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arXiv:1912.06748 · Physical Review X

Spectrum-Wide Quantum Criticality at the Surface of Class AIII Topological Phases: An “Energy Stack” of Integer Quantum Hall Plateau Transitions

Björn Sbierski, Jonas F. Karcher, Matthew S. Foster

Published May 1, 2020• 23 citations
arXivHTMLPDF

Abstract

In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT).In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales.Normally, QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field.In this paper we show that QHPT states can be realized throughout an energy continuum, i.e., as an "energy stack" of critical states wherein each state in the stack exhibits QHPT phenomenology.The stacking occurs without fine-tuning at the surface of a class AIII topological phase, where it is protected by U(1) and (anomalous) chiral or time-reversal symmetries.Spectrum-wide criticality is diagnosed by comparing numerics to universal results for the longitudinal Landauer conductance and wave function multifractality at the QHPT.Results are obtained from an effective 2D surface field theory and from a bulk 3D lattice model.We demonstrate that the stacking of quantum-critical QHPT states is a robust phenomenon that occurs for AIII topological phases with both odd and even winding numbers.The latter conclusion may have important implications for the still poorly understood logarithmic conformal field theory believed to describe the QHPT.

DOI
10.1103/physrevx.10.021025

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In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT).In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales.Normally, QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field.In this paper we show that QHPT states can be realized throughout an energy continuum, i.e., as an "energy stack" of critical states wherein each state in the stack exhibits QHPT phenomenology.The stacking occurs without fine-tuning at the surface of a class AIII topological phase, where it is protected by U(1) and (anomalous) chiral or time-reversal symmetries.Spectrum-wide criticality is diagnosed by comparing numerics to universal results for the longitudinal Landauer conductance and wave function multifractality at the QHPT.Results are obtained from an effective 2D surface field theory and from a bulk 3D lattice model.We demonstrate that the stacking of quantum-critical QHPT states is a robust phenomenon that occurs for AIII topological phases with both odd and even winding numbers.The latter conclusion may have important implications for the still poorly understood logarithmic conformal field theory believed to describe the QHPT.
We present numerical evidence that most two-dimensional surface states of a bulk topological superconductor (TSC) sit at an integer quantum Hall plateau transition. We study TSC surface states in class CI with quenched disorder. Low-energy (finite-energy) surface states were expected to be critically delocalized (Anderson localized). We confirm the low-energy picture, but find instead that finite-energy states are also delocalized, with universal statistics that are independent of the TSC winding number, and consistent with the spin quantum Hall plateau transition (percolation).
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
We present numerical evidence that most two-dimensional surface states of a bulk topological superconductor (TSC) sit at an integer quantum Hall plateau transition. We study TSC surface states in class CI with quenched disorder. Low-energy (finite-energy) surface states were expected to be critically delocalized (Anderson localized). We confirm the low-energy picture, but find instead that finite-energy states are also delocalized, with universal statistics that are independent of the TSC winding number, and consistent with the spin quantum Hall plateau transition (percolation).
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.
We numerically investigate critically delocalized wave functions in models of two-dimensional Dirac fermions, subject to vector potential disorder. These describe the surface states of three-dimensional topological superconductors, and can also be realized through long-range correlated bond randomness in artificial materials like molecular graphene. A ``frozen'' regime can occur for strong disorder in these systems, wherein a single wave function presents a few localized peaks separated by macroscopic distances. Despite this rarefied spatial structure, we find robust correlations between eigenstates at different energies, at both weak and strong disorder. The associated level statistics are always approximately Wigner-Dyson. The system shows generalized Chalker (quantum critical) scaling, even when individual states are quasilocalized in space. We confirm analytical predictions for the density of states and multifractal spectra. For a single Dirac valley, we establish that finite energy states show universal multifractal spectra consistent with the integer quantum Hall plateau transition. A single Dirac fermion at finite energy can therefore behave as a ``Quantum Hall critical metal.'' For the case of two valleys and non-Abelian disorder, we verify predictions of conformal field theory. Our results for the non-Abelian case imply that both delocalization and conformal invariance are topologically protected for multivalley topological superconductor surface states.