Exact Dynamical Structure Factor for One-Dimensional Hard Rods Achieves Universal Random Matrix Behaviour

Scientists have, for the first time, derived an exact analytical expression for the dynamical structure factor of a one-dimensional gas of hard rods. Oleksandr Gamayun from the London Institute for Mathematical Sciences, Royal Institution, and Miłosz Panfil from the Faculty of Physics, University of Warsaw, alongside their colleagues, detail how their findings hold true for any state of the system, including crucial finite temperature and ground states. This research is significant because it not only verifies fundamental physical relations like the f-sum rule and detailed balance, but also uncovers a hidden fermionic structure and, remarkably, links the static limit to universal functions mirroring the level spacing distribution of the Gaussian Unitary Ensemble , offering a complete and exact characterisation of dynamic correlations in a strongly correlated one-dimensional system.

Hard Rod Gas Reveals Hidden Fermionic Structure

Scientists have obtained an exact analytic expression for the dynamical structure factor of a one-dimensional gas of hard rods, a significant achievement in understanding strongly correlated quantum systems .This research, detailed in a recent publication, delivers a result valid for arbitrary many-body states, encompassing both finite temperature scenarios and crucial ground state analyses. The team meticulously demonstrated that their expression adheres to fundamental physical principles, including the f-sum rule and the principle of detailed balance, confirming its robustness and accuracy. Crucially, the study unveils a hidden fermionic structure underlying the correlator, offering new insights into the system’s quantum mechanical behaviour.
The core of this breakthrough lies in the calculation of the dynamic structure factor, a key quantity describing how density fluctuations propagate within the hard rod gas. Researchers employed a sophisticated approach leveraging the quantum integrability of the model, allowing for exact computations previously inaccessible in strongly interacting systems. This involved determining the exact form-factor of the density operator, a complex mathematical undertaking that required careful consideration of the system’s collective behaviour and the interplay between particle statistics. This unexpected connection to random matrix theory suggests deep underlying mathematical relationships and provides a powerful tool for analysing the system’s behaviour. The work builds upon the classical hard rod gas model, extending its principles to the quantum realm and offering a complementary approach to the well-studied Lieb-Liniger model. This advancement is particularly relevant for understanding systems like superfluid 4He and Rydberg atoms, where similar quantum phenomena are observed.

The computation involved overcoming significant technical hurdles, notably the summation over Cauchy determinants and the collective dressing of quasimomenta due to particle interactions. By adapting methods from the study of integrable mobile impurities, the scientists successfully navigated these complexities, enabling the calculation of the spectral sum in the thermodynamic limit. This precise calculation provides access to both finite and large space-time regimes, crucial for interpreting scattering experiments and understanding universal low-energy descriptions. The research establishes a foundation for exploring the dynamics of strongly correlated quantum matter with unprecedented accuracy and detail, promising further advancements in the field.

Hard Rod Gas Dynamical Structure Factor Calculation

Scientists have derived an exact analytic expression for the dynamical structure factor of a one-dimensional gas of hard rods, valid for arbitrary states including finite temperature and ground states. This work builds upon the solvability of the Lieb-Liniger model, addressing the long-standing challenge of determining exact dynamic correlation functions, previously limited to large space-time regimes or perturbative approaches. The study pioneers a calculation of the dynamical structure factor using a quantum gas of hard rods as a model system, motivated by the need for an interacting microscopic model amenable to exact computation. Researchers employed a thermodynamic Bethe Ansatz to construct the equilibrium state, defining the particle density ρp(λ) as 1 − ρ0a / 2πn(λ), where ρ0 represents the average particle density and n(λ) is the Fermi-Dirac distribution, a crucial step in characterising the system’s behaviour. The Hamiltonian for the hard-rod gas, H = N∑j=1 p2j + N∑i The team overcame technical difficulties in evaluating the dynamic structure factor, specifically the summation over Cauchy determinants and the correlated nature of the sums over quasimomenta, utilising the Cauchy determinant C[μ,λ] = det(1/(μi − λj)). This innovative methodology revealed a hidden fermionic.

Hard Rod Gas Reveals Hidden Fermionic Structure

Scientists have obtained an exact analytic expression for the dynamical structure factor of a one-dimensional gas of hard rods, a significant achievement in understanding strongly correlated quantum systems. The research demonstrates the validity of this expression for arbitrary states, encompassing both finite temperature states and the crucial ground state, which the team analysed in detail. This work provides a full and exact characterisation of a dynamic correlation function within a strongly correlated interacting system, opening avenues for further investigation. The researchers computed the dynamic structure factor, S(x,t), for a system of hard rods with length ‘a’ and density ‘ρ0’, utilising a recently determined exact form-factor of the density operator. Measurements confirm that the Luttinger parameter, K, for the quantum hard rods is given by K = (1 −ρ0a)2, where ρ0 ≡N/L represents the average particle density.

The team successfully navigated the complexities of calculating the spectral sum in the thermodynamic limit, employing techniques adapted from integrable mobile impurity problems. Tests prove that the dynamic density-density correlation function, S(x,t), can be expressed as an integral over momentum P, with the integral ranging from negative infinity to positive infinity, and incorporating a factor dependent on the Luttinger parameter K and the particle density. Specifically, the equation derived is S(x,t) = ∞ ∫−∞ dP 2π √KP 2e−iPx (2sin(πν))2 ∞ ∫−∞ dseisP Dν(s,t), where ν = aP/(2π). This breakthrough delivers a powerful tool for analysing the dynamics of interacting quantum gases and has potential applications in understanding phenomena in superfluid 4He and Rydberg atom systems.

Hard Rod Gas Reveals Hidden Fermionic Structure

Scientists have derived an exact analytical expression for the dynamical structure factor of a one-dimensional gas of hard rods, a significant achievement in understanding strongly correlated quantum systems. This result holds true for any state of the system, encompassing both finite temperature states and the crucial ground state, which the researchers analysed with particular attention. The expression successfully satisfies established physical principles, including the f-sum rule and detailed balance, confirming its validity and internal consistency. Furthermore,. The authors note the formula’s applicability across all values of position, time, momentum, and frequency, allowing access to features beyond those captured by simpler, low-energy theories like the Luttinger liquid model or hydrodynamics.

This research establishes an important benchmark for studying strongly correlated quantum systems and suggests potential avenues for future investigation, such as deriving the linear and non-linear Luttinger liquid theory from a precise microscopic description. The authors acknowledge a limitation in that the model’s complexity may pose challenges for direct comparison with experimental data, but the exactness of the solution offers a valuable tool for theoretical exploration and validation of approximations. They express gratitude for discussions with several colleagues and acknowledge funding support from the National Science Centre, Poland.