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Cited by 905Topological Materials and PhenomenaQuantum and electron transport phenomenaAdvanced Condensed Matter Physics

Time reversal polarization and a Z 2 adiabatic spin pump

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Fu, Liang, Kane, C. L.·2006·Physical Review B
DOI 10.1103/physrevb.74.195312
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Abstract

We introduce and analyze a class of one-dimensional insulating Hamiltonians that, when adiabatically varied in an appropriate closed cycle, define a ``${Z}_{2}$ pump.'' For an isolated system, a single closed cycle of the pump changes the expectation value of the spin at each end even when spin-orbit interactions violate the conservation of spin. A second cycle, however, returns the system to its original state. When coupled to leads, we show that the ${Z}_{2}$ pump functions as a spin pump in a sense that we define, and transmits a finite, though nonquantized, spin in each cycle. We show that the ${Z}_{2}$ pump is characterized by a ${Z}_{2}$ topological invariant that is analogous to the Chern invariant that characterizes a topological charge pump. The ${Z}_{2}$ pump is closely related to the quantum spin Hall effect, which is characterized by a related ${Z}_{2}$ invariant. This work presents an alternative formulation that clarifies both the physical and mathematical meaning of that invariant. A crucial role is played by time reversal symmetry, and we introduce the concept of the time reversal polarization, which characterizes time reversal invariant Hamiltonians and signals the presence or absence of Kramers degenerate end states. For noninteracting electrons, we derive a formula for the time reversal polarization that is analogous to Berry's phase formulation of the charge polarization. For interacting electrons, we show that Abelian bosonization provides a simple formulation of the time reversal polarization. We discuss implications for the quantum spin Hall effect, and argue in particular that the ${Z}_{2}$ classification of the quantum spin Hall effect is valid in the presence of electron electron interactions.

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Crossref · 10.1103/physrevb.74.195312

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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 [66]

  • GdBiPt is a possible example of the antiferromagnetic topological insulator class.

    “In closing we discuss GdBiPt as a possible example of this topological class.”

    From Antiferromagnetic topological insulators (2010) · cites this paper as [19]

  • Transport experiments indicate that GdBiPt is a semiconductor with a narrow gap.

    “Transport experiments indicate that GdBiPt is a semiconductor with a narrow gap. 19,35”

    From Antiferromagnetic topological insulators (2010) · cites this paper as [19]

  • The recent proposals that many Heusler compounds may be topological insulators, together with the antiferromagnetic order in GdBiPt below 9 K, suggest a possible candidate for the state proposed here.

    “The recent proposals that many Heusler compounds may be topological insulators, 20,21 together with the antiferromagnetic order in GdBiPt below 9 K, 19 suggest a possible candidate for the state proposed here.”

    From Antiferromagnetic topological insulators (2010) · cites this paper as [19]

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Research works printed but not linked: 4

  1. 30 D.N. Sheng, Z.Y. Weng, L. Sheng and F.D.M. Haldane, cond-mat/0603054 (2006). · arXiv
  2. 31 S. Ryu and Y. Hatsugai, cond-mat/0601237 (2006). · arXiv
  3. 33 X.L. Qi, W.S. Wu and S.C. Zhang, cond-mat/0604071 (2006). · arXiv
  4. 37 R. Roy, cond-mat/0604211 (2006). · arXiv

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Authors
Fu, L., & Kane, C. L.
Affiliations
University of Pennsylvania (US)
ORCID
Venue
Physical Review B
Publisher
American Physical Society (APS)
Published
2006-11-08
Volume
74
Issue
19
ISSN
  • 1098-0121 (ISSN-L)
  • 1550-235X
DOI
10.1103/physrevb.74.195312
OpenAlex
W2064989124
Cited by
905
References
34
Citations by year
2026: 29 · 2025: 42 · 2024: 49 · 2023: 35 · 2022: 48 · 2021: 49 · 2020: 55 · 2019: 60 · 2018: 70 · 2017: 69 · 2016: 82 · 2015: 54 · 2014: 47 · 2013: 55 · 2012: 36
FWCI
9.65
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review B (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • ScholarlyCommons (University of Pennsylvania) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
Keywords
Z2 topological invariant, quantum spin Hall effect, time-reversal symmetry, Kramers degeneracy, spin-orbit coupling, Berry phase, Abelian bosonization, spin transport, electron-electron interaction

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