Antiferromagnetic topological insulators
Open accessAbstract
We consider antiferromagnets breaking both time-reversal $(\ensuremath{\Theta})$ and a primitive-lattice translational symmetry $({T}_{1/2})$ of a crystal but preserving the combination $S=\ensuremath{\Theta}{T}_{1/2}$. The $S$ symmetry leads to a ${\mathbb{Z}}_{2}$ topological classification of insulators, separating the ordinary insulator phase from the ``antiferromagnetic topological insulator'' phase. This state is similar to the ``strong'' topological insulator with time-reversal symmetry and shares with it such properties as a quantized magnetoelectric effect. However, for certain surfaces the surface states are intrinsically gapped with a half-quantum Hall effect $[{\ensuremath{\sigma}}_{xy}={e}^{2}/(2h)]$, which may aid experimental confirmation of $\ensuremath{\theta}=\ensuremath{\pi}$ quantized magnetoelectric coupling. Step edges on such a surface support gapless, chiral quantum wires. In closing we discuss GdBiPt as a possible example of this topological class.
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Written from the paper’s full text.
TL;DR
The paper defines an antiferromagnetic topological insulator (AFTI) class for crystals that break time-reversal Θ and a primitive lattice translation T₁/₂ but preserve S = ΘT₁/₂, leading to a Z₂ topological classification. It predicts surface-state and magnetoelectric signatures, including a half-quantum Hall effect on certain surfaces and chiral step-edge modes, and discusses GdBiPt as a possible candidate.
Key findings
- Insulators with unbroken S = ΘT₁/₂ have a Z₂ classification separating ordinary insulators from AFTIs; the AFTI is related to the strong topological insulator and has quantized magnetoelectric coupling θ = π.
- Type A surfaces preserve S and have gapless Dirac surface states, but these can be gapped by disorder; type F surfaces break S, are typically gapped, and exhibit a half-quantum Hall effect with σ_xy = e²/(2h).
- Step edges on type F surfaces support gapless chiral 1D quantum wires, proposed as an experimental signature for scanning tunneling measurements.
- GdBiPt is discussed as a possible AFTI candidate: it is an antiferromagnet below 9 K and a narrow-gap semiconductor, but whether its antiferromagnetic order falls in the S symmetry class requires further experiments such as neutron scattering.
Methods
- Theoretical symmetry and topological classification of band insulators preserving S = ΘT₁/₂, using Z₂ invariants, Berry connection/curvature, and Chern-Simons magnetoelectric response.
- Construction of two explicit effective Hamiltonians: model A from a strong topological insulator with staggered time-reversal-breaking term, and model B from magnetically induced spin-orbit coupling in a rock-salt/FCC structure.
- Numerical slab calculations of bulk and surface band structure, plus spatially resolved Hall conductance for model B slabs with 32 and 33 layers.
Limitations
- Stated: the authors cannot provide a good argument why the in-plane hopping t′ should be much less than the interplane hopping t in a real material for model B.
- Stated: GdBiPt’s antiferromagnetic order has not been confirmed to fall under the S symmetry class; neutron scattering and further experiments are required, and the material must be truly insulating through its antiferromagnetic transition.
- Stated: the AFTI surface state is not generally stable to disorder and is topological in a weaker sense than the strong 3D topological insulator; experimental confirmation of θ = π had not yet occurred.
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Claims the paper makes
0 of 26 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3
The presence of S symmetry forces the Chern numbers on all BZ planes to be zero.
“The presence of S symmetry forces the Chern numbers on all BZ planes to be zero.”
Quote in the paperNot a substantial claim - Findingreasoning§ I. INTRODUCTION · p. 1
The product S symmetry guarantees that the space-averaged θ is quantized to zero or π.
“The product S is also enough to guarantee that the space-averaged θ is quantized to zero or π , since θ is odd under S .”
Quote in the paperSubstantial claim - Findingnone given§ A. Relation to the time-reversal invariant topological insulator · p. 3
Upon doubling the unit cell in the a3 direction, the Brillouin zone halves by folding in the k3 direction.
“Upon doubling the unit cell in the a 3 direction, the Brillouin zone halves by folding in the k 3 direction.”
Quote in the paperSubstantial claim - Findingreasoning§ A. Relation to the time-reversal invariant topological insulator · p. 3
Adding an antiferromagnetic (Θ-breaking) parameter M to a strong topological insulator produces an AFTI.
“Adding an antiferromagnetic (Θ-breaking) parameter M to a STI produces an AFTI.”
Quote in the paperSubstantial claim - Findingnone given§ I. INTRODUCTION · p. 1
In the gapped surface case, there must be a half-integer quantum Hall effect.
“The bulk value θ = π allows either metallic surfaces or gapped surfaces, but in the gapped case there must be a half-integer quantum Hall effect.”
Quote in the paperSubstantial claim
- Findingreasoning§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2
For spin-1/2 particles, the combination S_k = Θ T1/2(k) satisfies S^2 = -e^{ik3}.
“while Θ 2 = -1 for spin-1/2 particles, S 2 = S -k S k = -e ik 3 .”
Quote in the paperSubstantial claim - Findingnone given§ Antiferromagnetic topological insulators · p. 1
The S symmetry (S = Θ T1/2) leads to a Z2 topological classification of insulators, separating the ordinary insulator phase from the antiferromagnetic topological insulator (AFTI) phase.
“The S symmetry leads to a Z 2 topological classification of insulators, separating the ordinary insulator phase from the 'antiferromagnetic topological insulator' (AFTI) phase.”
