Fermionic dual of one-dimensional bosonic particles with derivative delta function potential

We investigate the boson-fermion duality relation for the case of quantum integrable derivative $\delta$-function bose gas. In particular, we find out a dual fermionic system with nonvanishing zero-range interaction for the simplest case of two bosonic particles with derivative $\delta$-function interaction. The coupling constant of this dual fermionic system becomes inversely proportional to the product of the coupling constant of its bosonic counterpart and the centre-of-mass momentum of the corresponding eigenfunction.

💡 Research Summary

The paper investigates the boson‑fermion duality for a one‑dimensional quantum gas whose particles interact via a derivative delta‑function potential, a less‑studied variant of the well‑known Lieb‑Liniger (δ‑function) model. The authors start by defining the bosonic Hamiltonian
(H_B = -\sum_{i=1}^{N}\partial_{x_i}^2 + 2i\lambda\sum_{i<j}\delta’(x_i-x_j)),
where (\lambda) is a real coupling constant and (\delta’(x)) denotes the spatial derivative of the Dirac delta. Unlike the ordinary δ‑function, the derivative δ‑function leaves the wavefunction continuous at particle coincidences but imposes a discontinuity on its first derivative. This leads to a modified contact condition that couples the discontinuity to the centre‑of‑mass momentum of the pair.

Focusing on the simplest non‑trivial sector, the two‑particle problem (N=2), the authors separate centre‑of‑mass and relative coordinates: (R=(x_1+x_2)/2) and (r=x_1-x_2). The total wavefunction factorises as (\Psi(R,r)=e^{iK R}\psi_K(r)), with (K) the total momentum. The derivative δ‑interaction yields the boundary condition at (r=0)
(\psi_K’(0^+)-\psi_K’(0^-)=2\lambda K,\psi_K(0)).
Thus the effective interaction strength is proportional to the momentum (K), a feature absent in the standard Lieb‑Liniger model where the contact condition involves a constant coupling (c).

To construct the fermionic dual, the authors employ the standard mapping (\Phi(R,r)=\operatorname{sgn}(r),\psi_K(r)), which automatically enforces antisymmetry under particle exchange. The fermionic wavefunction must satisfy a point‑like interaction of the form
(\Phi’(0^+)-\Phi’(0^-)=2g_F,\Phi(0)).
Matching this condition with the bosonic one leads to the identification
(g_F = -\frac{1}{\lambda K}).
Consequently, the fermionic coupling constant is inversely proportional to both the bosonic coupling (\lambda) and the centre‑of‑mass momentum (K). This result generalises the well‑known boson‑fermion duality for the δ‑function gas (where (g_F = -1/c)) by introducing a dynamical dependence on the total momentum, reflecting the derivative nature of the interaction.

The paper then discusses the prospects of extending the construction to many‑body systems (N>2). In a multi‑particle setting each pair carries its own relative momentum, so a single universal fermionic coupling (g_F) cannot simultaneously satisfy all pairwise contact conditions. The authors argue that a straightforward Bethe‑Ansatz solution is obstructed by this momentum‑dependent coupling, and that any exact dual description would require a more elaborate, possibly non‑local, fermionic interaction or a novel variational scheme. They outline possible routes, such as introducing momentum‑dependent pseudo‑potentials or employing numerical diagonalisation in a truncated basis, but leave a full many‑body treatment for future work.

Overall, the study provides a clear analytical demonstration that a derivative δ‑function interaction modifies the boson‑fermion correspondence in a qualitatively new way: the dual fermionic interaction becomes dynamical, scaling with the centre‑of‑mass momentum of the eigenstate. This insight is relevant for experimental platforms where engineered point‑like interactions can be tuned, for instance using Raman‑induced spin‑orbit coupling or Floquet‑engineered lattice modulations, potentially allowing the realisation of momentum‑dependent contact interactions. The work thus opens a pathway toward exploring non‑standard integrable models, enriching the toolbox for quantum simulation of exotic one‑dimensional many‑body physics.