Mass-imbalance effect on the cluster formation in a one-dimensional Fermi gas with coexistent $s$- and $p$-wave interactions

We consider the mass-imbalance effect on the clustering in a one-dimensional two-component Fermi gas with coexistent even- and odd-wave interactions resulting in different configurations of clustering phases. We obtain the solutions of both stable two- and three-body cluster states with different mass ratios and configurations by solving the corresponding variational equations. We numerically map out phase diagrams consisting of the $s$- and $p$-wave pairing phases, and {trimer} phase with different configurations, in a plane of $s$- and $p$-wave pairing strengths. Within the explored ranges of $s$- and $p$-wave pairing strengths, the in-vacuum three-body states are always more deeply bound than the two-body ones. While for the in-medium case, the Cooper {trimer} phase dominates over the pairing phases when both $s$- and $p$-wave interactions are moderately strong. There is also a competition between different clustering configurations of three-body clustering.

💡 Research Summary

This paper investigates how mass imbalance influences cluster formation in a one‑dimensional two‑component Fermi gas where even‑parity (s‑wave) and odd‑parity (p‑wave) interactions coexist. The authors formulate a Hamiltonian that includes kinetic energy, an interspecies s‑wave contact interaction, and intraspecies p‑wave contacts for each component. The interaction strengths are renormalized in terms of the s‑wave scattering length a_s (taken attractive, a_s > 0) and a common p‑wave scattering length a_p. Because the two species have different masses (m_a ≠ m_b) but equal densities, their chemical potentials differ, introducing a natural mismatch that strongly affects pairing and clustering.

To explore possible bound states, the authors employ a variational approach built on top of an inert Fermi sea. They construct trial wave functions for five distinct configurations: (i) s‑wave a‑b pairing, (ii) p‑wave aa pairing, (iii) p‑wave bb pairing, (iv) aab trimer (two a atoms plus one b), and (v) abb trimer (two b atoms plus one a). The variational parameters Ω and auxiliary amplitudes Γ are introduced, leading to coupled equations (14)–(21) that must be solved for the lowest‑energy eigenvalue in each sector. Numerically, momentum space is discretized on a uniform grid (Δk = 0.01 k_F) with a cutoff Λ = 10 k_F, and the coupled equations are cast as a linear eigenvalue problem.

The study first examines the vacuum case (no Fermi sea). Using a representative mass ratio m_a = 2 m_b, the authors find that three‑body bound states are always more deeply bound than any two‑body pair across the explored ranges of a_s and a_p. This is attributed to the cooperative effect of the simultaneous s‑ and p‑wave attractions: a trimer can exploit both channels, whereas a pair can only benefit from one. Within the trimer sector, the aab configuration dominates when the p‑wave interaction is weak, while the abb configuration becomes energetically favorable as the p‑wave strength increases. The p‑wave pairing also shows a clear mass‑dependence: the heavier‑species bb pair binds more strongly than the lighter‑species aa pair, consistent with the analytic scaling E ∝ m_r/m_i derived from the variational equation.

A phase diagram in the (1/Λa_s, 1/Λa_p) plane is constructed for the vacuum case, confirming that the trimer region occupies the entire diagram for the chosen mass ratio. The diagram illustrates the competition between aab and abb trimers and the suppression of two‑body pairing by the deeper three‑body binding.

The authors then turn to the in‑medium situation, where the Fermi sea imposes Pauli blocking via step functions θ(|k|−k_F) in the variational equations. The chemical potentials µ_a and µ_b differ because of the mass imbalance, further influencing the energetics. In this setting, when both s‑ and p‑wave interactions are of moderate strength, the Cooper trimer (especially the aab configuration) overtakes the s‑wave and p‑wave pairing phases, establishing a “Cooper trimer” phase. This dominance arises because the medium‑induced blocking reduces the phase space for two‑body pairing more severely than for three‑body clustering, allowing the trimer to gain a relative energy advantage. The phase diagram in the medium shows a rich structure where pairing and trimer regions meet, and the location of the boundaries shifts with the mass ratio: larger m_a/m_b expands the aab‑trimer domain, while the opposite ratio favors the abb‑trimer.

Overall, the work demonstrates that (i) mass imbalance qualitatively changes the hierarchy of bound states, (ii) coexistence of s‑ and p‑wave attractions can stabilize three‑body clusters even when two‑body pairs are weakly bound, and (iii) in a many‑body environment the Cooper trimer can become the ground‑state instability over conventional pairing. The authors stress that their variational treatment captures the lowest‑energy cluster instability within a restricted subspace rather than a fully self‑consistent many‑body ground state. They suggest that experimental platforms such as heteronuclear mixtures (e.g., 6Li–133Cs) or multiorbital alkaline‑earth gases, where both s‑ and p‑wave Feshbach resonances can be tuned, are promising for realizing the predicted phases. Future theoretical extensions could include self‑consistent many‑body techniques, higher dimensions, lattice potentials, or spin‑orbit coupling to connect the findings with real unconventional superconductors.