Area law and universality in the statistics of subsystem energy

We introduce R\'enyi entropy of a subsystem energy as a natural quantity which closely mimics the behavior of the entanglement entropy and can be defined for all the quantum many body systems. For this purpose, consider a quantum chain in its ground state and then, take a subdomain of this system with natural truncated Hamiltonian. Since the total Hamiltonian does not commute with the truncated Hamiltonian, the subsystem can be in one of its eigenenergies with different probabilities. Using the fact that the global energy eigenstates are locally close to diagonal in the local energy eigenbasis, we argue that the R\'enyi entropy of these probabilities follows an area law for the gapped systems. When the system is at the critical point, the R\'enyi entropy follows a logarithmic behavior with a universal coefficient. Consequently, our quantity not only detects the phase transition but also determines the universality class of the critical point. Moreover we show that the largest defined probabilities are very close to the biggest Schmidt coefficients. We quantify this by defining a {\it{truncated}} Shannon entropy which its value is almost indistinguishable from the {\it{truncated}} von Neumann entanglement entropy. Compare to the entanglement entropy, our quantity has the advantage of being associated to a natural observable (subsystem Hamiltonian) that can be defined (and probably measured) easily for all the short-range interacting systems. We support our arguments by detailed numerical calculations performed on the transverse field XY-chain.