Interlayer Pairing Symmetry of Composite Fermions in Quantum Hall Bilayers

We study the pairing symmetry of the interlayer paired state of composite fermions in quantum Hall bilayers. Based on the Halperin-Lee-Read (HLR) theory, the effect of the long-range Coulomb interaction and the internal Chern-Simons gauge fluctuation is analyzed with the random-phase approximation beyond the leading order contribution in small momentum expansion, and we observe that the interlayer paired states with a relative angular momentum $l=+1$ are energetically favored for filling $\nu =\frac{1}{2}+\frac{1}{2}$ and $\frac{1}{4}+\frac{1}{4}$. The degeneracy between states with $\pm l$ is lifted by the interlayer density-current interaction arising from the interplay of the long-range Coulomb interaction and the Chern-Simons term in the HLR theory.

Figure 1

(a) Geometry of the bilayer system. The magnetic field $\mathit{B}$ is applied upward through the two layers with the distance $d$. An interlayer paired state with a relative angular momentum $l$ ives a winding phase $2\pi l$ when one moves a CF counterclockwise around another in the other layer. (b) Effective interaction for CFs. $\mu =0$ (1) at a vertex means a coupling between the density (current) fluctuation of CFs and the Chern-Simons gauge field.

Figure 2

Frequency dependence of (a)–(c) the effective coupling constants ${\lambda }_{\varphi ,m}^{(l)}$ and (d)–(f) the difference $\mathrm{\Delta }{\lambda }_{\varphi ,m}^{(l)}={\lambda }_{\varphi ,m}^{(l)}-{\lambda }_{\varphi ,m}^{(0)}$. We set the filling fraction in (a) and (d) to $\nu =\frac{1}{2}+\frac{1}{2}$, in (b) and (e) to $\nu =\frac{1}{4}+\frac{1}{4}$, and in (c) and (f) to $\nu =\frac{1}{6}+\frac{1}{6}$. The ratio of the Coulomb energy to the Fermi energy is $(e^2/ϵl_0)/{\epsilon }_F=1$ and the layer spacing is $k_{F}d=1$. At filling $\nu =\frac{1}{2}+\frac{1}{2}$ and $\frac{1}{4}+\frac{1}{4}$, the $l=+1$ state is favored for all frequencies. In contrast, the $l=0$ pairing is stable for low frequencies at $\nu =\frac{1}{6}+\frac{1}{6}$.

Figure 3

(a) Layer spacing dependence of $\mathrm{\Delta }{\lambda }_{\varphi ,m}^{(l)}$. We set $(e^2/ϵl_0)/{\epsilon }_F=1$ and ${\omega }_m=0$ at $\nu =\frac{1}{2}+\frac{1}{2}$. Reducing the spacing makes the interaction strength stronger. (b) Effective mass dependence of $\mathrm{\Delta }{\lambda }_{\varphi ,m}^{(l)}$. Note $m^{*}\propto (e^2/ϵl_0)/{\epsilon }_F$. We set $k_{F}d=1$ and ${\omega }_m=0$ at $\nu =\frac{1}{2}+\frac{1}{2}$. In both cases, the ordering of $\mathrm{\Delta }{\lambda }_{\varphi ,m}^{(l)}$ does not change. At $\nu =\frac{1}{2}+\frac{1}{2}$, the $l=+1$ pairing is favored at any cases. $\mathrm{\Delta }{\lambda }_{\varphi ,0}^{(+2)}$ identically vanishes for $\tilde{\varphi }=2$. See also Eq. (17).