Classification of crystalline insulators without symmetry indicators: Atomic and fragile topological phases in twofold rotation symmetric systems

Kooi, Sander H.; Miert, Guido van; Ortix, Carmine

doi:10.1103/physrevb.100.115160

Cited by 33 publications

(33 citation statements)

References 91 publications

(120 reference statements)

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“…Such a refined partitioning of energy bands has been recently applied to certain C 2 T -symmetric and PT -symmetric systems (C 2 is π rotation, T is time reversal, and P is space inversion) when symmetry indicators are not necessarily available. Indeed, information from the irreducible representations [21,23,25] and elementary band representations [26], may not be sufficient to diagnose the fragile criterion, rather similar to how they cannot detect Chern number, or the Kane-Mele Z 2 invariant [45] and the Z 2 nested Berry phases [46][47][48], in certain scenarios. In this context, the consideration of multiple spectral gaps recently provided new insights into the fragile band topology characterized by Wilson loop winding (Euler class) [36,46,49] and has led to the prediction of a new kind of reciprocal braiding of band nodes inside the momentum space [46,[50][51][52].…”

Section: Introductionmentioning

confidence: 99%

Geometric approach to fragile topology beyond symmetry indicators

2020

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We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on C 2 T-symmetric systems that have gained recent attention, for example, in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Plücker embedding, to induce windings in the band structure necessary to facilitate nontrivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.

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Section: Introductionmentioning

confidence: 99%

Geometric approach to fragile topology beyond symmetry indicators

2020

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“…This delocalized charge accumulation is also consistent with the profile of j 0 1 (θ) in Eq. (26). Now, let us consider the 3D fourth-order TI.…”

Section: B Numerical Calculationsmentioning

confidence: 99%

“…Fragile topology is one interesting characteristics of some TCIs discovered in recent theoretical studies [16][17][18][19][20][21][22][23][24][25][26][27]. In general, TCIs are not adiabatically deformable to atomic insulators without breaking symmetries, that is, they have an obstruction to constructing exponentially localized symmetric Wannier states [28][29][30].…”

Section: Introductionmentioning

confidence: 99%

“…Another intriguing property of some TCIs is that they do not follow the conventional bulk-boundary correspondence. Recently, it has been found that some TCIs have gapless states only in some subspace of the boundary such as hinges or corners [26,. In general, dD topological phases with gapless states in (d − k)D boundaries are called dD kth-order topological phases.…”

Section: Introductionmentioning

confidence: 99%

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Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and Wilson loop

2019

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We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A of the Altland-Zirnbauer classification. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Using this method, we show that among all the possible parity configurations of inversion-symmetric insulators, at least 17% of them have fragile topology in two dimensions while fragile TIs are less than 3% percent in three dimensions. Second, we study the bulk-boundary correspondence of fragile TIs protected by inversion symmetry. In particular, we generalize the notion of d-dimensional (dD) kth-order TIs, which is normally defined for 0 < k ≤ d, to the cases with k > d, and show that they all have fragile topology. In terms of the Dirac Hamiltonian, a dD kth-order TI has (k − 1) boundary mass terms. We show that a minimal fragile TI with the filling anomaly can be considered as the dD (d + 1)th-order TI, and all the other dD kth-order TIs with k > (d + 1) can be constructed by stacking dD (d + 1)th-order TIs. Although dD (d + 1)thorder TIs have no in-gap states, the boundary mass terms carry an odd winding number along the boundary, which induces localized charges on the boundary at the positions where the boundary mass terms change abruptly. In the cases with k > (d+1), we show that the net parity of the system with boundaries can distinguish topological insulators and trivial insulators. Also, by studying the Wilson loop and nested Wilson loop spectra, we show that all the spectral windings of the Wilson loop and nested Wilson loop should be unwound to resolve the Wannier obstruction of fragile TIs. By counting the minimal number of bands required to unwind the spectral winding of the Wilson loop and nested Wilson loop, we determine the minimal number of bands to resolve the Wannier obstruction, which is consistent with the prediction from our diagnosis method of fragile topology. Finally, we show that a (d + 1)D (k − 1)th-order TI can be obtained by an adiabatic pumping of dD kth-order TI, which generalizes the previous study of the 2D third-order TI.

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“…This is because Kramers' theorem inevitably doubles the electronic charges, making the real-space invariants partially, often completely, trivial. Progress can be made identifying (partial) Berry phase 29 invariants and/or using Wilson loops as topological indices [45][46][47][48][49] as exemplified by the bulk-dislocation charge correspondence of rotation-symmetric two-dimensional crystals 31 . This additional knowledge, however, does not completely determine the (fractional) quantized electronic charges at the crystal boundaries.…”

Section: Introductionmentioning

confidence: 99%

The bulk-corner correspondence of time-reversal symmetric insulators

2021

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The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.

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Classification of crystalline insulators without symmetry indicators: Atomic and fragile topological phases in twofold rotation symmetric systems

Cited by 33 publications

References 91 publications

Geometric approach to fragile topology beyond symmetry indicators

Geometric approach to fragile topology beyond symmetry indicators

Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and Wilson loop

The bulk-corner correspondence of time-reversal symmetric insulators

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