Speaker
Description
Dynamical two-point correlation functions are central objects in the study of quantum many-body systems. They encode the response to local perturbations, determine the transport coefficients, and in general provide the bridge between the microscopic quantum mechanical description and experimentally observable quantities. Obtaining their long-time, large-distance asymptotics for systems in thermal or non-thermal equilibrium, such as a generalised Gibbs ensemble, continues to be an open and difficult problem even for integrable models.
We present rigorous results for the Lieb—Liniger model in the limit of infinite repulsion (also known as the impenetrable Bose gas and the Tonks—Girardeau gas), covering a large class of non-thermal equilibrium conditions and extending previous results established for the thermal equilibrium case. Starting with exact representations of correlation functions as Fredholm determinants of an integrable integral operator, which depends parametrically on distance $x$, time $t$, and on the filling fraction characterizing the thermal or non-thermal equilibrium conditions, we perform a rigorous asymptotic analysis using Riemann—Hilbert techniques.
We identify two classes of filling fractions, characterized by the number of poles on the real axis (a generalization of Fermi points) that contribute, together with the unique saddle point, to the asymptotic expansion. For each class, we derive the asymptotic behaviour as a series in $x^{-1/2}$ as $x$ and $t$ go to infinity, for a fixed ratio $x / t$, with explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants. In the thermal equilibrium case, we verify our results by comparing them with the existing results in the literature and with numerical data, finding good agreement.
Based on joint work with Frank Göhmann, Karol K. Kozlowski, and Alexander Weiße.