Composite-fermion pairing at half-filled and quarter-filled lowest Landau level

The Halperin-Lee-Read Fermi sea of composite fermions at half-filled lowest Landau level is the realization of a fascinating metallic phase that is a strongly correlated “non-Fermi liquid” from the electrons' perspective. Remarkably, experiments have found that, as the width of the quantum well is increased, this state makes a transition into a fractional quantum Hall state, the origin of which has remained an important puzzle since its discovery more than three decades ago. We perform detailed and accurate quantitative calculations using a systematic variational framework for the pairing of composite fermions that closely mimics the Bardeen-Cooper-Schrieffer theory of superconductivity. Our calculations show that, (i) as the quantum-well width is increased, the single-component composite-fermion Fermi sea occupying the lowest symmetric subband of the quantum well undergoes an instability into a single-component $p$-wave paired state of composite fermions; (ii) the theoretical phase diagram in the quantum-well width–electron-density plane is in excellent agreement with experiments; (iii) a sufficient amount of asymmetry in the charge distribution of the quantum well destroys the fractional quantum Hall effect, as observed experimentally; and (iv) the two-component 331 state is energetically less favorable than the single-component paired state. Evidence for fractional quantum Hall effect has been seen in wide quantum wells also at quarter-filled lowest Landau level; here our calculations indicate an $f$-wave paired state of composite fermions. We further investigate bosons in the lowest Landau level at filling factor equal to one and show that a $p$-wave pairing instability of composite fermions, which are bosons carrying a single vortex, occurs for the short range as well as the Coulomb interaction, in agreement with exact diagonalization studies. The general consistency of the composite-fermion Bardeen-Cooper-Schrieffer approach with experiments lends support to the notion of composite-fermion pairing as the primary mechanism of fractional quantum Hall effects at even-denominator filling factors. Various experimental implications are mentioned.

Figure 1

The energy per particle for various candidate states at $\nu =1/2$ as a function of $\tilde{\mathrm{\Delta }}$ at density = $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$ and QW $\mathrm{width}=70\mathrm{nm}$ for a system of 32 particles. The torus geometry is used for the calculation. The BCS-$p(l=1)$ state clearly has lower energy than the CFFS. The energies are measured relative to the energy of the CFFS state ($E_{\text{CFFS}}$) in units of $e^2/\epsilon \ell$.

Figure 2

(upper panel) The energy as a function of the well width for several candidate states at density = $2.5\times {10}^{11}{\mathrm{cm}}^{-2}$. (lower panel) Energy as a function of density for a fixed well width = 60 nm. The energies are plotted with respect to the energy of the CFFS, in units of $e^2/\epsilon \ell$. The calculations are performed for $N=32$ particles on a torus.

Figure 3

This figure shows the minimum-energy state at $\nu =1/2$ (blue dots for the CFFS and red squares for the $p$-wave paired state; all other candidate states have higher energies) as a function of the electron density and the QW width. The calculation is performed for a system of 32 particles on a torus. The dashed line depicts the experimental phase boundary separating the compressible and the FQHE states at $\nu =\frac{1}{2}$ taken from Refs. [33, 34, 35, 36].

Figure 4

This figure shows the variation in the phase boundary separating the CFFS and the CF $p$-wave paired state at $\nu =1/2$ as a function of the degree of asymmetry of the charge distribution in the QW (defined in the text).

Figure 5

The energy $E/N$ of the CFFS, Pfaffian and the 331 states as a function of $1/N$ for QW width $w=70$ nm and electron density $\rho =3.0\times {10}^{11}{\mathrm{cm}}^{-2}$ at $\nu =1/2$. The spherical geometry is used for the calculation. The energies are plotted in units of $e^2/\epsilon \ell$. The energy of the 331 state plotted above is the interaction energy; it does not include the contribution $\frac{1}{2}{\mathrm{\Delta }}_{\mathrm{SAS}}$ per particle.