Quote in the paperSubstantial claim - Findingnone given§ I. INTRODUCTION · p. 1
Along some planar surfaces, AFTIs have gapless surface modes.
“Along some planar surfaces they have gapless surface modes, while along others the surface is gapped and there is a nonzero magnetoelectric coupling from the intrinsic material; an experimental signature in the latter case is the existence of one-dimensional (1D) metallic states along step edges on the surface.”
Quote in the paperSubstantial claim - Findingreasoning§ Antiferromagnetic topological insulators · p. 1
For certain surfaces of the AFTI, the surface states are intrinsically gapped.
“However, for certain surfaces the surface states are intrinsically gapped with a half-quantum Hall effect [ σ xy = e 2 / (2 h )], which may aid experimental confirmation of θ = π quantized magnetoelectric coupling.”
Quote in the paperSubstantial claim - Findingnone given§ Antiferromagnetic topological insulators · p. 1
On these gapped surfaces, there is a half-quantum Hall effect with σxy = e^2/(2h).
“However, for certain surfaces the surface states are intrinsically gapped with a half-quantum Hall effect [ σ xy = e 2 / (2 h )], which may aid experimental confirmation of θ = π quantized magnetoelectric coupling.”
Quote in the paperSubstantial claim - Findingnone given§ I. INTRODUCTION · p. 1
Starting from a nonmagnetic topological insulator on a bipartite Bravais lattice and adding antiferromagnetic order that doubles the unit cell, one of the three-dimensional topological invariants survives.
“A simple picture of the state we discuss is obtained by starting from a nonmagnetic topological insulator on a bipartite Bravais lattice, then adding antiferromagnetic order that doubles the unit cell. One of the three-dimensional topological invariants survives in this process.”
Quote in the paperNot a substantial claim - Findingown data§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2
The 3D Z2 invariant is independent of unit cell choice.
“In the Appendix, we show that the 3D Z 2 invariant is independent of unit cell choice.”
Quote in the paperNot a substantial claim - Findingreasoning§ I. INTRODUCTION · p. 1
In topological antiferromagnets, some surfaces have a gapped state just from the material's own time-reversal-breaking.
“In the topological antiferromagnets, some surfaces have a gapped state just from the material's own time-reversal-breaking, which suggests that experimental confirmation that θ = π , which has not yet occurred, may be easier in these materials, using the same techniques previously used to extract θ in Cr 2 O 3 .”
Quote in the paperSubstantial claim - Findingnone given§ Antiferromagnetic topological insulators · p. 1
Step edges on such a gapped surface support gapless, chiral quantum wires.
“Step edges on such a surface support gapless, chiral quantum wires.”
Quote in the paperNot a substantial claim - Findingown data§ 2. Effective Hamiltonian · p. 5
The model B has spatial inversion symmetry given by Π = [0 1; 1 0], which swaps layers A1 and A2.
“This model has spatial inversion symmetry, given by the operator Π = [ 0 1 1 0 ], which in effect swaps the layers A 1 and A 2.”
Quote in the paperSubstantial claim - Findingown data§ IV. SURFACE BAND STRUCTURE · p. 6
Model B's surface spectrum is completely disconnected from the bulk, forming a complete two-dimensional band structure.
“This model has the peculiar feature that the surface spectrum is completely disconnected from the bulk, that is, it forms a complete two-dimensional band structure.”
Quote in the paperNot a substantial claim - Findingown data§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
The surface spectrum of an AFTI depends on the surface cut, classified as type A/F.
“We have also demonstrated that the surface spectrum depends on the surface cut, classified as type A/F.”
Quote in the paperSubstantial claim - Findingnone given§ A. Construction from strong topological insulators · p. 4
In the absence of M, the doubled system has two parity (spatial inversion) centers, given by operators Π1 and Π2.
“In the absence of M this system also has two parity (spatial inversion) centers, given by the operators:”
Quote in the paperSubstantial claim - Findingnone given§ 2. Effective Hamiltonian · p. 5
The model B Hamiltonian is gapped in the bulk when |t'| < |t|, 1/√3 |λ|.
“which is gapped (in the bulk) when | t ′ | < | t | , 1 √ 3 | λ | .”
Quote in the paperSubstantial claim - Findingown data§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
In model B, adding a layer such that the two surfaces have the same Zeeman field switches the conductance on that surface from +1/2 to -1/2, so that the total slab conductance vanishes.
“adding a layer such that the two surfaces have the same Zeeman field switches the conductance on that surface from +1 / 2 to -1 / 2, so that the total slab conductance vanishes.”
Quote in the paperSubstantial claim - Findingreasoning§ 1. Motivation · p. 4
In the rhombus geometry, the spin-orbit coupling from X to Y vanishes because the cross products d1×d2 from the two paths cancel.
“In this geometry this coupling vanishes as the cross products d 1 × d 2 from the two paths cancel.”
Quote in the paperSubstantial claim0 of 2 sources checked - Findingown data§ IV. SURFACE BAND STRUCTURE · p. 6
For model B, a surface spectrum exists for all values of t′ and non-zero values of t and λ.
“As the dispersion shows, a surface spectrum exists for all values of t ′ and non-zero values of t and λ .”
Quote in the paperSubstantial claim - Findingreasoning§ A. Construction from strong topological insulators · p. 4
The product of the two inversion centers Π1 and Π2 results in a translation by half a unit cell: Π1Π2 = T1/2.
“Their product results in a translation by half a unit cell: Π 1 Π 2 = T 1 / 2 .”
Quote in the paperSubstantial claim - Findingown data§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
The paper found a Z2 classification within the S symmetry class that leads to the existence of an antiferromagnetic topological insulator (AFTI).
“In this paper, we have looked at the topological classification of materials breaking both time-reversal Θ and translational symmetry T 1 / 2 , but preserving the combination S = Θ T 1 / 2 , and found a Z 2 classification within the S symmetry class that leads to the existence of an antiferromagnetic topological insulator (AFTI).”
Quote in the paperSubstantial claim - Findingnone given§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
The paper provides a topological classification and experimental consequences for the combination of time-reversal symmetry and lattice symmetry Θ T1/2.
“We have provided a topological classification and experimental consequences for a particular combination of time-reversal symmetry and a lattice symmetry (Θ T 1 / 2 ).”
Quote in the paperSubstantial claim - Findingreasoning§ Appendix A: Invariance to choice of unit cell · p. 9
Different ways to choose the doubled unit cell are equivalent given a choice of structural cell.
“In particular, we show that different ways to choose the doubled unit cell are equivalent given a choice of structural cell.”
Quote in the paperSubstantial claim - Findingreasoning§ Appendix A: Invariance to choice of unit cell · p. 9
The Z2 invariant does not depend on how the unit cell is chosen.
“In other words, the Z 2 invariant does not depend on how we choose the unit cell.”
Quote in the paperNot a substantial claim- Rests on an assumption: Any choice of the effective Brillouin zone will give the same Z2 invariant.
- Findingown data§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
The paper's numerical calculations based on explicit band models agree with these results.
“Our numerical calculations based on explicit band models agree with these results.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. Relation to the time-reversal invariant topological insulator · p. 3
A non-trivial topological antiferromagnet can be constructed by taking a strong topological insulator and introducing a staggered magnetization that breaks time-reversal but preserves S.
“This gives a natural way to construct a non-trivial topological antiferromagnet - by taking a STI and introducing a staggered magnetization which breaks time-reversal but preserves S .”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
Experimental confirmation that θ=π may be easier in topological antiferromagnets than in conventional topological insulators, using techniques previously used to extract θ in Cr2O3.
“In the topological antiferromagnets, some surfaces have a gapped state just from the material's own time-reversal-breaking, which suggests that experimental confirmation that θ = π , which has not yet occurred, may be easier in these materials, using the same techniques previously used to extract θ in Cr 2 O 3 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] source
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- Interpretationreasoning§ I. INTRODUCTION · p. 1
The topological invariant involves explicitly the lattice operation T1/2, so it is sensitive to how this lattice operation is modified by a surface.
“Because the topological invariant involves explicitly the lattice operation T 1 / 2 , it is sensitive to how this lattice operation is modified by a surface, as mentioned above, and even the gapless surface state is not expected to be stable to disorder (in contrast to the conventional topological insulator).”
Quote in the paperSubstantial claim - Interpretationnone given§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3
The topological phase remains well-defined even when the single-particle invariant is not, in the case with electron-electron interactions.
“The topological phase remains well-defined even when the single particle invariant is not, in the case with electron-electron interactions.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
An experimental signature of the gapped surface case is the existence of 1D metallic states along step edges on the surface.
“Along some planar surfaces they have gapless surface modes, while along others the surface is gapped and there is a nonzero magnetoelectric coupling from the intrinsic material; an experimental signature in the latter case is the existence of one-dimensional (1D) metallic states along step edges on the surface.”
Quote in the paperSubstantial claim - Interpretationnone given§ I. INTRODUCTION · p. 1
The bulk value θ=π allows either metallic surfaces or gapped surfaces.
“The bulk value θ = π allows either metallic surfaces or gapped surfaces, but in the gapped case there must be a half-integer quantum Hall effect.”
Quote in the paperSubstantial claim - Interpretationcitation§ Antiferromagnetic topological insulators · p. 1
GdBiPt is a possible example of the antiferromagnetic topological insulator class.
“In closing we discuss GdBiPt as a possible example of this topological class.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [19] source
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- Premisecitation§ A. Construction from strong topological insulators · p. 4
The four-band model on a cubic lattice by Hosur et al. is in the strong topological phase when |t| < |m| < 3|t| and λ=0.
“This Hamiltonian is in the strong topological phase when | t | < | m | < 3 | t | and λ = 0, with the time-reversal operator represented by -iσ y K .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [28] source
Not checked in the free run.
- Interpretationreasoning§ I. INTRODUCTION · p. 1
The gapless surface state of the AFTI is not expected to be stable to disorder, in contrast to the conventional topological insulator.
“Because the topological invariant involves explicitly the lattice operation T 1 / 2 , it is sensitive to how this lattice operation is modified by a surface, as mentioned above, and even the gapless surface state is not expected to be stable to disorder (in contrast to the conventional topological insulator).”
Quote in the paperSubstantial claim - Interpretationnone given§ IV. SURFACE BAND STRUCTURE · p. 6
Any defect or impurity will break the translational and S symmetry on the type A surface, thereby opening a gap.
“However, any defect or impurity will break the translational and S symmetry on the surface, thereby opening a gap.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
Macroscopically, S symmetry implies a quantized magnetoelectric response ∂P/∂B|B=0 = θ/(2π) e2/h with θ=π for an AFTI.
“Macroscopically, S symmetry implies a quantized magnetoelectric response ∂P ∂B ∣ ∣ B =0 = θ 2 π e 2 h with θ = π for an AFTI.”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
The transitions between half-integer quantum Hall plateaus appear when the chemical potential passes through an extended state.
“The transitions between these plateaus appear when the chemical potential passes through an extended state.”
