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Journal of Physics A Mathematical and Theoretical

Mathematical calculation for exact solutions of infinitely strongly interacting Fermi gases in tight waveguides

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

Published Sep 7, 2009• 10 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/42/38/385210

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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 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.
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.
The two- and three-body correlation functions of the ground state of an optically trapped ultracold spin-12\frac{1}{2}21​ Fermi gas (SFG) in a tight waveguide [one-dimensional (1D) regime] are calculated in the plane of even- and odd-wave coupling constants, assuming a 1D attractive zero-range odd-wave interaction induced by a 3D ppp-wave Feshbach resonance, as well as the usual repulsive zero-range even-wave interaction stemming from 3D sss-wave scattering. The calculations are based on the exact mapping from the SFG to a ``Lieb-Liniger-Heisenberg'' model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, and indicate that the SFG should be stable against three-body recombination in a large region of the coupling constant plane encompassing parts of both the ferromagnetic and antiferromagnetic phases. However, the limiting case of the fermionic Tonks-Girardeau gas, a spin-aligned 1D Fermi gas with infinitely attractive ppp-wave interactions, is unstable in this sense. Effects due to the dipolar interaction and a Zeeman term due to a resonance-generating magnetic field do not lead to shrinkage of the region of stability of the SFG.
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.
Divergence-free pseudo-potentials for spatially even- and odd-wave interactions in spinor Fermi gases in tight atom waveguides are derived. The Fermi-Bose mapping method is used to relate the effectively one-dimensional fermionic many-body problem to that of a spinor Bose gas. Depending on the relative magnitudes of the even- and odd-wave interactions, the NNN-atom ground state may have total spin S=0S=0S=0, S=N∕2S=N∕2S=N∕2, and possibly also intermediate values, the case S=N∕2S=N∕2S=N∕2 applying near a ppp-wave Feshbach resonance, where the NNN-fermion ground state is space-antisymmetric and spin-symmetric. In this case the fermionic ground state maps to the spinless bosonic Lieb-Liniger gas. An external magnetic field with a longitudinal gradient causes a Stern-Gerlach spatial separation of the corresponding trapped Fermi gas with respect to various values of Sz{S}_{z}Sz​.
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.