Equivalent expression of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>topological invariant for band insulators using the non-Abelian Berry connection

Yu, Rui; Qi, Xiao Liang; Bernevig, Andrei; Fang, Zhong; Dai, Xi

doi:10.1103/physrevb.84.075119

Cited by 850 publications

(682 citation statements)

References 56 publications

Supporting

Mentioning

666

Contrasting

Unclassified

Order By: Relevance

“…Finally, we remark that the method of Wilson loops (synonymous [26] with the method of Wannier centers [27]) is actively being used in topologically classifying band insulators [26,27,[44][45][46]. The present work advances the Wilson-loop methodology by (i) relating it to group cohomology through Eq.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…Consequently, P ⊥ŷ P ⊥ (equivalently, log[W]) and H s share certain traits that are only well defined in the discrete part of the spectrum, and, moreover, these traits are robust in the continued presence of said symmetries. The trait that identifies the QSH phase (in both the Zak phases and the edgemode dispersion) is a zigzag connectivity where the spectrum is discrete; here, eigenvalues are well defined, and they are Kramers degenerate at timereversal-invariant momenta but otherwise singly degenerate, and, furthermore, all Kramers subspaces are connected in a zigzag pattern [26,44,45]. In the QSH example, it might be taken for granted that the representation (T 2 ¼ −I) of the edge symmetry is identical for both H s and W; the invariance of T 2 ¼ −I throughout the interpolation accounts for the persistence of Kramers degeneracies, and consequently for the entire zigzag topology.…”

Section: B Bulk-boundary Correspondence Of Topological Insulatorsmentioning

confidence: 99%

See 1 more Smart Citation

Topological Insulators from Group Cohomology

2016

View full text Add to dashboard Cite

We classify insulators by generalized symmetries that combine space-time transformations with quasimomentum translations. Our group-cohomological classification generalizes the nonsymmorphic space groups, which extend point groups by real-space translations; i.e., nonsymmorphic symmetries unavoidably translate the spatial origin by a fraction of the lattice period. Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology. In this sense, cohomology underlies band topology. Our claim is exemplified by the first theory of time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These insulators may be described as "piecewise topological," in the sense that subtopologies describe the different high-symmetry submanifolds of the Brillouin zone, and the various subtopologies must be pieced together to form a globally consistent topology. The subtopologies that we discover include a glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow topology (exemplified by our recently proposed KHgSb material class), and quantized non-Abelian polarizations. Our cohomological classification results in an atypical bulk-boundary correspondence for our topological insulators.

show abstract

Section: Summary Of Resultsmentioning

confidence: 99%

Section: B Bulk-boundary Correspondence Of Topological Insulatorsmentioning

confidence: 99%

Topological Insulators from Group Cohomology

2016

View full text Add to dashboard Cite

show abstract

“…c=-1 c=-2 c=2 c=1 c=0 of these two values is an integer [14][15][16]. If this integer is odd, the system is in a topological phase; if it is even, the phase is topologically trivial.…”

mentioning

confidence: 99%

Z 2 antiferromagnetic topological insulators with broken C 4 symmetry

2017

View full text Add to dashboard Cite

A two-dimensional topological insulator may arise in a centrosymmetric commensurate Néel antiferromagnet (AF), where staggered magnetization breaks both the elementary translation and time reversal, but retains their product as a symmetry. Fang et al., [Phys. Rev. B 88, 085406 (2013)] proposed an expression for a Z2 topological invariant to characterize such systems. Here, we show that this expression does not allow to detect all the existing phases if a certain lattice symmetry is lacking. We implement numerical techniques to diagnose topological phases of a toy Hamiltonian, and verify our results by computing the Chern numbers of degenerate bands, and also by explicitly constructing the edge states, thus illustrating the efficiency of the method.Physical phenomena, whose description involves topology, have been invariably attracting attention regardless of whether the word "topology" was actually used at the time: early examples involve topologically non-trivial stable defects such as dislocations in crystals, as well as vortices in superconductors and superfluids. Quantum Hall Effect and its remarkably precise conductance quantization [1] marked the advent of an entirely new class of phenomena, related not so much to the appearance in the sample of finite-size topological objects, but rather to the electron state of the entire sample changing its topology in a way, that could no longer be undone by a local perturbation. More recently, it was understood that, in fact, non-trivial topology may appear even in zero magnetic field: the fact that a commonplace band insulator may find itself in distinct electron states that cannot be continuously transformed one into another without a phase transition, came as a major surprise [2,3].These phenomena invite the question of classifying topologically distinct states of matter: labeling each state by a set of discrete indices in such a way as to have different sets for any two phases that cannot be continuously transformed one into another without the system undergoing a phase transition. In the general setting, the problem remains unsolved.In fact, open questions are present even in a noninteracting description of systems that are believed to admit a Z 2 ("even-odd") classification, and thus have only one topologically trivial and one topologically non-trivial phase, commonly called topological. Here, we address one such question, that has recently attracted attention: diagnosing the topological phase of a Z 2 insulating Néel antiferromagnet.To put the subsequent presentation in context, we recapitulate the key results for the prototypical Z 2 system: a paramagnetic topological insulator. Fu and Kane [4] have shown that the Z 2 invariant for such a system can be defined via the so-called sewing matrix w(k) mn :where the |Ψ n,k is the Bloch eigenstate of the n-th band at momentum k. The w(k) mn turns out to be of particular interest at special momenta Γ i such that −Γ i = Γ i + G, with G a reciprocal lattice vector. Such Γ i are now commonly called the "time reversal-i...

show abstract

“…29,36,[41][42][43] Here we will follow Ref. 29 and we will argue here that this new formulations bring certain numerical advantages which open the possibility of directly computing the Z 2 invariants for systems with extremely large unit cells, particularly for disordered samples (as opposed to indirectly inferring the Z 2 invariants from other type of calculations such as transport simulations of the surface states).…”

mentioning

confidence: 99%

Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of theZ2invariant

Leung

Prodan

2012

Phys. Rev. B

View full text Add to dashboard Cite

We study the effect of strong disorder in a 3-dimensional topological insulators with time-reversal symmetry and broken inversion symmetry. Firstly, using level statistics analysis, we demonstrate the persistence of delocalized bulk states even at large disorder. The delocalized spectrum is seen to display the levitation and pair annihilation effect, indicating that the delocalized states continue to carry the Z2 invariant after the onset of disorder. Secondly, the Z2 invariant is computed via twisted boundary conditions using an efficient numerical algorithm. We demonstrate that the Z2 invariant remains quantized and non-fluctuating even after the spectral gap becomes filled with dense localized states. In fact, our results indicate that the Z2 invariant remains quantized until the mobility gap closes or until the Fermi level touches the mobility edges. Based on such data, we compute the phase diagram of the Bi2Se3 topological material as function of disorder strength and position of the Fermi level.

show abstract

Equivalent expression ofZ2topological invariant for band insulators using the non-Abelian Berry connection

Cited by 850 publications

References 56 publications

Topological Insulators from Group Cohomology

Topological Insulators from Group Cohomology

Z 2 antiferromagnetic topological insulators with broken C 4 symmetry

Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of theZ2invariant

Contact Info

Product

Resources

About