Topological quantum materials
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Introduction Topological quantum materials have nontrivial topological invariants in electronic structures. These materials show exotic properties and have created many exciting opportunities for materials science research. As the topological numbers in these systems remain unchanged under continuous deformation, they are only altered through structural or time-reversal symmetry breaking. In addition to unique new properties, their electronic and spin states associated with their topology have shown robustness against external perturbations. The first experimentally observed topological states were in the quantum Hall effect (QHE), where a bulk insulating phase induced by Landau levels is differentiated from an ordinary one by a nontrivial topological number known as the Chern invariant (i.e., the Landau index in this case). The Chern invariant can be understood physically in terms of a geometric phase, or the Berry phase of the Bloch wave functions u m (k ) where k is the wavevector. When u m (k ) is transported around a closed loop in k, a gauge-invariant Berry phase γ = i u m |∇k |u m dk is acquired, where the argument of the integral A m = i u m |∇k |u m is Berry’s connection. Using the Stokes theorem, γ can also be expressed as a surface integral of yet another quantity, which is the Berry curvature (or Berry field) Fm, defined as the curl of the Berry connection
(i.e., Fm = ∇ × A m).The Chern invariant is an integer, counting the total flux of the Berry field in the Brillouin zone, cm =
1 Fm d 2k. 2π
total
(1)
N
= m =1 cm is the summation The total Chern number c over all the occupied bands, which is invariant during adiabetic modifications. Thouless et al. showed that quantized conductance in QHE was a direct consequence of the quantized Chern number (Landau index):1,2 σ xy = c totale 2 h , where e is the electron charge and h is the Planck constant. Since the quantization of σxy is dictated by such nontrivial topology, the values captured in experiments have been robust and exact, leading to their widespread adoption as a reliable resistance standard equal to the Klitzing constant (h e2, ∼25.8 kΩ). While the QHE (Figure 1a) requires an external magnetic field to form the Landau levels, a Chern invariant is a generic feature that can be implemented even without the magnetic field. In thin-film ferromagnets with perpendicular anisotropy and a large spin–orbit interaction, an anomalous Hall conductivity emerges at zero fields, a phenomenon known as the anomalous Hall effect (AHE). In line with quantized Hall conductivity in the QHE, it was natural to investigate the presence of an analogous quantum anomalous Hall effect (QAHE) at zero fields. Initially, QAHE
Kang L. Wang, Departments of Electrical and Computer Engineering, Physics, and Astronomy, and Materials Science and Engineering, University of California, Los Angeles, USA; [email protected] Yingying Wu, Department of Electrical and Computer Engineering, University of California, Los Angeles, USA; [email protected] Christopher Eckberg, US Ar
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