Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids

The discontinuity of guiding-center Hall viscosity (a bulk property) at the edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a nonuniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. The total Hall viscosity has two distinct contributions: a “trivial” contribution associated with the geometry of the Landau orbits and a nontrivial contribution associated with guiding-center correlations. We describe a relation between the guiding-center edge-dipole moment and “momentum polarization,” which relates the guiding-center part of the bulk Hall viscosity to the “orbital entanglement spectrum (OES).” We observe that using the computationally more-onerous “real-space entanglement spectrum” just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the nontrivial information is completely contained in the OES, which also exposes a fundamental topological quantity, $\gamma$ = $\tilde{c}-\nu$, the difference between the “chiral stress-energy anomaly” (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids and vanishes in uncorrelated integer quantum Hall fluids.

Figure 1

The gray area represents the Hall fluid. In general, the drift velocity depends on the distance from the edge. Each edge follows an equipotential line.

Figure 2

Occupation profile ($•$) for the Laughlin $\nu =1/3$ state and the uniform occupation profile ${\overline{n}}_m=\nu$ ($\circ$) on a cylinder with circumference $L=15$ (${\ell }_B=1$).

Figure 3

$N=2$ and $N=4$ root occupations for (a) $\nu =1/2$ and (b) $\nu =1/3$ Laughlin states with two different circumferences $L$ and $2L$. The two bold arrows denote two Fermi momenta. For $\nu =1/2$, there are $4=2\times 2$ and $8=2\times 4$ states, respectively, between the two Fermi momenta. For $\nu =1/3$, there are $6=3\times 2$ and $12=3\times 4$ states, respectively.

Figure 4

$\nu =1/2$ Laughlin-state density profile. Red dots stand for $N=14$ and blue dots $N=15$. It plots data obtained with different $L=15$ to 24 with increments of $0.5$ (in units of ${\ell }_B$). The data points for $N=14$ are shifted up by 1/10. The horizontal lines are 1/2 and $1/2+1/10$. The linear fit of ${log}_e(k=2\pi /L)$ versus ${log}_{e}n\left(k\right)$ gives ${log}_{e}n\left(k\right)=0.963{log}_{e}k-0.010$ with the norm of residues $0.003$.

Figure 5

$\nu =1/3$ Laughlin-state density profile. Red dots stand for $N=14$ and blue dots $N=15$. $L=$ 12.5 to 22 with increments of 0.5. The data points for $N=14$ are shifted up by 1/10. The horizontal lines are $1/3+1/10$. The linear fit of ${log}_e(k=3\pi /L)$ versus ${log}_{e}n\left(k\right)$ gives ${log}_{e}n\left(k\right)=1.853{log}_{e}k-0.609$ with the norm of residues $0.017$.

Figure 6

$\nu =1/4$ Laughlin-state density profile. Blue dots stand for $N=11$. $L=$ 12.5 to 22 with increments of 0.5. The horizontal line is 1/4. The linear fit of ${log}_e(k=4\pi /L)$ versus ${log}_{e}n\left(k\right)$ gives ${log}_{e}n\left(k\right)=2.722{log}_{e}k-0.530$ with the norm of residues $0.028$.

Figure 7

$\nu =2/2$ Moore-Read-state density profile. Red dots stand for $N=18$ and blue dots $N=20$. $L=$ 13 to 20 with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$. The horizontal lines are 1 and $1+1/10$. The linear fit of ${log}_e(k=2\pi /L)$ versus ${log}_{e}n\left(k\right)$ gives ${log}_{e}n\left(k\right)=1.076{log}_{e}k+0.598$ with the norm of residues $0.002$.

Figure 8

$\nu =2/4$ Moore-Read-state density profile. Red dots stand for $N=18$ and blue dots $N=20$. $L=$ 16 to 19.5 with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$. The horizontal lines are 1/2 and $1/2+1/10$. The linear fit of ${log}_e(k=3\pi /L)$ versus ${log}_{e}n\left(k\right)$ gives ${log}_{e}n\left(k\right)=1.879{log}_{e}k-0.054$, with the norm of residues $0.002$.

Figure 9

$\Delta N\left(k\right)$ for the $\nu =1/2$ Laughlin state. Red dots stand for $N=14$ and blue dots $N=15$. It plots data obtained with $L=$ 15 to 24 with increments of 0.5. The data points for $N=14$ are shifted up by $1/10$.

