Z2Topological Term, the Global Anomaly, and the Two-Dimensional Symplectic Symmetry Class of Anderson Localization

Abstract

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 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 Z2 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 Z2 topological insulators in the symplectic symmetry class.

Universal scaling of current fluctuations in disordered graphene

“…Some of the predictions made so far are conflicting on this point. Using a supersymmetric nonlinear sigma model, 30 Ref. 23 suggests the existence of a universal minimal conductivity, whereas recent numerical simulations find that σ increases in a logarithmic fashion with the system size, 24 while the beta function exists (single parameter scaling) and always remains positive.…”

Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

“…24 In the T -invariant symplectic case, the unpaired single Dirac flavor avoids the usual spinorbit metal-insulator transition, 9 remaining delocalized even for strong disorder due to a topological term. 10 The generic case of broken-T with all three disorder potentials non-zero realizes the unitary class A, and is believed to flow under renormalization to the plateau transition in the integer quantum Hall effect. 24,29 (See Sec.…”

“…9 In the absence of time-reversal symmetry breaking, Anderson localization is prohibited on the surface of a topological insulator. 7,10 The symplectic metal possesses a finite (non-critical) average density of states at charge neutrality, and a distinct τ (q) spectrum. For a large effective diffusion constant D (induced for a Dirac fermion subject to sufficiently strong disorder, 9 or for a weakly disordered system examined on large length scales), the lowest order result for the multifractal spectrum appears in Eqs.…”

“…In particular, these states are protected from Anderson localization, 8 even in the presence of a "strong" impurity potential, so long as time-reversal invariance is preserved. 9,10 With its 2D Dirac band pinned to an exposed surface, a 3D TI is ideally suited to local probes such as scanning tunneling microscopy (STM). In spectroscopic mode, an STM captures an areal map of the local density of states (LDOS).…”

“…18,[26][27][28] These appear at strong coupling (many impurities) for dirty Dirac fermions. 9,10,24,29 We consider the case of a single flavor Dirac surface band, relevant to (e.g.) the TIs Bi 2 Se 3 and Bi 2 Te 3 (Refs.…”