Entanglement-Entropy for Groundstates, Low-lying and Highly Excited Eigenstates of General (Lattice) Hamiltonians

We investigate the behavior of entanglement-entropy on a broad scale, that is, a large class of systems, Hamiltonians and states describing the interaction of many degrees of freedom. It is one of our aims to show which general characteristics are responsible for the different types of quantitative behavior of entantglement-entropy. Our main lesson is that what really matters is the degree of degeneracy of the spectrum of certain nearby reference Hamiltonians. For calculational convenience we study primarily systems defined on large but finite regions of regular lattices. We show that general vector states, being not related to some short-range Hamiltonian do not lead in the generic case to an area-like behavior of entanglement-entropy. The situation changes if eigenstates of a Hamiltonian with short-range interactions are studied. We found three broad classes of eigenstates. Global groundstates typically lead to entanglement-entropies of subvolumes proportional to the area of the dividing surface. Macroscopically excited (vector)states have in the generic case an entanglement-entropy which is proportional to the enclosed subvolume and, furthermore, display a certain Gibbsian behavior. Low-lying excited states, on the other hand, lead to an entanglement-entropy which is usually proportional to the logarithm of the enclosed subvolume times the area of the dividing surface .