Signature of criticality in angular momentum resolved entanglement of scalar fields in d>1

The scaling of entanglement entropy with subsystem size fails to distinguish between the gapped and the gapless ground states of a scalar field theory in $d>1$ dimensions. We show that the scaling of the angular momentum resolved entanglement entropy S_ with the subsystem radius R can clearly distinguish between these states. For a massless theory with momentum cutoff Λ, S_~ln0.16em0ex[ΛR/] for ΛR, while S_~R⁰ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector kF, S_~ln0.16em0ex[kFR] for kFR. We show how this leads to an ``area-log'' scaling of total entanglement entropy of fermions, while the extra factor of leads to a leading area law even for massless bosons. At finite temperatures, we find that there is a crossover in the scaling of S_ from the $T=0$ logarithmic scaling to a high T linear scaling S_~πTR/3. The logarithmic scaling exists for larger subsystem sizes for larger values of . We provide estimates of temperatures and subsystem sizes where this critical scaling can be seen in experiments on ultracold atoms.