Abstract:
I begin by briefly reviewing various experimental results on the pseudogap phenomena in underdoped cuprates. I argue that, taken together, all of these lead to a picture of singlet pairing above $T_c$. I then explore the idea that the pseudogap is a normal state precursor of the superconducting gap due to local, dynamic pairing correlations in a state without long range phase coherence. Early work on simple model systems which exhibit pseudogap anomalies in the normal state of 2D superconductors in a low density, small pair size regime is reviewed and critically re-examined in view of more recent developments. I also describe recent studies of how the underlying d-wave superconducting ground state affects the anisotropy of the pseudogap and the destruction of the Fermi surface.

Abstract:
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of \emph{non-relativistic} quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\zeta(\omega)$ and $\eta(\omega)$ respectively, to derive exact, non-perturbative results. Our results include: a microscopic connection between the shear viscosity $\eta$ and the normal fluid density $\rho_n$; sum rules for $\zeta(\omega)$ and $\eta(\omega)$ and their evolution through the BCS-BEC crossover; universal high-frequency tails for $\eta(\omega)$ and the dynamic structure factor $S({\bf q}, \omega)$. We use our sum rules to show that, at unitarity, $\zeta(\omega)$ is identically zero and thus relate $\eta(\omega)$ to density-density correlations. We predict that frequency-dependent shear viscosity $\eta(\omega)$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.

Abstract:
We show that there are qualitative differences between the temperature dependence of the spin and charge correlations in the normal state of the 2D attractive Hubbard model using quantum Monte Carlo simulations. The one-particle density of states shows a pseudogap above $\tc$ with a depleted $N(0)$ with decreasing $T$. The susceptibility $\cs$ and the low frequency spin spectral weight track $N(0)$, which explains the spin-gap scaling: $1/T_1T \sim \cs(T)$. However the charge channel is dominated by collective behavior and the compressibility $dn/d\mu$ is $T$-independent. This anomalous ``spin-charge separation'' is shown to exist even at intermediate $|U|$ where the momentum distribution $n(\bk)$ gives evidence for degenerate Fermi system.

Abstract:
The evolution of the ground state and the excitation spectrum of the two and three dimensional attractive Hubbard model is studied as the system evolves from a Cooper pair regime for weak attraction to a composite boson regime for a strong attraction.

Abstract:
Recent experiments on a $\2d$ Fermi gas find an undamped breathing mode at twice the trap frequency over a wide range of parameters. To understand this seemingly scale-invariant behavior in a system with a scale, we derive two exact results valid across the entire BCS-BEC crossover at all temperatures. First, we relate the shift of the mode frequency from its scale-invariant value to $\gamma_d \equiv (1+2/d)P-\rho(\partial P/\partial\rho)_s$ in $d$ dimensions. Next, we relate $\gamma_d$ to dissipation via a new low-energy bulk viscosity sum rule. We argue that $\2d$ is special, with its logarithmic dependence of the interaction on density, and thus $\gamma_2$ is small in both the BCS and BEC regimes, even though $P - 2\varepsilon/d$, sensitive to the dimer binding energy that breaks scale invariance, is not.

Abstract:
We construct a phenomenological superfluid Fermi liquid theory for a two-dimensional d-wave superconductor on a square lattice, and study the effect of quasiparticle interactions on the superfluid density. Using simple models for the dispersion and the Landau interaction function, we illustrate the deviation of these results from those for the isotropic superfluid. This allows us to reconcile the value and doping dependence of the superfluid density slope at low temperature obtained from penetration depth measurements, with photoemission data on nodal quasiparticles.

Abstract:
We argue based on theoretical considerations and analysis of experimental data that quasiparticle excitations near the nodes determine the low temperature properties in the superconducting state of cuprates. Quantum effects of phase fluctuations are shown to be quantitatively important, but thermal effects are small for $T \ll T_c$. An anisotropic superfluid Fermi liquid phenomenology is presented for the effect of quasiparticle interactions on the temperature and doping dependence of the low $T$ penetration depth.

Abstract:
We show that the universal $1/k^4$ tail in the momentum distribution of dilute Fermi gases implies that the spectral function $A(\kk,\omega)$ must have weight below the chemical potential for large momentum $k \gg k_F$, with observable consequences in RF spectroscopy experiments. We find that this incoherent spectral weight is centered about $\omega \simeq - \epsilon(\kk)$ in a range of energies of order $v_F k$. This "bending back" in the dispersion, while natural for superfluids, is quite surprising for normal gases. This universal structure is present in the hard-sphere gas as well as the Fermi liquid ground state of the highly imbalanced, attractive gas near unitarity. We argue that, even in the BCS superfluid, this bending back at large $k$ is dominated by interaction effects which do not reflect the pairing gap.

Abstract:
The crossover from weak coupling Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of tightly bound pairs, as a function of the attractive interaction in Fermi systems, has long been of interest to theoretical physicists. The past decade has seen a series of remarkable experimental developments in ultracold Fermi gases that has realized the BCS-BEC crossover in the laboratory, bringing with it fresh new insights into the very strongly interacting unitary regime in the middle of this crossover. In this review, we start with a pedagogical introduction to the crossover and then focus on recent progress in the strongly interacting regime. While our focus is on new theoretical developments, we also describe three key experiments that probe the thermodynamics, transport and spectroscopy of the unitary Fermi gas. We discuss connections between the unitary regime and other areas of physics -- quark-gluon plasmas, gauge-gravity duality and high temperature superconductivity -- and conclude with open questions about strongly interacting Fermi gases.

Abstract:
We investigate the crossover from BCS pairing to molecular BEC in an atomic gas with two fermion species with masses $m_\up \ne m_\dn$ tuned through a Feshbach resonance. We present results for the T=0 equation of state as a function of the scattering length including the effects of Gaussian fluctuations about the mean field ground state. We compute the ground state energy as a function of $m_\up/m_\dn$ at unitarity and find excellent agreement with the quantum Monte Carlo result for $m_\up/m_\dn = 6.67$ for a $^{40}$K-$^6$Li mixture. We show that the dimer scattering length in the BEC limit as a function of $m_\up/m_\dn$ compares well with exact four-body results of Petrov {\it et al}. We also derive the condition for trapping frequencies to obtain an unpolarized gas in a harmonic trap.