High-momentum tails as magnetic-structure probes for strongly correlated $\text{SU}\left(\kappa \right)$ fermionic mixtures in one-dimensional traps

A universal $k^{-4}$ 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 $\text{SU}\left(\kappa \right)$ 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 $\kappa$ components of the gas and the Young tableaux for the $S_N$ 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)].

Figure 1

Normalized momentum distributions $n_{\nu }\left(k\right)/N_{\nu }$ (top and middle panels), in units of $a_{\mathrm{ho}}$, and the function $k^4{n}_{\nu }\left(k\right)/N_{\nu }$ (bottom panel), in units of $a_{\mathrm{ho}}^{-3}$ as a function of $k{a}_{\mathrm{ho}}$, for the case of balanced mixtures with $N=6$. (Top panel) All curves correspond to the mixture with $\kappa =3$. The dotted line corresponds to $g_{1\mathrm{D}}=0$, the long-dashed line to $g_{1\mathrm{D}}=1\hslash \omega a_{\mathrm{ho}}$, the short-dashed line to $g_{1\mathrm{D}}=10\hslash \omega a_{\mathrm{ho}}$, and the continuous line to $g_{1\mathrm{D}}\to \infty .$ (Middle and bottom panels) From the highest curve to the lowest, $\kappa =6$ (red), $\kappa =3$ (coral), and $\kappa =2$ (orange). The horizontal lines in the bottom panel correspond to the exact solution for the contacts given in Eq. (A4).

Figure 2

Exact and MPS results for ${\mathcal{C}}_{\nu }$ for balanced mixtures with different number of fermions (up to $N=12)$, components, and interaction strengths, as functions of the parameter ${\alpha }_0/2=a_{\mathrm{ho}}/\left(\right|a_{1\mathrm{D}}\left|N^{1/2}\right)$. The blue circles correspond to $\kappa =2$, the orange squares to $\kappa =3$, the green diamonds to $\kappa =4$, the red up-triangles to $\kappa =5$, and the violet down-triangles to $\kappa =6$. The data are compared with the perturbative LDA expression (13) to first and second order in $1/{\alpha }_0$ (dashed and continuous lines, respectively). The inset shows the data collapse on the same $\kappa$-dependent curve when the $N^{5/2}$ dependence of ${\mathcal{C}}_{\nu }$ is taken away; the main panel then displays the very weak dependence on $\kappa$ once the limiting value at $g_{1\mathrm{D}}\to \infty$ of Eq. (12) is also factorized away.

Figure 3

The interaction energy parameter $K$ (or equivalently ${\mathcal{C}}_{\mathrm{tot}})$ for different mixtures: $5+1,4+2,3+3,3+2+1$, and $2+2+2$. The different colors correspond to different symmetries characterized by their Young's tableaux defined in the main text and Table 1: $Y_{-15}$ (black), $Y_{-9}$ (red), $Y_{-5}$ (green), $Y_{-3,-8}$ (blue), $Y_{-3,4}$ (magenta), $Y_0$ (gray), $Y_3$ (orange). For the sake of visibility, the symbols are shifted with the same order of appearence (from left to right) than the colors.

Figure 4

Numerical evidence for the existence of a generalized Tan's relation in a fermionic multicomponent balanced mixture at finite interactions: $g_{1\mathrm{D}}=1\hslash \omega a_{\mathrm{ho}}$ (violet), $10\hslash \omega a_{\mathrm{ho}}$ (dark violet), $100\hslash \omega a_{\mathrm{ho}}$ (orange). As in Figs. 1 and 1, we display the case of $\kappa =3$ and $N=6$, with the momentum distributions normalized to the species' population and plotted in units of $a_{\mathrm{ho}}$ versus the momentum in units of $1/a_{\mathrm{ho}}$. The solid curves are the momentum distributions $n_{\nu }\left(k\right)$ obtained by MPS numerics, while the dashed straight lines are ${\mathcal{C}}_{\nu }{k}^{-4}$ where no free parameter is adjusted and ${\mathcal{C}}_{\nu }$ are extracted from the interaction energies measured in the same simulations.

Figure 5

Contact ${\mathcal{C}}_{\nu }/N^2$, in units of $a_{\mathrm{ho}}^{-3}$, as a function of the temperature $T$, in units of $\hslash \omega /k_{\mathrm{B}}$, for the case of balanced mixtures with $N=6$. The lines correspond to large-temperature behavior given in Eq. (18) and the points to the exact solution at $T=0$ given in Eq. (12). From top to bottom: $\kappa =6$ (red), $\kappa =3$ (coral), and $\kappa =2$ (orange).

Figure 6

Normalized density distributions $n\left(x\right)/N$ in units of $1/a_{\mathrm{ho}}$, as a function of $x/a_{\mathrm{ho}}$, for the case of balanced mixtures with $N=6$ and $\kappa =3$. From the highest curve to the lowest: $g_{1\mathrm{D}}=0,g_{1\mathrm{D}}=1\hslash \omega a_{\mathrm{ho}},g_{1\mathrm{D}}=10\hslash \omega a_{\mathrm{ho}},g_{1\mathrm{D}}=100\hslash \omega a_{\mathrm{ho}}$, and $g_{1\mathrm{D}}\to \infty$.