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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• Prethermalization in the PXP Model under Continuous Quasiperiodic Driving
• Momentum-space entanglement after smooth quenches
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Quantum Pontus-Mpemba Effects in Real and Imaginary-time Dynamics
• General theory for discrete symmetry-breaking equilibrium states
• Symmetry-prohibited thermalization after a quantum quench
• Extensive increase of entropy in quantum quench
• The time crystal phase emerges from the qubit network under unitary random operations
• Spin Drude weight for the integrable XXZ chain with arbitrary spin
• Coherent charge and spin oscillations induced by local quenches in nanowires with spin-orbit coupling
• Finding local integrals of motion in quantum lattice models in the thermodynamic limit
• Probing Off-diagonal Eigenstate Thermalization with Tensor Networks
• Information Dynamics in a Model with Hilbert Space Fragmentation
• Evaluating thermal expectation values by almost ideal sampling with Trotter gates
• Quantum thermalization must occur in translation-invariant systems at high temperature
• Emergent eigenstate solution for generalized thermalization
• New equilibrium ensembles for isolated quantum systems
• Engineering Hierarchical Symmetries
• Universality in the tripartite information after global quenches: spin flip and semilocal charges
• Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers
• Signatures of Quantum Phase Transitions in Driven Dissipative Spin Chains
• Conserved densities of hard rods: microscopic to hydrodynamic solutions
• Shortcuts to adiabaticity in open quantum critical systems
• Persistence of power-law correlations in nonequilibrium steady states of gapped quantum spin chains
• Quasicondensation and off-diagonal long-range order of hard-core bosons during a free expansion
• Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions
• Thermalization/Relaxation in integrable and free field theories: an Operator Thermalization Hypothesis
• 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
• Critical quantum dynamics of observables at eigenstate transitions
• 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
• A hydrodynamic approach to Stark localization
• Out-of-equilibrium Eigenstate Thermalization Hypothesis
• Non-stationarity and Dissipative Time Crystals: Spectral Properties and Finite-Size Effects
• Kinetics of Quantum Reaction-Diffusion systems
• Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction
• Few-body precursors of topological frustration
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium
• Full counting statistics after quantum quenches as hydrodynamic fluctuations
• Hierarchy of timescales in a disordered spin-$1/2$ XX ladder
• Entanglement of Stationary States in the Presence of Unstable Quasiparticles
• Subsystem Evolution Speed as Indicator of Relaxation
• On the Q operator and the spectrum of the XXZ model at root of unity
• Delocalization in a partially disordered interacting many-body system
• Open-system dynamics in local Lindbladians with chaotic spectra
• Quantum quench and thermalization to GGE in arbitrary dimensions and the odd-even effect
• Thermalization is typical in large classical and quantum harmonic systems
• On computing non-equilibrium dynamics following a quench
• Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers
• 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
• Boundary-induced classical Generalized Gibbs Ensemble with angular momentum
• Universality in Non-Equilibrium Quantum Systems
• Scale invariant distribution functions in quantum systems with few degrees of freedom
• Thermodynamic symmetry resolved entanglement entropies in integrable systems
• Configurable photonic simulator for quantum field dynamics
• Ballistic particle transport and Drude weight in gases
• Rationally independent free fermions with local hopping
• The second law of thermodynamics from concave energy in classical mechanics
• Bistabilities and domain walls in weakly open quantum systems
• Exact results, transient generalized Gibbs ensembles, and analytic approximations for spacetime propagators of massive, real scalar fields in one spatial dimension
• Basis dependence of eigenstate thermalization
• Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices
• Temperature and conditions for thermalization after canonical quenches
• Robust quantum many-body scars in lattice gauge theories
• 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
• Effects of intertube dipole-dipole interactions in nearly integrable one-dimensional ${}^{162}$Dy gases
• Absence of local conserved charges of the Fredkin spin chain and its truncated versions
• Fluctuation Relations for Heat Exchange in the Generalized Gibbs Ensemble
• Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems
• Non-ergodic one-magnon magnetization dynamics of the kagome lattice antiferromagnet
• Efficient computation of average subsystem Bures distance between fermionic Gaussian states
• Effective metal-insulator non-equilibrium quantum phase transition
• Odd Entanglement Entropy and Logarithmic Negativity for Thermofield Double States
• Nonergodic dynamics of the one-dimensional Bose-Hubbard model with a trapping potential
• Deforming integrable models of AdS${}_3$ strings
• Tunable Hilbert space fragmentation and extended critical regime
• Star-topology Registers: NMR and Quantum Information Perspectives
• Generalized uncertainty relation between thermodynamic variables in quantum thermodynamics
• The Bloch theorem in the presence of an additional conserved charge
• Theoretical study on thermalization in isolated quantum systems
• Persistent-current states originating from the Hilbert space fragmentation in momentum space
• Quantum transitions, ergodicity, and quantum scars in the coupled top model
• Fourth-neighbour two-point functions of the XXZ chain and the Fermionic basis approach
• Rényi negativities in non-equilibrium open free-boson chains
• Generalized Gibbs Ensemble from Eigenstate Entanglement Hamiltonian
• Macroscopic Quantum States and Universal Correlations in a Disorder-Order Interface Propagating over a 1D Ground State
• Pauli weight requirement of the matrix elements in time-evolved local operators: dependence beyond the equilibration temperature