This dissertation presents the theoretical study of two-component fermionic gases. It firstly studies the BCS-BEC crossover for the system of the two-component fermionic gas in optical lattices. If the system were loaded into deep three-dimensional optical lattices, a tight-binding model with the lowest band is applicable. If the band is more than half-filled and by increasing the attractive interactions, then there exists a novel crossover, from a paired BCS superfluid to a BEC of holes, back to the BCS superfluid, and finally to a conventional BEC regime of diatomic molecules. If the band is less than half-filled, the crossover is similar to that without optical lattices. If the band is fully filled, a quantum phase transition from band insulator to BCS-BEC superfluid occurs. Then, this dissertation studies a model describing the one-dimensional Feshbach resonance. Because this model satisfies the Yang-Baxter equations, it was assumed that the Bethe Ansatz could solve this model. However, this dissertation demonstrates that it is not integrable by calculating the chemical potential from the Bethe Ansatz, and by studying the scattering between the bosons. To the researcher's knowledge, this model is the first continuum model that satisfies the Yang-Baxter equations, but is not integrable. After that, this dissertation studies the quenches in the system of paired fermionic superfluids. If the system is slightly driven from its equilibrium state, the asymptotic behaviors of this small oscillation are calculated with both s- and p-wave interactions, and in both the BCS and the BEC regimes, under the mean field collisionless approximation.
We study the thermodynamics of a two-species Feshbach-resonant atomic Fermi gas in a periodic potential, focusing in a deep optical potential where a tight binding model is applicable. We show that for a more than half-filled band the gas exhibits a reentrant crossover with decreased detuning (increased attractive interaction), from a paired BCS superfluid to a Bose-Einstein condensate (BEC) of molecules of holes, back to the BCS superfluid, and finally to a conventional BEC of diatomic molecules. This behavior is associated with the nonmonotonic dependence of the chemical potential on detuning and the concomitant Cooper-pair or molecular size, larger in the BCS and smaller in the BEC regimes. For a single filled band we find a quantum phase transition from a band insulator to a BCS-BEC superfluid, and map out the corresponding phase diagram.
In this thesis, we discuss ultracold quantum gases both in continuum and optical lattices. For the continuum Fermi gases in BCS-BEC crossover, we present an effective field theory study on the recently discovered puzzling damping phenomena on the BCS side of the crossover. We find that in contrast to the previous proposed pair-breaking mechanism of damping, the damping process is due to the interaction between superfluid phonons and thermally excited fermionic quasi particles. Results from our effective field theory are compared quantitatively with experiments, showing a good agreement. For the ultracold fermionic atoms in optical lattices, we propose two novel quantum phases. Firstly, we show that a novel superconducting pairing occurs for spin-imbalanced Fermi gases with the spin up and down Fermi levels lying within the px- and s- orbital bands of a quasi-one-dimensional optical lattice. The pairs condense at a finite momentum equal to the sum of the two Fermi momenta of spin up and down fermions, and form a p-orbital condensate. The phase diagram shows that the p-orbital pair condensate occurs in a wide range of fillings. Secondly, we study instabilities of single-species fermionic atoms in the p-orbital bands in two-dimensional square optical lattices. From the nearly-perfect nesting Fermi surfaces, charge density wave and orbital density wave orderings with stripe or checkerboard patterns are found for attractive and repulsive interactions, respectively. The superconducting phase, usually expected of attractively interacting fermions, is strongly suppressed. We also use field theory to analyze the possible liquid crystal phases in our system. For bosons, we study ultracold bosonic atoms loaded in a one-dimensional optical lattice of two-fold p-orbital degeneracy at each site, and find an anti-ferro-orbital, a homogeneous px Mott insulator phase and two kinds of superfluid phases distinguished by the orbital ordering.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS-BEC crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc. As a result, the maximum Tc is obtained away the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature
Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent
t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of
Tc. As a result, the maximum
Tc is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
The superfluidity and pairing phenomena in ultracold atomic Fermi gases have been of great interest in recent years, with multiple tunable parameters. Here we study the BCS-BEC crossover behavior of balanced two-component Fermi gases in a one-dimensional optical lattice, which is distinct from the simple three-dimensional (3D) continuum and a fully 3D lattice often found in a condensed matter system. We use a pairing fluctuation theory which includes self-consistent feedback effects at finite temperatures and find widespread pseudogap phenomena beyond the BCS regime. As a consequence of the lattice periodicity, the superfluid transition temperature
Tc decreases with pairing strength in the BEC regime, where it approaches asymptotically ${T}_{\mathrm{c}}=\ensuremath{\pi}an/2m$, with
a being the
s-wave scattering length, and
n (
m) the fermion density (mass). In addition, the quasi-two dimensionality leads to fast-growing (absolute value of the) fermionic chemical potential $\ensuremath{\mu}$ and pairing gap $\mathrm{\ensuremath{\Delta}}$, which depends exponentially on the ratio
d/a. Importantly,
Tc at unitarity increases with the lattice constant
d and hopping integral
t. The effect of the van Hove singularity on
Tc is identified. The superfluid density exhibits
T3/2 power laws at low
T, away from the extreme BCS limit. These predictions can be tested in future experiments.
We model fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction
U and nearest neighbor hopping
t. Our goal is to understand the crossover from BCS superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large-
N theory with Sp(2N) symmetry to study the fluctuations beyond mean field theory. At T=0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density
ns is found to have non-trivial
U/t dependence in this lattice system. We show that the transition temperature
Tc scales with the energy gap in the weak coupling limit but crosses over to a
t2/U scaling in the BEC limit, where phase fluctuations controlled by
ns determine
Tc. We also find, quite contrary to our expectations, that in the strong coupling limit, the large-
N theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-
N theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-
N approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-
N approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level.