Classification of topological quantum matter with symmetries

By Ching-Kai Chiu, J. C. Teo, A. Schnyder, S. Ryu. 2015

2,180 citations recorded by the source corpus.

Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. Both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time reversal, as well as spatial symmetries, such as reflection, are considered. Furthermore, the classification of gapless modes localized on topological defects is surveyed. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K theory, and nonlinear sigma models describing the Anderson (de)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, a brief overview of recent results and open questions concerning the topological classifications of interacting systems is also provided.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/f725ad47e34ebc59bb0e0e4954c03a1d498eaea6

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Gauging spatial symmetries and the classification of topological crystalline phases (2016, background): “There are some topological crystalline phases that have been argued to be robust to disorder, so long as the spatial symmetry is respected on average [40].”
Geometry of quantum phase transitions (2019, methodology): “They are at the core of the characterisation of topological phase transitions [34, 35, 36], and have been employed in the description and detection of QPT, both theoretically [31, 37, 38, 32, 39, 40, 76] and experimentally [41].”
Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle (2018, background): “The winding number of the Hamiltonian along a loop C is defined to be the winding number of n(k) [52]:NC = 1 2π ∮”
Symmetry-protected topological photonic crystal in three dimensions (2015, neutral): “Spatial symmetries [9, 11, 12, 44, 45] other than the glide reflection are to be studied in the rich context of 230 space groups and 1651 magnetic groups for any bosonic particles.”
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions (2016, background): “[28] is suitable for the context of probing the global-symmetry protected states, so-called Symmetry Protected Topological states [45] (SPTs, see [10, 46, 47] for recent reviews).”
Topological acoustics (2022, background): “internal or spatial symmetries [212], while the effect of a”
Non-Hermitian Chern Bands. (2018, neutral): “Moreover, the topology of this 1D model requires a chiral symmetry [85], which is often fragile in real systems.”
Topological characterization of chiral models through their long time dynamics (2017, background): “A classification of topological insulators in terms of their discrete symmetries and their spatial dimensionality has been obtained in the celebrated periodic table of topological insulators and superconductors [2].”