Bound energy, entanglement and identifying critical points in 1D long-range Kitaev model

We investigate the entanglement structure of a bipartite quantum system through the lens of quantum thermodynamics in the absence of conformal symmetry. Specifically, we consider the long-range Kitaev model, where the pairing interaction decays as a power law with exponent $\alpha$, with broken conformal symmetry for $\alpha<3/2$. We analytically show that the bound energy, a quantum thermodynamical quantity, is linearly proportional to the square of entanglement entropy per unit system size for $\alpha=1$ where conformal symmetry is broken. We further show that for all values of $\alpha$, bound energy, in the thermodynamic limit, shows a pronounced minimum at the critical point, which enables the identification of $\mu=1$.