Unitary Limit Research Articles

High-temperature expansion for interacting fermions

We present a general method for the high-temperature expansion of the self-energy of interacting particles. Though the method is valid for fermions and bosons, we illustrate it for spin one half fermions interacting via a zero range potential, in the Bose Einstein Condensate - Bardeen Cooper Schrieffer (BEC-BCS) crossover. The small parameter of the expansion is the fugacity z. Our results include terms of order z and z^2, which take into account respectively two and three body correlations. We give results for the high temperature expansion of Tan's contact at order z^3 in the whole BEC-BCS crossover. We apply our method to calculate the spectral function at the unitary limit. We find new structures which were overlooked by previous approaches, which included only two body correlations. This shows that including three-body correlations can play an important role in the structures of the spectral function.

Transition and Damping of Collective Modes in a Trapped Fermi Gas between BCS and Unitary Limits near the Phase Transition.

We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion’s scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.

Ab initio Lattice Results for Fermi Polarons in Two Dimensions.

We investigate the attractive Fermi polaron problem in two dimensions using nonperturbative Monte Carlo simulations. We introduce a new Monte Carlo algorithm called the impurity lattice Monte Carlo method. This algorithm samples the path integral in a computationally efficient manner and has only small sign oscillations for systems with a single impurity. As a benchmark of the method, we calculate the universal polaron energy in three dimensions in the scale-invariant unitarity limit and find agreement with published results. We then present the first fully nonperturbative calculations of the polaron energy in two dimensions and density correlations between the impurity and majority particles in the limit of zero-range interactions. We find evidence for a smooth crossover transition from fermionic quasiparticle to molecular state as a function of the interaction strength.

Unitary limit in crossed Andreev transport

One of the most promising approaches for generating spin- and energy-entangled electron pairs is splitting a Cooper pair into the metal through spatially separated terminals. Utilizing hybrid systems with the energy-dependent barriers at the superconductor/normal metal (NS) interfaces, one can achieve a practically 100% efficiency outcome of entangled electrons. We investigate a minimalistic one-dimensional model comprising a superconductor and two metallic leads and derive an expression for an electron-to-hole transmission probability as a measure of splitting efficiency. We find the conditions for achieving 100% efficiency and present analytical results for the differential conductance and differential noise.

RKKY interaction and local density of states for a triangular triple quantum dot system

By means of the numerical renormalization group technique, we study the local density of states (LDOS) for a triangular triple quantum dot system, with two dots connected in parallel to the conduction leads. We find the location of the Ruderman–Kittel–Kasuya–Yosida (RKKY) peak identified in the LDOS could be illustrated as JRKKY=aΓ2/U+bt22/U, with U being the on-site Coulomb repulsion, Γ the dot-lead coupling, and t2 the hopping between the connected dots and the side dot. When the hopping between two connected dots t1 turns on, the spectrum weight of the RKKY peaks decreases due to the competition between the direct and the RKKY interactions. As t1 increases beyond a critical point t1c, two connected dots form a spin singlet, and decouple from both the side dot and the conduction leads, thus the Kondo and RKKY peaks could not be found. For t1<t1c, the conductance reaches to the unitary limit, while for t1≥t1c, it drops to zero.

Impurity in a Bose-Einstein condensate: Study of the attractive and repulsive branch using quantum Monte Carlo methods

We investigate the properties of an impurity immersed in a dilute Bose gas at zero temperature using quantum Monte-Carlo methods. The interactions between bosons are modeled by a hard sphere potential with scattering length $a$, whereas the interactions between the impurity and the bosons are modeled by a short-range, square-well potential where both the sign and the strength of the scattering length $b$ can be varied by adjusting the well depth. We characterize the attractive and the repulsive polaron branch by calculating the binding energy and the effective mass of the impurity. Furthermore, we investigate structural properties of the bath, such as the impurity-boson contact parameter and the change of the density profile around the impurity. At the unitary limit of the impurity-boson interaction, we find that the effective mass of the impurity remains smaller than twice its bare mass, while the binding energy scales with $\hbar^2n^{2/3}/m$, where $n$ is the density of the bath and $m$ is the common mass of the impurity and the bosons in the bath. The implications for the phase diagram of binary Bose-Bose mixtures at small concentrations are also discussed.

One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries. II. N Particles

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly–unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.

One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the $N$ particle case.

