Chern numbers, quaternions, and Berry's phases in Fermi systems

Avron, J. E.; Sadun, Lorenzo; Segert, Jan; Simon, Barry

doi:10.1007/bf01218452

Cited by 129 publications

(210 citation statements)

References 32 publications

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“…The determinant line bundle possesses over TRIM an oriented real structure. The latter originates from a quaternionic structure in the valence band bundle imposed by the time-reversal symmetry [48]. The real orientations of the determinant bundle at different TRIM are constrained by the topology of the whole time-reversal symmetric vector bundle in a way described by the Kane-Mele invariant.…”

Section: An Intrinsic Point Of View On the 2 Invariantmentioning

confidence: 99%

“…-E is a n-dimensional quaternionic vector space; -E is a 2n-dimensional complex vector space, endowed with a -antilinear operation J : E → E satisfying J 2 = −1 (called a « quaternionic structure map » (see [48])); -E is a 4n-dimensional real vector space endowed with three maps I, J, K satisfying I 2 = J 2 = K 2 = −1 and I J = −J I = K. Those applications correspond to the multiplication by i, j, and k. Conversely, if one considers a 2n-dimensional complex vector space, a map such as J endows it with a quaternionic structure [40]. In this case, we call a quaternionic basis an indexed family (e i ) i∈I of vectors that form a basis of E as a quaternionic vector space (i.e.…”

Section: Appendix B2 Vector Spaces and Quaternionsmentioning

confidence: 99%

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An introduction to topological insulators

Fruchart

Carpentier

2013

Comptes Rendus. Physique

159

126

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Electronic bands in crystals are described by an ensemble of Bloch wave functions indexed by momenta defined in the first Brillouin Zone, and their associated energies. In an insulator, an energy gap around the chemical potential separates valence bands from conduction bands. The ensemble of valence bands is then a well defined object, which can possess non-trivial or twisted topological properties. In the case of a twisted topology, the insulator is called a topological insulator. We introduce this notion of topological order in insulators as an obstruction to define the Bloch wave functions over the whole Brillouin Zone using a single phase convention. Several simple historical models displaying a topological order in dimension two are considered. Various expressions of the corresponding topological index are finally discussed. RésuméLes bandes électroniques dans un cristal sont définies par un ensemble de fonctions d'onde de Bloch dépendant du moment défini dans la première zone de Brillouin, ainsi que des énergies associées. Dans un isolant, les bandes de valence sont séparées des bandes de conduction par un gap en énergie. L'ensemble des bandes de valence est alors un objet bien défini, qui peut en particulier posséder une topologie non triviale. Lorsque cela se produit, l'isolant correspondant est appelé isolant topologique. Nous introduisons cette notion d'ordre topologique d'une bande comme une obstruction à la définition des fonctions d'ondes de Bloch à l'aide d'une convention de phase unique. Plusieurs modèles simples d'isolants topologiques en dimension deux sont considérés. Différentes expressions des indices topologiques correspondants sont finalement discutées.Keywords: topological insulator, topological band theory, quantum anomalous Hall effect, quantum spin Hall effect, Chern insulator, Kane-Mele insulator Mots-clés : isolant topologique, théorie des bandes topologique, effet Hall quantique anomal, effet Hall quantique de spin, isolant de Chern, isolant de Kane-Mele

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Section: An Intrinsic Point Of View On the 2 Invariantmentioning

confidence: 99%

Section: Appendix B2 Vector Spaces and Quaternionsmentioning

confidence: 99%

An introduction to topological insulators

Fruchart

Carpentier

2013

Comptes Rendus. Physique

159

126

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“…(Here I am summarizing some results from a forthcoming paper with Kam Tuen Law [8]. Earlier, related work includes [9,10]. )…”

Section: The General Casementioning

confidence: 99%

“…The usual procedure is to cover the circle with two overlapping patches, say over −π − δ < φ < δ and −δ < φ < π + δ, with δ a small positive number. On each patch we can use the ψ + of equation (9). In this formulation, the geometric matrix is just the identity, but there is a nontrivial transition factor at the overlap, since the wave functions on the two patches differ by a sign around φ = ±π.…”

Section: Back To the Vortexmentioning

confidence: 99%

Introduction toquantum matter

Wilczek

2012

Phys. Scr.

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“…Furthermore, as far as the present set of elementary particles goes, confined quarks would be difficult to have as experimental references, and leptons are written as Weyl fermions in the Standard Model and as such 6 There is no logical necessity for having three reference particles, other than a convenience for analyzing experimental data within the bounds of the possibly approximate conservation laws of baryon number, lepton number and strangeness (or, as Weinberg proposes [115], electric charge, since strangeness conservation is now known to be approximate).…”

Section: Reference Particlesmentioning

confidence: 99%

The Pin Groups in Physics: C, P and T

et al. 2001

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A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. Pin(1,3) is to O (1,3) what Spin(1,3) is to SO (1,3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations -provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence of two Pin groups is relevant to physics:• In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3,1).• If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1,3) and Pin(3,1).Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; it makes possible clear definitions of intrinsic parities and time reversal; it clarifies colloquial, but literally meaningless, statements. Given the difference between the Pin group and the Spin group it is useful to distinguish their representations, as groups of transformations on "pinors" and "spinors", respectively.The Pin(1,3) and Pin(3,1) fermions are twin-like particles whose behaviors differ only under space or time reversal.A section on Pin groups in arbitrary spacetime dimensions is included.

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Chern numbers, quaternions, and Berry's phases in Fermi systems

Abstract: Yes, but some parts are reasonably concrete.

Cited by 129 publications

References 32 publications

An introduction to topological insulators

An introduction to topological insulators

Introduction toquantum matter

The Pin Groups in Physics: C, P and T

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