Hydrodynamic projections and the emergence of linearised Euler equations in one-dimensional isolated systems
arXiv:2011.00611 · doi:10.1007/s00220-022-04310-3
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 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.
arXiv commentv1: 64 pages. v2: 70 pages; results and proofs for quantum spin chains clarified
References in corpus (57)
- Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of Lattice Hard-Core Bosons
- Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
- Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
- Lieb-Robinson bounds and the generation of correlations and topological quantum order
- Generalized Gibbs ensemble in integrable lattice models
- Quench dynamics and relaxation in isolated integrable quantum spin chains
- Experimental Observation of a Generalized Gibbs Ensemble
- Emergent hydrodynamics in integrable quantum systems out of equilibrium
- Spectral Gap and Exponential Decay of Correlations
- Transport and conservation laws
- How Algebraic Bethe Ansatz works for integrable model
- Complete Generalized Gibbs Ensemble in an interacting Theory
- Finite-temperature transport in one-dimensional quantum lattice models
- Lieb-Robinson Bounds and the Exponential Clustering Theorem
- Generalized HydroDynamics on an Atom Chip
- Non-stationary coherent quantum many-body dynamics through dissipation
- Quasilocal charges in integrable lattice systems
- Yang-Yang thermodynamics on an atom chip
- Families of quasi-local conservation laws and quantum spin transport
- Integrability and ideal conductance at finite temperatures
- Conformal field theory out of equilibrium: a review
- Kinetic equation for a dense soliton gas
- Lecture notes on Generalised Hydrodynamics
- Generalized hydrodynamics in strongly interacting 1D Bose gases
- Locality in Quantum and Markov Dynamics on Lattices and Networks
- Drude Weight for the Lieb-Liniger Bose Gas
- Structure of the conservation laws in integrable spin chains with short range interactions
- Locality of temperature
- Ergodic properties of a generic non-integrable quantum many-body system in thermodynamic limit
- From Interacting Particles to Equilibrium Statistical Ensembles
- Normal Typicality and von Neumann's Quantum Ergodic Theorem
- Strictly linear light cones in long-range interacting systems of arbitrary dimensions
- Superuniversality of superdiffusion
- The thermodynamic limit of the Whitham equations
- Exact large-scale correlations in integrable systems out of equilibrium
- Generalized hydrodynamics of classical integrable field theory: the sinh-Gordon model
- Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
- Thermalization and pseudolocality in extended quantum systems
- On Generalized Gibbs Ensembles with an infinite set of conserved charges
- Thermodyamic bounds on Drude weights in terms of almost-conserved quantities
- Hydrodynamics in lattice models with continuous non-Abelian symmetries
- Integrability of a deterministic cellular automaton driven by stochastic boundaries
- Time-dependent matrix product ansatz for interacting reversible dynamics
- Quantum invariants of motion in a generic many-body system
- Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field
- Quantum spin systems on infinite lattices
- Some Properties of Correlations of Quantum Lattice Systems in Thermal Equilibrium
- Hydrodynamic Limit for an Hamiltonian System with Boundary Conditions and Conservative Noise
- Rigorous bounds on dynamical response functions and time-translation symmetry breaking
- On the Algebra of Fluctuation in Quantum Spin Chains
- Revisiting the Mazur bound and the Suzuki equality
- Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators
- Hydrodynamic limit for a disordered harmonic chain
- Hydrodynamic Limit for an Anharmonic Chain under Boundary Tension
- Lieb-Robinson Bounds and Quasi-locality for the Dynamics of Many-Body Quantum Systems
- Undular diffusion in nonlinear sigma models
- Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit
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- Correlation functions and transport coefficients in generalised hydrodynamics
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- 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
- Diffusion and superdiffusion from hydrodynamic projections
- The hydrodynamic theory of dynamical correlation functions in the XX chain
- Hydrodynamics and the eigenstate thermalization hypothesis
- The Space of Integrable Systems from Generalised -Deformations
- Diffusive hydrodynamics from long-range correlations
- Free energy fluxes and the Kubo-Martin-Schwinger relation
- Popcorn Drude weights from quantum symmetry
- An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics
- 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