The entanglement area law for gapped ground states in two dimensions

Theorem 1.1: for fixed $q,R,J,\Delta$ there is $C$ such that for every finite induced domain $\Lambda\subset\mathbb Z^2$ and every Hamiltonian of range $R$ and term norm $J$ with a unique ground vector $\Omega$ and $H-E_0\ge\Delta(1-|\Omega\rangle\langle\Omega|)$, every $A\subseteq\Lambda$ has $S_\Omega(A)\le C|\partial_\Lambda A|$. Corollary 1.2 gives the vertex-boundary forms. The companion proves that on open $L\times L$ grids with nearest-neighbour interactions the ground state has a PEPS approximation of bond dimension $CL^c$ and global error $L^{-1}$. It does not treat dimensions three and higher, Renyi entropies, or degenerate ground states, and gives no explicit constant.

Theorem 1.1: for fixed $q,R,J,\Delta$ there is $C$ such that for every finite induced domain $\Lambda\subset\mathbb Z^2$ and every Hamiltonian of range $R$ and term norm $J$ with a unique ground vector $\Omega$ and $H-E_0\ge\Delta(1-|\Omega\rangle\langle\Omega|)$, every $A\subseteq\Lambda$ has $S_\Omega(A)\le C|\partial_\Lambda A|$. Corollary 1.2 gives the vertex-boundary forms. The companion proves that on open $L\times L$ grids with nearest-neighbour interactions the ground state has a PEPS approximation of bond dimension $CL^c$ and global error $L^{-1}$. It does not treat dimensions three and higher, Renyi entropies, or degenerate ground states, and gives no explicit constant.