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In the absence of spin-orbit coupling, the conventional dogma of Andersonlocalization asserts that all states localize in two dimensions, with a glaringexception: the quantum Hall plateau transition (QHPT). In that case, thelocalization length diverges and interference-induced quantum-critical spatialfluctuations appear at all length scales. Normally QHPT states occur only atisolated energies; accessing them therefore requires fine-tuning of theelectron density or magnetic field. In this paper we show that QHPT states canbe realized throughout an energy continuum, i.e. as an "energy stack" ofcritical states wherein each state in the stack exhibits QHPT phenomenology.The stacking occurs without fine-tuning at the surface of a class AIIItopological phase, where it is protected by U(1) and (anomalous) chiral ortime-reversal symmetries. Spectrum-wide criticality is diagnosed by comparingnumerics to universal results for the longitudinal Landauer conductance andwave function multifractality at the QHPT. Results are obtained from aneffective 2D surface field theory and from a bulk 3D lattice model. Wedemonstrate that the stacking of quantum-critical QHPT states is a robustphenomenon that occurs for AIII topological phases with both odd and evenwinding numbers. The latter conclusion may have important implications for thestill poorly-understood logarithmic conformal field theory believed to describethe QHPT.

Spectrum-wide quantum criticality at the surface of class AIII topological phases: An "energy stack" of integer quantum Hall plateau transitions