U(1) X U(1) XI Z(2) Chern-Simons theory and Z(4) parafermion fractional quantum Hall states
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Barkeshli-2010-U(1) X U(1) XI Z(2).pdf
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Author(s)
Barkeshli, Maissam
Wen, Xiao-Gang
Date Issued
January 2010
Journal
Physical Review B
Publisher
American Physical Society
Citation
Maissam Barkeshli and Xiao-Gang Wen (2010). U(1) X U(1) XI Z(2) Chern-Simons theory and Z(4) parafermion fractional quantum Hall states. Phys. rev. B 81: 045323/1-18. © 2010 The American Physical Society
Version
Final published version
Abstract
We study U(1)×U(1)⋊Z2 Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U(1)×U(1)⋊Z2 Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the U(1)×U(1)⋊Z2 Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l=3 U(1)×U(1)⋊Z2 Chern-Simons theory is the low-energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction ν=2/2k−3. The U(1)×U(1)⋊Z2 theory is more useful than an alternative SU(2)4×U(1)∕U(1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the ν=2∕3 phase diagram.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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DOI of Published Version
https://doi.org/10.1103/PhysRevB.81.045323
