Abstract We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-1/2 Heisenberg chain.
Abstract We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-1/2 Heisenberg chain.
We study quenches in integrable spin-1/2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit.
We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.
In the present thesis we study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench in presence of disorder. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive, one-body versus many-body) and on the localization properties of the final Hamiltonian. We then show an extension of the Wang-Landau algorithm which allows the computation of weighted distributions associated to quantum quenches, like the diagonal and the GGE ensemble expectation-value distributions. We present results on three one-dimensional models, the Anderson model, a disordered one-dimensional fermionic chain with long-range hopping, and the disordered Ising/XY spin chain.
In this Thesis I present the results of four research projects I carried out during my Ph.D. training at SISSA. The broad topic of my research was the mechanism of relaxation in isolated, integrable quantum systems. Roughly speaking, integrable quantum systems are one-dimensional lattice models or field theories that feature an extensively large number of conservation laws beyond energy and momentum conservation, which greatly influence their dynamical behaviour. Integrable systems have enjoyed a considerable interest in the last decade, mainly due to the development of ultracold atomic techniques. These techniques made it possible to implement experimentally a wealth of one-dimensional quantum models whose significance had been purely theoretical before. The theoretical interest in integrable systems stems in part from the fact that in many aspects they are exactly solvable. Exact analytical formulas are available for their spectrum and their equilibrium thermodynamic properties, creating an incentive to study analytically their non-equilibrium behaviour as well. Exact results for the dynamics of integrable systems may serve as a benchmark for confirming theories in statistical mechanics or testing approximative methods. To this day, computing analytically the entire time evolution of observables in a genuinely interacting integrable system is prohibitively difficult in general. In this Thesis, I concentrate on the description of the system at long times after it is brought out of equilibrium. Homogeneous, translationally invariant integrable systems are expected to relax to a stationary state, in which local observables are described by a generalization of the Gibbs ensemble that takes into consideration all the conservation laws beyond the energy. Two of the projects presented here concern this relaxed state. In Chapter 2, a method is presented for computing the Rényi entropies of the Generalized Gibbs Ensemble describing the relaxed state in the integrable Heisen- berg XXZ spin chain. Rényi entropies provide information about the relaxed state and its entanglement properties, going beyond the ordinary von Neumann thermo- dynamic entropy. The method presented there is very general; in principle it works in any integrable model and any translationally invariant thermodynamic macrostate. An alternative to the GGE in describing relaxed states after a quantum quench is the Quench Action approach. This approach is based on the explicit analytical knowledge of the overlaps between the initial state and the Hamiltonian eigenstates. In Chapter 3, this approach is generalized to a spin-1 integrable spin chain, the SU(3) invariant Lai–Sutherland model. The evolution of entanglement is then studied using the quasiparticle picture, which features ballistically propagating classical particle pairs carrying entanglement. If the requirement of translational invariance of the initial state is relaxed, various transport phenomena arise in integrable systems. The evolution of slowly varying, smoothly distributed states is described by a generalization of hydrodynamics, which takes into consideration the infinitely many local conservation laws. At the scale of non-diffusive, ballistic transport, the main feature of this generalized hydrodynamics (GHD) is the existence of continuity equations for the occupation number of each mode. Further two projects are presented in this Thesis which are concerned with GHD.I n Chapter 4, a numerical method is presented for computing the evolution of entanglement entropies within GHD. This method combines the quasiparticle picture of entanglement with a molecular dynamics simulation of GHD, the so-called fleagas. Numerical evidence is shown that this method reproduces previously known analytical results in GHD. Its versatility is demonstrated by computing another entanglement-related quantity, the mutual information, for which no analytical results are available. Finally, in Chapter 5, the GHD framework is generalized to a system of spin-1/2 fermions with repulsive contact interaction, the Yang–Gaudin model. It is shown that at low temperatures, the dynamics of the system is described by the superposition of two uncoupled, inhomogeneous conformal field theories. This fact can be interpreted as a separation of the spin and the charge degrees of freedom of the system.
What is the long-time behavior of a quantum system after it is brought out of equilibrium? In a generic case, it has been proposed that an effective thermalization occurs, and the local properties of the system can then be described by a thermal Gibbs ensemble. However, in integrable systems, with infinitely many conservation laws, the answer is far more complicated. It was shown in several special cases that the long-time steady state can be exactly described by the so called complete generalized Gibbs ensemble. The latter can be obtained by maximizing the entropy with constraints imposed by the conservation of energy, but also by additional carefully chosen subset of the integrals of motion. In this work, the authors analyze more general initial states, focusing on integrable XXZ Heisenberg spin chains. They find that the complete generalized Gibbs ensemble indeed correctly describes the system at long times after the quench.
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.