Topological field theory of time-reversal invariant insulators
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We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.
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Three-dimensional topological insulators are nonmagnetic insulators with novel surface states that are a consequence of the nontrivial topology of electronic wavefunctions in the bulk.
“Topological insulators (TIs) in three dimensions are nonmagnetic insulators with novel surface states (SSs) that are a consequence of the nontrivial topology of electronic wavefunctions in the bulk of the materials ( 1-5 ).”
From Topological Phase Transition and Texture Inversion in a Tunable Topological Insulator (2011) · cites this paper as [4]
Spins tangentially arranged on such a circular Fermi surface lead to a Berry's phase of π, which is also a measure of the topological invariant and axion angle (ν0=1=θ/π) of the system.
“Spins tangentially arranged on such a circular FS lead to a Berry's phase of π , which is also a measure of the topological invariant and the axion angle ( ν 0 = 1 = θ/π ) of the system ( 1, 4, 6 ).”
From Topological Phase Transition and Texture Inversion in a Tunable Topological Insulator (2011) · cites this paper as [4]
In the 'topological' limit where all occupied bands have the same energy -1 and all unoccupied bands have the same energy +1, the ordinary terms beyond the Berry phase disappear.
“these 'ordinary' terms disappear in either the 'topological' limit of (70) where all occupied bands have the same energy -1 and all unoccupied bands have the same energy +1”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [70]
The three-dimensional topological insulator is connected to the orbital contribution to the magnetoelectric polarizability dP/dB.
“The three-dimensional topological insulator turns out to be connected to a basic electromagnetic property of solids: the orbital contribution to the magnetoelectric polarizability dP/dB (70,71,72).”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [70]
A time-reversal breaking perturbation, such as magnetic impurities, can induce scattering between the states |k↑> and -|k↓> and open up a local energy gap at the Dirac point.
“Theoretically, it was pointed out [5, 30, 31] that a time-reversal breaking perturbation, such as magnetic impurities, can induce scattering between the states | ⃗ k ↑〉 and -| ⃗ k ↓〉 and open up a local energy gap at the Dirac point.”
From Experimental Demonstration of Topological Surface States Protected by Time-Reversal Symmetry (2009) · cites this paper as [5]
The total conductance of a slab is always an integer.
“Note that the total conductance of a slab is always an integer, as required. 10,11,33”
From Antiferromagnetic topological insulators (2010) · cites this paper as [11]
The helical Dirac fermion on the surface of Bi2Se3 owes its existence to a non-zero topological number ν0 given by (-1)^ν0 = Π_kT Π_{m=1}^N ξ_{2m}(kT).
“The helical Dirac fermion on the surface of Bi 2 Se 3 owes its existence to a non-zero topological number ν 0 given by (-1) ν 0 = Π k T Π N m =1 ξ 2 m ( k T )”
From A tunable topological insulator in the spin helical Dirac transport regime (2009) · cites this paper as [4]
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“The action is Sθ = 1 2π e2 h ∫ dtdrθ(r, t)E ·B. (21) This is similar to the formalism used in describing the electromagnetic response of topological insulators (Essin et al., 2009; Qi et al., 2008; Wu et al., 2016).”
Method
- Authors
- Qi, X.-L., Hughes, T. L., & Zhang, S.-C.
- Affiliations
- Stanford University (US)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2008-11-24
- Volume
- 78
- Issue
- 19
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W1514707913
- Cited by
- 3,329
- References
- 6
- Citations by year
- 2026: 98 · 2025: 157 · 2024: 161 · 2023: 183 · 2022: 213 · 2021: 253 · 2020: 282 · 2019: 201 · 2018: 191 · 2017: 178 · 2016: 237 · 2015: 202 · 2014: 194 · 2013: 210 · 2012: 212
- FWCI
- 68.36
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- Bulletin of the American Physical Society
- 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
- topological field theory, Chern-Simons theory, second Chern number, topological magnetoelectric effect, quantum spin Hall insulator, topological insulators, dimensional reduction, fine-structure constant
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