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

TL;DR

This paper derives an exact analytic expression for the dynamic structure factor $S(P,\omega)$ of a one-dimensional quantum gas of hard rods, valid for arbitrary many-body states. It builds on an exact density form-factor framework and recasts the spectral sum as a Fredholm determinant $\mathcal{D}_{\nu}(s,t)$ with kernel $V(\lambda,\mu)$, enabling nonperturbative, high-precision calculation across finite $P$ and $\omega$ and through the thermodynamic limit. The authors establish fundamental checks, including the $f$-sum rule and detailed balance (KMS) relations, reveal a hidden free-fermionic structure, and show that in the zero-temperature static limit the determinant reduces to sine-kernel structures linked to Gaussian Unitary Ensemble level-spacing statistics; at high temperature the results reproduce classical hard-rod correlations. Overall, the work provides a exact, nonperturbative benchmark for dynamics in strongly correlated integrable systems and builds bridges to random-matrix theory and Luttinger-liquid descriptions, with potential to illuminate how microscopic integrability manifests in low-energy effective theories.

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.

Figures (4)

• Figure 1: The comparison of the Fourier transformed DSF in the thermodynamic limit and the finite system for zero temperature, $k_F=\pi \rho_0$, and $a\rho_0=1/4$.
• Figure 2: Function $S_n(x,t)$ for $n>0$ can be interpreted as contribution to the correlation function involving scattering with $n$ particles. Here two sources of the correlation between $(0,0)$ and $(x,t)$ are shown. First, the particle starting at $(0,0)$ scatters with two other particles. The signal reaches at time $t$ point $x$ in for hard rods of length $a$ and point $x-2a$ for hard-core particles. Similarly, for $n<0$, the scattering happens at position larger than $x$ and the signal reaches point $x$ or $x+a$ for hard rods and hard-core particles respectively. This explains the shift in the argument $S_n(x - na, t)$ in \ref{['Sxt11']}.
• Figure 3: Plots of the k-level spacing probability densities ($k=0,1,2,3,4$) for the zero temperature (solid lines), and for $T=10$, $a=0.5$, $\rho_0=1$ (dashed lines); The inset shows the classical result \ref{['pcl']}.
• Figure 4: The static structure factor at $\rho_0=1$, $a=0.5$, and the various temperature shown in the legends. The classical answer (magenta line) corresponds to \ref{['pcl']}. The "Luttinger" is the answer obtained in the Luttinger model approximation in Eq. \ref{['Luttinger']}.