Application of the theory of C*-algebras to the emergence of hydrodynamics in quantum many-body systems

TL;DR

This thesis rigorously derives hydrodynamic behavior from microscopic quantum many-body dynamics using C*-algebra methods, demonstrating universality and establishing foundational results on ergodicity, correlation decay, and diffusion bounds.

Contribution

It provides a rigorous, general framework connecting microscopic quantum dynamics to emergent hydrodynamics, including new bounds on diffusion and ergodic properties.

Findings

Proved ergodicity and decorrelation in quantum lattice models.

Established a Boltzmann-Gibbs principle at the Euler scale.

Derived a positive lower bound on diffusion constants in quantum spin chains.

Abstract

This Ph.D. thesis reports on progress in rigorously establishing hydrodynamic principles from the microscopic Hamiltonian dynamics of quantum many-body systems in a general, non-model-specific manner. Using the C*-algebra framework of statistical mechanics, we treat systems directly in the thermodynamic limit, primarily focusing on quantum lattice models where tools such as Lieb-Robinson bounds yield rigorous statements. We thus provide a proof-of-principle that large-scale behaviours can indeed be seen as emerging from microscopic dynamics, with mathematical proof. We first report on ergodicity results in short-range models with exponentially decaying or finite-range interactions. We show that time-averaged observables converge to their ensemble averages and decorrelate from all other observables almost everywhere within the light-cone defined by Lieb-Robinson bounds. This relaxation property indicates the loss of information at large scales, from which we prove a Boltzmann-Gibbs principle: at the Euler scaling limit of large time and distance, observables project onto hydrodynamic modes (extensive conserved quantities), within correlation functions. These results hold independently of microscopic details, capturing the physical idea that such details are lost at large space-time scales. Regarding finer scales of hydrodynamics, we discuss rigorous lower bounds on the strength of diffusion. We establish a general result on the clustering of n-th order connected correlations within C* dynamical systems. These results are applied to obtain a strictly positive lower bound on the diffusion constant of chaotic open quantum spin chains with nearest-neighbor interactions. This thesis underlines the universality of hydrodynamic principles, provides a framework for establishing them rigorously, and sets the stage for future progress toward the goal of proving the hydrodynamic equations.