Bilayer mapping of the paired quantum Hall state: Instability toward anisotropic pairing

One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $\nu =5/2$ is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole-symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, $d$, roughly equal to the magnetic length, $l_{\mathrm{B}}$. Specifically, by using exact diagonalization in the torus geometry, we show that the exact $5/2$ state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding $d/l_{\mathrm{B}}$ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at $d\gg l_{\mathrm{B}}$, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at $d\gg l_{\mathrm{B}}$ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic $p_x$- or $p_y$-wave pairing instead of $p_x\pm i{p}_y$-wave pairing in the MR Pfaffian/anti-Pfaffian state.

Figure 1

Schematic diagram for the bilayer mapping, via which the exact $5/2$ state at a given Haldane pseudopotential variation, $\delta V_1^{\left(1\right)}$, is mapped onto the antisymmetrized bilayer ground state at a corresponding interlayer distance, $d$.

Figure 2

Square of overlap, ${\left|\mathcal{O}\right|}^2$, between the antisymmetrized ground state of the BQH system, $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$, and three trial states, ${\Psi }_{\mathrm{trial}}$: (i) the fully pseudospin-polarized CF sea state at half filling, ${\Psi }_{{}^2{\mathrm{CFS}|}_{S_{\max }}}$; (ii) the MR Pfaffian state, ${\Psi }_{\mathrm{MR}}$; and (iii) the antisymmetrized product state of two CF seas at quarter filling with pseudospin-up and -down, $\mathcal{A}{\Psi }_{{}^4{\mathrm{CFS}}_{\uparrow }}\otimes {\Psi }_{{}^4{\mathrm{CFS}}_{\downarrow }}$. Here ${\Psi }_{\mathrm{BQH}}\left[d\right]$ is obtained via ED of the BQH system in the torus geometry with the total number of electrons $N=12$ and the number of pseudospin-up and -down electrons $N_{\uparrow }=N_{\downarrow }=6$. The aspect ratio of the rectangular unit call is set as $r_a=0.98$. All calculations are performed at pseudomomentum $\mathbf{K}=(6,6)$. Note that ${\Psi }_{{}^4{\mathrm{CFS}}_{\uparrow }}\otimes {\Psi }_{{}^4{\mathrm{CFS}}_{\downarrow }}$ is obtained as the exact ground state of the BQH system at sufficiently large interlayer distance, say, $d/l_{\mathrm{B}}=10$.

Figure 3

Square of overlap (a)–(d) between the exact $5/2$ state, ${\Psi }_{5/2}\left[\delta V_1^{\left(1\right)}\right]$, and the antisymmetrized bilayer ground state, $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$, as a function of $\delta V_1^{\left(1\right)}/V_1^{\left(1\right)}$ and $d/l_{\mathrm{B}}$. Here ${\Psi }_{5/2}\left[\delta V_1^{\left(1\right)}\right]$ is obtained via ED of the SLL Hamiltonian in Eq. (2), $H_{1\mathrm{LL}}$, in the $N=12$ system. Meanwhile, $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$ is obtained by applying the antisymmetrization operator onto the exact ground state of the BQH system, ${\Psi }_{\mathrm{BQH}}\left[d\right]$, which is, in turn, obtained via ED of the BQH Hamiltonian, $H_{\mathrm{BQH}}$, in Eq. (7) in the $N=12$ system with $N_{\uparrow }=N_{\downarrow }=6$. While the overall pattern is similar, the square of overlap is improved substantially by applying the PH symmetrization operator, ${\mathcal{S}}_{\mathrm{PH}}$, onto $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$, the results of which are shown in (e)–(h). For convenience, several contour lines are drawn, along which the square of overlap takes the constant values of $77\%,81\%,85\%,89\%,93\%$, and $97\%$ from low to high values. All calculations are performed at pseudomomentum $\mathbf{K}=(6,6)$ in the torus geometry. The aspect ratio, $r_a$, of the rectangular unit cell is varied from 1.0 [(a) and (e)], to 0.98 [(b) and (f)], to 0.8 [(c) and (g)], and to 0.5 [(d) and (h)]. Note that, among all pseudomomentum sectors, the global ground state of $H_{1\mathrm{LL}}$ occurs at $\mathbf{K}=(6,6)$ in almost the entire range of $\delta V_1^{\left(1\right)}$ except for few sporadic windows regardless of $r_a$. Even when it does not, the global ground state tends to occur at one of the three quasidegenerate sectors among $\mathbf{K}=(6,6),(6,0)$, and $(0,6)$. The quasidegeneracy is most pronounced in the regime of $0\lesssim \delta V_1^{\left(1\right)}/V_1^{\left(1\right)}\lesssim 0.1$. The path following $\delta V_1^{\left(1\right)}=0$ is denoted by white solid lines, where the interaction is at the pure Coulomb point. The path following $d/l_{\mathrm{B}}=2$ is denoted by white dashed lines, where $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]\simeq {\Psi }_{\mathrm{MR}}$.

