Critical Behaviors of Anderson Transitions in Three Dimensional Orthogonal Classes with Particle-hole Symmetries*

From transfer-matrix calculation of localization lengths and their finite-size scaling analyses, we evaluate critical exponent of the Anderson transition in three dimensional (3D) orthogonal class with particle-hole symmetry, class CI, as ν = 1.16 [ .14, .18 ]. We further study disorder-driven quantum phase transitions in the 3D nodal line Dirac semimetal model, which belongs to class BDI, and estimate critical exponent as ν = 0.80 [ .78, .82 ]. From a comparison of the critical exponents, we conclude that a disorder-driven re-entrant insulator-metal transition from the topological insulator phase in the class BDI to the diffusive metal phase belongs to the same universality class as the Anderson transition in the 3D class BDI. We also discover a topological quantum critical point (TQCP) between one-dimensional (1D) class BDI topological insulator and Anderson insulator and showed that the re-entrant transition in the 3D model is connected to the TQCP in its 1D limit. We argue that an infinitesimally small disorder drives the nodal line Dirac semimetal in the clean limit to the diffusive metal.