We study the long-time, large-distance asymptotic behaviour of dynamical two-point correlation functions of the Lieb–Liniger model in the limit of infinite repulsion (the one-dimensional impenetrable Bose gas). Starting with exact representations of correlation functions in terms of Fredholm determinants of an integrable integral operator, we perform the rigorous asymptotic analysis, using Riemann–Hilbert techniques. The integral operator
depends parametrically on time \(t\), distance \(x\) and on the filling fraction — a function that characterizes the thermal or non-thermal equilibrium conditions. We consider a large class of non-thermal equilibrium conditions, extending previous results established for the thermal equilibrium case. The long-time and large-distance asymptotic behaviour is derived for two classes of filling fractions. These classes are characterized by the number of poles on the real axis (a generalization of Fermi points) that, together with the unique saddle point, contribute to the asymptotic expansion. For each class, we derive the long-time, large-distance symptotic behaviour as a series in \(x\)^\(−1/2\) as \(x\) and \(t\) go to infinity for a fixed ratio \(x/t\). We provide explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants in terms of special functions and simple integrals. For the impenetrable Bose gas in thermal equilibrium, we verify the derived asymptotic expansions by comparing them with the existing results in the literature and with numerical data.
This talk is based on joint work with Frank Göhmann, Karol K. Kozlowski, and Alexander Weiße.