Figure 4

Similar plot to Fig. 3 for the square of overlap between the exact $5/2$ state, ${\Psi }_{5/2}\left[\delta V_1^{\left(1\right)}\right]$, and the antisymmetrized bilayer ground state, $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$, at $\mathbf{K}=(0,6)$, whose ground state at $\delta V_1^{\left(1\right)}=0$ is globally the third-lowest energy state at $r_a=0.98$ and the second-lowest energy state at $r_a=0.8$. Note that the global ground state occurs at $\mathbf{K}=(6,6)$. See Fig. 6 for the exact 5/2 energy spectra at $\delta V_1^{\left(1\right)}=0$. As before, several contour lines are drawn, along which the square of overlap takes the constant values of $81\%,85\%,89\%,93\%$, and $97\%$ from low to high values. Note that the PH parity of the exact 5/2 state changes at the first-order-like phase boundary.

Figure 5

Square of overlap, ${\left|\mathcal{O}\right|}^2$, between the exact $5/2$ state, ${\Psi }_{5/2}\left[\delta V_1^{\left(1\right)}\right]$, and the PH-symmetry-restored antisymmetrized bilayer ground state, ${\mathcal{S}}_{\mathrm{PH}}\left\{\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]\right\}$, at (a) $d/l_{\mathrm{B}}=2$ and (b) 5 as a function of $\delta V_1^{\left(1\right)}/V_1^{\left(1\right)}$ for various $r_a$. All calculations are performed at pseudomomentum $\mathbf{K}=(6,6)$.

Figure 6

Exact energy spectra of the $N=12$ system in the half-filled SLL at the Coulomb point, i.e., $\delta V_1^{\left(1\right)}=0$ for $r_a=$ (a) 0.98, (b) 0.8, and (c) 0.5 as a function of pseudomomentum magnitude $K={[{(2\pi K_x/a)}^2+{(2\pi K_y/b)}^2]}^{1/2}$ in units of $1/l_{\mathrm{B}}$. Note that there is a threefold quasidegeneracy among the ground states in $\mathbf{K}=(6,6),(6,0)$, and $(0,6)$, which are enclosed by solid ellipses to guide the eye. There are additional quasidegenerate excited copies of the ground states at near-unity $r_a$, which are enclosed by a dashed ellipse in (a).

Figure 7

Square of “self-overlap,” ${\left|\mathcal{O}\right|}^2$, between the PH-symmetry-restored antisymmetrized bilayer ground states with respect to the variation of (a) the aspect ratio of the unit cell, $r_a$, (b) the anisotropy ratio of band mass, $r_m$, and (c) the anisotropy ratio of dielectric tensor, $r_d$, at $d/l_{\mathrm{B}}=2$ and 5. Specifically, (a) shows the square of overlap, ${\left|\mathcal{O}\right|}^2$, between the PH-symmetry-restored antisymmetrized bilayer ground state in the square unit cell, ${\mathcal{S}}_{\mathrm{PH}}\left\{\mathcal{A}{\Psi }_{\mathrm{BQH}}[d;r_a=1]\right\}$, and that in the rectangular unit cell at general $r_a,{\mathcal{S}}_{\mathrm{PH}}\left\{\mathcal{A}{\Psi }_{\mathrm{BQH}}[d;r_a]\right\}$. Panels (b) and (c) show the similar square of overlap as functions of $r_m$ and $r_d$, respectively. Note that, in the study of the band mass as well as the dielectric tensor anisotropy, all states are obtained in the square unit cell.

Figure 8

Square of overlap, ${\left|\mathcal{O}\right|}^2$, between the PH-symmetry-restored antisymmetrized bilayer ground state at unity $r_a,{\mathcal{S}}_{\mathrm{PH}}\left\{\mathcal{A}{\Psi }_{\mathrm{BQH}}[d;r_a=1]\right\}$, and its “rotational-symmetry-restored” version at general $r_a,\frac{1}{\mathcal{N}}{\mathcal{S}}_{\mathrm{PH}}\{\mathcal{A}{\Psi }_{\mathrm{BQH}}[d;r_a]\pm \mathcal{A}{\Psi }_{\mathrm{BQH}}[d;r_a^{-1}]\}$, where $1/\mathcal{N}$ is the normalization constant. Here $d/l_{\mathrm{B}}=5$. This provides a compelling piece of evidence for spontaneous breaking of the discrete rotational symmetry at large $d/l_{\mathrm{B}}$.

Figure 9

Square of overlap between the exact $5/2$ state, ${\Psi }_{5/2}\left[\delta V_1^{\left(1\right)}\right]$, and the antisymmetrized bilayer ground state, $\mathcal{A}{\Psi }_{\mathrm{BQH}}\left[d\right]$, as a function of $\delta V_1^{\left(1\right)}/V_1^{\left(1\right)}$ and $d/l_{\mathrm{B}}$ at $\mathbf{K}=(5,0)$ [(a) and (c)] and $\mathbf{K}=(5,5)$ [(b) and (d)] in the $N=10$ system. Here the aspect ratio $r_a$ is set to be unity. Note that at $r_a=1$ the global ground state occurs at $\mathbf{K}=(5,0)$ and $(0,5)$ near the Coulomb point, while the lowest-energy state at $\mathbf{K}=(5,5)$ is the third excited state following two degenerate groups of the first excited states at $\mathbf{K}=(0,1),(1,0),(0,9),(9,0)$ and the second excited states at $\mathbf{K}=(1,1),(1,9),(9,1),(9,9)$. At $r_a=0.7$, the lowest-energy state at $\mathbf{K}=(5,5)$ becomes the first excited state, while the global ground state occurs at $\mathbf{K}=(5,0)$.