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arXiv:2608.02007

Benchmarking the fermionic quasi-1D many-body problem

Ekaterina Gradova, Frédéric Chevy

Published Aug 3, 2026• 0 citations
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Abstract

We investigate the validity of effective one-dimensional models for quasi-1D fermionic systems by benchmarking a coupled-channel approach against the exact low-energy theory derived from the underlying three-dimensional problem in the weakly and strongly attractive limits. We show that reproducing the exact two-body scattering amplitude is insufficient to construct the correct effective low-energy theory of quasi-1D fermions. In the weakly attractive regime, it does not capture the emergent three-body interaction induced by transverse excitations. In the strongly attractive regime, it yields an atom-dimer scattering length with an incorrect dependence on the three-dimensional scattering length. These results demonstrate the limitations of effective one-dimensional descriptions based solely on two-body physics and highlight the need to explicitly include few-body correlations in quasi-1D systems.

DOI
10.48550/arxiv.2608.02007

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We investigate the validity of effective one-dimensional models for quasi-1D fermionic systems by benchmarking a coupled-channel approach against the exact low-energy theory derived from the underlying three-dimensional problem in the weakly and strongly attractive limits. We show that reproducing the exact two-body scattering amplitude is insufficient to construct the correct effective low-energy theory of quasi-1D fermions. In the weakly attractive regime, it does not capture the emergent three-body interaction induced by transverse excitations. In the strongly attractive regime, it yields an atom-dimer scattering length with an incorrect dependence on the three-dimensional scattering length. These results demonstrate the limitations of effective one-dimensional descriptions based solely on two-body physics and highlight the need to explicitly include few-body correlations in quasi-1D systems.
Full three-dimensional (3D) calculations of small two-component Fermi gases under highly elongated confinement, in which unlike fermions interact through short-range potentials with variable atom-atom sss-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.
We study a one-dimensional system of two-component fermions in the limit of strong attractive particle-particle interactions. First, we analyze scattering in the corresponding few-body problem, which is analytically solvable via Bethe ansatz. This allows us to engineer effective interactions between the system's effective degrees of freedom: fermions and bosonic dimers (tightly bound pairs of fermions). We argue that, although these interactions are strong, the resulting effective problem can be mapped onto a weakly interacting one, paving the way for the use of perturbation theory. This finding simplifies studies of many-fermion systems under confinement that are beyond reach of state-of-the-art numerical methods. We illustrate this statement by considering an impurity atom in a Fermi gas.
In this article we discuss the accuracy of effective one-dimensional (1D) theories used to describe the behavior of ultracold atomic ensembles confined in quantum wires by a harmonic trap. We derive, within a fully many-body approach, the effective Hamiltonian describing this class of systems and we calculate the beyond-mean-field corrections to the energy of the ground state arising from virtual transitions towards the excited state of the confining potential. We find that, due to the Pauli principle, effective finite-range corrections are one order of magnitude larger than effective three-body interactions. By comparing to exact solutions of the purely 1D problem, we conclude that a 1D effective theory provides a good description of the ground state of the system for a rather large range of interaction parameters.
We derive one-dimensional effective Hamiltonians for spin-orbit-coupled Fermi gases confined in quasi-one-dimensional trapping potentials. For an energy regime around the two-body bound-state energy, the effective Hamiltonian takes a two-channel form, where the populations in transverse excited levels are described by dressed molecules in the closed channel. For an energy regime slightly above the continuum threshold, the effective Hamiltonian takes a single-channel form, where low-energy physics is governed by the one-dimensional interacting strength determined by three-dimensional scattering length and transverse confinement. We further discuss the effect of spin-orbit coupling and effective Zeeman field on the position of confinement-induced resonances and show that these resonances can be understood as Feshbach resonances between the threshold of the transverse ground state and the two-body bound state associated with the transverse excited states. We expect that the shift of confinement-induced resonances can be observed under present experimental technology at attainable temperatures.
We investigate the properties of an impurity immersed in an ensemble of spin-polarized fermions confined in a tight quantum wire. We use a non-perturbative variational approach that accounts for virtual transverse excitations and regularizes the zero-range interaction. We compute the polaron's energy, effective mass, and spectral weight, and benchmark our results against the exactly solvable Yang-Gaudin model of a purely 1D system. While the two models agree in the weakly interacting regime, we find significant deviations in the strongly attractive limit, including a divergence of the effective mass and evidence for a polaron-to-molecule transition inherited from the underlying 3D physics, which is absent in the purely 1D description. Our work quantifies the accuracy of quasi-1D systems as analog quantum simulators and highlights the emergence of beyond-1D physics even in nominally 1D settings.
Two of the most popular quantum-mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a one-dimensional (1D) double-well potential, a model of atoms in a quasi-1D optical lattice, or electrons of a hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density-functional theory (DFT) functional and two-site Hubbard models. A 1D adiabatic local-density approximation kernel is constructed for use in time-dependent density-functional theory (TDDFT) and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably to the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with nonadiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.