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 $S_{\ell }$ with the subsystem radius $R$ can clearly distinguish between these states. For a massless theory with momentum cutoff $\Lambda , S_{\ell }\sim ln[\Lambda R/\ell ]$ for $\Lambda R\gg \ell$, while $S_{\ell }\sim R^0$ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector $k_F, S_{\ell }\sim ln\left[k_{F}R\right]$ for $k_{F}R\gg \ell$. We show how this leads to an "area-log" scaling of total entanglement entropy of fermions, while the extra factor of $\ell$ leads to a leading area law even for massless bosons. At finite temperatures, we find that there is a crossover in the scaling of $S_{\ell }$ from the $T=0$ logarithmic scaling to a high $T$ linear scaling $S_{\ell }\sim \pi TR/3$. The logarithmic scaling exists for larger subsystem sizes for larger values of $\ell$. We provide estimates of temperatures and subsystem sizes where this critical scaling can be seen in experiments on ultracold atoms.