Universal relation between thermal entropy and entanglement entropy in conformal field theories

Inspired by the holographic computation of large interval entanglement entropy of two-dimensional (2D) conformal field theory at high temperature, it was proposed that the thermal entropy is related to the entanglement entropy as ${S}_{th}=\underset{l\ensuremath{\rightarrow}0}{\mathrm{lim}}({S}_{EE}(R\ensuremath{-}l)\ensuremath{-}{S}_{EE}(l))$. In this paper, we prove this relation for 2D CFT with a discrete spectrum in two different ways. Moreover, we discuss this relation for a 2D noncompact free scalar, which is a gapless CFT with continuous spectrum. We show that it could be recovered, after appropriately regularizing the theory.