Turtles all the way down: Spectrum-wide quantum criticality in topological, and maybe high-Tc superconductors

Quantum criticality typically occurs only with fine-tuning. In the context of Anderson localization, this is tuning to the mobility edge. In topological Anderson insulators such as the quantum Hall effect, criticality is realized only at the quantum Hall plateau transition (QHPT).

In a trio of recent papers [1-3], we have used exact diagonalization to revisit the problem of 2D Dirac fermions, subject to special kinds of “pure gauge” disorder. These include abelian and nonabelian vector potentials, and stress tensor disorder ("quenched gravitational disorder QGD") [3]. These models arise naturally as theories for dirty surface states of 3D topological superconductors (TSCs) in classes CI, AIII, or DIII. Alternatively, these dirty Dirac fermions can be realized in 2D by fine-tuning the details of microscopic disorder potentials, as might be relevant for the high-Tc cuprate superconductors [3]. Another application is to the chiral (pure-flat band) Hamiltonian for twisted bilayer graphene. 

Conventional wisdom predicted only Anderson localization (CI, AIII) or weak antilocalization (DIII) for finite-energy states of these dirty Dirac models. Localization would render meaningless the notion of "topological protection." This led us to suspect that conventional arguments might fail here. 

Indeed, we find no evidence of localization anywhere in these 2D Dirac models. Most remarkably, we instead find that most of the finite-energy states form a “stack” of critical wave functions with universal, energy-independent statistics. For two cases, the stacked states appear to correspond precisely to the isolated, universal states that appear at the spin and charge QHPTs. This is evidence for a new, deep link between 2D QHPTs transitions in classes (C, A), and surface states of bulk topological superconductors in (CI, AIII). For quenched gravitational disorder (DIII), a new class of stacked criticality is identified, which may correspond to the thermal QHPT. Our most intriguing result concerns the gravitational dirt. When the latter is restricted to quenched nematic fluctuations, the predicted phenomenology appears similar to STM studies of BSCCO [3].

[1] S. A. A. Ghorashi, Y. Liao, and M. S. Foster, Phys. Rev. Lett. 121, 016802 (2018).
[2] B. Sbierski, J. F. Karcher, and M. S. Foster, Phys. Rev. X 10, 021025 (2020).
[3] S. A. A. Ghorashi, J. F. Karcher, S. M. Davis, and M. S. Foster, Phys. Rev. B 101, 214521 (2020).