Quantum Hall edges with hard confinement: Exact solution beyond Luttinger liquid

We consider a Laughlin droplet in a confining potential which is very steep but also weak compared to the ultra-short-ranged interparticle interactions. We find that the eigenstates have a Jack polynomial structure, and have an energy spectrum which is extremely different from the well-known Luttinger liquid edge.

Figure 1

A comparison of exact diagonalization data and our analytic result for filling fraction $\nu =\frac{1}{2}$ and system size $N=12$ in the ${\lambda }_1=1$ branch. For the analytics we fit ${\mathcal{Z}}_{N-1}/{\mathcal{Z}}_N$ using these data (and check the form against Monte Carlo integration). The right panel demonstrates the convergence of ratios of certain energies (which should be exact without the problematic ${\mathcal{Z}}_{N-1}/{\mathcal{Z}}_N$ factors). Circles represent exact diagonalization data and dumbbells represent Monte Carlo data with a size corresponding to the total error in the result. The dotted line is an extrapolation to $N\to \infty$ which uses the first two subleading corrections (i.e., we fit to some function $a+b/N+c/N^{\frac{3}{2}}$). The horizontal dashed lines are the “exact” predictions for the thermodynamic limit behavior.

Figure 2

A comparison of exact diagonalization and analytics for the ${\lambda }_1=2$ band for $\nu =\frac{1}{2}$ and $N=12$ alongside the finite size scaling of energy ratios. Once again, the $N\to \infty$ extrapolation is a fit of the first two subleading corrections. Here, the $1/N^{\frac{3}{2}}$ terms are large enough to be important.