  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 .

High Energy Physics - Theory;Statistical Mechanics;General Relativity and Quantum Cosmology;Quantum Physics