We present an analytical treatment of a single ↓ atom within a Fermi sea of ↑ atoms, when the interaction is strong enough to produce a bound state, dressed by the Fermi sea. Our method makes use of a diagrammatic analysis, with the involved diagrams taking only into account at most two particle-hole pairs excitations. The agreement with existing Monte-Carlo results is excellent. In the BEC limit our equation reduces exactly to the Skorniakov and Ter-Martirosian equation. We present results when ↑ and ↓ atoms have different masses, which is of interest for experiments in progress.

Using a Fermi-liquid approach, we provide a comprehensive treatment of the current and current noise through a quantum dot whose low-energy behavior corresponds to an SU͑N͒ Kondo model, focusing on the case N = 4 relevant to carbon nanotube dots. We show that for general N, one needs to consider the effects of higher-order Fermi-liquid corrections even to describe low-voltage current and noise. We also show that the noise exhibits complex behavior due to the interplay between coherent shot noise, and noise arising from interaction-induced scattering events. We also treat various imperfections relevant to experiments, such as the effects of asymmetric dot-lead couplings.ductance and the shot noise as well as details on their derivation. In Sec. V, we summarize our main results for the conductance and shot noise of a SU͑N͒ Kondo quantum dot, and conclude. II. MODEL DESCRIPTION A. Kondo HamiltonianWe give here a compact synopsis of the quantum-dot model we study and how it gives rise to Kondo physics. The dot connected to the leads is described by the following Anderson Hamiltonian 29 . ͑1͒ c L/R,k is the annihilation operator for an electron of spin =1...N and energy k = បv F k ͑measured from the Fermi energy F ͒ confined on the left/right lead. d is the electron operator of the dot and n = d † d the corresponding density. U denotes the charging energy, d the single-particle energy on the dot and t L/R denotes the tunneling-matrix elements from the dot to the left/right lead. The general case of asymmetric leads contacts is parametrized by t L = t cos , t R = t sin with = ͓0, / 2͔. = / 4 recovers the symmetric case. The rotation in the basis of leads electrons

We present a simple derivation of the energy formula found by Tan, relative to the single channel hamiltonian relevant for ultracold Fermi gases. This derivation is generalized to particles with different masses, to arbitrary mixtures, and to two-dimensional space. We show how, in a field theoretical approach, the 1/k 4 tail in the momentum distribution and the energy formula arise in a natural way. As a specific example, we consider quantitative calculations of the energy, from different formulas within the ladder diagrams approximation in the normal state. The comparison of the results provides an indication on the quality of the approximation.

We present a field theoretic method for the calculation of the second and third virial coefficients b2 and b3 of 2-species fermions interacting via a contact interaction. The method is mostly analytic. We find a closed expression for b3 in terms of the 2 and 3-body T-matrices. We recover numerically, at unitarity, and also in the whole BEC-BCS crossover, previous numerical results for the third virial coefficient b3

We show that the study of the collective oscillations in a harmonic trap provides a very sensitive test of the equation of state of a Fermi gas near a Feshbach resonance. Using a scaling approach, whose high accuracy is proven by comparison with exact hydrodynamic solutions, the frequencies of the lowest compressional modes are calculated at T=0 in terms of a dimensionless parameter characterizing the equation of state. The predictions for the collective frequencies, obtained from the equations of state of mean-field BCS theory and of recent Monte Carlo calculations, are discussed in detail.

We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length a B in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length a F , and the already known result a B = 0.60 a F is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three bbb and four bbbb resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion bbf, two bosons plus two fermions bf ↑ bf ↓ , and three bosons plus one fermion bbbf.

We present an exact diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length aB in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF and the already known result aB = 0.60 aF is recovered exactly. Then we make use of our approach to obtain new results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound state energies for three bbb and four bbbb resonantly interacting bosons in two dimensions. We consider also the case of resonant interaction between fermions and bosons and we obtain the exact bound state energies of two bosons plus one fermion bbf , two bosons plus two fermions bf ↑ bf ↓ , and three bosons plus one fermion bbbf .

The authors have identified two errors in their recent Letter [1]. Although the results for the SU(4) symmetry [and more generally for SUðNÞ] should be modified, the predictions for the SU(2) case are still correct. Moreover, the conclusion that scattering produces a net shot noise increase for SU(4) remains valid. Apart from these two errors the core of the Letter, in particular, the theoretical framework, is unchanged.