We present steps towards an ab initio derivation of generalised hydrodynamics in quantum integrable models, starting from the Bethe wave functions, and explained on the example of the repulsive Lieb-Liniger model. This includes an identification of the generalised hydrodynamics quasi-particles as wave packets in the quantum model. These wave packets evolve according to a classical particle model and collect two-particle scattering shifts similar to solitons in integrable PDEs. We then discuss potential routes to obtain the generalised hydrodynamics equation for average conserved densities in long-wavelength states from this description. As part of this, we provide an explicit formula for the action of the spectral phase-space density operator on Bethe wave functions, and show that it generates local conserved densities.
We present steps towards an ab initio derivation of generalised hydrodynamics in quantum integrable models, starting from the Bethe wave functions, and explained on the example of the repulsive Lieb-Liniger model. This includes an identification of the generalised hydrodynamics quasi-particles as wave packets in the quantum model. These wave packets evolve according to a classical particle model and collect two-particle scattering shifts similar to solitons in integrable PDEs. We then discuss potential routes to obtain the generalised hydrodynamics equation for average conserved densities in long-wavelength states from this description. As part of this, we provide an explicit formula for the action of the spectral phase-space density operator on Bethe wave functions, and show that it generates local conserved densities.
Abstract We present steps towards an ab initio derivation of generalized hydrodynamics (GHD) in quantum integrable models, starting from the Bethe wave functions, and explained on the example of the repulsive Lieb–Liniger model. This includes an identification of the GHD quasi-particles as wave packets in the quantum model. These wave packets evolve according to a classical particle model and collect two-particle scattering shifts similar to solitons in integrable PDEs. We then discuss potential routes to obtain the GHD equation for average conserved densities in long-wavelength states from this description. As part of this, we provide an explicit formula for the action of the spectral phase-space density operator on Bethe wave functions, and show that it generates local conserved densities.
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.
We show that the equations of generalized hydrodynamics (GHD), a hydrodynamic theory for integrable quantum systems at the Euler scale, emerge in full generality in a family of classical gases, which generalize the gas of hard rods. In this family, the particles, upon colliding, jump forward or backward by a distance that depends on their velocities, reminiscent of classical soliton scattering. This provides a "molecular dynamics" for GHD: a numerical solver which is efficient, flexible, and which applies to the presence of external force fields. GHD also describes the hydrodynamics of classical soliton gases. We identify the GHD of any quantum model with that of the gas of its solitonlike wave packets, thus providing a remarkable quantum-classical equivalence. The theory is directly applicable, for instance, to integrable quantum chains and to the Lieb-Liniger model realized in cold-atom experiments.
Abstract Generalised hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables, and the GHD equation can be seen as a 2+1 2 + 1 -dimensional classical field theory. In its 1+1 1 + 1 -dimensional reduction corresponding to the case without external forces, we further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.
Generalised hydrodynamics (GHD) is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from the delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more. Yet it has only been applied to (1+1)-dimensional systems and generalisation to higher dimensions of space is non-trivial. We study the Boussinesq equation which, while generally considered to be less physically relevant than the KdV equation, is interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili (KP) equation, a prototypical and universal example of a nonlinear integrable PDE in (2+1) dimensions. We follow a heuristic approach inspired by the Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq soliton gas. Such approach allows for a statistical mechanics interpretation of the Boussinesq soliton gas that comes naturally with the GHD picture. This is to be seen as a first step in the construction of the KP soliton gas, yielding insight on some classes of solutions from which we may be able to build an intuition on how to devise a more general theory. This also offers another perspective on the construction of anisotropic bidirectional soliton gases previously introduced phenomenologically by Congy et al (2021).
Generalised hydrodynamics (GHD) is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from the delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more. Yet it has only been applied to (1+1)-dimensional systems and generalisation to higher dimensions of space is non-trivial. We study the Boussinesq equation which, while generally considered to be less physically relevant than the KdV equation, is interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili (KP) equation, a prototypical and universal example of a nonlinear integrable PDE in (2+1) dimensions. We follow a heuristic approach inspired by the Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq soliton gas. Such approach allows for a statistical mechanics interpretation of the Boussinesq soliton gas that comes naturally with the GHD picture. This is to be seen as a first step in the construction of the KP soliton gas, yielding insight on some classes of solutions from which we may be able to build an intuition on how to devise a more general theory. This also offers another perspective on the construction of anisotropic bidirectional soliton gases previously introduced phenomenologically by Congy et al (2021).