Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

We study different aspects of quantum von Neumann and R\'enyi entanglement entropy of one dimensional long-range harmonic oscillators that can be described by well-defined non-local field theories. We show that the entanglement entropy of one interval with respect to the rest changes logarithmically with the number of oscillators inside the subsystem. This is true also in the presence of different boundary conditions. We show that the coefficients of the logarithms coming from different boundary conditions can be reduced to just two different universal coefficients. We also study the effect of the mass and temperature on the entanglement entropy of the system in different situations. The universality of our results is also confirmed by changing different parameters in the coupled harmonic oscillators. We also show that more general interactions coming from general singular Toeplitz matrices can be decomposed to our long-range harmonic oscillators. Despite the long-range nature of the couplings we show that the area law is valid in two dimensions and the universal logarithmic terms appear if we consider subregions with sharp corners. Finally we study analytically different aspects of the mutual information such as its logarithmic dependence to the subsystem, effect of mass and influence of the boundary. We also generalize our results in this case to general singular Toeplitz matrices and higher dimensions.