Entanglement and symmetries in many-body quantum systems

As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian.