Sufficient Condition for Entanglement Area Laws in Thermodynamically Gapped Spin Systems

We consider general locally interacting arbitrary-dimensional lattice spin systems that are gapped for any system size. We show under reasonable conditions that nondegenerate ground states of such systems obey the entanglement area law. In so doing, we offer an intuitive picture on how a spectral gap restricts the correlations that a ground state can accommodate and leads to such a special feature.

Figure 1

The ($n-1$)-spin and $n$-spin systems differ only locally. The reduced density matrices of the ground states ${\rho }_{\backslash n,l_0}^{(n-1)}$ and ${\rho }_{\backslash n,l_0}^{(n)}$ are obtained by tracing out a local neighborhood of spin $n$. In this illustration, $l_0=3$.

Figure 2

Starting from a system corresponding to region $A^{\prime }$, the system is gradually enlarged while retaining its shape as far as possible.

Figure 3

(a) Energy spectrum of $H_s^{(n)}(\lambda )$. (b) By adding a coupling term, the ground state degeneracy is lifted.