Generalized Gibbs ensemble in integrable lattice models

Abstract

The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin-spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.

Review article for special issue of JSTAT on non-equilibrium dynamics in integrable systems

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• Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials
• Gaussian-state Ansatz for the non-equilibrium dynamics of quantum spin lattices
• Quantum many-body scars in spin-1 Kitaev chain with uniaxial single-ion anisotropy
• Quantum master equation from the eigenstate thermalization hypothesis
• Thermalization of a Lipkin-Meshkov-Glick model coupled to a bosonic bath
• Universality in the tripartite information after global quenches: (generalised) quantum XY models
• Observation of slow relaxation due to Hilbert space fragmentation in strongly interacting Bose-Hubbard chains
• Zero modes of local operators in 2d CFT on a cylinder
• Chaotic and quantum dynamics in driven-dissipative bosonic chains
• Fluctuation Theorem for Quasi-Integrable Systems
• Excitations in the Yang-Gaudin Bose gas
• Scaling at the OTOC Wavefront: free versus chaotic models
• Machine learning algorithms based on generalized Gibbs ensembles
• Quasiparticle Picture for Entanglement Hamiltonians in Higher Dimensions
• Exact many-body scars based on pairs or multimers in a chain of spinless fermions
• Critical quench dynamics of random quantum spin chains: Ultra-slow relaxation from initial order and delayed ordering from initial disorder
• Steady off-diagonal long-range order state in a half-filled dimerized Hubbard chain induced by a resonant pulsed field
• High-low pressure domain wall for the classical Toda lattice
• Localization and integrability breaking in weakly interacting Floquet circuits
• A remark on the notion of independence of quantum integrals of motion in the thermodynamic limit
• Conserved Quantities from Entanglement Hamiltonian
• Quantum reaction-limited reaction-diffusion dynamics of noninteracting Bose gases
• Non-thermal eigenstates and slow relaxation in quantum Fredkin spin chains
• Signature of Generalized Gibbs Ensemble Deviation from Equilibrium: Negative Absorption Induced by a Local Quench
• $T\overline{T}$ deformations and the width of fundamental particles
• Generalized thermalization for integrable system under quantum quench
• Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure
• Relaxation to Gaussian Generalized Gibbs Ensembles in Quadratic Bosonic Systems in the Thermodynamic Limit
• Nonequilibrium phenomena in nonlinear lattices: from slow relaxation to anomalous transport
• Constrained thermalisation and topological superconductivity
• Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory
• Quench dynamics of Rényi negativities and the quasiparticle picture
• Thermalization and hydrodynamics in an interacting integrable system: the case of hard rods
• Breakdown of Measurement-Induced Phase Transitions Under Information Loss
• Probing chaos in the spherical p-spin glass model
• Hopf algebras and solvable unitary circuits
• The relevant excitations for the one-body function in the Lieb-Liniger model
• Subsystem entropy in 2d CFT and KdV ETH
• The Adiabatically Deformed Ensemble: Engineering Non-Thermal States of Matter
• Local integrals of motion in dipole-conserving models with Hilbert space fragmentation
• Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices
• Connecting scrambling and work statistics for short-range interactions in the harmonic oscillator
• Long-time equilibration can determine transient thermality
• Generalized hydrodynamics of integrable quantum circuits
• Quantum memory at an eigenstate phase transition in a weakly chaotic model
• Quantum quench in a driven Ising chain
• Integrability of Goldilocks quantum cellular automata
• Construction of the steady state density matrix and quasilocal charges for the spin-1/2 XXZ chain with boundary magnetic fields
• Single-quasiparticle eigenstate thermalization
• Tracking locality in time evolution of disordered systems
• Intensity pseudo-localized phase in the glassy random laser
• Prethermalization in the PXP Model under Continuous Quasiperiodic Driving
• Exact volume-law entangled zero-energy eigenstates in a large class of spin models
• Key Observable for Linear Thermalization
• Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations
• Thermalization of a Closed Sachdev-Ye-Kitaev System in the Thermodynamic Limit
• Atypical energy eigenstates in the Hubbard chain and quantum disentangled liquids
• Accuracy of time-dependent GGE under weak dissipation
• General interaction quenches in a Luttinger liquid
• Non-Markovianity of subsystem dynamics in isolated quantum many-body systems
• The second law of thermodynamics from concavity of energy eigenvalues
• Momentum-space entanglement after smooth quenches
• Symmetry-prohibited thermalization after a quantum quench
• Finding local integrals of motion in quantum lattice models in the thermodynamic limit
• Coherent charge and spin oscillations induced by local quenches in nanowires with spin-orbit coupling
• Information Dynamics in a Model with Hilbert Space Fragmentation
• Extensive increase of entropy in quantum quench
• Quantum thermalization must occur in translation-invariant systems at high temperature
• General theory for discrete symmetry-breaking equilibrium states
• Quantum Pontus-Mpemba Effects in Real and Imaginary-time Dynamics
• Evaluating thermal expectation values by almost ideal sampling with Trotter gates
• Engineering Hierarchical Symmetries
• Emergent eigenstate solution for generalized thermalization
• Universality in the tripartite information after global quenches: spin flip and semilocal charges
