Bose-Fermi mixture in one-dimensional optical lattices with hard-core interactions

We study a mixture of $N_{b}$ bosons with point hard-core boson-boson interactions and $N_{f}$ noninteracting spinless fermions with point hard-core boson-fermion interactions in 1D optical lattice with external harmonic confine potential. Using an extended Jordan-Winger transformation (JWT) which maps the hard-core Bose-Fermi mixture into two component noninteracting spinless fermions with hard-core interactions between them, we get the ground states of the system. Then we determine in details the one particle density matrix, density profile, momentum distribution, the natural orbitals and their occupations based on the constructed ground state wavefunctions. We also discuss the ground state properties of the system with large but finite interactions which lead to the lift of ground degeneracy. Our results show that, although the total density profile is almost not affected, the distributions for bosons and fermions strongly depend on the relative strengthes of boson-boson interactions and boson-fermion interactions.

💡 Research Summary

The paper investigates a one‑dimensional (1D) mixture of $N_{b}$ bosons and $N_{f}$ spin‑less fermions confined in an optical lattice with an additional harmonic trap. Both the boson‑boson (BB) and boson‑fermion (BF) interactions are taken to be point‑like hard‑core (i.e., infinite repulsion), while the fermions are otherwise non‑interacting. Under these conditions the bosons behave as hard‑core bosons (HCB) and the fermions as ordinary spin‑less fermions. The authors’ central methodological tool is an extended Jordan‑Wigner transformation (JWT). In the standard JWT a 1D hard‑core boson field can be mapped onto a spin‑less fermion field, thereby eliminating the BB interaction. Here the transformation is applied in two stages: first the bosonic operators are constrained to the hard‑core subspace and then mapped onto a second species of spin‑less fermions; the original fermions are left unchanged. The resulting model consists of two non‑interacting fermionic components that nevertheless retain a hard‑core constraint between the two species, which exactly reproduces the original BB and BF hard‑core repulsions.

Because the transformed Hamiltonian is that of free fermions in a harmonic potential, its single‑particle eigenfunctions $\phi_{n}(x)$ and eigenenergies $E_{n}$ are known analytically. The many‑body ground state is constructed as a Slater determinant of the lowest $N_{b}+N_{f}$ orbitals, with the first $N_{b}$ orbitals assigned to the “bosonic” fermions and the remaining $N_{f}$ to the original fermions. This wavefunction provides an exact representation of the ground state of the original Bose‑Fermi mixture in the hard‑core limit.

From the exact ground state the authors compute several key observables:

1. One‑particle density matrix $\rho^{\sigma}_{1}(x,x’)$ for each component $\sigma=b,f$, obtained by tracing out all other degrees of freedom.
2. Density profiles $n_{\sigma}(x)=\rho^{\sigma}{1}(x,x)$, showing that the total density $n{\text{tot}}(x)=n_{b}(x)+n_{f}(x)$ is essentially dictated by the external trap and is only weakly modified by the interactions. In contrast, the individual bosonic and fermionic densities respond strongly to the relative strength of $U_{bb}$ and $U_{bf}$.
3. Momentum distributions $n_{\sigma}(k)$ obtained by Fourier transforming the off‑diagonal part of $\rho^{\sigma}_{1}$. Bosons display a pronounced low‑momentum peak characteristic of quasi‑condensation, while fermions exhibit a broader distribution with a sharp drop near the effective Fermi momentum.
4. Natural orbitals and occupations by diagonalizing $\rho^{\sigma}_{1}$. For the bosonic component the leading natural orbital carries a macroscopic fraction of the particles (a “quasi‑condensate”), whereas the fermionic occupations are spread over many orbitals, reflecting Pauli exclusion.

The paper also goes beyond the strict hard‑core limit by considering large but finite interaction strengths $U_{bb}$ and $U_{bf}$. Using perturbative arguments the authors show that the massive ground‑state degeneracy present at infinite repulsion is lifted, leading to small energy splittings that depend on the ratio $U_{bb}/U_{bf}$. When $U_{bb}>U_{bf}$ the bosons are pushed toward the trap centre and the fermions are expelled toward the edges; the opposite occurs when $U_{bf}>U_{bb}$, producing an interleaved spatial pattern. Despite these rearrangements the total density remains almost unchanged, highlighting the subtle interplay between inter‑species and intra‑species hard‑core constraints.

Finally, the authors discuss experimental feasibility. In ultracold‑atom setups the BB and BF interaction strengths can be tuned independently via Feshbach resonances and species‑selective optical lattices, while the harmonic confinement is readily implemented. In‑situ imaging can resolve the component‑specific density profiles, and time‑of‑flight measurements give access to the momentum distributions. Moreover, the natural‑orbital occupations could be probed with quantum‑gas‑microscope techniques, offering a direct test of the theoretical predictions.

In summary, the work provides an exact analytical solution for a 1D hard‑core Bose‑Fermi mixture in a trap, elucidates how component‑resolved observables depend on interaction ratios, and outlines how finite‑interaction effects lift degeneracies. These results constitute a valuable benchmark for future numerical studies and for experimental investigations of multi‑component strongly correlated quantum gases in low dimensions.