Thermal Area Law in Long-Range Interacting Systems

The area law of the bipartite information measure characterizes one of the most fundamental aspects of quantum many-body physics. In thermal equilibrium, the area law for the mutual information universally holds at arbitrary temperatures as long as the systems have short-range interactions. In systems with power-law decaying interactions, $r^{-\alpha}$ ($r$: distance), conditions for the thermal area law are elusive. In this work, we aim to clarify the optimal condition $\alpha> \alpha_c$ such that the thermal area law universally holds. A standard approach to considering the conditions is to focus on the magnitude of the boundary interaction between two subsystems. However, we find here that the thermal area law is more robust than this conventional argument suggests. We show the optimal threshold for the thermal area law by $\alpha_c= (D+1)/2$ ($D$: the spatial dimension of the lattice), assuming a power-law decay of the clustering for the bipartite correlations. Remarkably, this condition encompasses even the thermodynamically unstable regimes $\alpha < D$. We verify this condition numerically, finding that it is qualitatively accurate for both integrable and non-integrable systems. Unconditional proof of the thermal area law is possible by developing the power-law clustering theorem for $\alpha > D$ above a threshold temperature. Furthermore, the numerical calculation for the logarithmic negativity shows that the same criterion $\alpha > (D+1)/2$ applies to the thermal area law for quantum entanglement.