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 ${S}_{\ensuremath{\ell}}$ with the subsystem radius $R$ can clearly distinguish between these states. For a massless theory with momentum cutoff $\mathrm{\ensuremath{\Lambda}}, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[\mathrm{\ensuremath{\Lambda}}R/\ensuremath{\ell}]$ for $\mathrm{\ensuremath{\Lambda}}R\ensuremath{\gg}\ensuremath{\ell}$, while ${S}_{\ensuremath{\ell}}\ensuremath{\sim}{R}⁰$ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector ${k}_F, {S}_{\ensuremath{\ell}}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}[{k}_FR]$ for ${k}_FR\ensuremath{\gg}\ensuremath{\ell}$. We show how this leads to an ``area-log'' scaling of total entanglement entropy of fermions, while the extra factor of $\ensuremath{\ell}$ leads to a leading area law even for massless bosons. At finite temperatures, we find that there is a crossover in the scaling of ${S}_{\ensuremath{\ell}}$ from the $T=0$ logarithmic scaling to a high $T$ linear scaling ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\pi}TR/3$. The logarithmic scaling exists for larger subsystem sizes for larger values of $\ensuremath{\ell}$. We provide estimates of temperatures and subsystem sizes where this critical scaling can be seen in experiments on ultracold atoms.