Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions
Abstract
Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highlyinteracting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of DijkgraafWitten twisted discrete gauge theories. Other bosonic TQFTs beyond the DijkgraafWitten description and all fermionic TQFTs (namely the spin TQFTs) are either higherform gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic SymmetryProtected Topological states (SPTs). We analytically calculate both the Abelian and nonAbelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can onetoone characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar AharonovBohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(BrownKervaire), SatoLevine and others. We propose a new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular S^{xy} and T^{xy} matrices, and the partition function on RP^{3} manifold, to identify all ν ∈Z_{8} classes of 2+1 dimensional gauged Z_{2}Isingsymmetric Z_{2}^{f} fermionic Topological Superconductors (realized by stacking ν layers of a pair of chiral and antichiral pwave superconductors [ p + ip and p  ip], where boundary supports nonchiral MajoranaWeyl modes) with continuum spinTQFTs.
 Publication:

Annals of Physics
 Pub Date:
 September 2017
 DOI:
 10.1016/j.aop.2017.06.019
 arXiv:
 arXiv:1612.09298
 Bibcode:
 2017AnPhy.384..254P
 Keywords:

 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Geometric Topology
 EPrint:
 41 pages, 9 figures, 3 tables. On 2+1D and 3+1D TQFTs / spin TQFTs. v4 journal: Remarks and Refs added