Quantum Correlations in Field Theory and Integrable Systems

Abstract

In this PhD thesis we investigate some properties of one-dimensional quantum systems, focusing on two important aspects of integrable models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will therefore be 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 theory with non perturbative solutions. We derive its bipartite Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap is analysed, both analytically and numerically. In particular it is worth mentioning that the numerical analysis of the entropies of the XYZ model presented in this dissertation has never been published in literature. In the second part of the thesis we study the dynamics of correlators after a quantum quench, preparing the system in a squeezed coherent initial state. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, the latter studied by means of the semi-classical theory, the former by a form-factor approach. Moreover in the last chapter we outline a general result about the dynamics of correlation functions of local observables after a quantum quench. In particular we show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model at long times (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations). For the Transverse Field Ising Chain this result means that its dynamics is described by a Generalized Gibbs Ensemble.

PhD thesis, 176 pages. v2: References added

References in corpus (28)

• Thermalization and its mechanism for generic isolated quantum systems
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice
• Non-equilibrium coherence dynamics in one-dimensional Bose gases
• Quantum Quench in the Transverse Field Ising Chain
• Quench dynamics and non equilibrium phase diagram of the Bose-Hubbard model
• The Luttinger model following a sudden interaction switch-on
• Entanglement spectrum in one-dimensional systems
• Dephasing and the steady state in quantum many-particle systems
• Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field
• Strong and weak thermalization of infinite non-integrable quantum systems
• The Statistics of the Work Done on a Quantum Critical System by Quenching a Control Parameter
• Quench dynamics across quantum critical points
• Quantum quench dynamics of the Luttinger model
• Absence of Thermalization in Nonintegrable Systems
• Effective thermal dynamics following a quantum quench in a spin chain
• Dynamical Correlations after a Quantum Quench
• Entanglement entropy of two disjoint blocks in XY chains
• Long time dynamics following a quench in an integrable quantum spin chain: local versus non-local operators and effective thermal behavior
• Quantum quench dynamics of the sine-Gordon model in some solvable limits
• Time evolution of correlations in strongly interacting fermions after a quantum quench
• Unusual Corrections to Scaling in Entanglement Entropy
• Quantum relaxation after a quench in systems with boundaries
• Exact infinite-time statistics of the Loschmidt echo for a quantum quench
• On $c=1$ critical phases in anisotropic spin-1 chains
• Entanglement in spatially inhomogeneous many-fermion systems
• Essential singularity in the Renyi entanglement entropy of the one-dimensional XYZ spin-1/2 chain
• Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at $\Delta$=1/2