Figure 10

$\Delta N\left(k\right)$ for the $\nu =1/3$ Laughlin state. Red dots stand for $N=14$ and blue dots $N=15$. $L=$ 12.5 to 22 with increments of 0.5. The data points for $N=14$ are shifted up by $1/10$.

Figure 11

$\Delta N\left(k\right)$ for the $\nu =1/4$ Laughlin state. Blue dots stand for $N=11$. $L=12.5$ to 22 with increments of $0.5$.

Figure 12

$\Delta N\left(k\right)$ for the $\nu =2/2$ Moore-Read state. Red dots stand for $N=18$ and blue dots $N=20$. $L=13$ to 20 with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$.

Figure 13

$\Delta N\left(k\right)$ for the $\nu =2/4$ Moore-Read state. Red dots stand for $N=18$ and blue dots $N=20$. $L=16$ to $19.5$ with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$.

Figure 14

$p\left(k\right)/L$ in units of $-e/4\pi$ for the $\nu =1/2$ Laughlin state. Calculated from Fig. 4. Red dots stand for $N=14$ and blue dots $N=15$. It plots data obtained with $L=$ 15 to 24 with increments of 0.5. The data points for $N=14$ are shifted up by $1/10$. The horizontal lines are 1/4 and $1/4+1/10$.

Figure 15

$p\left(k\right)/L$ in units of $-e/4\pi$ for the $\nu =1/3$ Laughlin state. Calculated from Fig. 5. Red dots stand for $N=14$ and blue dots $N=15$. $L=12.5$ to 22 with increments of 0.5. The data points for $N=14$ are shifted up by $1/10$. The horizontal lines are 1/3 and $1/3+1/10$.

Figure 16

$p\left(k\right)/L$ in units of $-e/4\pi$ for the $\nu =1/4$ Laughlin state. Calculated from Fig. 6. Blue dots stand for $N=11$. $L=12.5$ to 22 with increments of $0.5$. The horizontal line is 3/8.

Figure 17

$p\left(k\right)/L$ in units of $-e/4\pi$ for the $\nu =2/2$ Moore-Read state. Calculated from Fig. 7. Red dots stand for $N=18$ and blue dots $N=20$. $L=13$ to 20 with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$. The horizontal lines are 1/2 and $1/2+1/10$.

Figure 18

$p\left(k\right)/L$ in units of $-e/4\pi$ for the $\nu =2/4$ Moore-Read state. Calculated from Fig. 8. Red dots stand for $N=18$ and blue dots $N=20$. $L=16$ to $19.5$ with increments of 0.5. The data points for $N=18$ are shifted up by $1/10$. The horizontal lines are 1/2 and $1/2+1/10$.

Figure 19

Dots represent $\langle \Delta M_L\rangle$ of the 1/3 Laughlin (red), 1/5 Laughlin (blue), and 2/4 Moore-Read (green) states for different values of circumference $L$ calculated from the orbital entanglement spectra with the truncation level equal to 12. Here each orbital cut is a vacuum cut, $h_{\alpha }=0$. Each line represents the dominant value $L^2/\left(8{\pi }^2{\ell }_B^2\right)\times (-s/q)$, where $-s/q=1/3,2/5,1/2$, respectively.

Figure 20

The plot of subleading term $\gamma /24$ for the 1/3 Laughlin state. Red dots, vacuum cut with truncation level 12; blue dots, quasihole cut with truncation level 12; green dots, vacuum cut with truncation level 13. The horizontal line represents 1/36.

Figure 21

The plot of subleading term $\gamma /24$ for the 1/5 Laughlin state. Red dots, vacuum cut with truncation level 13; blue dots, one quasihole cut with truncation level 12; green dots, two quasihole cut with truncation level 12; black dots, vacuum cut with truncation level 16. The horizontal line represents 1/30.

Figure 22

The plot of subleading term $\gamma /24$ for the 2/4 Moore-Read state. Red dots, vacuum cut with truncation level 11; blue dots, one quasihole cut with truncation level 9; green dots, isolated fermion cut with truncation level 9; black dots, vacuum cut with truncation level 12. The horizontal line represents 1/24.