Quantum spin Hall effect and topological insulators

Mapping Intimacies ◽

10.1093/oso/9780198787075.003.0017 ◽

2017 ◽

Author(s):

S. Murakami ◽

T. Yokoyama

Keyword(s):

Topological Insulators ◽

Surface States ◽

Two Dimensions ◽

Quantum Spin ◽

Three Dimensions ◽

Pure Spin ◽

Topological Order ◽

Edge States ◽

Edge Surface ◽

Quantum Spin Hall

This chapter begins with a description of quantum spin Hall systems, or topological insulators, which embody a new quantum state of matter theoretically proposed in 2005 and experimentally observed later on using various methods. Topological insulators can be realized in both two dimensions (2D) and in three dimensions (3D), and are nonmagnetic insulators in the bulk that possess gapless edge states (2D) or surface states (3D). These edge/surface states carry pure spin current and are sometimes called helical. The novel property for these edge/surface states is that they originate from bulk topological order, and are robust against nonmagnetic disorder. The following sections then explain how topological insulators are related to other spin-transport phenomena.

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Designing photonic topological insulators with quantum-spin-Hall edge states using topology optimization

Nanophotonics ◽

10.1515/nanoph-2019-0057 ◽

2019 ◽

Vol 8 (8) ◽

pp. 1363-1369 ◽

Cited By ~ 6

Author(s):

Rasmus E. Christiansen ◽

Fengwen Wang ◽

Ole Sigmund ◽

Søren Stobbe

Keyword(s):

Topology Optimization ◽

Band Gap ◽

Design Problem ◽

Topological Insulators ◽

Quantum Spin ◽

Edge States ◽

Unit Cells ◽

Improved Performance ◽

Inverse Design Problem ◽

Quantum Spin Hall

AbstractDesigning photonic topological insulators (PTIs) is highly non-trivial because it requires inversion of band symmetries around the band gap, which was so far done using intuition combined with meticulous trial and error. Here we take a completely different approach: we consider the design of PTIs as an inverse design problem and use topology optimization to maximize the transmission through an edge mode past a sharp bend. Two design domains composed of two different but initially identical C6ν-symmetric unit cells define the geometrical design problem. Remarkably, the optimization results in a PTI reminiscent of the shrink-and-grow approach to quantum-spin-Hall PTIs but with notable differences in the crystal structure as well as qualitatively different band structures and with significantly improved performance as gauged by the band-gap sizes, which are at least 50% larger than in previous designs. Furthermore, we find a directional β-factor exceeding 99% and very low losses for sharp bends. Our approach allows the introduction of fabrication limitations by design and opens an avenue towards designing PTIs with hitherto-unexplored symmetry constraints.

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TOPOLOGICAL QUANTUM PHASE TRANSITION BETWEEN QUANTUM SPIN HALL STATE AND QUANTUM ANOMALOUS HALL STATE

International Journal of Modern Physics B ◽

10.1142/s0217979211100096 ◽

2011 ◽

Vol 25 (17) ◽

pp. 2323-2340 ◽

Cited By ~ 4

Author(s):

LAN-FENG LIU ◽

SU-PENG KOU

Keyword(s):

Phase Transitions ◽

Quantum Phase Transitions ◽

Normal Band ◽

Quantum Spin ◽

Quantum Phase ◽

Topological Order ◽

Edge States ◽

Topological Quantum ◽

The Masses ◽

Quantum Spin Hall

In this paper, starting from a lattice model of topological insulators, we study the quantum phase transitions among different quantum states, including quantum spin Hall state, quantum anomalous Hall state and normal band insulator state by calculating their topological properties (edge states, quantized spin Hall conductivities, and the number of zero mode on a π-flux). We find that at the topological quantum phase transitions (TQPTs), the topological "order parameter" — spin Chern number will jump. And since the masses of the nodal fermions will change sign, the third derivative of ground-state energy is nonanalytic. In addition, we discuss the finite temperature properties and the stability of the TQPTs.

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