Pre-Thermalization in Quantum Spin Chains

The results presented in this PhD Thesis aim at understanding thermalization dynamics and, strictly speaking, pre-thermalization in both open and closed quantum spin chains driven out-of-equilibrium. In order to make analytical progress, we consider perturbations of the Quantum Ising Chain (QIC), one of the simplest integrable models, whose non-equilibrium dynamics has been studied in great details in the last few years. In Chapter 2 we introduce for the non-technical reader the main recent ideas behind quantum thermalization and relaxation in integrable and non-integrable quantum many body systems, while Chapter 3 contains an introduction to the current understanding of pre-thermalization. In Chapter 4 we start our study considering a QIC perturbed by a time dependent noise in the direction of the transverse magnetic field [6] [7]. We generalize the concept of pre-thermalization to a noisy quantum many body system driven away from equilibrium simultaneously by a quantum quench of the transverse field and by the noise, highlighting the mechanism behind pre-thermalization and the subsequent relaxation towards the thermal state in a wide class of physically relevant observables. Our main achievement is to show that pre-thermalization occurs quite generally up to time scales where the noise is dominated by the non-equilibrium dynamics induced by the quench of the transverse field, while for longer times pre-thermal plateaux disappear and the noise leaves different signatures in different observables. In Chapter 5 we consider the more challenging problem of pre-thermalization in a closed non-integrable quantum spin chain [8]. We show using a mapping of a weakly non-integrable spin chain into a hard core boson model, that pre-thermalization is successfully captured by a low-density description of the theory, which results in a integrable, bosonic model, that constitutes an excellent description for the out-of-equilibrium dynamics up to intermediate time scales. On the other hand, in order to investigate the late time dynamics we resort to a diagrammatic approach, which serves for a two-fold scope. First, we focus on the effect of the leading inelastic processes, realizing that they constitute only a sub-leading correction to the pre-thermal plateau, found within the effective low-energy description; and secondly, we show from a Quantum Boltzmann equation perspective how inelastic processes are the fundamental ingredient in order to have thermalization in the late time dynamics. The main technical tool used to derive the results in Chapter 4 and 5 is the Keldysh contour technique which is reviewed in the Appendix. This PhD Thesis is based on the following works: 1) J. Marino, A. Silva, Relaxation, pre-thermalization and diffusion in a noisy Quantum Ising Chain, Phys. Rev. B 86, 060408 (Rapid Communications) (2012). 2) M. Marcuzzi, J. Marino, A. Gambassi, A. Silva, Pre-thermalization in a non-integrable quantum spin chain after a quench, arXiv:1307.3738 (2013) (submitted to Phys. Rev. Lett.). 3) J. Marino, A. Silva, Non-Equilibrium Dynamics of a Noisy Quantum Ising Chain: statistics of the work and prethermalization after a sudden quench of the transverse field, arXiv:1309.7595 (2013) (submitted to Phys. Rev. B).