Single-parameter variational wave functions for quantum Hall bilayers

Bilayer quantum Hall states have been shown to be described by a BCS-paired state of composite fermions. However, finding a qualitatively accurate model state valid across all values of the bilayer separation is challenging. Here, we introduce two variational wave functions, each with a single variational parameter, which can be thought of as a proxy for the BCS order parameter. Studying systems of up to $9+9$ electrons in a spherical geometry using Monte Carlo methods, we show that the ground state can be accurately described by these single-parameter variational states. In addition, we provide a numerically exact wave function for the Halperin-111 state in terms of composite fermions.

Figure 1

Variational results for $\mathrm{\Delta }$ optimization. (a) Overlaps with three representative states [the two model states and the (energy-optimized) ground state at $d/{\ell }_B=1]$ for $-2\le {log}_{10}\left(\mathrm{\Delta }\right)\le 2$ in a $6+6$ system. The CFL is described by the limit $\mathrm{\Delta }\to 0$ (BCS regime), while the 111 state is well described by the limit $\mathrm{\Delta }\to \infty$ (BEC regime). At intermediate distances $d\sim {\ell }_B$ we have the best overlap with a state having $\mathrm{\Delta }\sim 1$. (b) Orbital occupation probability for variational states. Occupation probabilities are analytical results evaluated according to Eq. (2). In the limit $\mathrm{\Delta }\to 0$ only the lowest two CF shells are filled, while at $\mathrm{\Delta }\to \infty$ all CF shells have equal occupation.

Figure 2

Variational results for $\alpha$ optimization. (a) Overlaps with three representative states [the two model states and the (energy-optimized) ground state at $d/{\ell }_B=1]$ for $-5\le \alpha \le 10$ in a $6+6$ system. The CFL state has maximum overlap with the $\alpha \to -\infty$ state, and the 111 state has maximum overlap with the $\alpha \to \infty$ state. At intermediate $d\sim {\ell }_B$, the optimum value is $\alpha \sim 1$. (b) Orbital occupation probability for variational states. For $\alpha \sim 1$ we have roughly equal occupation of all CF orbitals (corresponding to $\mathrm{\Delta }\to \infty$). The overlap with the 111 state has a local maximum at this value, however, for larger system sizes this local maximum decreases showing that the 111 state is truly captured by the $\alpha \to \infty$ limit [see Fig. 3]. In the limit $\alpha \to \infty$ which corresponds to the 111 state, only the highest CF orbital is occupied.

Figure 3

Overlaps of the $\alpha$ trial state for different system sizes and interlayer separations. The maximum overlap of $\alpha$-ansatz trial state Eq. (3) (a) with ${\mathrm{\Psi }}_{111}$ for different system sizes $N_1$ and (b) for different interlayer distances $d$ for $6+6$ electrons. The one-parameter ansatz captures the 111 state accurately for all system sizes shown and captures the state at intermediate distances $d$ well too. We also show the optimum value of $\alpha$ as a function of interlayer separation. The optimum $\alpha$ value increases as $d$ decreases and jumps discontinuously to its maximum value in the optimization range ($\alpha =10$) around $d\sim 0.5{\ell }_B$.