Quantum Correlations in Field Theory and Integrable Systems

In this thesis we will investigate some properties of one-dimensional quantum systems. From a theoretical point of view quantum models in one dimension are particularly interesting because they are strongly interacting, since particles cannot avoid each other in their motion, and you we can never ignore collisions. Yet, integrable models often generate new and non-trivial solutions, which could not be found perturbatively. In this dissertation we shall focus on two important aspects of integrable one- dimensional models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will be therefore devoted to the study of the entanglement entropy in one- dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting model. We will derive its Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap will be analysed. In the second part of the thesis we will instead study the dynamics of correlators after a quantum quench , which represent a powerful tool to measure how perturbations and signals propagate through a quantum chain. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, which will be both studied by means of a semi-classical approach. Moreover in the last chapter we will demonstrate a general result about the dynamics of correlation functions of local observables after a quantum quench in integrable systems. In particular we will show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations).

Quantum Integrability in Non-Linear Sigma Models related to Gauge/String Correspondences

The Thermodynamic Bethe Ansatz analysis is carried out for the extended-CP^N class of integrable 2-dimensional Non-Linear Sigma Models related to the low energy limit of the AdS_4xCP^3 type IIA superstring theory. The principal aim of this program is to obtain further non-perturbative consistency check to the S-matrix proposed to describe the scattering processes between the fundamental excitations of the theory by analyzing the structure of the Renormalization Group flow. As a noteworthy byproduct we eventually obtain a novel class of TBA models which fits in the known classification but with several important differences. The TBA framework allows the evaluation of some exact quantities related to the conformal UV limit of the model: effective central charge, conformal dimension of the perturbing operator and field content of the underlying CFT. The knowledge of this physical quantities has led to the possibility of conjecturing a perturbed CFT realization of the integrable models in terms of coset Kac-Moody CFT. The set of numerical tools and programs developed ad hoc to solve the problem at hand is also discussed in some detail with references to the code.

Aspects of Integrability in Gauge/String Correspondence

In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences. We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality. In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic (1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences. In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side. One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size. At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood by the rapidly evolving developments of this extremely exciting research field.

Correlations and Quantum Dynamics of 1D Fermionic Models: New Results for the Kitaev Chain with Long-Range Pairing

In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance as a power law. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range, (ii) purely algebraically. In the latter the entanglement entropy is found to diverge logarithmically. Most interestingly, along the critical lines, long-range pairing breaks also the conformal symmetry. This can be detected via the dynamics of entanglement following a quench. In the second part of the thesis we studied the evolution in time of the entanglement entropy for the Ising model in a transverse field varying linearly in time with different velocities. We found different regimes: an adiabatic one (small velocities) when the system evolves according the instantaneous ground state; a sudden quench (large velocities) when the system is essentially frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations (also as a function of the velocity). Finally, we discussed the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase.