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Physical Review A

One-dimensional harmonically confined SU(N) fermions

M. C. Gordillo, Ferran Mazzanti, Jordi Boronat

Published Aug 5, 2019• 5 citations
Publisher

Abstract

We study the momentum distributions and spatial correlations of few harmonically confined SU($N$) fermions using quantum Monte Carlo methods. In our study, we vary the spin degeneracy $N$ from 2 to 6 and the total number of particles from 6 to 18. Only balanced mixtures, with the same number of atoms per spin type, and repulsive unlike-spin contact interactions are considered. Going from $N=2$ to $N=6$, with the same occupancy of each spin state, we observe an increase of atom-atom correlations. This effect is particularly significant in the momentum distributions, which show fatter tails at large $k$ ($k\ensuremath{\sigma}\phantom{\rule{0.16em}{0ex}}>\phantom{\rule{0.16em}{0ex}}5, \ensuremath{\sigma}$ being the oscillator length) when $N$ grows, in agreement with experimental findings. Those tails also show the expected ${k}^{\ensuremath{-}4}$ decay related to the Tan contact for different values of the spin degeneracy. According to our results, the local spin ordering and the spin-spin correlations are mainly determined by $N$ via the Pauli exclusion principle, with minor influences from the particle-particle interactions, irrespective of the total number of confined atoms.

DOI
10.1103/physreva.100.023603

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We study the momentum distributions and spatial correlations of few harmonically confined SU(NNN) fermions using quantum Monte Carlo methods. In our study, we vary the spin degeneracy NNN from 2 to 6 and the total number of particles from 6 to 18. Only balanced mixtures, with the same number of atoms per spin type, and repulsive unlike-spin contact interactions are considered. Going from N=2N=2N=2 to N=6N=6N=6, with the same occupancy of each spin state, we observe an increase of atom-atom correlations. This effect is particularly significant in the momentum distributions, which show fatter tails at large kkk (k\ensuremathσ>5,\ensuremathσk\ensuremath{\sigma}\phantom{\rule{0.16em}{0ex}}>\phantom{\rule{0.16em}{0ex}}5, \ensuremath{\sigma}k\ensuremathσ>5,\ensuremathσ being the oscillator length) when NNN grows, in agreement with experimental findings. Those tails also show the expected k\ensuremath−4{k}^{\ensuremath{-}4}k\ensuremath−4 decay related to the Tan contact for different values of the spin degeneracy. According to our results, the local spin ordering and the spin-spin correlations are mainly determined by NNN via the Pauli exclusion principle, with minor influences from the particle-particle interactions, irrespective of the total number of confined atoms.
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Abstract Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box. We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles. Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps). In non-interacting systems, the characteristic length scale of the oscillations is set by ( 2 k F ) − 1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations of period ( 4 k F ) − 1 . Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix. While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number. We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface. In addition, we determine the spatially varying on-site density–density correlation, which in turn yields Tan’s contact density and, upon integration, Tan’s contact. As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.
A universal ${k}^{\ensuremath{-}4}$ decay of the large-momentum tails of the momentum distribution, fixed by Tan's contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. Here we consider a repulsive multicomponent Fermi gas under harmonic confinement, as in the experiment of G. Pagano et al. [Nat. Phys. 10, 198 (2014)], realizing a gas with tunable $\text{SU}(\ensuremath{\kappa})$ symmetry. We exploit an exact solution at infinite repulsion to show a direct correspondence between the value of the Tan's contact for each of the $\ensuremath{\kappa}$ components of the gas and the Young tableaux for the SN{S}_{N}SN​ permutation symmetry group identifying the magnetic structure of the ground state. This opens a route for the experimental determination of magnetic configurations in cold atomic gases, employing only standard (spin-resolved) time-of-flight techniques. Combining the exact result with matrix-product-state simulations, we obtain the Tan's contact at all values of repulsive interactions. We show that a local-density approximation (LDA) on the Bethe-ansatz equation of state for the homogeneous mixture is in excellent agreement with the results for the harmonically confined gas. At strong interactions, the LDA predicts a scaling behavior of the Tan's contact. This provides a useful analytical expression for the dependence on the number of fermions, number of components, and interaction strength. Moreover, using a virial approach, we study the Tan's contact behavior at high temperature and in the limit of infinite interactions, and we show that it increases with the temperature and the number of components. At zero temperature, we predict that the weight of the momentum distribution tails increases with interaction strength and the number of components if the population per component is kept constant. This latter property was experimentally observed in G. Pagano et al. [Nat. Phys. 10, 198 (2014)].
Ultracold fermions with SU(N >2) symmetry offer a unique opportunity to study quantum dynamics and interaction effects in the large spin systems that have no analogue in condensed matter physics. In this poster, we present a set of experiments on the measurement of contact parameters and collective excitations with a SU(N) Fermi gas of 173Yb atoms. First, we explore the short-range interaction effect via Tan’s contact parameters in a multi-component Fermi gas with SU(N >2) symmetry. The s-wave contact parameter is experimentally measured by recording the high-momentum tail of weakly interacting fermions. For a tunable number of spin component N with a fixed number of atoms per component, we verify the linear increase in the contact with N providing experimental confirmation of SU(N) interactions. Furthermore, we explore the momentum distribution of SU(N) fermions at the low momentum regime. Next, we measure collective excitations of a harmonically trapped twodimensional SU(N) Fermi gas. Various collective modes are investigated with a tunable number of spin component N showing a decrease in the ratio of quadruple and dipole mode with N. Our work will pave the way for the experimental study of interacting SU(N) Fermi gases with large spin.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.