The Z_2 network model for the quantum spin Hall effect: two-dimensional Dirac fermions, topological quantum numbers, and corner multifractality

Abstract

The quantum spin Hall effect shares many similarities (and some important differences) with the quantum Hall effect for the electric charge. As with the quantum (electric charge) Hall effect, there exists a correspondence between bulk and boundary physics that allows 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, the so-called Z_2 network model, a Dirac Hamiltonian in two dimensions. In the clean limit of this Dirac Hamiltonian, we show that the bulk Kane-Mele Z_2 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 non-linear 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 Z_2 network model. Finally, we improve the quality and extend the numerical study of boundary multifractality in the Z_2 topological insulator. We show that the hypothesis of two-dimensional conformal invariance at the metal-insulator transition is verified within the accuracy of our numerical results.

31 pages, 9 figures, submitted to New J Phys

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• Network Models of Photonic Floquet Topological Insulators
• Measurement of a topological edge invariant in a microwave network
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• Momentum space metric, non-local operator, and topological insulators
• Disorder-induced metal-insulator transitions in three-dimensional topological insulators and superconductors
• Spin-directed network model for the surface states of weak three-dimensional ${\mathbb{Z}}_2^{}$ topological insulators
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• Thermal metal-insulator transition in a helical topological superconductor
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• Helical Majorana edge mode in a superconducting antiferromagnetic quantum spin Hall insulator
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• Conductance distributions in disordered quantum spin-Hall systems
• Exact exponents for the spin quantum Hall transition in the presence of multiple edge channels
• Localized surfaces of three dimensional topological insulators
• Critical exponent for the quantum spin Hall transition in Z_2 network model