Dynamic structure factor of quantum hard rods from exact form-factors

TL;DR

This work investigates the one-dimensional quantum hard-rod gas, an integrable system solvable by the coordinate Bethe Ansatz, to derive exact density form factors and develop a semi-analytical route for dynamic and static structure factors. A compact Cauchy-determinant form for the density form factors reduces computational cost and enables precise spectral-sum calculations of $S(k,\omega)$, benchmarked against the $f$-sum rule and TG limits. The study confirms Tomonaga–Luttinger liquid universality, derives explicit Luttinger-liquid prefactors from umklapp form factors via Barnes $G$-functions, and demonstrates a transition from TL liquid behavior to a densely packed, quasi-crystalline regime as $K$ decreases. These results furnish accurate benchmarks for numerical methods and deepen understanding of correlations in strongly interacting 1D quantum fluids, with potential links to deformations of spin chains and holographic-type deformations.

Abstract

We study the quantum hard-rods model and obtain compact analytical expressions for density form factors, and a semi-analytical treatment for dynamic and static structure factors calculations, greatly reducing computational complexity. We identify conditions under which these form factors vanish and analyze real-space correlations, confirming the model's Tomonaga-Luttinger liquid behavior. The results reveal universal features of low energy physics of a gapless quantum fluid and its relation to Luttinger liquid theory, providing precise benchmarks for numerical simulations. This work establishes quantum hard rods as an important testbed for theories of strongly correlated one-dimensional systems.

Figures (7)

• Figure 1: Illustration of quantum numbers $I_j$ corresponding to the ground state (top row) and the two fundamental excitations modes (middle rows) for the system of $N=9$ particles. Filled (empty) points mark non-zero (zero) values. The umklapp excitations obtained by moving a particle from one to the other Fermi edge are illustrated in the bottom row. They can be seen as a limiting case of the hole excitation.
• Figure 2: Form-factors between the ground state and a state containing either a single particle (circles) excitation or a single hole excitation (squares), both of fixed momentum $k_F$, as a function of the system size $L$ with density $\rho_0=1$. For $K=1$ form-factors squared scale as $1/L^2$ (shown with a black line) which is expected in a free theory. Instead for other values of $K$ the scalling is irrational with hole form-factors scaling with a smaller power while particle form-factors with a larger power.
• Figure 3: Color maps of the dynamic structure factor, presented on a logarithmic scale as ${\rm log}[S(k,\omega)]$ are shown for three representative values of the Luttinger parameter $K$, with $\rho_0=1$ and for finite number $N$ of particles as indicated in the plots. The red dashed lines indicate the dispersion relations of the two fundamental modes \ref{['fundamental_modes']}, and for $K=0.26$ the dispersion relation of the second hole-mode is also shown. For $K=0.26$, finite-size effects manifest as stripes, also visible in the fixed-momentum cuts of Fig \ref{['fig:dsf_fixed']}; incomplete summation of high-$k$ excitations, despite including over $10^9$ states (see Tab. \ref{['tab:states']}), produces bright patches.
• Figure 4: Fixed-momentum cuts of the dynamic structure factor $S(k,\omega)$ are shown for various $k$ and system sizes $N$. The lineshapes are characterized by a peak in the vicinity of the lower threshold and a long high-energy tail. We observe that finite-size effects for $N=64$ are essentially below the resolution of the figure, except in the vicinity of the peak, where such effects are expected to be more pronounced. The dashed black lines is $S_{\rm mTG}(k,\omega)$, the DSF of the modified Tonks-Girardeau gas, see Eq. \ref{['DSF_guess']}.
• Figure 5: Static structure factor $S(k)$ of the system with $N=16$ and representative values of $K$ as indicated in legend. The corresponding in color dashed lines represent asymptotic behavior $S(k\sim0) \approx |k|/v_s$ known from TL theory. The only red dashed line represents known TG gas behavior when $K\to 1$.