Entanglement entropy scaling of composite Fermi liquids, Bose metals, and non-Fermi liquids on the lattice

Quantum phases characterized by surfaces of gapless excitations in momentum space are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\sim L^{d-1} \log L$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wavefunction of the $\nu=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wavefunction on various finite-size geometries, we argue that the prefactor of the leading $L \log L$ term is equivalent to that in the analogous free fermion wavefunction. We thus conclude that the ``Widom formula'' likely holds even in this non-Fermi liquid CFL state. Furthermore, we calculate and analyze the entanglement entropy scaling of various other U(1) Bose metal and non-Fermi liquid states built from fermionic slave particles with Fermi surfaces. Overall, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wavefunction sign structure and $S\sim L \log L$ entanglement scaling in such highly entangled gapless phases.