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arXiv:2509.00536

Ground State Energy of Dilute Fermi Gases in 1D

Johannes Agerskov, Robin Reuvers, Jan Philip Solovej

Published Aug 30, 2025• 0 citations
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Abstract

We study the one-dimensional spin-$J$ Fermi gas, interacting through a general repulsive 2-body potential, and prove asymptotics of the ground state energy in the dilute limit. The asymptotic behaviour is given in terms of the ground state energy of a spin chain, which is the Heisenberg antiferromagnet in the case of spin-1/2 fermions. We determine similar asymptotics for the Lieb-Liniger-Heisenberg model, showing that a product trial state used by Girardeau is not optimal.

DOI
10.48550/arxiv.2509.00536

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We study the one-dimensional spin-JJJ Fermi gas, interacting through a general repulsive 2-body potential, and prove asymptotics of the ground state energy in the dilute limit. The asymptotic behaviour is given in terms of the ground state energy of a spin chain, which is the Heisenberg antiferromagnet in the case of spin-1/2 fermions. We determine similar asymptotics for the Lieb-Liniger-Heisenberg model, showing that a product trial state used by Girardeau is not optimal.
Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin 1/21/21/2 fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.
The ground and excited states of a one-dimensional (1D) spin-1/2 Fermi gas (SFG) with both attractive zero-range odd-wave interactions and repulsive zero-range even-wave interactions are mapped exactly to a 1D Lieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, such that the complete SFG and LLH energy spectra are identical. The ground state in the ferromagnetic phase is given exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in the antiferromagnetic phase by a variational method combining Bethe ansatz solutions of the LL and 1D Heisenberg models. There are excitation branches corresponding to LL type I and II phonons and spin waves, the latter behaving quadratically for small wave numbers in the ferromagnetic phase and linearly in the antiferromagnetic phase.
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin 1/2 fermions, this is E∼E0+(ℏ2/2m)2πNρaE \sim E^0 + (\hbar^2/2m) 2 πN ρaE∼E0+(ℏ2/2m)2πNρa, where E0E^0E0 is the energy of the non-interacting system and ρρρ is the density.
The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.