${\mathbb{Z}}_2$ Topological Term, the Global Anomaly, and the Two-Dimensional Symplectic Symmetry Class of Anderson Localization

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a ${\mathbb{Z}}_2$ topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The ${\mathbb{Z}}_2$ topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This ${\mathbb{Z}}_2$ topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of ${\mathbb{Z}}_2$ topological insulators in the symplectic symmetry class.

Figure 1

The energy eigenvalue spectrum in the vicinity of the band center for the Dirac kernel $D^{\prime }[Q_t]$, Eq. (4b), is computed numerically as a function of the parameter $0\le t\le 1$ for $(a_0\Delta {)}^{-1}=1$ and $N=11$. The field $Q_t$ interpolates between $Q_i$ when $t=0$ and $Q_f$ when $t=1$. (e) and (f): Same as in (c) and (d), respectively, except for the addition of a small perturbation so as to lift any accidental or quasi degeneracies.