Quantized response and topology of magnetic insulators with inversion symmetry

Turner, Ari M.; Zhang, Yi; Mong, Roger S. K.; Vishwanath, Ashvin

doi:10.1103/physrevb.85.165120

Cited by 291 publications

(343 citation statements)

References 41 publications

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“…(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. With the recent emergence of Floquet topological phases, an interesting direction would be to consider further extending Eq.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…On the other hand, W topologies encode these bulk symmetries, and are therefore more reliable in a topological classification. In some cases [26,46,55,56], these bulk symmetries enable W topologies that have no boundary analog. Simply put, some topological phases do not have robust boundary states, a case in point being the Z topology of 2D inversion-symmetric insulators [26].…”

Section: B Bulk-boundary Correspondence Of Topological Insulatorsmentioning

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: B Bulk-boundary Correspondence Of Topological Insulatorsmentioning

confidence: 99%

Topological Insulators from Group Cohomology

2016

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“…Therefore A (0) a = 0, which meansĀ (0) a,11 comes only from the gauge twist, i.e.,Ā (0) a,11 = C a, 11 . It follows that ξ 1 = φ 1 = ±π for such a special branch choice, and θ VL = ±π according to Eq.…”

Section: Time-reversal Symmetrymentioning

confidence: 99%

“…Not surprisingly, in the presence of either time-reversal (T ) or inversion (P) symmetry, the ME responses coming from the spin terms and from the Kubo-like orbital terms all vanish. However, there can still be an exotic isotropic ME response, which vanishes in an ordinary insulator but takes values of ±e 2 /2h in T -respecting strong topological insulators 8,9 and in P-respecting axion insulators, 10,11 arising from the Chern-Simons term. [11][12][13] This Chern-Simons coupling is conventionally parametrized by a dimensionless phase angle θ via…”

Section: Introductionmentioning

confidence: 99%

Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling

Liu

Vanderbilt

2015

Phys. Rev. B

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We propose a new method for calculating the Chern-Simons orbital magnetoelectric coupling, conventionally parametrized in terms of a phase angle θ. According to previous theories, θ can be expressed as a 3D Brillouinzone integral of the Chern-Simons 3-form defined in terms of the occupied Bloch functions. Such an expression is valid only if a smooth and periodic gauge has been chosen in the entire Brillouin zone, and even then, convergence with respect to the k-space mesh density can be difficult to obtain. In order to solve this problem, we propose to relax the periodicity condition in one direction (say, the kz direction) so that a gauge discontinuity is introduced on a 2D k plane normal to kz. The total θ response then has contributions from both the integral of the Chern-Simons 3-form over the 3D bulk BZ and the gauge discontinuity expressed as a 2D integral over the k plane. Sometimes the boundary plane may be further divided into subregions by 1D "vortex loops" which make a third kind of contribution to the total θ, expressed as a combination of Berry phases around the vortex loops. The total θ thus consists of three terms which can be expressed as integrals over 3D, 2D and 1D manifolds. When time-reversal symmetry is present and the gauge in the bulk BZ is chosen to respect this symmetry, both the 3D and 2D integrals vanish; the entire contribution then comes from the vortex-loop integral, which is either 0 or π corresponding to the Z2 classification of 3D time-reversal invariant insulators. We demonstrate our method by applying it to the Fu-Kane-Mele model with an applied staggered Zeeman field.

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“…For our interest, we consider the topological phase of this model to be protected by inversion symmetry 67,68 , where the inversion operator I acts on the lattice fermions as…”

Section: A Coupled Wire Construction Of a Fermionic Diracmentioning

confidence: 99%

Bosonic analog of a topological Dirac semimetal: Effective theory, neighboring phases, and wire construction

2016

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We construct a bosonic analog of a two-dimensional topological Dirac Semi-Metal (DSM). The low-energy description of the most basic 2D DSM model consists of two Dirac cones at positions ±k0 in momentum space. The local stability of the Dirac cones is guaranteed by a composite symmetry Z T I 2 , where T is timereversal and I is inversion. This model also exhibits interesting time-reversal and inversion symmetry breaking electromagnetic responses. In this work we construct a bosonic version by replacing each Dirac cone with a copy of the O(4) Nonlinear Sigma Model (NLSM) with topological theta term and theta angle θ = ±π. One copy of this NLSM also describes the gapless surface termination of the 3D Bosonic Topological Insulator (BTI). We compute the time-reversal and inversion symmetry breaking electromagnetic responses for our model and show that they are twice the value one gets in the DSM case matching what one might expect from, for example, a bosonic Chern insulator. We also investigate the stability of the BSM model and find that the composite Z T I 2 symmetry again plays an important role. Along the way we clarify many aspects of the surface theory of the BTI including the electromagnetic response, the charges and statistics of vortex excitations, and the stability to symmetry-allowed perturbations. We briefly comment on the relation between the various descriptions of the O(4) NLSM with θ = π used in this paper (a dual vortex description and a description in terms of four massless fermions) and the recently proposed dual description of the BTI surface in terms of 2 + 1 dimensional Quantum Electrodynamics with two flavors of fermion (N = 2 QED3). In a set of four Appendixes we review some of the tools used in the paper, and also derive some of the more technical results.

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Quantized response and topology of magnetic insulators with inversion symmetry

Cited by 291 publications

References 41 publications

Topological Insulators from Group Cohomology

Topological Insulators from Group Cohomology

Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling

Bosonic analog of a topological Dirac semimetal: Effective theory, neighboring phases, and wire construction

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