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 gapped and gapless ground state of a scalar field theory in $d>1$ dimensions. We show that the scaling of angular momentum resolved entanglement entropy $S_{\mathrm{\ell }}$ with the subsystem radius $R$ can clearly distinguish between these states. For a massless theory with momentum cut-off $\mathrm{\Lambda }$, $S_{\mathrm{\ell }}\sim \ln [\mathrm{\Lambda }R\mathrm{/}\mathrm{\ell }]$ for $\mathrm{\Lambda }R\gg \mathrm{\ell }$, while $S_{\mathrm{\ell }}\sim R^0$ for the massive theory. In contrast, for a free Fermi gas with Fermi wave vector $k_F$, $S_{\mathrm{\ell }}\sim \ln [k_{F}R]$ for $k_{F}R\gg \mathrm{\ell }$. We show how this leads to an ``area-log’’ scaling of total entanglement entropy of Fermions, while the extra factor of $\mathrm{\ell }$ leads to a leading area law even for massless Bosons.