<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>topology in nonsymmorphic crystalline insulators: Möbius twist in surface states

Shiozaki, Ken; Sato, Masatoshi; Gomi, Kiyonori

doi:10.1103/physrevb.91.155120

Cited by 178 publications

(170 citation statements)

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“…By our cohomological classification of quasimomentum submanifolds through Eq. (1), we provide a unifying framework to classify chiral topological insulators [39] and topological insulators with robust edge states protected by space-time symmetries [1,2,4,7,8,18,19,25,[40][41][42][43]. Our framework is also useful in classifying some topological insulators without edge states [26,55,56]; one counterexample that eludes this framework may nevertheless by classified by bent Wilson loops [46] rather than the straight Wilson loops of this work.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…The advantage of this analogy is that the Wilson bands may be interpolated [35,38] [39], and all topological insulators with robust edge states protected by space-time symmetries. Here, we refer to topological insulators with either symmorphic [1,2,8] or nonsymmorphic spatial symmetries [4,7,19,40], the time-reversal-invariant quantum spin Hall phase [25], and magnetic topological insulators [18,[41][42][43]. These case studies are characterized by extensions of G ∘ by TABLE I.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…In our companion work [21], we have identified a criterion on the surface group that characterizes all robust surface states that are protected by space-time symmetries [1,2,4,7,8,18,19,25,[40][41][42][43]54]. Our criterion introduces the notion of connectivity within a submanifold (M) of the surface-Brillouin torus and generalizes the theory of elementary band representations [57,62].…”

Section: Discussion and Outlookmentioning

confidence: 99%

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Topological Insulators from Group Cohomology

2016

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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.

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Section: Discussion and Outlookmentioning

confidence: 99%

Section: Summary Of Resultsmentioning

confidence: 99%

Section: Discussion and Outlookmentioning

confidence: 99%

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Topological Insulators from Group Cohomology

2016

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“…This has been the case of all Dirac semimetals discovered so far. Below we show that a nonsymmorphic symmetry [29,[53][54][55][56][57][58][59][60][61] protects the crossing and with it the topological surface states.…”

Section: Introductionmentioning

confidence: 99%

Topological semimetals with helicoid surface states

et al. 2016

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Riemann surfaces are geometric constructions in complex analysis that may represent multi-valued holomorphic functions using multiple sheets of the complex plane. We show that the energy dispersion of surface states in topological semimetals can be represented by Riemann surfaces generated by holomorphic functions in the two-dimensional momentum space, whose constant height contours correspond to Fermi arcs. This correspondence is demonstrated in the recently discovered Weyl semimetals and leads us to predict new types of topological semimetals, whose surface states are represented by double-and quad-helicoid Riemann surfaces. The intersection of multiple helicoids, or the branch cut of the generating function, appears on high-symmetry lines in the surface Brillouin zone, where surface states are guaranteed to be doubly degenerate by a glide reflection symmetry. We predict the heterostructure superlattice [(SrIrO3)2(CaIrO3)2] to be a topological semimetal with double-helicoid Riemann surface states.Introduction The study of topological semimetals [1] has seen rapid progress since the theoretical proposal of a three-dimensional Weyl semimetal in a magnetic phase of pyrochlore iridates [2]. In general, topological semimetals are materials where the conduction and the valence bands cross in the Brillouin zone and the crossing cannot be removed by perturbations preserving certain crystalline symmetry such as the lattice translation. Bloch states in the vicinity of the band crossing possess a nonzero topological index, e.g., the Chern number in case of Weyl semimetals. The nontrivial topology gives rise to anomalous bulk properties of topological semimetals such as the chiral anomaly [3][4][5]. Several classes of topological semimetals have been theoretically proposed so far, including Weyl, [2,[6][7][8][9][10][11][12][13], Dirac [14][15][16][17] and nodal line semimetals [7,[17][18][19][20][21][22][23][24][25][26][27][28][29][30], some among which have been experimentally observed [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].Surface states of topological semimetals have attracted much attention. On the surface of a Weyl semimetal, the Fermi surface consists of open arcs connecting the projection of bulk Weyl points onto the surface Brillouin zone [2], instead of closed loops. The presence of Fermi arcs on the surface is a remarkable property that directly reflects the nontrivial topology of the bulk, and plays a key role in the experimental identification of Weyl semimetals [32,33,36]. In contrast, as shown by recent theoretical works [18,21,23,24,28,48,49], existing Dirac and nodal line semimetals do not have robust Fermi arcs that are stable against symmetry-allowed perturbations. Therefore, the general condition for protected Fermi arcs and their existence in topological semimetals beyond Weyl remain open questions.In this work, we report the discovery of a new topological semimetal phase in a wide variety of nonsymmorphic crystal structures with the glide reflection sym-

