Abelian topological order of $\nu =2/5$ and $3/7$ fractional quantum Hall states in lattice models

Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $\nu =2/5$ and $3/7$ states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $\nu =2/5$ and $3/7$ states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.

Figure 1

Von Neumann entanglement entropy $S_{\text{vN}}$ against cylinder circumference $L_y$ in units of magnetic length, $l_B={\left(2\pi n_{\varphi }\right)}^{-1/2}$, for the fermionic hierarchy states at (a) $\nu =1/3$, (b) $\nu =2/5$, and (c) $\nu =3/7$. In each case, we use nearest-neighbor interactions ($\kappa =1$). (a),(b) The threshold where the $R^2$ value first exceeds 0.99 is marked with a green dashed line. All of the points above this threshold are used to construct the line of best fit. In (c), we present the complete data set with our largest system size $L_y=14$. In all cases, we obtain points based on the systematic procedure outlined in Appendix pp1 with ${\chi }_{\text{max}}=3000$. The complete data sets are shown in Fig. 7, and the points circled in red are studied in Fig. 2.

Figure 2

Case studies of the points circled in red from Fig. 1. The $\nu =2/5$ state is obtained at $n_{\varphi }=1/7, (L_x,L_y)=(1,10)$, and $\chi =800$. The $\nu =3/7$ state is obtained at $n_{\varphi }=1/10, (L_x,L_y)=(1,14)$, and $\chi =2000$. (a),(e) Average charge on the left half of the cylinder $\langle Q_{\mathrm{L}}\rangle$ as a function of the external flux ${\mathrm{\Phi }}_x$ adiabatically inserted along the cylinder axis, as shown in the inset (not to scale). The charge pumping was performed at the reduced bond dimensions of $\chi =400$ and 500, respectively. (b),(f) Momentum-resolved entanglement spectrum, showing the entanglement energies ${\epsilon }_{\alpha }$ as a function of the eigenvalues corresponding to the translation of Schmidt states around the cylinder $k_a$, additionally colored according to their $U\left(1\right)$ charge sector. In (b) we shift the spectrum, such that $k_a\to {\tilde{k}}_a$, to emphasize the edge modes. (c),(g) Average density $\langle {\rho }_i\rangle$ and (d),(h) two-point correlation function $\langle :{\rho }_0{\rho }_i:\rangle$ for each site in the MPS unit cell. Note that the dimensions of the MPS unit cell are given in units of the lattice constant in this figure.

Figure 3

Tuning the interaction range for the area law plots in Fig. 1. In each case we tune (a),(b) the corresponding colored points from Figs. 1 and 1, and (c) the four colored points from Fig. 1 that are closest to the line of best fit. The data for $\kappa =1,2,3$ is colored in black, green, and red, respectively, and data for intermediate values of $\kappa$ is colored blue. The value of the topological entanglement entropy predicted from the Abelian theory, $\gamma =ln\left(\sqrt{s}\right)$, is additionally marked with a dashed line. In all cases, we increase the bond dimension of these runs appropriately such that we maintain the same error threshold. Consequently, the maximum bond dimension is ${\chi }_{\text{max}}=4000$ for each filling factor. The $n_{\varphi }=1/4$ and $1/5, \kappa =3$ data points at $\nu =1/3$ filling are marked as outliers.

Figure 4

Case studies from Fig. 2 as a function of $\kappa$. The results for the $\nu =2/5$ state in (a),(b) are obtained at $\chi =800$, 2000, 2000 for $\kappa =1,2,3$, respectively, whereas the results for the $\nu =3/7$ state in (c),(d) are obtained at $\chi =2000$ for both $\kappa =1$ and 2. These bond dimensions correspond to the values required for the area law plot in Fig. 3. (a),(c) Average charge on the left half of the cylinder $\langle Q_{\mathrm{L}}\rangle$ as a function of the external flux ${\mathrm{\Phi }}_x$ adiabatically inserted along the cylinder axis. The charge pumping was performed at the reduced bond dimensions of $\chi =400$ for $\nu =2/5$ at $\kappa =1$ and $\chi =500$ for all other states. (b),(d) Momentum-resolved entanglement spectra, showing the entanglement energies ${\epsilon }_{\alpha }$ as a function of the eigenvalues corresponding to the translation of Schmidt states around the cylinder $k_a$. Where necessary, we shift the spectra to emphasize the edge modes.

Figure 5

Tuning the interaction strength for the $\kappa =1$ area law plots in Fig. 3. The topological entanglement entropy data for the (a) $\nu =1/3$, (b) $2/5$, and (c) $3/7$ series are colored blue, orange, and green, respectively. On the same axes, we plot the $R^2$ values of the corresponding area law plots with black crosses. The value of the topological entanglement entropy predicted from the Abelian theory, $\gamma =ln\left(\sqrt{s}\right)$, is marked with a dashed line. In all cases, the bond dimensions are the same as those for the $\kappa =1$ plots in Fig. 3.

Figure 6

Extrapolation of the entanglement entropy $S_{\mathrm{vN}}$ in the $\chi \to \infty$ limit for the fermionic Hofstadter model at $\nu =1/3$, with flux density $n_{\varphi }=1/3$ and cylinder circumference $L_y=6$. (a) $S_{\mathrm{vN}}$ as a function of ${\chi }^{-1}$, where $\chi \in [50,100,150,\cdots ,1100]$. (b) A closeup of the last two points shown in (a), including the equation of the straight line through these points and the ${lim}_{\chi \to \infty }\left(S_{\mathrm{vN}}\right)$ estimate with error bars shown in red. (c) The change in entropy as the bond dimension is incremented, $\mathrm{\Delta }{S}_{\mathrm{vN}}$, for the data points in (a).

Figure 7

The complete data sets for the: (a) bosonic and (b) fermionic Laughlin (zeroth hierarchy) states with $<0.1\%$ error, (c) complete bosonic, (d) complete fermionic, (e) $<0.1\%$ error fermionic first hierarchy states, and (f) the fermionic second hierarchy state. (Top panels) Von Neumann entanglement entropy $S_{\text{vN}}$ plotted as a function of cylinder circumference $L_y$ in units of magnetic length, $l_B={\left(2\pi n_{\varphi }\right)}^{-1/2}$. The line of best fit is drawn through the set of points above the smallest $L_y/l_B$ that yields $R^2>0.99$ for $<0.1\%$ error data and $R^2=R_{\text{max}}^2$ otherwise. This cutoff is marked with a green dashed line. (Middle panels) The $y$ intercept of the line of best fit drawn through all points greater than or equal to $L_y/l_B$, denoted as $-{\gamma }_{\ge }$. The Abelian theory prediction, $\gamma =ln\left(\sqrt{s}\right)$, is marked in red. (Bottom panels) The square of Pearson's correlation coefficient for the line of best fit drawn through all points greater than or equal to $L_y/l_B$, denoted as $R_{\ge }^2$. The corresponding threshold, either $R^2=0.99$ or $R_{\text{max}}^2$, is marked with a red dashed line.