Density Matrix Renormalization Group Study of Incompressible Fractional Quantum Hall States

We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions $\nu =1/3$ and $\nu =5/2$ for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by carefully comparing with existing numerical results on smaller systems, that DMRG is a highly effective numerical tool for studying incompressible FQH states.

Figure 1

(a) Ground-state and variational energies at $\nu =1/3$ obtained with DMRG and exact diagonalization (ED). The extrapolated values are obtained using a quadratic function of $1/N_e$. Variational results are barely visible because they lie on top of the DMRG energies at this scale. (b) Finite-size scaling using a renormalized magnetic length ${\ell }_0$, following Refs. 19, 20.

Figure 2

(a) Ground-state energies at $\nu =5/2$ for DMRG and exact diagonalization. (b) Finite-size scaling using a renormalized magnetic length ${\ell }_0$, following Refs. 19, 20.

Figure 3

Size dependence of the excitation energies and gap at $\nu =5/2$, obtained with DMRG.

Figure 4

Pair correlation function for (a) $N_e=18$, calculated using exact diagonalization and DMRG (indistinguishable from each other), and the Moore-Read state; and (b) $N_e=20$, 22 using DMRG, and the Moore-Read state for $N_e=22$, using Monte Carlo calculations.