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“…Recently, the nonsymmorphic-symmetry-protected topological quantum states, including topological insulators and semimetals were under intensive research [42][43][44][45][46][47][48][49][50][51][52] . Among them, KHgX (X=As, Sb, Bi) compounds were proposed as the first family of nonsymmorphic topological insulator [49][50][51] .…”

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confidence: 99%

Observation of the topological surface state in the nonsymmorphic topological insulator KHgSb

et al. 2017

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, KHgX are insulating in the bulk but possess robust gapless surface states (see Fig. 1a), forming the unique "hourglass Fermions"on the (010) surface. The surface fermion contains four branches (quadruplets) dispersions and unbreakable zigzag chain-like patterns 49 (Fig. 1b). These surface states can also be understood as two copies of surface states of weak topological insulators 53 . The nonsymmorphic glide mirror protects these two copies from annihilating with each other inside the mirror plane. In order to visualize the intriguing hourglass fermion surface states and confirm the nonsymmorphic topological insulator nature of KHgX, ARPES is the natural experimental tool.In this work, we report comprehensive ARPES study on the electronic structures of KHgSb on both (001) and (010) Basic information of KHgSbHigh quality KHgSb crystals was synthesized by the flux method (see Appendix for details).The crystal structure of KHgSb is shown in Fig. 1c with space group P63/mmc and the lattice constants a=b=4.78 Å , c=10.225 Å . The in-plane Hg and Sb atoms show strong bonding, forming in-plane honeycomb lattices. The off-plane K atoms sit above the center of each honeycomb, sandwiched loosely by the two adjacent layers and serve as their inversion center.The natural cleavage surface is along the (001) and (010) surface (parallel to the diagonal of the in-plane honeycomb and preserves the glide reflection). The bulk Brillouin zone (BZ) and the surface BZs of both (001) and (010) surfaces are shown in Fig. 1d. Fig. 1e, f illustrate the broad Fermi surface mapping across multiple BZs on the (001) and (010) surfaces, respectively, confirming the cleavage surfaces with correct lattice parameters (the momentum direction of kx is defined along Γ − X; ky along Γ ̅ − K ̅ and kz along Γ − Z, respectively, see Fig. 1d). The core level photoemission spectra (Fig. 1g) show sharp characteristic Sb4d, K3p and Hg5d levels. The electronic structures on the (001) surfaceWe first focus on the electronic structures on the (001) According to the band structure calculation 49, KHgSb is a fully gapped nonsymmorphic crystalline topological insulator with no surface states residing on the (001) surfaces. In order to prove this, we tuned the Fermi surface by introducing potassium atoms in situ onto the sample surface so that the absorbed potassium atoms on the surface would donate free electrons.After K-dosing, we could clearly observe the bottom of the conduction band and the top of the valence band simultaneously along both (Fig. 2g(i)-(ii)). An indirect band gap of ~200 meV is observed with no signatures of in-gap surface states (Fig. 2h), agreeing with the calculations (Fig. 2e). The electronic structures on the (010) surfaceNext we demonstrate the detailed electronic structure on the (010) surface in Figure 3. Fig. 3a shows the stacking CECs. After inspecting CECs ranging from Eb=0~400meV, one could observe (see Fig. 3a,b) the quasi-one-dimensional Fermi surface disperses into two pieces when going to higher binding energies until ...

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Z2topology in nonsymmorphic crystalline insulators: Möbius twist in surface states

Cited by 178 publications

References 49 publications

Topological Insulators from Group Cohomology

Topological Insulators from Group Cohomology

Topological semimetals with helicoid surface states

Observation of the topological surface state in the nonsymmorphic topological insulator KHgSb

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