Second-Harmonic Bulk Response of Fractional Quantum Hall Fluids

Abstract

We develop a bulk description for the second-harmonic Hall response of fractional quantum Hall fluids with broken inversion and rotational symmetries, distinct from edge theories based on nonlinear edge-mode dispersion. By projecting a tilted massive Dirac model onto a single low-energy band, we construct a composite-boson Chern--Simons theory for the lowest Landau level that retains projection-induced nonlinear electromagnetic couplings. The nonlinear Hall resistivity vanishes in the dc limit, while its finite-frequency response retains the characteristic filling-factor dependence $ρ^{(2)}_{TLL}\proptoν^{-2}$, where $ν$ is the Landau-level filling factor. At low frequency, its real and imaginary parts scale as $(ω/ω_c)^2$ and $ω/ω_c$, respectively, where $ω_c$ is the cyclotron frequency, and the response is strongly anisotropic with respect to the crystalline tilt axis. Notably, the Kohn-mode resonances present in the second-order conductivity cancel in the nonlinear Hall resistivity. These results establish finite-frequency nonlinear transport as a bulk probe of fractional quantum Hall fluids without requiring nonlinear edge-mode dispersion.