Hearing the shape of a Dirac drum: Dual quantum Hall states on curved surfaces

Abstract

The geometry of a physical system is intimately related to its spectral properties, a concept colloquially referred to as "hearing the shape of a drum". Three-dimensional topological insulator nanowires in a strong magnetic field $B$ generally host Dirac-type quantum Hall (QH) surface states. The surface itself is shaped by spatial variations of the wires' cross section, yielding a curved geometrical background, the "drum", with imprints in the corresponding QH spectra. We show that the latter are composed of two different classes. The first one is asymptotically insensitive to the surface shape, scaling as $B^{1/2}$, like regular planar QH states. Instead, the second has an asymptotic $B$-field dependence intimately related to the wire geometry. We further demonstrate that an (axial-symmetric) curved nanowire surface possesses a reciprocal partner surface, such that the respective QH spectra are dual to each other upon exchanging angular momentum and magnetic flux. Notably, a cone-shaped nanowire, and the Corbino geometry as its limiting case, has a reciprocal partner with a dual QH spectrum that is $B$-field independent, with corresponding non-magnetic QH-type states. We support our analytical findings by numerical results for $B$-field ranges and wire geometries within reach of current experiment.

References in corpus (28)

• Topological Insulators
• Target Space Duality in String Theory
• Model Hamiltonian for Topological Insulators
• Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe
• Viscosity of Quantum Hall Fluids
• An Introduction to T-Duality in String Theory
• Hall viscosity, orbital spin, and geometry: paired superfluids and quantum Hall systems
• Surface States of Topological Insulators
• Electromagnetic and gravitational responses of two-dimensional non-interacting electrons in background magnetic field
• Landau Levels in the Presence of Topological Defects
• Unexpected edge conduction in HgTe quantum wells under broken time reversal symmetry
• The surface states of topological insulators - Dirac fermion in curved two dimensional spaces
• Probing Spin Helical Surface States in Topological HgTe Nanowires
• Geometric adiabatic transport in quantum Hall states
• Emergent Conformal Symmetry of Quantum Hall States on Singular surfaces
• Spin and Charge Structure of the Surface States in Topological Insulators
• Observation of Zeeman effect in topological surface state with distinct material dependence
• Electronic States of Wires and Slabs of Topological Insulators: Quantum Hall Effects and Edge Transport
• Highly Anisotropic Transport in the Integer Quantum Hall Effect
• Magnetoconductance, Quantum Hall Effect, and Coulomb Blockade in Topological Insulator Nanocones
• Aharonov-Bohm interference and phase-coherent surface-state transport in topological insulator rings
• 2D-Dirac surface states and bulk gap probed via quantum capacitance in a 3D topological insulator
• CdTe-HgTe core-shell nanowire growth controlled by RHEED
• Landau level quantization of Dirac electrons on the sphere
• Perfect transmission and Aharanov-Bohm oscillations in topological insulator nanowires with nonuniform cross section
• Graphene in a Strong Magnetic Field: Massless Dirac Particles vs. Skyrmions
• Theory of magnetotransport in shaped topological insulator nanowires
• Central charge from adiabatic transport of cusp singularities in the quantum Hall effect