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First-principles calculations for topological quantum materials


Discoveries of topological states and topological materials have reshaped our understanding of physics and materials over the past 15 years. First-principles calculations have had an important role in bridging the theory of topology and experiments by predicting realistic topological materials. In this Review, we offer an overview of the first-principles methodology on topological quantum materials. First, we unify different concepts of topological states in the same band inversion scenario. We then discuss the topology using first-principles band structures and newly established topological materials databases. We stress challenges in characterizing symmetry-independent Weyl semimetals and calculating topological surface states, closing with an outlook on the exciting transport and optical phenomena induced by the topology.

Key points

  • Heuristically, the simple but intuitive band inversion scenario, which is easily accessible for first-principles calculations, can unify different topological states and rationalize their topological boundary states.

  • Quantitatively, topological invariants and symmetry indicators distinguish topological phases from atomic insulators.

  • The Wilson loop reveals the bulk topology and also the surface dispersion profile.

  • The surface state topology is uniquely determined by the bulk state topology. Surface band dispersion changes as the specific surface condition is varied.

  • The band structure topology leads to interesting anomalous transport and nonlinear optical phenomena.

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Fig. 1: Band inversion, gapless boundary states and the bulk–boundary correspondence.
Fig. 2: 2D topological states and the Wilson loop.
Fig. 3: Illustration of the bulk and surface states for different topological phases.
Fig. 4: Topological insulators.
Fig. 5: Topological crystalline insulators and higher-order topological insulators.
Fig. 6: Weyl semimetals.
Fig. 7: 3D topological phases.


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B.Y. acknowledges financial support by the Willner Family Leadership Institute for the Weizmann Institute of Science, the Benoziyo Endowment Fund for the Advancement of Science, Ruth and Herman Albert Scholars Program for New Scientists and the European Research Council (ERC) (ERC Consolidator Grant No. 815869, “NonlinearTopo”).

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B.Y. conceived the review. J.X. made calculations and wrote the manuscript with input from B.Y.

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Correspondence to Binghai Yan.

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Supplementary information


Space groups

Symmetry groups that include all crystal symmetries. There are 230 space groups in total for 3D crystals.

Irreducible representation

For a space group, a representation is a set of matrices, each of which responds to a symmetry operation. The relation of symmetry operations is equivalent to the calculation of matrices. The irreducible, block-diagonal form of the matrix representation is called irreducible representation or irrep.

Wyckoff sites

In a space group, Wyckoff positions denote the symmetry-allowed positions, including sites and multiplicity, where atoms can be found.

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Xiao, J., Yan, B. First-principles calculations for topological quantum materials. Nat Rev Phys 3, 283–297 (2021).

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