Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy and Generalized Hydrodynamics

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.

Automated summary

In this study, we investigate fermions on a continuous one-dimensional interval subjected to weak repulsive two-body interactions. We demonstrate the possibility of perturbatively constructing an extensive number of mutually compatible conserved charges for any interaction potential. The densities of these charges at higher orders are generally nonlocal and only become spatially localized under certain compatibility conditions. We prove that the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials are the only solutions to the first of these conditions. We utilize our construction to show the emergence of generalized hydrodynamics from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy and argue for the robustness of generalized hydrodynamics under nonintegrable perturbations in the weak interaction regime.