Fermi wave vector for the partially spin-polarized composite-fermion Fermi sea

The fully spin-polarized composite-fermion (CF) Fermi sea at the half-filled lowest Landau level has a Fermi wave vector $k_{\mathrm{F}}^{*}=\sqrt{4\pi {\rho }_e}$, 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 $k_{\mathrm{F}}^{*}=\sqrt{4\pi {\rho }_e}$ for $\nu <1/2$ but $k_{\mathrm{F}}^{*}=\sqrt{4\pi {\rho }_h}$ 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 $k_{\mathrm{F}}^{*\uparrow ,\downarrow }=\sqrt{4\pi {\rho }_e^{\uparrow ,\downarrow }}$, where ${\rho }_e^{\uparrow }={\rho }_e^{\downarrow }={\rho }_e/2$, which implies that the residual interactions between composite fermions do not cause a nonperturbative correction for spin-singlet CFFS either. Our results suggest the natural conjecture that for arbitrary spin polarization the CF Fermi wave vectors are given by $k_{\mathrm{F}}^{*\uparrow }=\sqrt{4\pi {\rho }_e^{\uparrow }}$ and $k_{\mathrm{F}}^{*\downarrow }=\sqrt{4\pi {\rho }_e^{\downarrow }}$.

Figure 1

(a) Pair-correlation function $g\left(r\right)$ as a function of $r/\ell$, where $r$ is the arc distance on the sphere, obtained using the projected wave functions of Eq. (1). The solid lines are fits using Eq. (3) for oscillations in an intermediate range of $r$. The curves (except for 14/29) have been shifted up or down by multiples of 0.02 to avoid clutter. (b) Thermodynamic value of $k_{\mathrm{F}}^{*}\ell$ as a function of $\nu$ from arc fits (solid vertical bars) and chord fits (dashed vertical bars), slightly shifted horizontally for clarity. The mean-field value $k_{\mathrm{F}}^{*\mathrm{MF}}\ell =\sqrt{\nu }$ is shown for reference.

Figure 2

Same as in Fig. 1 but for the unprojected wave functions. Also shown for reference is the mean-field value $k_{\mathrm{F}}^{*\mathrm{MF}}\ell =\sqrt{\nu }$ corresponding to composite fermions made from electrons.

Figure 3

Thermodynamic extrapolation of the Fermi wave vector $k_{\mathrm{F}}^{*}\ell$ for the projected spin-singlet Jain wave function at various filling factors along the sequence $n/(2n\pm 1)$ and the $\nu =1/2$ CF Fermi sea (bottom-most panel). The open (solid) symbols correspond to the values obtained from the chord (arc) distance on the sphere and the dashed (thick) lines show linear fits to these values as a function of $1/N$.

Figure 4

Same as Fig. 3 but for the thermodynamic extrapolation of the Fermi wave vector $k_{\mathrm{F}}^{*\mathrm{un}}\ell$ for the unprojected Jain wave function defined in Eq. (4) at various filling factors along the sequence $n/(2n+1)$.

Figure 5

Thermodynamic extrapolation of the Coulomb ground-state energies in the spherical geometry for projected (left and center panels) and unprojected (right panel) spin-singlet states in the sequence $n/(2n\pm 1)$ with even $n$. The extrapolated energies are listed in Table 1.

Figure 6

(a) The pair-correlation function obtained from the projected Jain wave functions of fully spin-polarized composite fermions, as a function the arc distance on the sphere. The thick lines are fits to $g\left(r\right)=1+A{\left(r\sqrt{4\pi {\rho }_{\mathrm{e}}}\right)}^{-\alpha }sin\left(2k_{\mathrm{F}}^{*}r+\theta \right)$ in an intermediate range of $r$ where oscillations are seen. The curves (except for $6/13$) have been shifted up or down by multiples of 0.02 for ease of viewing. (b) Thermodynamic value of $k_{\mathrm{F}}^{*}\ell$ as a function of $\nu$ obtained from linear and quadratic fits to the arc and chord data. The mean-field values $\sqrt{2\nu }$ (blue) and $\sqrt{2(1-\nu )}$ (green) are shown for reference.

Figure 7

Thermodynamic extrapolation of the Fermi wave vector $k_{\mathrm{F}}^{*}\ell$ for the projected fully polarized Jain wave function at various filling factors along the sequence $n/(2n+1)$ and the $\nu =1/2$ CF Fermi sea (bottom-most panel). The open (solid) symbols correspond to the values obtained from the chord (arc) distance on the sphere and the thick and dashed lines show linear and quadratic fits, respectively, to these values as a function of $1/N$.

Figure 8

Thermodynamic extrapolation of the Coulomb ground-state energies in the spherical geometry for the fully polarized Jain states in the sequence $n/(2n+1)$ (see Table 2 for the extrapolated energies).

Figure 9

Static structure factor obtained in the spherical geometry using Eq. (C5) with the composite-fermion wave function (red dots). The inset shows a comparison in the small wave-vector limit with predictions of the topological terms [Eq. (C6)] (blue circles) and Dirac composite-fermion theory [Eq. (C7)] (green diamonds).