Conformal field theory out of equilibrium: a review

Abstract

We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.

50 pages + 10 pages of references, 5 figures. v2: minor modifications. Review article for special issue of JSTAT on nonequilibrium dynamics in integrable quantum systems

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Cited by in corpus (144)

• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Quench dynamics and relaxation in isolated integrable quantum spin chains
• Emergent hydrodynamics in integrable quantum systems out of equilibrium
• Finite-temperature transport in one-dimensional quantum lattice models
• Quantum quenches in 1+1 dimensional conformal field theories
• Hydrodynamic Diffusion in Integrable Systems
• Quasilocal charges in integrable lattice systems
• Bethe-Boltzmann Hydrodynamics and Spin Transport in the XXZ Chain
• Soliton gases and generalized hydrodynamics
• Solvable Hydrodynamics of Quantum Integrable Systems
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• Nonequilibrium quantum dynamics and transport: from integrability to many-body localization
• Diffusion in generalized hydrodynamics and quasiparticle scattering
• Microscopic origin of ideal conductivity in integrable quantum models
• The Quench Action
• Ballistic transport in the one-dimensional Hubbard model: the hydrodynamic approach
• Lecture notes on Generalised Hydrodynamics
• Analytic solution of the Domain Wall non-equilibrium stationary state
• Transport in out-of-equilibrium XXZ chains: non-ballistic behavior and correlation functions
• A note on generalized hydrodynamics: inhomogeneous fields and other concepts
• Dynamics of hard rods with initial domain wall state
• A geometric viewpoint on generalized hydrodynamics
• Entanglement and matrix elements of observables in interacting integrable systems
• Higher-Order Hydrodynamics in 1D: a Promising Direction and a Null Result
• Signatures of rare states and thermalization in a theory with confinement
• Correlation functions and transport coefficients in generalised hydrodynamics
• From Interacting Particles to Equilibrium Statistical Ensembles
• Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods
• Determination of the Nonequilibrium Steady State Emerging from a Defect
• Non-Equilibrium Dynamics and Weakly Broken Integrability
• Exact large-scale correlations in integrable systems out of equilibrium
• Thermalization and light cones in a model with weak integrability breaking
• Diffusion in deterministic interacting lattice systems
• Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids
• Suppression of transport in non-disordered quantum spin chains due to confined excitations
• Inhomogeneous quenches in the transverse field Ising chain: scaling and front dynamics
• Open Quantum Symmetric Simple Exclusion Process
• Fluctuations in ballistic transport from Euler hydrodynamics
• Transport in the sine-Gordon field theory: from generalized hydrodynamics to semiclassics
• Transport fluctuations in integrable models out of equilibrium
• Entanglement spreading in non-equilibrium integrable systems
• Ballistic front dynamics after joining two semi-infinite quantum Ising chains
• Lower Bounding Diffusion Constant by the Curvature of Drude Weight
• Entanglement dynamics in Rule 54: Exact results and quasiparticle picture
• Quantum Quenches in the Luttinger model and its close relatives
• On thermality of CFT eigenstates
• Spin-charge separation effects in the low-temperature transport of 1D Fermi gases
• Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics
• The Large D Limit of Einstein's Equations
• Heavy Operators and Hydrodynamic Tails
• Energy transport in an integrable parafermionic chain via generalized hydrodynamics
• Fluctuating hydrodynamics, current fluctuations and hyperuniformity in boundary-driven open quantum chains
• Domain-wall melting in the spin-$1/2$ XXZ spin chain: emergent Luttinger liquid with fractal quasi-particle charge
• Front dynamics and entanglement in the XXZ chain with a gradient
• Universal broadening of the light cone in low-temperature transport
• Ballistic macroscopic fluctuation theory
• Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54
• The holographic dual of a Riemann problem in a large number of dimensions
• Locally quasi-stationary states in noninteracting spin chains
• Finite-time universality in nonequilibrium CFT
• Hydrodynamic projections and the emergence of linearised Euler equations in one-dimensional isolated systems
• Time evolution of the Luttinger model with nonuniform temperature profile
• Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems
• Dynamics of charge fluctuations from asymmetric initial states
• Approximate light cone effects in a non-relativistic quantum field theory after a local quench
• Quasi locality of the GGE in interacting-to-free quenches in relativistic field theories
• Fragmentation and emergent integrable transport in the weakly tilted Ising chain
• Generalized hydrodynamics: a perspective
• Low-Temperature Transport in Out-of-Equilibrium XXZ Chains
• Scaling of temporal entanglement in proximity to integrability
• Full Counting Statistics of Charge in Chaotic Many-body Quantum Systems
• Time evolution of entanglement for holographic steady state formation
• Edge singularities and quasi-long-range order in non-equilibrium steady states
