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

Dimitrios Ampelogiannis

TL;DR

This work provides a universal, rigorous framework linking microscopic quantum lattice dynamics to emergent hydrodynamics using C*-algebraic statistical mechanics. It establishes space-time relaxation via space-like clustering and almost-everywhere ergodicity along space-time rays, and introduces a hydrodynamic projection principle that reduces Euler-scale correlations to the space of conserved quantities. By deriving clustering for higher-order cumulants (both classical and free) from two-point clustering, it yields rigorous diffusion bounds in open quantum spin chains and extends hydrodynamic concepts to oscillatory modes. The results demonstrate that large-scale transport is governed by universal hydrodynamic modes, independent of microscopic detail, and provide a principled path toward proving hydrodynamic equations from first principles. The framework unifies ergodicity, clustering, and cumulant decay to support the Boltzmann-Gibbs principle and offers a rigorous basis for diffusion bounds in chaotic quantum systems.

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.

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

TL;DR

This work provides a universal, rigorous framework linking microscopic quantum lattice dynamics to emergent hydrodynamics using C*-algebraic statistical mechanics. It establishes space-time relaxation via space-like clustering and almost-everywhere ergodicity along space-time rays, and introduces a hydrodynamic projection principle that reduces Euler-scale correlations to the space of conserved quantities. By deriving clustering for higher-order cumulants (both classical and free) from two-point clustering, it yields rigorous diffusion bounds in open quantum spin chains and extends hydrodynamic concepts to oscillatory modes. The results demonstrate that large-scale transport is governed by universal hydrodynamic modes, independent of microscopic detail, and provide a principled path toward proving hydrodynamic equations from first principles. The framework unifies ergodicity, clustering, and cumulant decay to support the Boltzmann-Gibbs principle and offers a rigorous basis for diffusion bounds in chaotic quantum systems.

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.
Paper Structure (65 sections, 41 theorems, 353 equations, 1 figure)

This paper contains 65 sections, 41 theorems, 353 equations, 1 figure.

Table of Contents

  1. Set up
  2. Introduction
  3. Motivation
  4. Hydrodynamics
  5. Organisation of the Thesis
  6. Mathematical Set-up: the algebraic approach
  7. Introduction to the C*-algebra framework
  8. Quantum spin lattice description
  9. Quasi-local C*-algebras
  10. Interactions and Dynamics
  11. Lieb-Robinson bounds and quasi-locality
  12. Equilibrium: KMS states
  13. Clustering
  14. Results
  15. Relaxation properties: Clustering and Ergodicity
  16. ...and 50 more sections

Key Result

Proposition 2.1.4

Given a state $ω \in E_{\mathfrak{U}}$ of a unital C$^*$ algebra $\mathfrak{U}$ there exists a (unique, up to unitary equivalence) triple $(H_ω,π_ω,Ω_ω)$ where $H_ω$ is a Hilbert space with inner product $\langle \cdot, \cdot \rangle$, $π_ω$ is a representation of the C$^*$ algebra by bounded operat If additionally we have a group $G$ of automorphisms $\{ τ_g \}_{g \in G}$ of $\mathfrak{U}$ and $ω

Figures (1)

  • Figure 1: Fluid cell approximation doyon_lecture_2020

Theorems & Definitions (81)

  • Definition 2.1.1: C$^*$-algebra
  • Definition 2.1.2: State
  • Definition 2.1.3: C$^*$ dynamical system
  • Proposition 2.1.4: GNS representation
  • Definition 2.2.1: Interaction
  • Definition 2.2.2: Quantum lattice C$^*$-dynamical system
  • Definition 2.2.3: Short-range quantum spin lattice
  • Example 2.2.4
  • Theorem 2.3.1: Lieb-Robinson bound
  • Lemma 2.3.2
  • ...and 71 more