Universal crossovers between entanglement entropy and thermal entropy

We postulate the existence of universal crossover functions connecting the universal parts of the entanglement entropy to the low temperature thermal entropy in gapless quantum many-body systems. These scaling functions encode the intuition that the same low energy degrees of freedom which control low temperature thermal physics are also responsible for the long range entanglement in the quantum ground state. We demonstrate the correctness of the proposed scaling form and determine the scaling function for certain classes of gapless systems whose low energy physics is described by a conformal field theory. We also use our crossover formalism to argue that local systems which are "natural" can violate the boundary law at most logarithmically. In particular, we show that several non-Fermi liquid phases of matter have entanglement entropy that is at most of order $L^{d-1}\log{(L)}$ for a region of linear size $L$ thereby confirming various earlier suggestions in the literature. We also briefly apply our crossover formalism to the study of fluctuations in conserved quantities and discuss some subtleties that occur in systems that spontaneously break a continuous symmetry.