Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3+1)-dimensional twisted gauge theory

Article

关键词: QUANTUM HALL STATES;    STATISTICS;    SYSTEMS;    COMPUTATION;    DEGENERACY;    BOUNDARIES;    ANYONS;   

DOI  :  10.1103/PhysRevB.91.035134

来源: SCIE

【 摘 要 】

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group G and a 4-cocycle twist omega(4) of G's cohomology group H-4(G,R/Z) in three-dimensional space and one-dimensional time (3 + 1D). We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The 3 + 1D twisted gauge theory can be characterized by a representation of a modular transformation group, SL(3,Z). We express the SL(3,Z) generators S-xyz and T-xy in terms of the gauge group G and the 4-cocycle omega(4). As we compactify one of the spatial directions z into a compact circle with a gauge flux b inserted, we can use the generators S-xy and T-xy of an SL(2,Z) subgroup to study the dimensional reduction of the 3D topological order C-3D to a direct sum of degenerate states of 2D topological orders C-b(2D) in different flux b sectors: C-3D = circle plus C-b(b)2D. The 2D topological orders C-b(2D) are described by 2D gauge theories of the group G twisted by the 3-cocycle omega(3(b)), dimensionally reduced from the 4-cocycle omega(4). We show that the SL(2,Z) generators, S-xy and T-xy, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into a three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

【 授权许可】

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