On the Hydrodynamic Approximation of Quantum Integrable Models

E. V. Kopanev, S. V. Kryukov, M. A. Sukhoruchkin

Friedrich Hübner

Generalized hydrodynamics is a framework to study the large scale dynamics of integrable models, special fine-tuned one-dimensional many-body systems that possess an infinite number of local conserved quantities. Unlike classical models, where the microscopic origins of generalized hydrodynamics are better understood, in quantum models it can only be derived using the hydrodynamic formalism. Using the paradigmatic and experimentally relevant repulsive Lieb-Liniger model as an example, this thesis introduces a new viewpoint on the dynamics of quantum integrable models by introducing so-called semi-classical Bethe models. These classical integrable models act as an intermediate description between the microscopic quantum realm and the macroscopic generalized hydrodynamics. After introducing these models and discussing their properties, we study the generalized hydrodynamics equation using new tools and show that solutions to the Euler generalized hydrodynamics equation of the Lieb-Liniger model exist, are unique and do not develop gradient catastrophes. Finally, we discuss new insights into the physics governing the diffusive correction, which, contrary to prior belief, is not described by a Navier-Stokes-like equation. Focusing on the main intuitive ideas, the thesis aims to provide a self-contained overview over these exciting new developments on generalized hydrodynamics.

Romain Vasseur, Vir B. Bulchandani, Christoph Karrasch +1 more

Eldad Bettelheim

Abstract The Whitham approach is a well-studied method to describe non-linear integrable systems. Although approximate in nature, its results may predict rather accurately the time evolution of such systems in many situations given initial conditions. A similarly powerful approach has recently emerged that is applicable to quantum integrable systems, namely the generalized hydrodynamics approach. This paper aims at showing that the Whitham approach is the semiclassical limit of the generalized hydrodynamics approach by connecting the two formal methods explicitly on the example of the Lieb–Liniger model on the quantum side to the non-linear Schrödinger equation on the classical side in the c → 0 limit, c being the interaction parameter. We show how quantum expectation values may be computed in this limit based on the connection established here which is mentioned above.

Irène Burghardt, Klaus B. Møller, Keith H. Hughes

Jacek Pawełczyk

Abstract We derive thermodynamic (TBA) and general hydrodynamic (GHD) equations corrected by virtual processes for integrable QFT on large but finite size space circle. Obtained TBA’s are solved numerically for the sinh-Gordon model. Complicated Euler scale GHD equations are expanded explicitly for small occupation ratio of virtual quasiparticles. The spectrum of velocities for the linear approximation to GHD is numerically calculated.