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Modular anomalies in (2+1)- and (3+1)-dimensional edge theories

Author(s): Park, Moon Jip; Fang, Chen; Bernevig, Bogdan A.; Gilbert, Matthew J

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Abstract: The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a D-dimensional system under modular transformation, or modular anomaly, signals the presence of a (D + 1)-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a (2 + 1)-D and a (3 + 1)-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in (2 + 1)-D and (3 + 1)-D are modular invariant. However, we show in (3 + 1)-D that when the edge theory is coupled to a background U(1) gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a (4 + 1)-D bulk. Using the modular anomaly, we find that the edge theory of a (4 + 1)-D insulator with space-time inversion symmetry (PT) and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.
Publication Date: 16-Jun-2017
Electronic Publication Date: 16-Jun-2017
Citation: Park, Moon Jip, Fang, Chen, Bernevig, B Andrei, Gilbert, Matthew J. (2017). Modular anomalies in (2+1)- and (3+1)-dimensional edge theories. PHYSICAL REVIEW B, 95 (10.1103/PhysRevB.95.235130
DOI: doi:10.1103/PhysRevB.95.235130
ISSN: 2469-9950
EISSN: 2469-9969
Type of Material: Journal Article
Journal/Proceeding Title: PHYSICAL REVIEW B
Version: Final published version. Article is made available in OAR by the publisher's permission or policy.

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