We study the unitary equilibration of a chain of interacting spins described by quantum mechanical XV-model in a transverse field. Using the exact solubility of the model we compute the behavior of the Loschmidt echo along with other observables when the system prepared far away from equilibrium in a quantum quench. In a local system like the XV-model there is a maximal speed of information in the system, bounded by the Lieb-Robinson bound. We argue that in the process of unitary equilibration observables fluctuate from their most probable value on a time scale given by the linear size of the system and the speed of interaction. In this thesis, we demonstrate that this time scale is universal and not dependent on the specific details of the Hamiltonian nor the initial state of the system. Furthermore, we provide evidence that suggests that our claims can be generalized to non integrable local systems as well.
We study the unitary equilibration of a chain of interacting spins described by quantum mechanical XV-model in a transverse field. Using the exact solubility of the model we compute the behavior of the Loschmidt echo along with other observables when the system prepared far away from equilibrium in a quantum quench. In a local system like the XV-model there is a maximal speed of information in the system, bounded by the Lieb-Robinson bound. We argue that in the process of unitary equilibration observables fluctuate from their most probable value on a time scale given by the linear size of the system and the speed of interaction. In this thesis, we demonstrate that this time scale is universal and not dependent on the specific details of the Hamiltonian nor the initial state of the system. Furthermore, we provide evidence that suggests that our claims can be generalized to non integrable local systems as well.
Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly solvable example of a quasifree system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double-peaked, universal form.
We study the structure of the revivals in an integrable quantum many-body system, the transverse field XY spin chain, after a quantum quench. The time evolutions of the Loschmidt echo, the magnetization, and the single-spin entanglement entropy are calculated. We find that the revival times for all of these observables are given by integer multiples of ${T}_{\mathrm{rev}}\ensuremath{\simeq}L/{v}_{\mathrm{max}}$, where L is the linear size of the system and vmax is the maximal group velocity of quasiparticles. This revival structure is universal in the sense that it does not depend on the initial state and the size of the quench. Applying nonintegrable perturbations to the XY model, we observe that the revivals are robust against such perturbations: they are still visible at time scales much larger than the quasiparticle lifetime. We therefore propose a generic connection between the revival structure and the locality of the dynamics, where the quasiparticle speed vmax generalizes into the Lieb-Robinson speed vLR.
In this work we investigate the equilibration dynamics after a sudden Hamiltonian quench of a quantum spin system initially prepared in a thermal state. To characterize the equilibration we evaluate the Loschmidt echo, a global measure for the degree of distinguishability between the initial and time-evolved quenched states. We present general results valid for small quenches and detailed analysis of the quantum XY chain. The result is that quantum criticality manifests, even at small but finite temperatures, in a universal double-peaked form of the echo statistics and poor equilibration for sufficiently relevant perturbations. In addition, for this model we find a tight lower bound on the Loschmidt echo in terms of the purity of the initial state and the more easily evaluated Hilbert-Schmidt inner product between initial and time-evolved quenched states. This bound allows us to relate the time-averaged Loschmidt echo with the purity of the time-averaged state, a quantity that has been shown to provide an upper bound on the variance of observables.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
Abstract We consider a local instantaneous quantum dynamical process (QDP) that disturbs the background unitary Hamiltonian dynamics of a spin chain. We consider both a non-unitary incoherent QDP and a coherent unitary QDP intervention of the dynamical evolution of the spin chain. To track the effect of QDP on the dynamics, we investigate the Loschmidt echo, which is quite sensitive to whether the background dynamics is integrable or not. For the integrable case, namely the Heisenberg model, the Loschmidt echo depends on the parameters corresponding to the QDP as well as the time of occurrence of QDP. The probability of reviving the system to its initial state is higher for non-unitary/incoherent QDPs occurring at large time intervals. In the case of a unitary/coherent QDP, some amount of the probability of reviving the state is lost. For the non-integrable background dynamics we consider a kicked Harper model. It exhibits a decaying behaviour when contrasted with integrable dynamics. The decay rate is slower when the corresponding classical Hamiltonian is non-chaotic.
In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.