The exact dynamical structure factor of one-dimensional hard rods and its universal random matrix behavior

TL;DR

This paper derives an exact expression for the dynamical structure factor of 1D hard rods, revealing universal random matrix behavior and fermionic structure, applicable across various many-body states including finite temperature and ground state.

Contribution

It provides the first exact analytic formula for the dynamical structure factor of 1D hard rods, connecting it to universal random matrix theory and uncovering hidden fermionic features.

Findings

Exact expression valid for arbitrary many-body states

Obeys fundamental sum rules and detailed balance

In the static limit, relates to Gaussian Unitary Ensemble level spacing

Abstract

We obtain an exact analytic expression for the dynamical structure factor of one-dimensional quantum gas of hard rods. Our result is valid for arbitrary many-body state of the system, with finite temperature states and the ground state being important special cases that we analyse in detail. We demonstrate that the expression obeys fundamental relations such like the f-sum rule and the detailed balance. We also reveal the hidden fermionic structure behind the correlator. In the static limit we show that it can be written in terms of universal functions which, at zero temperature, coincide with the level spacing distribution function of the Gaussian Unitary Ensemble. Our work provides a full and exact characterisation of a dynamic correlation function in a strongly correlated interacting quantum many-body system.