Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene

Yang-Zhi Chou, Matthew S. Foster

arXiv:1402.6797Physical Review BApr 30, 201428 citations

Nature

Designer Dirac fermions and topological phases in molecular graphene

Kenjiro K. Gomes, Warren Mar, Wonhee Ko, F. Guinea, Hari C. Manoharan

Published Mar 1, 2012• 741 citations

Publisher

DOI: 10.1038/nature10941

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Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene

Yang-Zhi Chou, Matthew S. Foster

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.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.

arXiv:1402.6797Physical Review BApr 30, 201428 citations

{\mathbb{Z}}_{2}

invariant and also study the phase transition using entanglement entropy. We use low-energy Hamiltonians and numerical result from model systems to show this effect. These transitions are observed in realistic materials, including strained graphene and buckled honeycomb group V (Sb/As).