Investigation of thennΛbound state in pionless effective theory

The possibility of an $nn\Lambda$ bound state is investigated in the framework of pionless effective field theory at leading order. A system of coupled integral equations are constructed in the spin-isospin basis, of which numerical solutions are investigated. In particular, we make use of the limit cycle behavior, i.e., cyclic singularities of coupled integral equations of the system, which would be associated with the formation of a three-body bound state, so-called the Efimov state, in the unitary limit. Furthermore, we find that, when the sharp momentum cutoff introduced in the integral equations is taken significantly larger than the hard scale of the effective theory, the coupling of a three-body contact interaction becomes cyclically singular indicating the onset of Efimov-like bound state formation. However, the paucity of empirical information to determine the parameters of the theory precludes a definitive conclusion on the existence of such a bound state. As a simple test of the feasibility of the $nn\Lambda$ bound system in nature, we explore the cutoff dependence of the theory, and uncertainties of the present study are discussed as well.

Four-boson bound states from a functional renormalization group

We use the functional renormalisation group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector and we introduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show qualitatively similar behaviour to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.

Isospin quartic term in the kinetic energy of neutron-rich nucleonic matter

The energy of a free gas of neutrons and protons is well known to be approximately isospin parabolic with a negligibly small quartic term of only $0.45$ MeV at the saturation density of nuclear matter $\rho_0=0.16/\rm{fm}^3$. Using an isospin-dependent single-nucleon momentum distribution including a high (low) momentum tail (depletion) with its shape parameters constrained by recent high-energy electron scattering and medium-energy nuclear photodisintegration experiments as well as the state-of-the-art calculations of the deuteron wave function and the equation of state of pure neutron matter near the unitary limit within several modern microscopic many-body theories, we show for the first time that the kinetic energy of interacting nucleons in neutron-rich nucleonic matter has a significant quartic term of $7.18\pm2.52\,\rm{MeV}$. Such a large quartic term has significant ramifications in determining the equation of state of neutron-rich nucleonic matter using both terrestrial and astrophysical observables.

Omega-Omega interaction from 2+1-flavor lattice quantum chromodynamics

We investigate the interaction between $\Omega$ baryons in the $^1S_0$ channel from 2+1 flavor lattice QCD simulations. On the basis of the HAL QCD method, the $\Omega\Omega$ potential is extracted from the Nambu-Bethe-Salpeter wave function calculated on the lattice by using the PACS-CS gauge configurations with the lattice spacing $a\simeq 0.09$ fm, the lattice volume $L\simeq 2.9$ fm and the quark masses corresponding to $m_\pi \simeq 700$ MeV and $m_\Omega \simeq 1970$ MeV. The $\Omega\Omega$ potential has a repulsive core at short distance and an attractive well at intermediate distance. Accordingly, the phase shift obtained from the potential shows moderate attraction at low energies. Our data indicate that the $\Omega\Omega$ system with the present quark masses may appear close to the unitary limit where the scattering length diverges.

Nonlocal quantum gravity and M-theory

We construct a ultraviolet completion of the bosonic sector of 11-dimensional supergravity motivated by string field theory. We start from a general class of theories characterized by an entire nonpolynomial form factor which allows one to avoid new poles in the propagator and improves the high-energy behavior of the loops amplitudes. Comparing these models with effective string field theory, a unique form factor is selected out. In view of this, we modify 10-dimensional supergravity and finally get a ultraviolet completion of 11-dimensional supergravity by an oxidation process. The result is a candidate for a finite and unitary particle-field limit of M-theory.

Interferometric resonance signatures of Majorana bound states

We calculate the current noise power spectrum in a nanoscopic interferometer consisting of a Majorana bound state (MBS) and a localized spin. We show that for large voltage (though less than the superconducting gap) several strong resonance peaks appear at frequencies that depend on the Zeeman splitting of the localized spin and on its tunneling to the localized spin. We also evaluate the differential conductance and find the unitary limit peak at zero voltage as well as peaks at voltages corresponding to the resonances. We propose that detection of the resonances and related peaks in the differential conductance provide as strong support for the presence of an MBS.

Spectrum of three-body bound States in a finite volume.

The spectrum of a bound state of three identical particles with a mass m in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by ΔE=c(κ^{2}/m)(κL)^{-3/2}|A|^{2}exp(-2κL/sqrt[3]), where κ is the bound-state momentum, L is the box size, |A|^{2} denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and c is a numerical constant. The formula is valid for κL≫1.

Monte Carlo Calculations for Fermi Gases in the Unitary Limit with a Zero-Range Interaction

An ultracold Fermi gas with a zero-range attractive potential in the unitary limit is investigated using variational and diffusion Monte Carlo methods. Previous calculations have used a finite range interactions and extrapolate the results to zero-range. Here we extend the quantum Monte Carlo method to directly use a zero-range interaction without extrapolation. We employ a trial wave function with the correct boundary conditions, and modify the sampling procedures to handle the zero-range interaction. The results are reliable and have low variance.

