Effective mass and tricritical point for lattice fermions localized by a random mass

This is a numerical study of quasiparticle localization in symmetry class $BD$ (realized, for example, in chiral $p$-wave superconductors), by means of a staggered-fermion lattice model for two-dimensional Dirac fermions with a random mass. For sufficiently weak disorder, the system size dependence of the average (thermal) conductivity $\ensuremath{\sigma}$ is well described by an effective mass ${M}_{\text{eff}}$, dependent on the first two moments of the random mass $M(\mathbit{r})$. The effective mass vanishes linearly when the average mass $\overline{M}\ensuremath{\rightarrow}0$, reproducing the known insulator-insulator phase boundary with a scale invariant dimensionless conductivity ${\ensuremath{\sigma}}_{c}=1/\ensuremath{\pi}$ and critical exponent $\ensuremath{\nu}=1$. For strong disorder a transition to a metallic phase appears, with larger ${\ensuremath{\sigma}}_{c}$ but the same $\ensuremath{\nu}$. The intersection of the metal-insulator and insulator-insulator phase boundaries is identified as a repulsive tricritical point.