Quote in the paperSubstantial claim- Rests on an assumption: The unitary symmetry class only has extended states at isolated values of the chemical potential; in general the surface state will have zero diagonal conductivity, with half-integer quantum Hall plateaus.
- Premisecitation§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
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”
Quote in the paperNot a substantial claim0 of 3 sources checked - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 8
The chiral mode at a type F surface step edge should give an observable signature in a tunneling experiment.
“Such a mode should give an observable signature in a tunneling experiment (Fig.7).”
Quote in the paperNot a substantial claim - Premisecitation§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The sign of the Hall conductance depends on the sign of the effective Dirac mass, which here is set by the direction of the Zeeman field at the surface.
“The sign of the Hall conductance depends on the sign of the effective Dirac mass, 10,31,32 which here is set by the direction of the Zeeman field at the surface.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [10] source
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- cites [31]
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- cites [32]
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- Interpretationnone given§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 8
At the interface between two integer quantum Hall domains whose conductance C differs by 1, there will be a chiral boundary mode with conductance e2/h.
“Now, at the interface between two integer quantum Hall domains whose conductance C differs by 1, there will be a chiral boundary mode with conductance e 2 /h , which can be thought of as 'half a quantum wire.'”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
Since both top and bottom surfaces of a slab have half-integer plateaus, the total quantum Hall effect when diagonal conductivity is zero is always integral.
“Note that since both top and bottom surfaces of a slab will have half-integer plateaus, the total quantum Hall effect when diagonal conductivity is zero is always integral, as required for a single-electron two-dimensional system.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The half QHE effect follows from the gapped Dirac dispersion of the surface states.
“This effect follows from the gapped Dirac dispersion of the surface states.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
An AFTI can be constructed by stacking QH layers with alternating Chern number ±1 (Hall conductivity ±e2/h).
“By analogy, we can construct an AFTI by stacking QH layers, with alternating Chern number of ± 1 (Hall conductivity ± e 2 /h ), as shown in Fig. 5.”
Quote in the paperSubstantial claim- Rests on an assumption: The QSH insulator may be constructed by combining two copies of a QH layer with opposite spin and Chern number ±n.
- Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
There are an odd number of Dirac cones on a clean type A surface.
“There are an odd number of Dirac cones on a clean type A surface, analogous to the STI.”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
In the presence of a sufficient number of random defects, the surface electronic states are described by the unitary symmetry class because of broken time-reversal symmetry.
“In the presence of a sufficient number of random defects, we expect that the surface electronic states are described by the unitary symmetry class because of the broken time-reversal symmetry.”
Quote in the paperSubstantial claim- Rests on an assumption: The unitary symmetry class only has extended states at isolated values of the chemical potential; in general the surface state will have zero diagonal conductivity, with half-integer quantum Hall plateaus.
- Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 8
In model B, the bulk gap would close while rotating the magnetization.
“(In model B, the bulk gap would close while rotating the magnetization.)”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The layers in the stacked QH model couple to one another, making the Chern number ill-defined on a per-layer basis.
“Just like the QSH case, we can expect the layers to couple to one another, in a way that makes the Chern number ill-defined on a per-layer basis.”
Quote in the paperSubstantial claim - Interpretationnone given§ IV. SURFACE BAND STRUCTURE · p. 6
If the bulk and surface spectrum of a type F surface are fully gapped, then the surface will exhibit the half-integer quantum Hall effect.
“If the bulk and surface spectrum are fully gapped ( i.e. , not a semi-metal), then the surface will exhibit the half-integer quantum Hall effect, to be discussed in the next section.”
Quote in the paperSubstantial claim- Rests on an assumption: The bulk electronic band structure of an AFTI must be gapped to allow the topological distinction between the trivial and non-trivial phase.
- Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
Any long-wavelength probe of the system will be unable to discern the individual QH layers.
“In other words, any longwavelength probe of the system will be unable to discern the individual QH layers.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The Z2 invariant for an AFTI is computed from the k3=0 plane and S|k3=0 in the same way the QSH invariant is computed from the 2D BZ and Θ.
“As described in Sec. II, the Z 2 invariant is computed from the two-dimensional plane k 3 = 0 and the symmetry operator S | k 3 =0 in precisely the way that the quantum spin Hall (QSH) invariant is computed from the two-dimensional BZ and Θ.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 8
Since the sign of the surface conductance C changes when the antiferromagnetic moment is rotated by π, the surface (or bulk) gap must close at some magnetization orientation.
“Since the sign of the surface conductance C changes during this process, the surface (or bulk) gap must close at some magnetization orientation.”
Quote in the paperSubstantial claim- Rests on an assumption: For a magnetic field parallel to a STI surface, the Dirac cone shifts in momentum space but no gap opens; any infinitesimal component out of the plane opens a gap, so going from B out of the surface to B into the surface must close the surface gap.
- Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The AFTI can be considered adiabatically connected to the staggered QH layer model.
“Once again, it is appropriate to consider the AFTI to be adiabatically connected the staggered QH layer model.”
Quote in the paperSubstantial claim - Interpretationnone given§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
In the most basic picture, an AFTI can be obtained from adding a staggered magnetization to a strong topological insulator (STI).
“In the most basic picture, an AFTI can be obtained from adding a staggered magnetization to a strong topological insulator (STI).”
Quote in the paperNot a substantial claim - Interpretationreasoning§ 1. Motivation · p. 5
Magnetization of intermediate sites M1 and M2 induces a spin-orbit interaction between sites X and Y.
“Hence the magnetization of intermediate sites M 1 and M 2 induces a spin-orbit interaction between the sites X and Y .”
