Singlet quantum Hall effect and Chern-Simons theories

In this paper, we present a theory of the singlet quantum Hall effect (SQHE). We show that the Halperin-Haldane SQHE wave function can be written in the form of a product of a wave function for charged semions in a magnetic field and a wave function for the chiral spin liquid of neutral spin-1/2 semions. We introduce a field-theoretic model in which the electron operators are factorized in terms of charged spinless semions (holons) and neutral spin-1/2 semions (spinons). Holons and spinons are described in terms of a spinless charged fermion field coupled to U(1) (charge) Chern-Simons gauge field and a spin-1/2 neutral fermion coupled to SU(2) (spin) Chern-Simons gauge fields, respectively. Only the holons couple to the magnetic field. The physics that we find agrees with the results obtained using the Haldane-Halperin wave function: the spectrum of excitations has a gap and the quantum Hall conductance σ_xy equals ν/2π, where ν is the filling fraction. The entire spectrum of physical states is shown to factorize into a charge and a spin contribution. Our picture makes the SU(2) spin symmetry manifest. The spin sector of the wave function is shown to behave like a conformal block of primary fields of the SU(2) Wess-Zumino-Witten model. The conformal dimensions of primary fields unambiguously dictates the semion statistics of the spinons. We find a generalization of the Fock cyclic condition for singlet semion wave functions.