The ℤ₂ network model for the quantum spin Hall effect: Two-dimensional Dirac fermions, topological quantum numbers and corner multifractality

The quantum spin Hall effect shares many similarities (and some important differences) with the quantum Hall effect for electric charge. As with the quantum (electric charge) Hall effect, there exists a correspondence between bulk and boundary physics that allows one to characterize the quantum spin Hall effect in diverse and complementary ways. In this paper, we derive from the network model that encodes the quantum spin Hall effect, namely the so-called ℤ₂ network model, a Dirac Hamiltonian in two dimensions. In the clean limit of this Dirac Hamiltonian, we show that the bulk Kane-Mele ℤ₂ invariant is nothing but the SU(2) Wilson loop constructed from the SU(2) Berry connection of the occupied Dirac-Bloch single-particle states. In the presence of disorder, the nonlinear sigma model (NLSM) that is derived from this Dirac Hamiltonian describes a metal-insulator transition in the standard two-dimensional symplectic universality class. In particular, we show that the fermion doubling prevents the presence of a topological term in the NLSM that would change the universality class of the ordinary two-dimensional symplectic metal-insulator transition. This analytical result is fully consistent with our previous numerical studies of the bulk critical exponents at the metal-insulator transition encoded by the ℤ₂ network model. Finally, we improve the quality and extend the numerical study of boundary multifractality in the ℤ₂ topological insulator. We show that the hypothesis of two-dimensional conformai invariance at the metal-insulator transition is verified within the accuracy of our numerical results.