• correlation functions
• entanglement
• integrability
• quantum quench
• solvable models

The phenomenon of entanglement is probably the most fundamental characteristic distinguishing the quantum from the classical world. It was one of the first aspects of quantum physics to be studied and discussed, and after more than 75 years from the publication of the classical papers by Einstein, Podolsky and Rosen and by Schrodinger, the interest in the properties of entanglement is still growing.
The quantum nature of entanglement makes difficult any intuitive description, and it is better to consider directly what it implies. Entanglement means that the measurement of an observable of a subsystem may affect drastically and instantaneously the possible outcome of a measurement on another part of the system, no matter how far apart it is spatially. The weird and fascinating aspect is that the first measurement affects the second one with infinite speed.
After about 30 years from the appearance of concept of entanglement Bell published one of its most famous works in which he showed that the entanglement forbids an explanation of quantum randomness via hidden variables, unraveling the EPR paradox, once and for all. But only 15 years later, when the Hawking radiation has been put in relation with the entanglement entropy, it has been realized that entanglement could provide unexpected information. The interest in understanding the properties of entangled states has received an impressive boost with the advent of “quantum information”, in nineties. For quantum information the entanglement is a resource, indeed quantum (non-local) correlations are fundamental for quantum teleportation or for enhancing the efficiency of quantum protocols.
The progress made in quantum information for quantifying the entanglement has found important applications in the study of extended quantum systems. In this context the entanglement entropy becomes an indicator of quantum phase transitions, and its behavior at different subsystem sizes and geometries uncovers universal quantities characterizing the critical points. In comparison with quantum correlation functions, the entanglement entropy measures the fundamental properties of critical neighborhoods in a “cleaner” way, e.g. the simple (linear) dependence of entanglement entropy on the central charge in a conformal system.

The first part of the thesis fits into this last genre of research: The entanglement between a subsystem and the rest in the ground state of a 1D system is investigated. In particular the dependence of the entanglement entropies on the geometry of the subsystem and on boundary conditions is widely discussed.

The second part of the thesis is focused on non-equilibrium dynamics. The issue of equilibration of quantum systems has been firstly posed in a seminal paper by von Neumann in 1929, but for long time it remained only an academic problem. Indeed, in solid
state physics there are many difficulties in designing experiments in which the system's parameters can be tuned. Moreover the genuine quantum features of systems could not be preserved for large enough times, because of dissipation and decoherence. Consequently, the research on quantum non-equilibrium problems blew over. Only in the last decade, the many-body physics of ultracold atomic gases overcame these problems: these are highly tunable systems, weakly coupled to the environment, so that quantum coherence is preserved for large times. In fact, a unique feature of many-body physics of cold atoms is the possibility to “simulate” quantum systems in which both the interactions and external potentials can be modified dynamically. In addition, the experimental realization of low-dimensional structures has unveiled the role that dimensionality and conservation laws play in quantum non-equilibrium dynamics. These aspects were addressed recently in a fascinating experiment on the time evolution of non-equlibrium Bose gases in one dimension, interpreted as the quantum equivalent of Newton's cradle.
One of the most important open problems is the characterization of a system that evolves from a non-equilibrium state prepared by suddenly tuning an external parameter. This is commonly called quantum quench and it is the simplest example of out-of-equilibrium dynamics. The time-dependence of the various local observables could be theoretically calculated from first principles, but in general this is a too hard task that cannot be solved even by the most powerful computers (incidentally, this is also the reason why quantum computers can be extremely more effective than classical ones). Insights can be obtained exploiting the most advanced mathematical techniques for low-dimensional quantum systems to draw very general conclusions about the quantum quenches. For example, if for very large times local observables become stationary (even though the entire system will never attain equilibrium), one could describe the system by an effective stationary state that can be obtained without solving the too complicated non-equilibrium dynamics. This is an intriguing aspect of quantum quenches that led to a vigorous research for clarifying the role played by fundamental features of the system, first of all integrability, that is to say the existence of an infinite number of conservation laws. The common belief is that in non-integrable systems (i.e. with a finite number of conservation laws) the stationary state can be described by a single parameter, that is an effective temperature encoding the loss of information about non local observables. Eventually the state at late times is to all intents and purposes equivalent to a thermal one with that temperature. This interesting picture opens the way for a quantum interpretation of thermalization as a local effective description in closed systems.
When there are many (infinite) conserved quantities, as in integrable systems, the effective temperature is not sufficient to describe the system's features at late times. It is widely believed that the behavior of local observables could be explained by generalizations of the celebrated Gibbs ensemble. In the thesis, this hypothesis has been tested and proved for the paradigm of systems undergoing quantum phase transitions: the quantum Ising model. For quenches within the ordered phase of the Ising model, an analytic formula that describes the evolution of the equal-time two-point correlation function of the order parameter has been obtained.