Hydrodynamic projections and the emergence of linearised Euler equations in one-dimensional isolated systems

Abstract

One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space-time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the $C^{\ast }$ algebra description of statistical mechanics by the Gelfand-Naimark-Segal construction. Using Araki's exponential clustering and the Lieb-Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.

v1: 64 pages. v2: 70 pages; results and proofs for quantum spin chains clarified

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• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Experimental Observation of a Generalized Gibbs Ensemble
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Generalized hydrodynamics in strongly interacting 1D Bose gases
• From Interacting Particles to Equilibrium Statistical Ensembles
• Superuniversality of superdiffusion
• Hydrodynamics in lattice models with continuous non-Abelian symmetries
• Some Properties of Correlations of Quantum Lattice Systems in Thermal Equilibrium
• Revisiting the Mazur bound and the Suzuki equality
• Undular diffusion in nonlinear sigma models

Cited by in corpus (22)

• Correlation functions and transport coefficients in generalised hydrodynamics
• Ballistic macroscopic fluctuation theory
• Current operators in integrable models: A review
• Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems
• Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems
• A Yang-Baxter integrable cellular automaton with a four site update rule
• Translation symmetry restoration under random unitary dynamics
• Hydrodynamic gauge fixing and higher order hydrodynamic expansion
• The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations
• The hydrodynamic theory of dynamical correlation functions in the XX chain
• Hydrodynamics and the eigenstate thermalization hypothesis
• Diffusive hydrodynamics from long-range correlations
• Free energy fluxes and the Kubo-Martin-Schwinger relation
• An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics
• Popcorn Drude weights from quantum symmetry
• Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
• Long-time dynamics in quantum spin lattices: ergodicity and hydrodynamic projections at all frequencies and wavelengths
• Entanglement Rényi Entropies from Ballistic Fluctuation Theory: the free fermionic case
• Almost everywhere ergodicity in quantum lattice models
• Spin Drude weight for the integrable XXZ chain with arbitrary spin
• Universal efficiency boost in prethermal quantum heat engines at negative temperature
• Derivation of Euler equations from quantum and classical microscopic dynamics