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arXiv:0806.1773 · Physical Review Letters

Exact Solution for Infinitely Strongly Interacting Fermi Gases in Tight Waveguides

Liming Guan, Shu Chen, Yupeng Wang, Zhong-Qi Ma

Published Apr 22, 2009• 116 citations
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Abstract

We present an exact analytical solution of the fundamental systems of quasi-one-dimensional spin-1/2 fermions with infinite repulsion for an arbitrary confining potential. The eigenfunctions are constructed by the combination of Girardeau's hard-core contacting boundary condition and group theoretical method, which guarantees the obtained states to be simultaneously the eigenstates of S and S_{z} and satisfy antisymmetry under odd permutation. We show that the total ground-state density profile behaves like the polarized noninteracting fermions, whereas the spin-dependent densities display different properties for different spin configurations. We also discuss the splitting of the ground states for large but finite repulsion.

DOI
10.1103/physrevlett.102.160402

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We present an exact analytical solution of the fundamental systems of quasi-one-dimensional spin-1/2 fermions with infinite repulsion for an arbitrary confining potential. The eigenfunctions are constructed by the combination of Girardeau's hard-core contacting boundary condition and group theoretical method, which guarantees the obtained states to be simultaneously the eigenstates of S and S_{z} and satisfy antisymmetry under odd permutation. We show that the total ground-state density profile behaves like the polarized noninteracting fermions, whereas the spin-dependent densities display different properties for different spin configurations. We also discuss the splitting of the ground states for large but finite repulsion.
Exact analytical solutions of the fundamental systems of N quasi-one-dimensional spin-1/2 fermions with infinite delta repulsion in an arbitrary confining potential were presented in our previous letter (Guan et al 2009 Phys. Rev. Lett. 102 160402). The solutions are the simultaneous eigenstates of the Hamiltonian H and the total spin operators S 2 and S z , which fulfil Girardeau's hard-core contacting boundary condition and are antisymmetric under odd permutations among fermions. They are approximate solutions when the coefficient in the delta repulsion is large but finite, and according to the Lieb–Mattis theorem, the solution with the lowest S is the ground state for the system. A detailed mathematical calculation for the exact solutions is presented. The property of the spin-dependent reduced one-body density is discussed.
We present an exact analytical solution of the fundamental system of quasi-one-dimensional spin-1 bosons with infinite delta repulsion. The eigenfunctions are constructed from the wave functions of noninteracting spinless fermions, based on Girardeau's Fermi-Bose mapping. We show that the spinor bosons behave like a compound of noninteracting spinless fermions and noninteracting distinguishable spins. This duality is especially reflected in the spin densities and the energy spectrum. We find that the momentum distribution of the eigenstates depends on the symmetry of the spin function. Furthermore, we discuss the splitting of the ground state multiplet in the regime of large but finite repulsion.
A Fermi-Bose mapping method is used to determine the exact ground states of several models of mixtures of strongly interacting ultracold gases in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). We detail the case of a Bose-Fermi mixture with TG boson-boson (BB) and boson-fermion (BF) interactions. Exact results are given for density profiles in a harmonic trap, single-particle density matrices, momentum distributions, and density-density correlations. Since the ground state is highly degenerate, we analyze the splitting of the ground manifold for large but finite BB and BF repulsions.
We consider a mixture of one-dimensional strongly interacting Fermi gases with up to six components, subjected to a longitudinal harmonic confinement. In the limit of infinitely strong repulsions we provide an exact solution which generalizes the one for the two-component mixture. We show that an imbalanced mixture under harmonic confinement displays partial spatial separation among the components, with a structure which depends on the relative population of the various components. Furthermore, we provide a symmetry characterization of the ground and excited states of the mixture introducing and evaluating a suitable operator, namely the conjugacy class sum. We show that, even under external confinement, the gas has a definite symmetry which corresponds to the most symmetric one compatible with the imbalance among the components. This generalizes the predictions of the Lieb–Mattis theorem for a Fermionic mixture with more than two components.
We present the exact solution for the many-body wavefunction of a one-dimensional mixture of bosons and spin-polarized fermions with equal masses and infinitely strong repulsive interactions under external confinement. Such a model displays a large degeneracy of the ground state. Using a generalized Bose-Fermi mapping, we find the solution for the whole set of ground-state wave functions of the degenerate manifold and we characterize them according to group-symmetry considerations. We find that the density profile and the momentum distribution depends on the symmetry of the solution. By combining the wave functions of the degenerate manifold with suitable symmetry and guided by the strong-coupling form of the Bethe-Ansatz solution for the homogeneous system, we propose an analytic expression for the many-body wave function of the inhomogeneous system which well describes the ground state at finite, large, and equal interaction strengths, as validated by numerical simulations.
The Fermi-Bose mapping method for one-dimensional Bose and Fermi gases with zero-range interactions is generalized to an anyon-fermion mapping and applied to exact solution of several models of ultracold gases with anyonic exchange symmetry in tight waveguides: anyonic Calogero-Sutherland model, anyons with point hard-core interaction (anyonic Tonks-Girardeau gas), and spin-aligned anyon gas with infinite zero-range odd-wave attractions (attractive anyonic Tonks-Girardeau, or AATG, gas). It is proved that for even N>or=4 there are states of the AATG gas on a ring, with anyonic phase slips which are odd integral multiples of pi/(N-1), of energy lower than that of the corresponding fermionic ground state. A generalization to a spinor Fermi gas state with anyonic symmetry under purely spatial exchange enables energy lowering by the same mechanism.