Quote in the paperSubstantial claim - Premisecitation§ IV. SURFACE BAND STRUCTURE · p. 6
Breaking time-reversal while preserving inversion can give a topological phase whose surface states are gapped.
“it is known that breaking time-reversal but preserving inversion can give a topological phase whose surface states are gapped. 30”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [30] source
Not checked in the free run.
- Premisenone given§ IV. SURFACE BAND STRUCTURE · p. 6
The k3=0 plane in the bulk BZ carries a non-trivial topological (QSH) phase.
“Since the plane carries a non-trivial topological (QSH) phase, its boundary must be gapless.”
Quote in the paperNot a substantial claim - Interpretationcitation§ IV. SURFACE BAND STRUCTURE · p. 6
These transitions can be regarded as a realization of the two-dimensional quantum Hall effect in zero net field discussed by Haldane.
“These transitions can be regarded as a realization of the two-dimensional quantum Hall effect in zero net field discussed by Haldane. 10”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] source
Not checked in the free run.
- Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
In the stacked QH model, the Hall conductance in the bulk averages to zero.
“In the stacked QH model, the Hall conductance in the bulk averages to zero, as the conductance of any individual layer is cancelled by neighboring layers of opposite type.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7
The half QHE on type F surfaces can be viewed as the root of the bulk magnetoelectric coupling θ = π.
“If one views the antiferromagnet as a STI with timereversal breaking term opening a surface gap, then the half QHE can be viewed as the root of the bulk magnetoelectric coupling θ = π .”
Quote in the paperSubstantial claim- Rests on an assumption: The QSH insulator may be constructed by combining two copies of a QH layer with opposite spin and Chern number ±n.
- Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
The Berry phase of a constant energy curve in the type A surface spectrum is 0 or π due to S symmetry.
“The Berry phase of this curve φ = ∮ γ Tr [ A ] is ambiguous by integer multiples of 2 π , so S symmetry forces this to be 0 or π , for the same reason it forced θ = π in Sec. II B.”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6
A type A surface parallel to a3 is gapless.
“We can see why the surface (parallel to a 3 ) is gapless by looking at the k 3 = 0 line on the surface spectrum, which is the boundary of the k 3 = 0 plane in the bulk BZ.”
Quote in the paperSubstantial claim- Rests on an assumption: The bulk electronic band structure of an AFTI must be gapped to allow the topological distinction between the trivial and non-trivial phase.
- Premisecitation§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
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”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [19] source
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- cites [35]
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- Interpretationnone given§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
The magnetoelectric coupling θ=π requires the half quantum Hall effect at the surface, provided the surface spectrum is gapped.
“The magnetoelectric coupling θ = π requires the half quantum Hall effect at the surface, provided the surface spectrum is gapped.”
Quote in the paperNot a substantial claim- Rests on an assumption: The bulk electronic band structure of an AFTI must be gapped to allow the topological distinction between the trivial and non-trivial phase.
- Premisereasoning§ Appendix A: Invariance to choice of unit cell · p. 9
Any local deformation to the surface that preserves A on the boundary will preserve the integral ∫F.
“any local deformation to the surface ( i.e. , one that preserves A on the boundary) will preserve the integral ∫ F .”
Quote in the paperSubstantial claim - Interpretationreasoning§ Appendix A: Invariance to choice of unit cell · p. 9
This guarantees that the boundary integral terms (∮A) in Eq. A1 are identical in the two cases.
“This guarantees that the boundary integral terms ( ∮ A ) in (Eq. A1) are identical in the two cases.”
Quote in the paperSubstantial claim - Premisecitation§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8
At least one related Heusler antiferromagnet (MnSbCu) is known to have antiferromagnetic ordering [alternating spin directions on (111) planes] which belongs in the S symmetry class.
“At least one related Heusler antiferromagnet (MnSbCu) is known to have antiferromagnetic ordering [alternating spin directions on (111) planes] which belongs in the S symmetry class. 36”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [36]
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- Classified as probably not a substantial claim. 8
- May combine two claims. 35
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2S symmetry gives no invariants in 1D or 2D.
“For example, S symmetry gives no invariants in 1D or 2D.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ A. Relation to the time-reversal invariant topological insulator · p. 3The planes k3=±π/2 map to k3^d=π, and γ^d_π is always zero.
“On the other hand, the planes k 3 = ± π/ 2 map to the plane k d 3 = π . Since ± π/ 2 are time-reversal conjugate and those planes (like all BZ planes, by assumption) have vanishing Chern numbers, it can be seen that γ d π is always zero.”
Quote in the paperSubstantial claim- The authors state this as an assumption: All Brillouin zone planes have vanishing Chern numbers.
May combine two claims.
Findingreasoning§ I. INTRODUCTION · p. 1Along other surfaces, the surface is gapped and there is a nonzero magnetoelectric coupling from the intrinsic material.
“Along some planar surfaces they have gapless surface modes, while along others the surface is gapped and there is a nonzero magnetoelectric coupling from the intrinsic material; an experimental signature in the latter case is the existence of one-dimensional (1D) metallic states along step edges on the surface.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2At the plane k3=π, S^2=+1 and there are no topological invariants associated with this plane.
“At the plane k 3 = π , by contrast, S 2 = +1 and there are no topological invariants associated with this plane.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ A. Relation to the time-reversal invariant topological insulator · p. 3The Z2 invariant describing the AFTI system is γ^d_0 = s, and at M=0 the system is a strong topological insulator provided the bulk gap does not close.
“The Z 2 invariant describing our system is γ d 0 = s and we have a STI at M = 0 (provided the bulk gap does not close).”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2The unit cell is effectively doubled in an antiferromagnet breaking T1/2 and Θ but preserving S, and T^2_{1/2} is the new lattice translation.
