Exact Density Profiles of 1D Quantum Fluids in the Thomas-Fermi Limit: Geometric Hierarchy to the Tonks-Girardeau Gas

TL;DR

This paper introduces a geometric framework for 1D quantum fluids in the Thomas-Fermi limit, unifying various interaction regimes through a discrete hierarchy and deriving universal sound velocity scaling laws.

Contribution

It develops a non-perturbative geometric hierarchy linking ideal, mean-field, and strongly correlated 1D quantum gases, and derives a universal sound velocity scaling law.

Findings

Density profiles form a discrete hierarchy across interaction regimes.

Universal sound velocity scaling law $c \,\propto\, \rho^{(1-q)/4}$ is established.

Framework links static geometry with dynamical excitations in many-body systems.

Abstract

We present a geometric framework for 1D quantum fluids across interaction regimes in the Thomas-Fermi limit. Based on the Linearization Principle via the $q$-logarithm, macroscopic density profiles form a discrete hierarchy: the ideal Bose gas ($q=1$), the mean-field Gross-Pitaevskii condensate ($q=-1$), and the strongly correlated Tonks-Girardeau gas ($q=-3$). We further derive a universal sound velocity scaling, $c \propto \rho^{(1-q)/4}$, valid in the interacting regimes ($q \le -1$). This establishes a non-perturbative link between static geometry and dynamical excitations in many-body systems.