Universal asymptotic eigenvalue distribution of density matrices and corner transfer matrices in the thermodynamic limit

We study the asymptotic behavior of the eigenvalue distribution of the corner transfer matrix ${(M}_{\mathrm{CTM}})$ and the density matrix ${(M}_{\mathrm{DM}})$ in the density-matrix renormalization group. We utilize the relationship $M_{\mathrm{DM}}{=M}_{\mathrm{CTM}}^4,$ which holds for noncritical systems in the thermodynamic limit. We derive the exact and universal asymptotic form of the $M_{\mathrm{DM}}$ eigenvalue distribution for a class of integrable models in the massive regime. For nonintegrable models, the universal asymptotic form is also verified by numerical renormalization group calculations.