Absence of a quantum limit to charge diffusion in bad metals

Good metals are characterised by diffusive transport of coherent quasi-particle states and the resistivity is much less than the Mott-Ioffe-Regel (MIR) limit, $\frac{ha}{e^{2}}$, where $a$ is the lattice constant. In bad metals, such as many strongly correlated electron materials, the resistivity exceeds the Mott-Ioffe-Regel limit and the transport is incoherent in nature. Hartnoll, loosely motivated by holographic duality (AdS/CFT correspondence) in string theory, recently proposed a lower bound to the charge diffusion constant, $D \gtrsim \hbar v_{F}^{2}/(k_{B}T)$, in the incoherent regime of transport, where $v_F$ is the Fermi velocity and $T$ the temperature. Using dynamical mean field theory (DMFT) we calculate the charge diffusion constant in a single band Hubbard model at half filling. We show that in the strongly correlated regime the Hartnoll's bound is violated in the crossover region between the coherent Fermi liquid region and the incoherent (bad metal) local moment region. The violation occurs even when the bare Fermi velocity $v_F$ is replaced by its low temperature renormalised value, $v_F^*$.The bound is satisfied at all temperatures in the weakly and moderately correlated systems as well as in strongly correlated systems in the high temperature region where the resistivity is close to linear in temperature. Our calculated charge diffusion constant, in the incoherent regime of transport, also strongly violates a proposed quantum limit of spin diffusion, $D_{s} \sim 1.3 \hbar/m$, where $m$ is the fermion mass, experimentally observed and theoretically calculated in a cold degenerate Fermi gas in the unitary limit of scattering.

Superfluid amplitude fluctuations aboveTcin a unitary Fermi gas

We study the transport properties of a Fermi gas with strong attractive interactions close to the unitary limit. In particular, we compute the spin diffusion lifetime of the Fermi gas due to superfluid fluctuations above the BCS transition temperature $T_c$. To calculate the spin diffusion lifetime we need the scattering amplitudes. The scattering amplitudes are dominated by the superfluid fluctuations at temperatures just above $T_c$. The normal scattering amplitudes are calculated from the Landau parameters. These Landau parameters are obtained from the local version of the induced interaction model for computing Landau parameters. We also calculate the leading order finite temperature correction to the diffusion lifetime. A calculation of the spin diffusion coefficient is presented in the end. Upon choosing a proper value of $F_0^a$, we are able to present a good match between the theoretical result and the experimental measurement which indicates the presence of the superfluid fluctuations near $T_c$.

Eilenberger theory for nuclear magnetic relaxation rate in superconducting vortex lattice state

On the basis of the Eilenberger theory, spatial variation of the local NMR relaxation rate ${T}_{1}^{\ensuremath{-}1}$ is quantitatively estimated in the vortex lattice state, to clarify the differences between the $s$-wave and the $d$-wave superconductors. We study the temperature and the magnetic field dependencies of ${T}_{1}^{\ensuremath{-}1}$ inside and outside of the vortex core, including influences of nonmagnetic impurity scatterings in the Born limit and in the unitary limit. These results are helpful to detect detailed characters of local electronic structures in the vortex lattice states via site-selective NMR experiments.

Anomalous Phonon Dispersion of an Ultracold $$^6\mathrm{{Li}}$$ 6 Li – $$^{40}\mathrm{{K}}$$ 40 K Mixture in a Square Optical Lattice

A necessary condition for the damping of the long-wavelength excitations of the superfluid phase (referred to as superfluid phonons) due to the three-particle process is to have an anomalous phonon dispersion. The existence of anomalous phonon dispersion has been confirmed in superfluid \(^4\mathrm {He}\). There are no experimental data suggesting that this phenomenon exists in superfluid Fermi gases. To the best of our knowledge, the existence of anomalous dispersion has been theoretically predicted only in atomic spin balanced Fermi gas close to the unitarity limit. The numerical results reported here suggest that the anomalous long-wavelength dispersion can be realized in mass and spin imbalanced atomic Fermi gases away from the unitary limit. In particular, the numerical solution of the Bethe–Salpeter equation in a weak-coupling regime shows that the long-wavelength part of the collective-mode dispersion of the superfluid Fulde–Ferrell phase of a mixture of population-imbalanced Lithium-6 and Potassium-40 atoms in a square lattice at some values of polarization, interacting strength and temperature initially bends upward before bending over.