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arXiv:1502.01706 · Physical Review A

Strongly interacting quantum gases in one-dimensional traps

Li Yang, Liming Guan, Han Pu

Published Apr 29, 2015• 77 citations
arXivHTMLPDF

Abstract

Under second-order degenerate perturbation theory, we show that the physics of $N$ particles with arbitrary spin confined in a one-dimensional trap in the strongly interacting regime can be described by superexchange interaction. An effective spin-chain Hamiltonian (non-translationally-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translationally-invariant Heisenberg model, where a transition between Heisenberg antiferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished by two different methods: the first is based on their distinct responses to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distributions. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniques.

DOI
10.1103/physreva.91.043634

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Under second-order degenerate perturbation theory, we show that the physics of NNN particles with arbitrary spin confined in a one-dimensional trap in the strongly interacting regime can be described by superexchange interaction. An effective spin-chain Hamiltonian (non-translationally-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translationally-invariant Heisenberg model, where a transition between Heisenberg antiferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished by two different methods: the first is based on their distinct responses to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distributions. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniques.
We present a general form of the effective spin-chain model for strongly interacting atomic gases with an arbitrary spin in one-dimensional (1D) traps. In particular, for high-spin systems the atoms can collide in multiple scattering channels, and we find that the resulting form of spin-chain model generically follows the same structure as that of the interaction potentials. This is a unified form working for any spin, statistics (Bose or Fermi), and confinement potentials. We adopt the spin-chain model to reveal both the ferromagnetic (FM) and antiferromagnetic (AFM) magnetic orders for strongly interacting spin-1 bosons in 1D traps. We further show that by adding the spin-orbit coupling, the FM and AFM orders can be gradually destroyed and eventually the ground state exhibits universal spin structure and contacts that are independent of the strength of spin-orbit coupling.
Strongly interacting one-dimensional (1D) Bose--Fermi mixtures form a tunable XXZXXZXXZ spin chain. Within the spin-chain model developed here, all properties of these systems can be calculated from states representing the ordering of the bosons and fermions within the atom chain and from the ground-state wave function of spinless noninteracting fermions. We validate the model by means of an exact diagonalization of the full few-body Hamiltonian in the strongly interacting regime. Using the model, we explore the phase diagram of the atom chain as a function of the boson-boson (BB) and boson-fermion (BF) interaction strengths and calculate the densities, momentum distributions, and trap-level occupancies for up to 17 particles. In particular, we find antiferromagnetic (AFM) and ferromagnetic (FM) order and a demixing of the bosons and fermions in certain interaction regimes. We find, however, no demixing for equally strong BB and BF interactions, in agreement with earlier calculations that combined the Bethe ansatz with a local-density approximation.
The highly controllable ultracold atoms in a one-dimensional (1D) trap provide a new platform for the ultimate simulation of quantum magnetism. In this regard, the N\'eel antiferromagnetism and the itinerant ferromagnetism are of central importance and great interest. Here we show that these magnetic orders can be achieved in the strongly interacting spin-1/2 trapped Fermi gases with additional ppp-wave interactions. In this strong-coupling limit, the 1D trapped Fermi gas exhibits an effective Heisenberg spin XXZXXZXXZ chain in the anisotropic ppp-wave scattering channels. For a particular ppp-wave attraction or repulsion within the same species of fermionic atoms, the system displays ferromagnetic domains with full spin segregation or the antiferromagnetic spin configuration in the ground state. Such engineered magnetisms are likely to be probed in a quasi-1D trapped Fermi gas of 40K^{40}\mathrm{K}40K atoms with very close sss-wave and ppp-wave Feshbach resonances.
One-dimensional spinor gases with strong $\ensuremath{\delta}$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain---in particular antiferromagnetic spin order---may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.
One of quantum physics' fundamental, but largely unsolved, problems is the computation of the correlation functions in many-body systems. We address this problem in the case of one-dimensional spinor gases with repulsive contact interactions in the presence of a confining potential. We take advantage of the fact that in the strong-coupling limit, the wave function factorizes with the charge degrees of freedom expressed as a Slater determinant of spinless fermions and the spin sector described by a spin chain of Sutherland type with exchange coefficients that depend only on the trapping potential. This factorization is also present in the expressions for the correlation functions. Still, analytical and numerical investigations were hindered by the fact that the local exchange coefficients and the charge component of the correlators are expressed as $N\ensuremath{-}1$ multidimensional integrals, with NNN the number of particles, which are notoriously hard to compute using conventional methods. We introduce a different approach to calculating these integrals that is extremely simple, scales polynomially with the number of particles, and is several orders of magnitude faster than the previous methods reported in the literature. This allows us to investigate the static and dynamic properties, temperature dependence, and nonequilibrium dynamics for systems with a larger number of particles than previously considered and discover new phenomena. We show that contrary to natural expectations, the momentum distribution of strongly interacting trapped spinor gases becomes narrower as we increase the temperature and derive simple determinant representations for the correlators in the spin incoherent regime that is valid for both equilibrium and nonequilibrium situations. For homogeneous systems of impenetrable particles at zero temperature, we analytically compute the large distance asymptotics of the correlators, finding that the leading term of the asymptotic expansion is proportional to ${e}^{\ensuremath{-}x{k}_{F}ln\ensuremath{\kappa}/\ensuremath{\pi}}$, with $\ensuremath{\kappa}$ the number of components of the system, which precludes any singularity of the momentum distribution.
We conduct a theoretical study of SU(N\mathrm{SU}(NSU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N\mathrm{SU}(NSU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these NNN-component systems. Lastly, we consider balanced SU(N\mathrm{SU}(NSU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4N=2,3,4. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of NNN-component fermions may be accurately inferred from the corresponding few-body systems of NNN distinguishable particles.