Fermi wave vector for the non-fully spin polarized composite-fermion Fermi sea

The fully spin polarized composite fermion (CF) Fermi sea at half filled lowest Landau level has a Fermi wave vector $k^*_{\rm F}=\sqrt{4\pi\rho_e}$, where $\rho_e$ is the density of electrons or composite fermions, supporting the notion that the interaction between composite fermions can be treated perturbatively. Away from $\nu=1/2$, the area is seen to be consistent with $k^*_{\rm F}=\sqrt{4\pi\rho_e}$ for $\nu<1/2$ but $k^*_{\rm F}=\sqrt{4\pi\rho_h}$ for $\nu>1/2$, where $\rho_h$ is the density of holes in the lowest Landau level. This result is consistent with particle-hole symmetry in the lowest Landau level. We investigate in this article the Fermi wave vector of the spin-singlet CF Fermi sea (CFFS) at $\nu=1/2$, for which particle-hole symmetry is not a consideration. Using the microscopic CF theory, we find that for the spin-singlet CFFS the Fermi wave vectors for up and down spin CFFSs at $\nu=1/2$ are consistent with $k^{*\uparrow,\downarrow}_{\rm F}=\sqrt{4\pi\rho^{\uparrow,\downarrow}_e}$, where $\rho^{\uparrow}_e=\rho^{\downarrow}_e=\rho_e/2$, which implies that the residual interactions between composite fermions do not cause a non-perturbative correction for non-fully spin polarized CFFS either. Our results suggest the natural conjecture that for arbitrary spin polarization the CF Fermi wave vectors are given by $k^{*\uparrow}_{\rm F}=\sqrt{4\pi\rho^{\uparrow}_e}$ and $k^{*\downarrow}_{\rm F}=\sqrt{4\pi\rho^{\downarrow}_e}$.