TY - JOUR

T1 - Building blocks of topological quantum chemistry

T2 - Elementary band representations

AU - Cano, Jennifer

AU - Bradlyn, Barry

AU - Wang, Zhijun

AU - Elcoro, L.

AU - Vergniory, M. G.

AU - Felser, C.

AU - Aroyo, M. I.

AU - Bernevig, B. Andrei

N1 - Publisher Copyright:
© 2018 American Physical Society.

PY - 2018/1/16

Y1 - 2018/1/16

N2 - The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time reversal in his theory of "elementary" band representations. In a recent paper [Bradlyn, Nature (London) 547, 298 (2017)NATUAS0028-083610.1038/nature23268] we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wave functions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

AB - The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time reversal in his theory of "elementary" band representations. In a recent paper [Bradlyn, Nature (London) 547, 298 (2017)NATUAS0028-083610.1038/nature23268] we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wave functions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

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U2 - 10.1103/PhysRevB.97.035139

DO - 10.1103/PhysRevB.97.035139

M3 - Article

AN - SCOPUS:85040628898

SN - 2469-9950

VL - 97

JO - Physical Review B

JF - Physical Review B

IS - 3

M1 - 035139

ER -