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 $k_{\mathrm{F}}^{\ast }=\sqrt{4\pi {\rho }_e}$, 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 $\nu =1\mathrm{/}2$, the area is seen to be consistent with $k_{\mathrm{F}}^{\ast }=\sqrt{4\pi {\rho }_e}$ for $\nu <1\mathrm{/}2$ 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 $\nu =1\mathrm{/}2$, 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 $\nu =1\mathrm{/}2$ 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$k^{*\downarrow}{\rm F}=\sqrt{4\pi\rho^{\downarrow}_e}$.