Wave Function and Emergent SU(2) Symmetry in the ${\nu }_T=1$ Quantum Hall Bilayer

We propose a trial wave function for the quantum Hall bilayer system of total filling factor ${\nu }_T=1$ at a layer distance $d$ to magnetic length $\ell$ ratio $d/\ell ={\kappa }_{c1}\approx 1.1$, where the lowest charged excitation is known to have a level crossing. The wave function has two-particle correlations, which fit well with those in previous numerical studies, and can be viewed as a Bose-Einstein condensate of free excitons formed by composite bosons and anticomposite bosons in different layers. We show the free nature of these excitons indicating an emergent SU(2) symmetry for the composite bosons at $d/\ell ={\kappa }_{c1}$, which leads to the level crossing in low-lying charged excitations. We further show the overlap between the trial wave function, and the ground state of a small size exact diagonalization is peaked near $d/\ell ={\kappa }_{c1}$, which supports our theory.

Figure 1

(a) The filling factor ${\nu }_1$ of ${\mathrm{\Psi }}_{1,\alpha }$ as a function of $\alpha$ for size $N=6$, 10, and 15 from MCMC in good agreement with ${\nu }_1=1-\alpha$. (b) The density profile $\rho (r)$ of ${\mathrm{\Psi }}_{1,\alpha }$ for different $\alpha$ and $N=6$. (c) The two-particle correlations $g_{ee}(r)$ and $g_{eh}(r)$ of ${\mathrm{\Psi }}_{1,\alpha }$ for different $\alpha$ and the interlayer correlation $g_{12}(r)$ plotted in the inset. (d) The low-lying charged excitation spectrum expected in our theory, where a level crossing occurs at both $d=0$ and $d/\ell ={\kappa }_{c1}$. (e)–(f) Interaction between two excitons formed by electrons and holes (e) or CBs and anti-CBs (f). (g)–(h) Illustration of the interlayer and intralayer bound states of two CB merons, which we expect to be excitations $E_2$ and $E_3$ in (d), respectively.

Figure 2

(a) Two-particle correlations $g_{11}(r)$ and $g_{12}(r)$ of state ${\mathrm{\Psi }}_{1,1/2}$ (solid blue lines) obtained on disk and those at $d/\ell ={\kappa }_{c1}$ reproduced from DMRG calculations on torus in Ref. [39] (dashed red lines). (b) Overlap of the ground state ${\mathrm{\Psi }}_{\mathrm{ED}}$ from ED calculations for $N=2$ electrons per layer with trial wave functions ${\mathrm{\Psi }}_{111}$, ${\mathrm{\Psi }}_{1/2}$ and ${\mathrm{\Psi }}_{1/2}^S$, respectively, as a function of $d/\ell$.