• Non-equilibrium quantum transport in presence of a defect: the non-interacting case
• Light-cone velocities after a global quench in a non-interacting model
• Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model
• Diffusion and signatures of localization in stochastic conformal field theory
• Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit
• Anomalous Dynamics and Equilibration in the Classical Heisenberg Chain
• Generalized hydrodynamics, quasiparticle diffusion, and anomalous local relaxation in random integrable spin chains
• Ballistic transport and boundary resistances in inhomogeneous quantum spin chains
• Dynamics of large deviations in the hydrodynamic limit: Non-interacting systems
• Scaling properties of work fluctuations after quenches at quantum transitions
• Thermal transport in $T\bar{T}$-deformed conformal field theories: from integrability to holography
• Energy Transport for Thick Holographic Branes
• Numerical Study of Continuous and Discontinuous Dynamical Phase Transitions for Boundary Driven Systems
• Semiclassical theory of front propagation and front equilibration following an inhomogeneous quantum quench
• Quantum dynamics of impenetrable SU(N) fermions in one-dimensional lattices
• Quenches in initially coupled Tomonaga-Luttinger Liquids: a conformal field theory approach
• Dynamics after quenches in one-dimensional quantum Ising-like systems
• Steady States of Holographic Interfaces
• Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems
• Linearized regime of the generalized hydrodynamics with diffusion
• Hydrodynamic Equations for the Toda Lattice
• Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states
• The hydrodynamic theory of dynamical correlation functions in the XX chain
• Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects
• Ballistic-to-diffusive transition in spin chains with broken integrability
• Symmetry Resolved Measures in Quantum Field Theory: a Short Review
• Global Quenches after Localised Perturbations
• Non-Equilibrium Quantum Spin Dynamics from Classical Stochastic Processes
• Scaling of fronts and entanglement spreading during a domain wall melting
• Exact large-scale fluctuations of the phase field in the sine-Gordon model
• Measurement-induced phase transitions in monitored infinite-range interacting systems
• Quench dynamics of correlated quantum dot proximitized to superconducting lead
• Emergent ballistic transport of Bose-Fermi mixtures in one dimension
• Hydrodynamics of massless integrable RG flows and a non-equilibrium c-theorem
• Non-equilibrium steady state formation in 3+1 dimensions
• Scaling and Universality at Ramped Quench Dynamical Quantum Phase Transition
• Non-equilibrium thermal transport and vacuum expansion in the Hubbard model
• Bridging two quantum quench problems -- local joining quantum quench and Möbius quench -- and their holographic dual descriptions
• Universal out-of-equilibrium dynamics of 1D critical quantum systems perturbed by noise coupled to energy
• Non-Equilibrium Thermodynamics and Conformal Field Theory
• Generalized-Hydrodynamic approach to Inhomogeneous Quenches: Correlations, Entanglement and Quantum Effects
• Efficient computation of cumulant evolution and full counting statistics: application to infinite temperature quantum spin chains
• Continuum approach to real time dynamics of 1+1D gauge field theory: out of horizon correlations of the Schwinger model
• Quantum simulation of the tricritical Ising model in tunable Josephson junction ladders
• Long-time dynamics in quantum spin lattices: ergodicity and hydrodynamic projections at all frequencies and wavelengths
• Meta-Schrödinger invariance
• High-low pressure domain wall for the classical Toda lattice
• Holographic systems far from equilibrium: a review
• Fundamental charges for dual-unitary circuits
• Spin fluctuations after quantum quenches in the S=1 Haldane chain: numerical validation of the semi-semiclassical theory
• Hayden-Preskill recovery in chaotic and integrable unitary circuit dynamics
• Relative entropy and entropy production for equilibrium states in pAQFT
• The Staircase Model: Massless Flows and Hydrodynamics
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Scaling and Universality at Noisy Quench Dynamical Quantum Phase Transitions
• Quantum quench in a driven Ising chain
• Generalized Clausius inequalities and entanglement production in holographic two-dimensional CFTs
• Full counting statistics of energy transfers in inhomogeneous nonequilibrium states of (1+1)D CFT
• Thermal-bath effects in quantum quenches within quantum critical regimes
• Quantum null energy condition in quenched 2d CFTs
• Generalised Hydrodynamics of Particle Creation and Decay
• Thermal modifications of mesons and energy-energy correlators from real-time simulations of a $U(1)$ lattice gauge theory
• Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit
• Gibbs state sampling via cluster expansions
• The massless thermal field and the thermal fermion bosonization in two dimensions
• Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models
• On computing non-equilibrium dynamics following a quench
• Tails of Instability and Decay: a Hydrodynamic Perspective
• Full counting statistics after quantum quenches as hydrodynamic fluctuations
• Fate of the Quasi-condensed State for Bias-driven Hard-Core Bosons in one Dimension
• Generalized stochastic spin-wave theory for open quantum spin systems