Fractional quantum Hall states at $\frac{1}{3}$ and $\frac{5}{2}$ filling: Density-matrix renormalization group calculations

In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\nu =1/3$ and $5/2$. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\nu =5/2$ for large systems with electron numbers up to $N_e=34$, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.

Figure 1

Error in the ground-state energy per electron $\delta {\epsilon }_0$ is shown at (a) $\nu =1/3$ and $N_e=20$, with 1600 ($◯$), 3200 ($\square$), and 5000 ($\diamond$) states kept, respectively. The triangle data (obtained by keeping 5000 states) illustrate a situation where DRMG runs into a local minimum during initial 12 sweeps, see text for details. The reference ground-state energy is $-$0.4210509, obtained by DMRG with at most 24 000 states kept. (b) $\nu =5/2$ and $N_e=24$ with 1600 ($◯$), 3200 ($\square$), and 5000 ($\diamond$) states kept, respectively. The data denoted by the $\times$ are obtained by keeping 8000, 10 000, and 14 000 states for each four sweeps starting from the first sweep and 16 000 states for the rest of sweeps. The reference ground-state energy is $-$0.3876150, obtained by DMRG with at most 30 000 states kept.

Figure 2

(a) Finite-size scaling of ground-state energy per electron at $\nu =1/3$ with a quadratic extrapolation. (b) Finite-size scaling of ground-state energy per electron at $\nu =1/3$ with a linear extrapolation. Energy unit is renormalized by $l_0^{(\infty )}$. The error bars in both figures are much smaller than the size of the symbols.

Figure 3

(a) Finite-size scaling of the ground-state energy at $\nu =5/2$ with a quadratic fitting. (b) Finite-size scaling of the ground-state energy at $\nu =5/2$ with a linear fitting. Energy unit is renormalized by $l_0^{(\infty )}$ and only $N_e⩾18$ is used in view of the strong finite-size oscillation.

Figure 4

The low-lying entanglement spectrum of $N_e=16$ for the partition $P[0|0]$ obtained by DMRG are shown.

Figure 5

The low-lying entanglement spectra of $N_e=34$ are shown for three partitions $P[0|0]$, $P[0|1]$, and $P[1|1]$.

Figure 6

Entanglement gap at $\Delta L=0$ vs. $1/N_e$ for three different partitions. Dashed blue line is a linear fit for $N_e⩾16$.