Thermal Area Law in Long-Range Interacting Systems

The area law for bipartite information measures states that the information shared between two subsystems scales with the boundary area, not the volume, and is essential for understanding the complexity of many-body systems. It applies to the ground states of various systems and can extend to states in thermal equilibrium with short-range interactions, known as the thermal area law. However, its applicability to long-range interactions remains largely unexplored.

In this work, we address this open question for systems with power-law decaying interactions, r^{-\alpha}, and rigorously demonstrate that the thermal area law holds when \alpha > (D+1)/2, where D represents the spatial dimension. We further show that this condition is not only sufficient but also optimal. This result surpasses the expected \alpha > D+1, which would be anticipated from a straightforward generalization of previous studies. By elucidating the role of power-law decaying bipartite correlations, we reveal the underlying physical mechanism driving the area law, advancing beyond conventional perspectives. Notably, this law remains valid even in thermodynamically unstable regimes where \alpha < D. Our numerical analysis confirms this criterion in the context of mutual information and further demonstrates that the same condition governs the thermal area law for quantum entanglement.