of the constituent atoms of the mixture.

Abstract:
We study ultra-cold neutral fermion superfluids in the presence of fictitious magnetic fields, as well as charged fermion superfluids in the presence of real magnetic fields. Charged fermion superfluids undergo a phase transition from type-I to type-II superfluidity, where the magnetic properties of the superfluid change from being a perfect diamagnet without vortices to a partial diamagnet with the emergence of the Abrikosov vortex lattice. The transition from type-I to type-II superfluidity is tunned by changing the scattering parameter (interaction) for fixed density. We also find that neutral fermion superfluids such as $^6$Li and $^{40}$K are extreme type-II superfluids, and that they are more robust to the penetration of a fictitious magnetic field in the BCS-BEC crossover region near unitarity, where the critical fictitious magnetic field reaches a maximum as a function of the scattering parameter (interaction).

Abstract:
The evolution from BCS to BEC superconductivity in the s-wave and d-wave channels is analyzed at zero temperature for a two-dimensional superconductor. Spectroscopic quantities for s-wave and d-wave systems present fundamental differences when particle density and attraction strength are varied. A detailed analysis of single quasiparticle properties (excitation spectrum, momentum distribution, spectral function and density of states) indicates that the evolution of these spectroscopic quantities in the d-wave case is not smooth, unlike the situation encountered for the s-wave system.

Abstract:
We analyze the zero temperature phase diagram for an asymmetric two-component Fermi gas as a function of mass anisotropy and population imbalance. We identify regions corresponding to normal, or uniform/non-uniform superfluid phases, and discuss topological quantum phase transitions in the Bardeen-Cooper-Schrieffer (BCS), unitarity and Bose-Einstein condensation (BEC) limits. Lastly, we derive the zero temperature low frequency and long wavelength collective excitation spectrum, and recover the Bogoliubov relation for weakly interacting dilute bosons in the BEC limit.

Abstract:
We discuss the possibility of a quantum phase transition in ultra-cold spin-polarized Fermi gases which exhibit a p-wave Feshbach resonance. We show that when fermionic atoms form a condensate that can be externally tuned between the BCS and BEC limits, the zero temperature compressibility and the spin susceptibility of the fermionic gas are non-analytic functions of the two-body bound state energy. This non-analyticity is due to a massive rearrangement of the momentum distribution in the ground state of the system. Furthermore, we show that the low temperature superfluid density is also non-analytic, and exhibits a dramatic change in behavior when the critical value of the bound state energy is crossed.

Abstract:
The authors discuss the possibility of coexistence of antiferromagnetism and triplet superconductivity as a particular example of a broad class of systems where the interplay of magnetism and superconductivity is important. This paper focuses on the case of quasi-one-dimensional metals, where it is known experimentally that antiferromagnetism is in close proximity to triplet superconductivity in the temperature versus pressure phase diagram. Over a narrow range of pressures, the authors propose an intermediate non-uniform phase consisting of alternating insulating antiferromagnetic and triplet superonducting stripes.