Z2classification of quantum spin Hall systems: An approach using time-reversal invariance
Open accessAbstract
We study the phases of Bloch insulators with time-reversal symmetry on the basis of the homotopy of the ground-state wave functions in momentum space and find that there are two topological classes characterized by a ${Z}_{2}$ invariant. The results are in agreement with a recent study based on counting the zeroes of a certain Pfaffian function related to the ground-state wave function. It is shown that there is a link between the formulation of the topological invariant presented here and the number of robust edge states. A formula is also provided which greatly simplifies the computation of the invariant in a large number of cases. The present study provides guidance for the search of systems which belong to the nontrivial topological class and also establishes a link between the quantum spin Hall effect and the integer quantum Hall effect.
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Kane and Mele showed that starting from the 2D time-reversal-invariant crystal, there are two topological classes; instead of an integer-valued topological invariant, there is a Z2 invariant with two values, even and odd.
“Kane and Mele showed that starting from the 2D T -invariant crystal, there are two topological classes; instead of an integer-valued topological invariant, there is a ' Z 2 invariant with two values, 'even' and 'odd'.”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [65]
The system exhibits a topologically nontrivial phase that is the photonic analogue of the quantum spin Hall effect, characterized by a quantized spin Chern number C_s = 2 and a Z2 invariant of 1.
“the system exhibits a topologically nontrivial phase which is the photonic analogue of the quantum spin Hall effect [ 43], characterized by a quantized spin Chern number [ 44] = ( ) = 2 and a Z 2 invariant [ 1, 5] of 1.”
From Photonic topological insulators (2013) · cites this paper as [5]
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- Authors
- Roy, R.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Roy, Rahul
- ORCID
- Roy, Rahul
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2009-05-21
- Volume
- 79
- Issue
- 19
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W1605377265
- Cited by
- 299
- References
- 16
- Citations by year
- 2026: 3 · 2025: 10 · 2024: 9 · 2023: 10 · 2022: 13 · 2021: 8 · 2020: 12 · 2019: 10 · 2018: 16 · 2017: 11 · 2016: 25 · 2015: 16 · 2014: 18 · 2013: 31 · 2012: 33
- FWCI
- 19.89
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
- Keywords
- quantum spin Hall effect, Z2 topological invariant, time-reversal symmetry, topological phases, integer quantum Hall effect, topological classification
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