We assume that the neutron-neutron potential is well-behaved and velocity-dependent. We can then apply perturbation theory to find the energy per particle of a neutron gas, in the range of Fermi wave numbers
0.5<kf<2 ${\mathrm{f}}^{\ensuremath{-}1}$. The energy through first order is found in closed form, or by a single numerical integration. We use two different velocity-dependent potentials adjusted to fit observed nucleon-nucleon
1S and
1D phase shifts. In the range of densities
0.5<kf<1 ${\mathrm{f}}^{\ensuremath{-}1}$, our two potentials give nearly the same energy/particle (within 0.5 Mev); our values tend to run an Mev below values found by Brueckner et al., for the Gammel-Thaler potential. Wider divergences appear at higher densities. Our values, and Brueckner's are higher than those found by Salpeter by a semiempirical approach. A crude estimate of the second-order energy for our potentials indicates that perturbation theory converges rapidly in the density range considered. Our results suggest that at moderately low densities the energy/particle in a many-body system is insensitive to the shape or nonlocal character of the assumed two-body potential.
One-dimensional gapped systems are often characterized by a ``hidden'' nonlocal order parameter, the so-called string order. Due to the gap, thermodynamic properties are robust against a weak higher-dimensional coupling between such chains or ladders. On the contrary, we find that the string order is not stable and decays for arbitrary weak interchain or interladder coupling. We investigate the vanishing of the order for three different systems: spin-1 Haldane chains, band insulators, and the transverse-field Ising model. Using perturbation theory and bosonization, we show that the fragility of the string order arises from nonlocal commutation relations between the nonlocal order parameter and the perturbation.
We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally nonlocal and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how generalized hydrodynamics emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that generalized hydrodynamics in the weak interaction regime is robust under nonintegrable perturbations.