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 - Funding Information:
B.B. and J.C. thank M. Zaletel and Judith Höller for fruitful discussions. M.G.V. would like to thank Gonzalo Lopez-Garmendia for help with computational work. B.B., J.C., and B.A.B. thank Adrian Po and Ashvin Vishwanath for helpful discussions. B.B., J.C., Z.W., and B.A.B. acknowledge the hospitality of the Donostia International Physics Center, where parts of this work were carried out. J.C. acknowledges the hospitality of the Kavli Institute for Theoretical Physics, and B.A.B. acknowledges the hospitality and support of the École Normale Supérieure and Laboratoire de Physique Théorique et Hautes Energies. The work of MVG was supported by FIS2016-75862-P and FIS2013-48286-C2-1-P national projects of the Spanish MINECO. The work of L.E. and M.I.A. was supported by the Government of the Basque Country (project IT779-13) and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). Z.W. and B.A.B. acknowledge the support of the NSF EAGER Award DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541, which were used to develop the initial theory and for further ab initio work. The development of the practical part of the theory, tables, and some of the code development was funded by Department of Energy de-sc0016239, the Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. J.C., B.B., and Z.W. contributed equally to the preparation of this work. APPENDIX A:
Funding Information:
The work of MVG was supported by FIS2016-75862-P and FIS2013-48286-C2-1-P national projects of the Spanish MINECO. The work of L.E. and M.I.A. was supported by the Government of the Basque Country (project IT779-13) and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). Z.W. and B.A.B. acknowledge the support of the NSF EAGER Award DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541, which were used to develop the initial theory and for further ab initio work. The development of the practical part of the theory, tables, and some of the code development was funded by Department of Energy de-sc0016239, the Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research.
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

VL - 97

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 3

M1 - 035139

ER -