We study representations of the Schr\"odinger algebra in terms of operators in nonrelativistic conformal field theories. We prove a correspondence between primary operators and eigenstates of few-body systems in a harmonic potential. Using the correspondence we compute analytically the energy of fermions at unitarity in a harmonic potential near two and four spatial dimensions. We also compute the energy of anyons in a harmonic potential near the bosonic and fermionic limits.Comment: 26 pages, 9 figures; added a comment on the convergence of epsilon expansion

The recent experimental realization of strongly imbalanced mixtures of ultracold atoms opens new possibilities for studying impurity dynamics in a controlled setting. In this paper, we discuss how the techniques of atomic physics can be used to explore new regimes and manifestations of Anderson's orthogonality catastrophe (OC), which could not be accessed in solid-state systems. Specifically, we consider a system of impurity atoms, localized by a strong optical-lattice potential, immersed in a sea of itinerant Fermi atoms. We point out that the Ramsey-interference-type experiments with the impurity atoms allow one to study the OC in the time domain, while radio-frequency (RF) spectroscopy probes the OC in the frequency domain. The OC in such systems is universal, not only in the long-time limit, but also for all times and is determined fully by the impurity-scattering length and the Fermi wave vector of the itinerant fermions. We calculate the universal Ramsey response and RF-absorption spectra. In addition to the standard power-law contributions, which correspond to the excitation of multiple particle-hole pairs near the Fermi surface, we identify a novel, important contribution to the OC that comes from exciting one extra particle from the bottom of the itinerant band. This contribution gives rise to a nonanalytic feature in the RF-absorption spectra, which shows a nontrivial dependence on the scattering length, and evolves into a true power-law singularity with the universal exponent 1=4 at the unitarity. We extend our discussion to spin-echo-type experiments, and show that they probe more complicated nonequilibirum dynamics of the Fermi gas in processes in which an impurity switches between states with different interaction strength several times; such processes play an important role in the Kondo problem, but remained out of reach in the solid-state systems. We show that, alternatively, the OC can be seen in the energy-counting statistics of the Fermi gas following a sudden quench of the impurity state. The energy distribution function, which can be measured in time-of-flight experiments, exhibits characteristic power-law singularities at low energies. Finally, systems in which the itinerant fermions have two or more hyperfine states provide an even richer playground for studying nonequilibrium impurity physics, allowing one to explore the nonequilibrium OC and even to simulate quantum transport through nanostructures. This provides a previously missing connection between cold atomic systems and mesoscopic quantum transport.

We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over ǫ = 4 − d, where d is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be µ/εF = 1 2 ǫ 3/2 + 1 16 ǫ 5/2 ln ǫ − 0.0246 ǫ 5/2 + · · · and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be ∆/µ = 2ǫ −1 −0.691+· · · . The minimum of the fermion dispersion curve is located at |p| = (2mε0) 1/2 where ε0/µ = 2 + O(ǫ). Extrapolation to d = 3 gives results consistent with Monte Carlo simulations.

We investigate a two-species Fermi gas in which one species is confined in a two-dimensional plane (2D) or one-dimensional line (1D) while the other is free in the three-dimensional space (3D). We discuss the realization of such a system with the interspecies interaction tuned to resonance. When the mass ratio is in the range 0.0351

We experimentally investigate the mix-dimensional scattering occurring when the collisional partners live in different dimensions. We employ a binary mixture of ultracold atoms and exploit a species-selective 1D optical lattice to confine only one atomic species in 2D. By applying an external magnetic field in proximity of a Feshbach resonance, we adjust the free-space scattering length to observe a series of resonances in mixed dimensions. By monitoring 3-body inelastic losses, we measure the magnetic field values corresponding to the mix-dimensional scattering resonances and find a good agreement with the theoretical predictions based on simple energy considerations. Degenerate atomic gases have provided quantum systems with unprecedented possibilities of manipulation and control, achieved by combining magnetic and optical potentials as well as scattering resonances. The capability to model and control tightly confining potentials sparked the experimental investigation on quantum systems of reduced dimensionality, since particles can be forced to occupy a single quantum level along specific directions. Spectacular achievements, such as the observation of the BKT crossover [1] in 2D and the realization of Tonks-Girardeau gases [2] in 1D, confirmed the importance of quantum gases as testbench for fundamental low-energy physical phenomena. Moreover, low dimensional ultracold atomic gases show further peculiar scattering properties leading to the appearance of confinement-induced resonances depending on the dimensionality of the system [3-6]. Interestingly, while much of the work done so far deals with well-defined dimensionality, systems composed of interacting parts living in different dimensions have received little attention and, besides recent theoretical analysis [7,8], experimental investigation is still lacking. Such mix-dimensional systems are relevant in several physical domains, ranging from cosmology to condensed matter physics. In brane theory, for example, particles and fields are confined to the ordinary 3D space and interact with gravitons that can propagate in extra dimensions [9].