“The unit cell is effectively doubled as a result and T 2 1 / 2 is the new lattice translation (which accounts for the notation).”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3The AFTI does not have microscopic time-reversal symmetry; the relevant symmetry operation is S.
“The antiferromagnetic topological insulator suggested here does not have time-reversal symmetry microscopically; the relevant symmetry operation is S .”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3In a three-dimensional system, the three Chern numbers are the only obstruction to finding a set of continuous wave functions in the Brillouin zone respecting Bloch boundary conditions, which guarantees the existence of a single-valued connection A.
“In a three-dimensional system, the three Chern numbers are the only obstruction to finding a set of continuous wave functions in the Brillouin zone (respecting Bloch boundary conditions). This guarantees the existence of a single-valued connection A for Eq. (7).”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ A. Relation to the time-reversal invariant topological insulator · p. 3Under the doubling process, the k3=0 and k3=π planes collapse onto the k3^d=0 plane, and the new invariant γ^d_0 is γ0+γπ=s.
“Under the doubling process, the k 3 = 0 and k 3 = π planes collapse onto the k d 3 = 0 plane. The new invariant γ d 0 is given as a sum γ 0 + γ π = s since the unitary transformation U does not affect any these topological invariants.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3Under time-reversal, θ changes sign; the translation operator T1/2 changes cs3 by an exact form and does not affect θ; together S symmetry implies θ = -θ + 2πn, quantizing θ to 0 or π.
“Under time-reversal | u k 〉 → Θ | u k 〉 , the quantities k → -k and cs 3 → -cs 3 , and θ changes sign. The translation operator T 1 / 2 = e i 2 k 3 [ 0 1 1 0 ] changes cs 3 by an exact form (total derivative) and does not affect θ . Together, S symmetry implies that θ = -θ +2 πn for some integer n , which quantizes θ to 0 (topologically-trivial phase) or π (topological insulator phase).”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ II. Z 2 TOPOLOGICAL INVARIANT · p. 2At the Brillouin zone plane k3=0, the Hamiltonian satisfies SH(k1,k2,0)S^{-1}=H(-k1,-k2,0) with (S|k3=0)^2=-1, leading to a Z2 topological classification of this two-dimensional system.
“At the Brillouin zone (BZ) plane k 3 = 0 the Hamiltonian satisfies SH ( k 1 , k 2 , 0) S -1 = H ( -k 1 , -k 2 , 0) with ( S | k 3 =0 ) 2 = -1. These properties lead to a Z 2 topological classification of this two-dimensional system, by analogy to the Z 2 invariant in the quantum spin Hall (QSH) effect 9”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9] source
Not checked in the free run.
May combine two claims.
Findingown data§ IV. SURFACE BAND STRUCTURE · p. 6Model B's surface parallel to a1 and a2 is type F and its excitations are exactly solvable.
“For model B, the surface parallel to a 1 and a 2 is type F and its excitations are exactly solvable, with dispersion”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7In model B with type F surfaces, when the Zeeman field on opposite surfaces points in opposite directions, the total conductance of the slab is C=1, with each surface having a net C=1/2.
“In this model, when the Zeeman field on opposite surfaces points in opposite directions (blue, upper curve) the total conductance of the slab is C = 1, with each surface having a net C = 1 / 2”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8Type A surfaces possess an antiferromagnetic order that preserves S symmetry, with associated gapless excitations that can be gapped by disorder.
“Type A surfaces possess an antiferromagnetic order that preserves S symmetry, with associated gapless excitations that can be gapped by disorder.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ 2. Effective Hamiltonian · p. 5The filled bands at momenta (0,0,0), (0,π,0), (π,0,0), (π,π,0) have parity -1, -1, -1, +1 respectively, so model B is in the non-trivial topological phase.
“The filled bands at the momenta ( k 1 , k 2 , k 3 ) = (0 , 0 , 0) , (0 , π, 0) , ( π, 0 , 0) , ( π, π, 0) have parity -1 , -1 , -1 , +1 respectively, so the model is in the non-trivial topological phase.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Findingnone given§ 2. Effective Hamiltonian · p. 5In the regime where t' is much smaller than t and λ, the model leads to a gap in the surface spectrum also.
“We are interested in the regime where t ′ is much smaller than t and λ , as this leads to a gap in the surface spectrum also.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ 2. Effective Hamiltonian · p. 5In the model with rock-salt structure and B sites antiferromagnetically ordered along (111) planes, the spin-orbit coupling between two A1 atoms on the same layer vanishes, while the spin-orbit coupling between A1 and A2 does not vanish.
“From Fig. 2, it is apparent that spinorbit coupling between two A 1's on the same layer vanishes by our argument earlier, as the intermediate sites have opposite magnetization. In contrast, the spin-orbit coupling between A 1 and A 2 does not vanish.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ 2. Effective Hamiltonian · p. 5In model B, λ is related to the parameter breaking time-reversal symmetry while protecting the bulk gap; turning λ to zero makes the model conducting rather than a strong topological insulator.
“In this model, λ is related to the parameter breaking time-reversal symmetry, at the same time protecting the bulk gap. If we turn the parameter λ down to zero, we will not get a STI at λ = 0, rather, the model becomes conducting.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ 1. Motivation · p. 5If spins on M1 and M2 are aligned oppositely in the ±z direction, there is no net magnetization in the rhombus, the symmetry between the two paths is restored, and there are no σz couplings between the two sites.
