Theory of arbitrarily polarized quantum Hall states

We propose a global model which accounts for all the observed quantum Hall states in terms of an abelian doublet of Chern-Simons gauge fields, with the strength of the Chern-Simons term given by a coupling matrix. The model is employed within the composite fermion picture. PACS numbers: 73.40.Hm, 11.15.Bt, 73.20.Dx Typeset using REVTEX 1 The quantum Hall effects in two dimensional systems are observed [1,2] at high magnetic fields (B ∼ 10T). In such a case, the spin degree of freedom of the electron gets frozen in the direction of the magnetic field (which is perpendicular to the plane of the system) and has no dynamical role. Thus one assumes that the electrons are spinless for these fully polarized quantum Hall (QH) states. The situation changes, however, when B becomes relatively small: the Zeeman splitting is now not that large (also partly due to small g value, g ∼ 0.4) and, as Halperin [3] has observed, these systems are not fully polarized. Indeed, experiments reveal that at relatively small values of B, the QH states at filling factors ν = 4 3 , 8 5 , 10 7 [4,5] and 2 3 [6,7] are unpolarized while the states at ν = 3 5 [7] and 7 5 [4] are partially polarized. Further, it is also known experimentally that the states which are at maximum polarization to start with pass over to partially polarized or unpolarized states as the Zeeman energy is lowered sufficiently — either by reducing the tilting angle of the magnetic field [4–7] or by decreasing the electron density [6]. In the vanishing Zeeman splitting (VZS) limit, it has been found from numerical computations [8] that the states with ν = 2/(2n + 1) are unpolarized and those of the Laughlin sequence [9] with ν = 1/(2n+ 1) are fully polarized, in the thermodynamic limit. Also the state at ν = 3 5 has been found to be partially polarized by an exact diagonalization study [10], in agreement with experiments. Wu, Dev and Jain [11] have studied this problem and constructed trial wave functions by employing the composite fermion model (CFM), proposed originally by Jain [12]. These trial wave functions are confirmed to be exact by numerical computation. They report that, in the VZS limit, all even numerator QH states are unpolarized and all those states with both the numerator and denominator (of ν) odd are partially/fully polarized. Further, Belkhir and Jain [13] have proposed that the CFM accommodates the sequence ν = 2n/(3n + 2) all of which are spin unpolarized. From the wave functions that they construct, they also interpret that these states possess a new feature where each composite fermion carries two different types of vortices — one of which seen by all electrons while the other is visible only to an electron of like spin. We present here a generalized and consistent study of QH states in an arbitrarily po2 larized state within the frame work of Chern-Simons (CS) theory, in the same spirit as the study of Lopez and Fradkin [14] for fully polarized states. To that end, we introduce a doublet of CS gauge fields