TY - JOUR

T1 - The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator

AU - Mera, Bruno

N1 - Funding Information:
B.M. acknowledges stimulating discussions with J. P. Nunes. B.M. is thankful for the support from SQIG—Security and Quantum Information Group, under the Fundação para a Ciência e a Tecnologia (FCT) Project UIDB/50008/2020, and European funds, namely, H2020 project SPARTA. B.M. acknowledges Projects QuantMining POCI-01-0145-FEDER-031826, PREDICT PTDC/CCI-CIF/29877/2017, and an internal IT Project QBigData PEst-OE/EEI/LA0008/2013, funded by FCT.
Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/10/29

Y1 - 2020/10/29

N2 - We provide an extensive look at Bott periodicity in the context of complex gapped topological phases of free fermions. In doing so, we remark on the existence of a product structure in the set of inequivalent phases induced by the external tensor product of vector bundles - a structure which has not yet been explored in condensed-matter literature. Bott periodicity appears in the form of a generalized Dirac monopole built out of a given phase, which is equivalent to the product of a Dirac monopole phase with that same given phase. The complex K-theory cohomology ring is presented as a natural way to store the information of these phases, with a grading corresponding to the number of Clifford symmetries modulo 2. The Künneth formula allows us to derive the result that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in one dimension allows one to generate the K-cohomology of the d-dimensional Brillouin zone. In particular, we find that the product of two SSH chains in independent momentum directions yields a two-dimensional Chern insulator. The results obtained relate the associated topological phases of charge-conserving band insulators and their topological invariants in all spatial dimensions in a unified way.

AB - We provide an extensive look at Bott periodicity in the context of complex gapped topological phases of free fermions. In doing so, we remark on the existence of a product structure in the set of inequivalent phases induced by the external tensor product of vector bundles - a structure which has not yet been explored in condensed-matter literature. Bott periodicity appears in the form of a generalized Dirac monopole built out of a given phase, which is equivalent to the product of a Dirac monopole phase with that same given phase. The complex K-theory cohomology ring is presented as a natural way to store the information of these phases, with a grading corresponding to the number of Clifford symmetries modulo 2. The Künneth formula allows us to derive the result that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in one dimension allows one to generate the K-cohomology of the d-dimensional Brillouin zone. In particular, we find that the product of two SSH chains in independent momentum directions yields a two-dimensional Chern insulator. The results obtained relate the associated topological phases of charge-conserving band insulators and their topological invariants in all spatial dimensions in a unified way.

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U2 - 10.1103/PhysRevB.102.155150

DO - 10.1103/PhysRevB.102.155150

M3 - Article

AN - SCOPUS:85095573019

SN - 2469-9950

VL - 102

JO - Physical Review B

JF - Physical Review B

IS - 15

M1 - 155150

ER -