Figure 6

The energies of the Jain CF state and the $\overline{3}\overline{2}111$ parton state at $\nu =6/13$ as a function of $1/N$ for three different systems. In all cases, the former has lower energy in the thermodynamic limit.

Figure 7

The energy per particle as a function of $\tilde{\mathrm{\Delta }}$ for several candidate states at $\nu =1/4$. The results are for the density = $1.5\times {10}^{11}{\mathrm{cm}}^{-2}$ and QW $\mathrm{width}=70\mathrm{nm}$ for a system of 32 particles. The torus geometry is used for the calculation. The BCS-$f(l=3)$ state clearly has lower energy than the other states considered in our study. The energies are quoted relative to the energy of the CFFS ($E_{\text{CFFS}}$) in units of $e^2/\epsilon \ell$.

Figure 8

(upper panel) Plot of per particle energy as a function of QW width at a fixed density of $20\times {10}^{10}{\mathrm{cm}}^{-2}$ at $\nu =1/4$. (lower panel) Plot of energy as a function of density for a fixed QW $\mathrm{width}=60\mathrm{nm}$ at $\nu =1/4$. The energies are plotted with respect to the energy of the CFFS ($E_{\text{CFFS}}$) state in units of $e^2/\epsilon \ell$. The calculations are performed for 32 particles on a torus.

Figure 9

This figure indicates the minimum-energy state at $\nu =1/4$ (blue dots for the CFFS state and black squares for the $f$-wave paired state; all other candidate states have higher energies) as a function of the density and the QW width for a system of 32 particles. The torus geometry has been used for the calculation. The solid line represents the approximate theoretical phase boundary. The stars mark the experimental onset of the FQHE as a function of the density for two QW widths [37].

Figure 10

The energy per particle for the bosonic BCS state with $l=1$ pairing at ${\nu }_b=1$ for two interactions: the Coulomb interaction and the contact interaction. The results are for a system of 12 particles on a torus. The energies are quoted relative to the energy of the bosonic CFFS state in units of $e^2/\epsilon \ell$. The minimum-energy state occurs at $\tilde{\mathrm{\Delta }}\approx 0.8$.

Figure 11

The energy per particle for the bosonic BCS state with $l=1$ pairing at ${\nu }_b=1/3$ for a system of 12 bosons on a torus. The energies are plotted relative to the energy of the bosonic CFFS. The minimum-energy state is obtained for $k_{\mathrm{cutoff}}=k_F$ for all values of $\tilde{\mathrm{\Delta }}$, indicating the absence of a pairing instability. The same result is obtained for BCS $l=-3$ and BCS $l=3$ states.

Figure 12

The phase diagram indicating the lowest energy state at $\nu =1/4$ as a function of density and quantum-well width. The solid line represents the approximate theoretical phase boundary. The stars indicate the parameter values where experiments show the onset of FQHE at $\nu =1/4$.

Figure 13

The energy $E/N$ for the CFFS and the 22111 states at $\nu =1/4$ as a function of $1/N$. The results are for a QW width $w=70$ nm and density $\rho =3.0\times {10}^{11}{\mathrm{cm}}^{-2}$. The spherical geometry is used for the calculation. The 22111 state has lower energy than the CFFS in the thermodynamic limit. The error bars are smaller than the size of the symbols. The energies are plotted in units of $e^2/\epsilon \ell$.

Figure 14

Thermodynamic extrapolation of the neutral (blue circles) and charge (red diamonds) gaps of the $V_0$ (top panel) and LLL Coulomb interactions (bottom panel) obtained in the spherical geometry using the disk (open symbols) and spherical (filled symbols) pseudopotentials for $N$ electrons at the bosonic ${\nu }_b=1$ MR Pf flux of $2Q=N-2$. The lines show a linear extrapolation of the gaps as a function of $1/N$ and the extrapolated gaps are quoted in the plots with the error in the extrapolation shown in the parentheses. These include gaps previously given in Ref. [77].