Entanglement in strongly fluctuating quantum many-body states

In this thesis, the scaling behavior of entanglement is investigated in quantum systems with strongly fluctuating ground states. We relate the reduced density matrices of quadratic fermionic and bosonic models to their Green's function matrices in a unified way, and calculate exactly the scaling of the entanglement entropy of finite systems in an infinite universe. In these systems, we observe quantum phase transition by tuning the parameters of the Hamiltonian. Our study shows that although in one dimension there is a unique relation between the quantum phase transition and the scaling behavior of entanglement, this is not necessarily true in higher dimensions.; By exactly solving a spinless fermionic system in two and three dimensions, we investigate the scaling behavior of the block entropy in critical and non-critical phases. We find that the scaling behavior of the block entropy is not exclusively controlled by the decay of the correlation function, and critical phases exist which exhibit the area law, i.e., the entropy of entanglement scales with the area of a given subsystem. We identify two regimes of scaling. The scaling of the block entropy crucially depends on the nature of the excitation spectrum of the system and on the topology of the Fermi surface. Noticeably, in the critical phases the scaling violates the area law and acquires a logarithmic correction only when a well-defined Fermi surface exists in the system. A more stringent criterion for the violation of the area law is thus conjectured, based on the co-dimension which describes the topology of gapless excitation in momentum space. According to this conjecture, logarithmic corrections to the area law appear only in critical phases with codimension d¯ = 1. For all other critical systems, as well as for the non-critical ones with finite ground-state entanglement, the area law holds.; In free bosons systems with a generic quadratic Hamiltonian, we verify that the scaling behavior of the block entropy in higher dimensions always follows the area law, and we explain why logarithmic corrections to area law are impossible.; Spin systems are harder to deal with. Few results can be obtained in higher dimensions. We study one candidate ground state, i.e. the resonant valence bond (RVB) state. In one dimension we find that: for short-range RVB states, the block entropy saturates to a constant, whereas the block entropy for long-range RVB states diverges logarithmically. We also prove that the block entropy in higher dimensions follows the area law for short-range RVB states. This is consistent with the one-dimensional relation between the quantum phase transition and scaling behavior of entanglement.