Exact solution for the degenerate ground-state manifold of a strongly interacting one-dimensional Bose-Fermi mixture

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.

Figure 1

Bosonic (magenta solid line) and fermionic (blue dashed line) density profiles (in units of $a_{\mathit{ho}}^{-1}=\sqrt{m\omega /\hslash }$) as a function of the spatial coordinate $x$ (in units of $a_{\mathit{ho}}$) for each BBFF wave function of the $N_B=N_F=2$ mixture. The total density (black solid thin line) is included in the first frame for reference.

Figure 2

Bosonic (magenta solid line) and fermionic (blue dashed line) density profiles (in units of $a_{\mathit{ho}}^{-1}$) as a function of the spatial coordinate $x$ (in units of $a_{\mathit{ho}}$) with the $Y$ (top) and $Y^{\prime }$ (bottom) symmetry for $N_B=N_F=2$.

Figure 3

Bosonic (magenta solid line) and fermionic (blue dashed line) momentum distributions (in units of $p_{\mathit{ho}}^{-1}=1/\sqrt{\hslash m\omega }$) as a function of $p$ (in units of $p_{\mathit{ho}}$) with the $Y$ (top) and $Y^{\prime }$ (bottom) symmetry for $N_B=N_F=2$. The first and second panels in each row coincide.

Figure 4

Bosonic (magenta solid line) and fermionic (blue dashed line) density profiles as obtained from dmrg simulations (in units of $a_{\mathit{ho}}^{-1}$) as a function of $x$ (in units of $a_{\mathit{ho}}$) at increasing interaction strength $U/t=1,10,{10}^2,{10}^4$ (from left to right and from top to bottom). The harmonic trap strength is $V/t=7\times {10}^{-6}$ and the number of lattice sites used in the simulation is $L=128$. In the last panel the analytical prediction from the generalized BA wave function (thin black solid line) is shown.

Figure 5

Bosonic (magenta solid lines) and fermionic (blue dashed lines) momentum distributions (in units of $p_{\mathit{ho}}^{-1}=1/\sqrt{\hslash m\omega }$) as a function of $p$ (in units of $p_{\mathit{ho}}$) from the numerical dmrg data at increasing interaction strength ($U/t=1,10,{10}^2,{10}^4$, from top to bottom and from left to right, thick lines). At increasing interactions the curves approach the analytical predictions from the generalized BA wave function for the bosonic (thin black solid lines) and fermionic (thin black dashed lines) momentum distributions, shown in each panel for reference (almost indistinguishable from the numerical data in the last panel). The other parameters used in the simulation are $V/t=7\times {10}^{-6}$ and $L=128$.