“If spins on M 1 and M 2 are aligned oppositely in the ± z direction, by contrast, there is no net magnetization in the rhombus and the symmetry between the two paths X → M ∗ → Y is restored. Rotating the system by π along the axis through points X and Y , taking M 1 to M 2 and vice versa, we see that there are no σ z couplings between the two sites.”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8Type F surfaces break S symmetry and are typically gapped, analogously to a Zeeman field on the surface of a STI.
“Type F surfaces break S symmetry and are typically gapped, analogously to the situation of a Zeeman field on the surface of a STI.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Antiferromagnetic topological insulators · p. 1The AFTI state is similar to the strong topological insulator with time-reversal symmetry and shares with it properties such as a quantized magnetoelectric effect.
“This state is similar to the 'strong' topological insulator with time-reversal symmetry, and shares with it such properties as a quantized magnetoelectric effect.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3The macroscopic θ angle remains quantized at 0 or π as long as the bulk gap does not close, so the AFTI is stable to sufficiently weak interactions.
“The macroscopic θ angle remains quantized (at 0 or π ) as long as the bulk gap does not close, so the AFTI is stable to sufficiently weak interactions.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based). May combine two claims.
Interpretationnone given§ B. Magnetoelectric effect and the Chern-Simons integral · p. 3The distinction between microscopic time-reversal symmetry and S symmetry should not affect the macroscopic response of the system to uniform fields (θ), although there could be short-wavelength components of αij.
“This distinction should not affect the macroscopic response of the system to uniform fields ( i.e. , θ ), although there could be short-wavelength components of α i j .”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 8Putting two slabs with different conductance together is equivalent to making a slab with a step edge on one surface, and the chiral mode will reside at this step edge.
“In the situation outlined above, putting the two slabs with different conductance together is equivalent to making a slab with a step edge on one surface, and the chiral mode will reside at this step edge.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationnone given§ IV. SURFACE BAND STRUCTURE · p. 6For a type F surface, S symmetry is broken on the surface and the usual protection for Dirac cones or conducting surfaces no longer exists.
“For a type F surface, S symmetry is broken on the surface and the usual protection for Dirac cones or conducting surfaces no longer exists.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7The QH layers at either end of the stack are not completely cancelled, so there is a half QHE at both surfaces.
“However, the QH layers at either end of the stack are not completely cancelled, there is a half QHE at both surface.”
Quote in the paperSubstantial claimMay combine two claims.
Premisecitation§ V. FERROMAGNETIC SURFACES AND HALF-INTEGER QUANTUM HALL EFFECT · p. 7A residual Z2 topological invariant remains in the QSH even when spin is not conserved, and the QSH can be considered adiabatically connected to the two-QH-layer model with n preserved mod 2.
“However, a residual Z 2 topological invariant remains, 9,29 and we can consider the QSH as being adiabatically connected to the two-QH-layer model, but with the topological invariant n only preserved mod2.”
Quote in the paperSubstantial claim0 of 2 sources checkedMay combine two claims.
Interpretationreasoning§ IV. SURFACE BAND STRUCTURE · p. 6In model A with staggered magnetization on a cubic lattice, {111} planes are type F, while {110} and {100} planes are type A.
“In model A with staggered magnetization on a cubic lattice, { 111 } planes are type F, while planes { 110 } and { 100 } are type A.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Appendix A: Invariance to choice of unit cell · p. 9Any single-valued unitary transformation or change of coordinates will not affect the obstruction of finding such a basis, and the Z2 invariant is independent of unit cell choice.
“For example, any (single-valued) unitary transformation or a change of coordinates will not affect the obstruction of finding such basis, and it is rather straightforward from the definition that the Z 2 invariant is independent of unit cell choice.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8The paper predicts the existence of chiral 1D quantum wires at type F surface step boundaries, an experimental signature verifiable via scanning tunneling measurements.
“Finally, we predict the existence of chiral 1D quantum wires at type F surface step boundaries, an experimental signature verifiable via scanning tunneling measurements.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Interpretationreasoning§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8Should GdBiPt be truly insulating through its antiferromagnetic transition, it suffices in principle to check if it is a strong topological insulator above the Néel temperature.
“Should the material be truly insulating ( i.e. , have a bulk gap) through its antiferromagnetic transition, it suffices in principle to check if it is a strong topological insulator above the N´ eel temperature.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Appendix A: Invariance to choice of unit cell · p. 9We cannot always deform one EBZ to the other; however, we can always decompose the new EBZ into the old EBZ plus planes with no boundaries.
“As Fig. 8(c) shows, we cannot always deform one EBZ to the other; however, we can always decompose the new EBZ into the old EBZ plus planes with no boundaries.”
Quote in the paperSubstantial claimMay combine two claims.
Premisereasoning§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8The Gd sites in GdBiPt form an FCC lattice and hence their antiferromagnetic interaction is frustrated.
“The Gd sites form an FCC lattice and hence their antiferromagnetic interaction is frustrated”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Interpretationcitation§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8Other combinations of time-reversal and crystal symmetries could lead to new topological materials beyond those in the exhaustive classification of topological insulators stable to disorder.
“Other such combinations of time-reversal and crystal symmetries could lead to new topological materials beyond those in the exhaustive classification of topological insulators stable to disorder. 22,37”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [22] source
Not checked in the free run.
- cites [37]
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone given§ Appendix A: Invariance to choice of unit cell · p. 9The curvature F is gauge invariant, but the connection A depends on a particular choice of gauge for the wave functions.
“The curvature F is 'gauge invariant' (does not depend on the choice of basis functions for occupied states), but A depends on a particular choice of gauge for the wave functions.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Appendix A: Invariance to choice of unit cell · p. 9The Z2 invariant can be viewed as an obstruction to finding a continuous basis respecting S symmetry in the entire Brillouin zone; the material is in a trivial phase if such a basis exists.
