  We investigate quantum many-body systems where all low-energy states are
entangled. As a tool for quantifying such systems, we introduce the concept of
the entanglement gap, which is the difference in energy between the
ground-state energy and the minimum energy that a separable (unentangled) state
may attain. If the energy of the system lies within the entanglement gap, the
state of the system is guaranteed to be entangled. We find Hamiltonians that
have the largest possible entanglement gap; for a system consisting of two
interacting spin-1/2 subsystems, the Heisenberg antiferromagnet is one such
example. We also introduce a related concept, the entanglement-gap temperature:
the temperature below which the thermal state is certainly entangled, as
witnessed by its energy. We give an example of a bipartite Hamiltonian with an
arbitrarily high entanglement-gap temperature for fixed total energy range. For
bipartite spin lattices we prove a theorem demonstrating that the entanglement
gap necessarily decreases as the coordination number is increased. We
investigate frustrated lattices and quantum phase transitions as physical
phenomena that affect the entanglement gap.
