Rényi entropy and subsystem distances in finite size and thermal states in critical XY chains

We study R\'enyi entropy and subsystem distances of one interval in finite size and thermal states in critical XY chains, focusing on critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, R\'enyi entropy, subsystem trace distance, Schatten two-distance and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we compare the spin chain numerical results with the analytical results in conformal field theories and find perfect matches in the continuum limit.