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Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as…
For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological…
Symmetry plays a key role in classifying topological phases. Recent theory shows that in the presence of gauge fields, the algebraic structure of crystalline symmetries needs to be projectively represented, which enables unprecedented…
Topological crystalline phases in electronic structures can be generally classified using the spatial symmetry characters of the valence bands and mapping them onto appropriate symmetry indicators. These mappings have been recently applied…
The recent discovery of topological insulators has revived interest in the topological properties of insulating band structures. In this work, we extend the topological classification of insulating band structures to include certain point…
Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery…
Topological crystalline insulators (TCI) possess electronic states protected by crystal symmetries, rather than time-reversal symmetry. We show that the transition metal oxides with heavy transition metals are able to support nontrivial…
Quantum states naturally represent symmetry groups, though often in a projective sense. Intriguingly, the projective nature of crystalline symmetries has remained underexplored until very recently. A series of groundbreaking theoretical and…
In condensed matter physics, symmetry profoundly governs the fundamentals of topological matter. The emergence of new topological phase is typically linked to the enrichment of symmetries. Different parity-time symmetry relations…
Topological phases of matter are classified based on symmetries, with nonsymmorphic symmetries like glide reflections and screw rotations being of particular importance in the classification. In contrast to extensively studied glide…
Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding…
The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are…
Symmetry and topology are two fundamental aspects of many quantum states of matter. Recently, new topological materials, higher-order topological insulators, were discovered, featuring, e.g., bulk-edge-corner correspondence that goes beyond…
Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B ${\bf 90}$, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$-theory, we…
The topological properties of materials are, until now, associated with the features of their crystalline structure, although translational symmetry is not an explicit requirement of the topological phases. Recent studies of hopping models…
Topological crystalline insulators (TCIs) are insulating materials whose topological property relies on generic crystalline symmetries. Based on first-principles calculations, we study a three-dimensional (3D) crystal constructed by…
Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a…
A lattice symmetry, if being nonsymmorphic, is defined by combining a point group symmetry with a fractional lattice translation that cannot be removed by changing the lattice origin. Nonsymmorphic symmetry has a substantial influence on…
Topological phases of matter have been extensively studied for their intriguing bulk and edge properties. Recently, higher-order topological insulators with boundary states that are two or more dimensions lower than the bulk states, have…
A peculiar feature of quantum states is that they may embody so-called projective representations of symmetries rather than ordinary representations. Projective representations of space groups-the defining symmetry of crystals-remain…