“We can also view the Z 2 invariant as an obstruction to finding a continuous basis (along with the appropriate Bloch periodic boundary conditions) for the wave functions respecting S symmetry in the entire Brillouin zone. 23 The material is in a trivial phase if such a basis exists.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [23] source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Study design and conductnone given§ Appendix A: Invariance to choice of unit cell · p. 9The boundary integral in the Z2 formula requires that the wave functions at k and -k be S-conjugate pairs.
“The boundary integral in the formula above requires that the wave functions at k and -k be S -conjugate pairs.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8The new AFTI state is topological in a weaker sense than the strong 3D topological insulator, because its surface state is dependent on the surface plane and not generally stable to disorder.
“The new AFTI state is topological in a weaker sense than the strong 3D topological insulator, because its surface state is dependent on the surface plane and not generally stable to disorder; in that respect it is similar to the weak topological insulator in 3D or the 'Hopf insulator.' 34”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [34]
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Interpretationcitation§ VI. CONCLUSIONS AND POSSIBLE RELEVANCE TO GdBiPt · p. 8The 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.”
Quote in the paperNot a substantial claim0 of 3 sources checkedMay combine two claims.
Interpretationreasoning§ Appendix A: Invariance to choice of unit cell · p. 9The momentum variables are related by ̃k3 = k3 + 2R·k, so we can always choose the EBZ for the new and old systems such that they share a common boundary, namely, the two lines satisfying R·k ∈ {0,π}.
“Since the momentum variables are related by ˜ k 3 = k 3 + 2 R · k , we can always choose the EBZ for the new and old systems such that they share a common boundary, namely, the two lines satisfying R · k ∈ { 0 , π } .”
Quote in the paperNot a substantial claim- Rests on an assumption: Any choice of the effective Brillouin zone will give the same Z2 invariant.
The authors state this as an assumption.
All Brillouin zone planes have vanishing Chern numbers.
“those planes (like all BZ planes, by assumption) have vanishing Chern numbers”
The paper relies on this without support.
The bulk electronic band structure of an AFTI must be gapped to allow the topological distinction between the trivial and non-trivial phase.
“The bulk electronic band structure of an AFTI must be gapped to allow the topological distinction between the trivial phase and the non-trivial phase.”
The paper relies on this without support.
The QSH insulator may be constructed by combining two copies of a QH layer with opposite spin and Chern number ±n.
“The QSH insulator may be constructed by combining two copies of a QH layer with opposite spin and Chern number ± n .”
The paper relies on this without support.
The unitary symmetry class only has extended states at isolated values of the chemical potential; in general the surface state will have zero diagonal conductivity, with half-integer quantum Hall plateaus.
“That symmetry class only has extended states at isolated values of the chemical potential; in general the surface state will have zero diagonal conductivity, with half-integer quantum Hall plateaus.”
The paper relies on this without support.
As in the STI, a π Berry phase implies that the Fermi surface encloses an odd number of Dirac cones.
“As in the STI, a π phase implies that the Fermi surface encloses an odd number of Dirac cones.”
The paper relies on this without support.
For a magnetic field parallel to a STI surface, the Dirac cone shifts in momentum space but no gap opens; any infinitesimal component out of the plane opens a gap, so going from B out of the surface to B into the surface must close the surface gap.
“For B parallel to the surface, the Dirac cone shifts in momentum space but no gap opens. Any infinitesimal component of B out of the plane will open a gap, hence going from B out of the surface to B into the surface must necessarily close the surface gap.”
The paper relies on this without support.
The number of Dirac cones in a STI depends on the surface plane, but there is always an odd number of such cones.
“(The number of Dirac cones in a STI also depends on the surface plane, but there is always an odd number of such cones.)”
The paper relies on this without support.
Any choice of the effective Brillouin zone will give the same Z2 invariant.
“Any choice of the EBZ will give the same Z 2 invariant.”
The paper relies on this without support.
S symmetry requires that the Chern number vanishes on all closed two-dimensional surfaces.
“S symmetry requires that the Chern number vanishes on all closed two-dimensional surfaces”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Similar behavior to that in simple topological insulators can occur in antiferromagnets.
“and in antiferromagnets (86).”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [86]
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Nakajima, Y., Hu, R., Kirshenbaum, K., Hughes, A., Syers, P., Wang, X., Wang, K., Wang, R., Saha, S. R., Pratt, D., et al. (2015) · Science Advances · doi:10.1126/sciadv.1500242 · Cited by 212 · see citing works
- Authors
- Mong, R. S. K., Essin, A. M., & Moore, J. E.
- Affiliations
- University of California, Berkeley (US)
- Lawrence Berkeley National Laboratory (US)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2010-06-17
- Volume
- 81
- Issue
- 24
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W2077013831
- Semantic Scholar
- 8a9a35ddb0dd3350d6402298b0f82d1132b94688
- Cited by
- 663
- References
- 30
- Citations by year
- 2026: 29 · 2025: 51 · 2024: 49 · 2023: 60 · 2022: 60 · 2021: 92 · 2020: 80 · 2019: 64 · 2018: 46 · 2017: 23 · 2016: 25 · 2015: 27 · 2014: 15 · 2013: 25 · 2012: 8
- FWCI
- 7.20
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- 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 › Magnetic properties of thin films
- Keywords
- antiferromagnetic topological insulator, time-reversal symmetry breaking, translational symmetry breaking, surface states, surface steps, antiferromagnets
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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