Nematic order and a new field theory of the quantum Hall effect

Motivated by recent experimental and theoretical studies of anisotropic versions of the fractional quantum Hall (FQH) effect, we construct an effective field theory for a continuous quantum phase transition between an isotropic FQH state and a nematic FQH state. The theory parallels earlier work on FQH ferromagnets. The $SO(3)$ order parameter $\mathbf{n}$ of the ferromagnet is replaced by the Landau-de Gennes nematic tensor order parameter $Q_{ab}$ which can be mapped to a $SO(2,1)$ Lorentz vector. We construct an analog of the $CP^1$ representation of a ferromagnet in terms of complex $SU(1,1)$ spinors. We identify these vector and spinor order parameters with the unimodular metric and zweibein fields appearing in Haldane's recent geometrical description of the FQH effect, where the metric field $g_{ab}$ is given by the matrix exponential of the nematic order parameter $Q_{ab}$. Our theory predicts that if the gap of the Girvin-MacDonald-Platzman collective mode can be made to collapse at zero momentum in a FQH system, an instability to a FQH nematic state should occur.