Classification of topological phases with finite internal symmetries in all dimensions
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Received: April 1, 2020 Accepted: August 12, 2020 Published: September 14, 2020
Classification of topological phases with finite internal symmetries in all dimensions
a
Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China b Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China c Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada d Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China f Department of Mathematics, Peking University, Beijing, 100871, China
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions. Keywords: Topological States of Matter, Anomalies in Field and String Theories, Anyons, Gauge Symmetry ArXiv ePrint: 2003.08898
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)093
JHEP09(2020)093
Liang Kong,a,b Tian Lan,c Xiao-Gang Wen,d Zhi-Hao Zhange,a and Hao Zhenga,b,f
Contents 1 Introduction
1 6 6 9 9 11 15
3 SPT/SET orders via boundary-bulk relation 3.1 0d SPT/SET orders 3.2 1d SPT/SET orders revisit 3.3 Condensation completion 3.4 2d SPT/SET orders 3.5 nd SPT/SET orders
18 19 20 24 31 38
A Multi-fusion n-categories
42
1
Introduction
The gapped liquid phases of matter is the simplest kind of the quantum matter in physics, and yet they contain very rich patterns of many-body quantum entanglement which require modern mathematics to describe. The notion of a gapped liquid phase is defined microscopically as an equivalence class of many-body states that are equivalent under local unitary transformations [12] and the stacking of product states [66, 81]. Gapped liquid phases without symmetries are called topological orders [49, 76, 77]. Gapped liquid phases with symmetries (i.e. 0-symmetries) inclu
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