How spectrum-wide quantum criticality protects surface states of\n topological superconductors from Anderson localization: Quantum Hall plateau\n transitions (almost) all the way down

Abstract

We review recent numerical studies of two-dimensional (2D) Dirac fermion\ntheories that exhibit an unusual mechanism of topological protection against\nAnderson localization. These describe surface-state quasiparticles of\ntime-reversal invariant, three-dimensional (3D) topological superconductors\n(TSCs), subject to the effects of quenched disorder. Numerics reveal a\nsurprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D\nquantum Hall effects in classes A, C, and D. Conventional arguments derived\nfrom the non-linear $\\sigma$-model picture imply that most TSC surface states\nshould Anderson localize for arbitrarily weak disorder (CI, AIII), or exhibit\nweak antilocalizing behavior (DIII). The numerical studies reviewed here\ninstead indicate spectrum-wide surface quantum criticality, characterized by\nrobust eigenstate multifractality throughout the surface-state energy spectrum.\nIn other words, there is an "energy stack" of critical wave functions. For\nclass AIII, multifractal eigenstate and conductance analysis reveals identical\nstatistics for states throughout the stack, consistent with the class A integer\nquantum-Hall plateau transition (QHPT). Class CI TSCs exhibit surface stacks of\nclass C spin QHPT states. Critical stacking of a third kind, possibly\nassociated to the class D thermal QHPT, is identified for nematic velocity\ndisorder of a single Majorana cone in class DIII. The Dirac theories studied\nhere can be represented as perturbed 2D Wess-Zumino-Novikov-Witten sigma\nmodels; the numerical results link these to Pruisken models with the\ntopological angle $\\vartheta = \\pi$. Beyond applications to TSCs, all three\nstacked Dirac theories (CI, AIII, DIII) naturally arise in the effective\ndescription of dirty $d$-wave quasiparticles, relevant to the high-$T_c$\ncuprates.\n