Quantum chromodynamics (QCD) at finite temperature (T ), baryon chemical potential (µB) and isospin chemical potential (µI) is studied in the strong coupling limit on a lattice with staggered fermions. With the use of large dimensional expansion and the mean field approximation, we derive an effective action written in terms of the chiral condensate and pion condensate as a function of T , µB and µI. The phase structure in the space of T and µB is elucidated, and simple analytical formulas for the critical line of the chiral phase transition and the tricritical point are derived. The effects of a finite quark mass (m) and finite µI on the phase diagram are discussed. We also investigate the phase structure in the space of T , µI and m, and clarify the correspondence between color SU(3) QCD with finite isospin density and color SU(2) QCD with finite baryon density. Comparisons of our results with those from recent Monte Carlo lattice simulations on finite density QCD are given.

We study a system of spinless fermions in two dimensions with a short-range interaction fine-tuned to a p-wave resonance. We show that three such fermions form an infinite tower of bound states of orbital angular momentum ℓ=±1 and their binding energies obey a universal doubly exponential scaling E(3)((n))∝exp(-2e(3πn/4+θ)) at large n. This "super Efimov effect" is found by a renormalization group analysis and confirmed by solving the bound state problem. We also provide an indication that there are ℓ=±2 four-body resonances associated with every three-body bound state at E(4)((n))∝exp(-2e(3πn/4+θ-0.188)). These universal few-body states may be observed in ultracold atom experiments and should be taken into account in future many-body studies of the system.

The character change of a superfluid state due to the variation of the attractive force is investigated in the relativistic framework with a massive fermion. Two crossovers are found. One is a crossover from the usual BCS state to the Bose-Einstein condensation (BEC) of bound fermion pairs. The other is from the BEC to the relativistic Bose-Einstein condensation (RBEC) of nearly massless bound pairs where antiparticles as well as particles dominate the thermodynamics. Possible realization of the BEC and RBEC states in the quark matter is also pointed out. PACS numbers: 74.20.Fg, 03.75.Nt, 11.10.Wx, Recently, new superfluid states in the ultracold gas of fermionic alkali atoms ( 40 K, 6 Li) were realized [1]. Using the Feshbach resonance, the long-standing idea of the crossover from the BCS state to the Bose-Einstein condensation (BEC) [2,3,4] has been extensively examined. The basic concept of the BCS-BEC crossover is as follows: As long as the attractive interaction between fermions is weak, the system exhibits the superfluidity characterized by the energy gap in the BCS mechanism. On the other hand, if the attractive interaction is strong enough, the fermions first form bound molecules (bosons). Then they start to condense into the bosonic zero-mode at some critical temperature. These two situations are smoothly connected without the phase transition.The possible realization of the BCS-BEC crossover in various systems has been theoretically investigated. These include the liquid 3 He [3], the trapped alkali atoms [5], and the nuclear matter [6]. One of the most striking features of the crossover is that the critical temperature in the BEC region is independent of the coupling for the attraction between fermions. This is because the increase of the coupling only affects the internal structure of the bosons, while the critical temperature is determined by the boson's kinetic energy. Thus, the critical temperature reaches a ceiling for the large coupling as long as the binding effect on the boson mass can be neglected. Even in the nuclear matter where the interaction is relatively strong, the binding energy of the deuteron is much smaller than the nucleon mass. This fact allows us to work within a nonrelativistic framework for describing such a crossover.It is interesting to ask how the situation changes in relativistic systems where the binding effect can not be neglected. The color superconducting phase in the dense quark matter [7,8] and the pion superfluid phase at finite isospin density [9] would be examples. In this article, we will show that there could be two crossovers in the relativistic superfluids. One is the ordinary BCS-BEC * E-mail: nishida@nt.phys.s.u-tokyo.ac.jp † E-mail: abuki@yukawa.kyoto-u.ac.jp crossover, where the critical temperature in the BEC region would not plateau because of the relativistic effect. The other is from the BEC state to the novel state, the relativistic BEC (RBEC), where the critical temperature increases to the order of the Fermi energy. In order to explore the BCS-BEC a...