• Spin Drude weight for the integrable XXZ chain with arbitrary spin
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Probing Off-diagonal Eigenstate Thermalization with Tensor Networks
• The time crystal phase emerges from the qubit network under unitary random operations
• New equilibrium ensembles for isolated quantum systems
• Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers
• Critical quantum dynamics of observables at eigenstate transitions
• Quasicondensation and off-diagonal long-range order of hard-core bosons during a free expansion
• Signatures of Quantum Phase Transitions in Driven Dissipative Spin Chains
• Persistence of power-law correlations in nonequilibrium steady states of gapped quantum spin chains
• Kinetics of Quantum Reaction-Diffusion systems
• Thermalization/Relaxation in integrable and free field theories: an Operator Thermalization Hypothesis
• Out-of-equilibrium Eigenstate Thermalization Hypothesis
• Conserved densities of hard rods: microscopic to hydrodynamic solutions
• A hydrodynamic approach to Stark localization
• Shortcuts to adiabaticity in open quantum critical systems
• Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions
• Anomalous charge transport in the sine-Gordon model
• Reaction-diffusion dynamics of the weakly dissipative Fermi gas
• Taking the temperature of quantum many-body scars
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• $\nu$-QSSEP: A toy model for entanglement spreading in stochastic diffusive quantum systems
• Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit
• Non-stationarity and Dissipative Time Crystals: Spectral Properties and Finite-Size Effects
• Thermodynamic symmetry resolved entanglement entropies in integrable systems
• Hierarchy of timescales in a disordered spin-$1/2$ XX ladder
• Boundary-induced classical Generalized Gibbs Ensemble with angular momentum
• Delocalization in a partially disordered interacting many-body system
• On the Q operator and the spectrum of the XXZ model at root of unity
• Timescales and necessary conditions for hydrodynamization in one-dimensional Bose gases
• Long-lived non-thermal states in pumped one-dimensional systems of hard-core bosons
• Realization of Two-dimensional Discrete Time Crystals with Anisotropic Heisenberg Coupling
• Full counting statistics after quantum quenches as hydrodynamic fluctuations
• Universality in Non-Equilibrium Quantum Systems
• Scale invariant distribution functions in quantum systems with few degrees of freedom
• Configurable photonic simulator for quantum field dynamics
• Ballistic particle transport and Drude weight in gases
• Quantum quench and thermalization to GGE in arbitrary dimensions and the odd-even effect
• Subsystem Evolution Speed as Indicator of Relaxation
• Few-body precursors of topological frustration
• On computing non-equilibrium dynamics following a quench
• Open-system dynamics in local Lindbladians with chaotic spectra
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium
• Entanglement of Stationary States in the Presence of Unstable Quasiparticles
• Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction
• Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers
• Thermalization is typical in large classical and quantum harmonic systems
• Robust quantum many-body scars in lattice gauge theories
• Exact results, transient generalized Gibbs ensembles, and analytic approximations for spacetime propagators of massive, real scalar fields in one spatial dimension
• Theoretical study on thermalization in isolated quantum systems
• Rationally independent free fermions with local hopping
• Generalized Gibbs Ensemble from Eigenstate Entanglement Hamiltonian
• Rényi negativities in non-equilibrium open free-boson chains
• The second law of thermodynamics from concave energy in classical mechanics
• Bistabilities and domain walls in weakly open quantum systems
• Effects of intertube dipole-dipole interactions in nearly integrable one-dimensional ${}^{162}$Dy gases
• Fluctuation Relations for Heat Exchange in the Generalized Gibbs Ensemble
• Loop Charges and Fragmentation in Pairwise Difference Conserving Circuits
• Additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras
• Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices
• Basis dependence of eigenstate thermalization
• Temperature and conditions for thermalization after canonical quenches
• Absence of local conserved charges of the Fredkin spin chain and its truncated versions
• Non-ergodic one-magnon magnetization dynamics of the kagome lattice antiferromagnet
• Efficient computation of average subsystem Bures distance between fermionic Gaussian states
• Persistent-current states originating from the Hilbert space fragmentation in momentum space
• Tunable Hilbert space fragmentation and extended critical regime
• Generalized uncertainty relation between thermodynamic variables in quantum thermodynamics
• The Bloch theorem in the presence of an additional conserved charge
• Odd Entanglement Entropy and Logarithmic Negativity for Thermofield Double States
• Nonergodic dynamics of the one-dimensional Bose-Hubbard model with a trapping potential
• Quantum transitions, ergodicity, and quantum scars in the coupled top model
• Effective metal-insulator non-equilibrium quantum phase transition
• Fourth-neighbour two-point functions of the XXZ chain and the Fermionic basis approach
• Macroscopic Quantum States and Universal Correlations in a Disorder-Order Interface Propagating over a 1D Ground State
• Deforming integrable models of AdS${}_3$ strings
• Pauli weight requirement of the matrix elements in time-evolved local operators: dependence beyond the equilibration temperature
• Star-topology Registers: NMR and Quantum Information Perspectives
• Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems