Anderson localization and the topology of classifying spaces

Abstract

We construct the generic phase diagrams encoding the topologically distinct localized and delocalized phases of noninteracting fermionic quasiparticles for any symmetry class from the tenfold way in one, two, and three dimensions. To this end, we start from a massive Dirac Hamiltonian perturbed by a generic disorder for any dimension of space and for any one of the ten symmetry classes from the tenfold way. The physics of Anderson localization is then encoded by a two-dimensional phase diagram that we deduce from the topology of the space of normalized Dirac masses. This approach agrees with previously known results and gives an alternative explanation for the even-odd effect in the one-dimensional chiral symmetry classes. We also give a qualitative explanation for the Gade singularity and Griffiths effects in the density of states using the first homotopy group of the normalized Dirac masses in two dimensions. Finally, this approach is used to analyze the stability of massless Dirac fermions on the surface of three-dimensional topological crystalline insulators.

45 pages, 6 figures

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Cited by in corpus (18)

• Delocalization Transition of Disordered Axion Insulator
• Boundary edge networks induced by bulk topology
• One-dimensional non-interacting topological insulators with chiral symmetry
• Mobility Edge in the Anderson model on partially disordered random regular graphs
• ${\mathcal{L}}^2$ localization landscape for highly-excited states
• Localized surfaces of three dimensional topological insulators
• Average-exact mixed anomalies and compatible phases
• Anderson Critical Metal Phase in Trivial States Protected by Average Magnetic Crystalline Symmetry
• Topologically localized insulators
• Marginally localized edges of time-reversal symmetric topological superconductors
• Quantized thermal and spin transports of dirty planar topological superconductors
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Topology of ultra-localized insulators and superconductors
• Instability of QBC systems to Topological Anderson Insulating phases
• Genuine topological Anderson insulator from impurity induced chirality reversal
• Chiral chains with two valleys and disorder of finite correlation length
• Universal localization-delocalization transition in chiral-symmetric Floquet drives
• Topological Tenfold Classification and the Entanglement of Two Qubits