Localization-delocalization transition in Hessian matrices of topologically disordered systems

Using the level-spacing (LS) statistics, we have investigated the localization-delocalization transitions (LDTs) in Hessian matrices of a simple fluid with short-ranged interactions. The model fluid is a prototype of topologically disordered systems and its Hessian matrices are recognized as an ensemble of Euclidean random matrices with elements subject to several kinds of constraints. Two LDTs in the Hessian matrices are found, with one in the positive-eigenvalue branch and the other in the negative-eigenvalue one. The locations and the critical exponents of the two LDTs are estimated by the finite-size scaling for the second moments of the nearest-neighbor LS distributions. Within numerical errors, the two estimated critical exponents are almost coincident with each other and close to that of the Anderson model (AM) in three dimensions. The nearest-neighbor LS distribution at each LDT is examined to be in a good agreement with that of the AM at the critical disorder. We conclude that the LDTs in the Hessian matrices of topologically disordered systems exhibit the critical behaviors of orthogonal universality class.