Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases

We show that strongly interacting multicomponent gases in one dimension realize an effective spin chain, offering an alternative simple scenario for the study of one-dimensional (1D) quantum magnetism in cold gases in the absence of an optical lattice. The spin-chain model allows for an intuitive understanding of recent experiments and for a simple calculation of relevant observables. We analyze the adiabatic preparation of antiferromagnetic and ferromagnetic ground states, and show that many-body spin states may be efficiently probed in tunneling experiments. The spin-chain model is valid for more than two components, opening the possibility of realizing SU(N) quantum magnetism in strongly interacting 1D alkaline-earth-metal or ytterbium Fermi gases.

Figure 1

Continuous (experimentally measurable) spin densities ${\rho }_{\uparrow ,\downarrow }\left(z\right)$ of the full model together with the discrete spin densities ${\rho }_{\uparrow ,\downarrow }^{\left(i\right)}$ of the spin-chain model for seven harmonically trapped spin-1/2 fermions $(N_{\uparrow }=4,N_{\downarrow }=3)$ in the antiferromagnetic state.

Figure 2

Magnetization of a spin-balanced AF spin chain consisting of 16 harmonically trapped particles for $G/J=0.05$ and $0.8$ ($G$ is the B-field gradient and $J={\sum }_i{J}_i/(N-1)$ is the average nearest-neighbor exchange). The symbols (shaded curves) denote the discrete (continuous) distributions.

Figure 3

Gap between the ground and first excited state of harmonically trapped spin-balanced spin-1/2 fermions for nonzero gradients ($G\ne 0$) around the resonance. While spin interactions dominate in the AF and F regimes, the B-field gradient dominates in the gray-shaded Stern-Gerlach (SG) regime, characterized by SG-like spin segregation. Inset: Spectrum of six spin-balanced spin-1/2 fermions as a function of $-J/G$.

Figure 4

Overlap between the F ground state of harmonically trapped spin-balanced spin-1/2 fermions expected for $-J/G=10$ and the state obtained after a linear sweep across the resonance starting with the AF ground state for $-J/G=-10$.

Figure 5

Mean occupation $\langle n_i\rangle$ of the harmonic-trap levels for the $(N_{\uparrow }=2,N_{\downarrow }=1)$ system in the Tonks regime $[g/(\hslash \omega l)=25]$ for the states $|0\rangle$, $|1\rangle$, and $|2\rangle$ (see text) of the ground-state multiplet.

Figure 6

Comparison between the exact-diagonalization results of the original continuous model and the results obtained from the effective spin model for $(N_{\uparrow }=2,N_{\downarrow }=1)$ harmonically trapped spin-1/2 fermions. Top: Spin densities of the effective spin-chain model (solid lines) and of an exact diagonalization of the full Hamiltonian for $g/\left(\hslash \omega l\right)=10$ in a harmonic trap (dashed and dash-dotted lines). Bottom: Ratio of energy differences of the ground-state multiplet as a function of $-1/g$ [solid (black) line with (red) circles]. The corresponding value of the spin-chain model is 3 (horizontal short-dashed line). The long-dashed line marks $-1\%$ deviation from this value.

Figure 7

The same as Fig. 6 for the $(N_{\uparrow }=3,N_{\downarrow }=1)$ system. Top: Spin densities of the effective spin-chain model (solid lines) and of an exact diagonalization of the full Hamiltonian for $g/\left(\hslash \omega l\right)=25$ in a harmonic trap (dashed and dash-dotted lines). Bottom: Ratio of energy differences of the ground-state multiplet as a function of $-1/g$ [solid (black) line with (red) circles]. The corresponding values of the spin-chain model are $1.982$ and $5.983$, respectively (horizontal short-dashed lines). The horizontal long-dashed lines mark $-1\%$ deviation from these values.

Figure 8

Mean occupation numbers $\langle n_i\rangle$ of the harmonic-oscillator orbitals of an $(N_{\uparrow }=3,N_{\downarrow }=2)$ Fermi system in the Tonks regime [$g/\left(\hslash \omega l\right)=30$, ground-state multiplet].