Investigating Anisotropic Quantum Hall States with Bimetric Geometry

We construct a low energy effective theory of anisotropic fractional quantum Hall (FQH) states. We develop a formalism similar to that used in the bimetric approach to massive gravity, and apply it to describe Abelian anisotropic FQH states in the presence of external electromagnetic and geometric backgrounds. We derive a relationship between the shift, the Hall viscosity, and a new quantized coupling to anisotropy, which we term anisospin. We verify this relationship by numerically computing the Hall viscosity for a variety of anisotropic quantum Hall states using the density matrix renormalization group. Finally, we apply these techniques to the problem of nematic order and clarify certain disagreements that exist in the literature about the meaning of the coefficient of the Berry phase term in the nematic effective action.

Figure 1

Hall viscosity ${\eta }_{22}^H$ as a function of anisotropy $\alpha$, for four different quantum Hall states. Data are obtained by introducing anisotropy into either the mass (blue squares) or interaction (green circles) part of the Hamiltonian. The lines correspond to Eq. (23), using the values of $s$ and $\varsigma$ given in Eqs. (27) and (28), but allowing $\xi$ to fluctuate to fit the data. The value at $\alpha =1$ is the shift ($\mathcal{}{\mathcal{S}}_N$). Data was obtained using system sizes $L=10–20$ and bond dimensions up to 5400. The data are plotted such that in the isotropic case they are equal to the shift.