The
ϵ expansion (expansion around four spatial dimensions) developed by Nishida and Son for a cold Fermi gas with infinite scattering length is extended to finite scattering length to study the Bose-Einstein condensate (BEC) to BCS crossover. A resummation of higher-order logarithms and a suitable extension of fermion coupling in
d dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}$, is computed to next-to-leading order in the
ϵ expansion as a function of $\ensuremath{\eta}=1∕(a{k}_{F})$, where
a is the scattering length and
kF is the Fermi momentum in a noninteracting system. Near the unitarity limit $\ensuremath{\mid}\ensuremath{\eta}\ensuremath{\mid}\ensuremath{\rightarrow}0$, we found $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.475\ensuremath{-}0.707\ensuremath{\eta}\ensuremath{-}0.5{\ensuremath{\eta}}^{2}$. Near the BEC limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.062∕\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{2}$, while near the BCS limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=1+0.707∕\ensuremath{\eta}$. Overall good agreement with quantum Monte Carlo results is found.
We study a weak interaction quench in a three-dimensional Fermi gas. We first show that, under some general assumptions on time-dependent perturbation theory, the perturbative expansion of the long-wavelength structure factor
S(q) is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular,
S(q) does develop an analytical component $\ensuremath{\sim}\mathrm{const}+O({q}^{2})$ at $\mathbf{q}\ensuremath{\rightarrow}\mathbf{0}$, as implied by thermalization, but, in contrast, it maintains a nonanalytic part $\ensuremath{\sim}|\mathbf{q}|$ characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity corresponds to persisting power-law decaying density-density correlations, whereas thermalization would predict only an exponential decay. We next consider the case of a dilute gas, where one can obtain nonperturbative results in the interaction strength but at lowest order in the density. We find that in the steady state the momentum distribution jump at the Fermi surface remains finite, though smaller than in equilibrium, up to second order in
kFf0, where
f0 is the scattering length of two particles in the vacuum. Both results question the emergence of a finite length scale in the quench dynamics as expected by thermalization.
Owing to their high tunability, ultracold atoms provide versatile platforms for investigating interacting systems.Here, the quantum statistics of particles is controlled by the choice of atomic species and the space geometry by the operation of optical lattices.In addition, the strength of the interparticle interaction can be tuned from weak to strong through the Feshbach resonance.One of the ultracold atomic systems which take maximum advantage of this tunability is a two-component Fermi gas near the two-body resonance, which is referred to as a resonant Fermi gas.Because the interaction is characterized by only one parameter, the s-wave scattering length, the interaction effect appears only through the scattering length near the resonance.Thus, the resonant Fermi gas shows universal behaviors independent of the details of the interaction and forms an ideal example of a strongly correlated quantum many-body system.This universality and the high tunability connect the study of ultracold atoms to various other fields of physics.In particular, a resonant Fermi gas in the unitarity limit, where the scattering length is tuned to infinity in three dimensions, shows remarkable universality because of the absence of length scales in the interaction.The emergence of the conformal invariance due to the absence of length scales is one representative example of the universality in the unitarity limit.The strong coupling, which is associated with conformality, motivates studies of the thermodynamic properties and the transport properties.For example, the system achieves an almost "perfect fluid" in the unitarity limit, where the bulk viscosity vanishes and the shear viscosity has a remarkably small ratio to the entropy density.Although the strong coupling leads to intriguing transport properties, it makes the calculations of transport coefficients difficult.This thesis investigates the transport properties of two-component Fermi gases near the two-body resonance in two and three dimensions.We particularly focus on their transport coefficients, such as the bulk viscosity, the shear viscosity, and the thermal conductivity, for an arbitrary scattering length.In order to carry out reliable analysis in the strongly correlated regime, such as near the unitarity limit, we need non-perturbative methods.One of the nonperturbative methods is to use the conformal symmetry that emerges in the unitarity limit.The application of this symmetry-based approach to hydrodynamics revealed the vanishing bulk viscosity in the unitarity limit.Another non-perturbative approach is an expansion in terms of the fugacity, called the quantum virial expansion.In the high-temperature and low-density regime, this expansion is valid because the fugacity is small.First of all, we focus on the bulk viscosity, which characterizes the dissipation caused by an expansion of the fluid volume.The vanishing bulk viscosity in the unitarity limit is intuitively understood because the entropy does not change before and after an isotropic expansion of the i List of Publications Publications relevant to this thesis 1.