Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum Hall systems

The Hall viscosity, a nondissipative transport coefficient analogous to Hall conductivity, is considered for quantum fluids in gapped or topological phases. The relation of the Hall viscosity to the mean orbital spin per particle $\overline{s}$ (discovered in previous work) is elucidated with the help of examples and of the geometry of shear transformations and rotations. For noninteracting particles in a magnetic field, there are several ways to derive the result (even at nonzero temperature), including standard linear response theory. Arguments for the quantization, and the robustness of $\overline{s}$ to small changes in the Hamiltonian that preserve rotational invariance, are given. Numerical calculations of adiabatic transport are performed to check the predictions for quantum Hall systems, with excellent agreement for trial states. The coefficient of $k^4$ in the static structure factor is also considered and shown to be exactly related to the orbital spin and robust to perturbations in rotation invariant systems.

Figure 1

The $\overline{s}$ of Laughlin states for various sizes, showing rapid convergence with size. Both boson ($\nu =1/2$) and fermion ($\nu =1/3$) cases are shown. $\tau =e^{i\pi /3}$ at the center of the circular path, corresponding to hexagonal geometry. The data for each case lie very close to the horizontal line that is the corresponding expected result.

Figure 2

Same as Fig. 1, but dependence on $\tau$ at the center of the circular path is shown. Writing $\tau =\left|\tau \right|\exp i\theta$, the horizontal axis is $\theta$, and the corresponding $\left|\tau \right|$ is shown for each point. The square geometry is at $\theta ={90}^{\circ }$.

Figure 3

Same as Fig. 1 but for $\nu =1$ (boson) MR state for various sizes.

Figure 4

Same as Fig. 1 but for $\nu =1/2$ (fermion) MR state for various sizes. Convergence here is slower than previous cases.

Figure 5

The $\overline{s}$ and overlap-squared with the $\nu =1/2$ Laughlin state as a $V_2$ pseudopotential is varied; $V_0=1$. Here $N=10$. The adiabatic curvature behaves erratically very near the transition.

Figure 6

The $\overline{s}$ of the first excited Landau level $\nu =1/2$ state as a function of the strength of an ultra-short-range three-body potential. For positive values $\overline{s}$ asymptotically approaches the MR value. For negative values it passes through the anti-Pfaffian value, indicated by the lower dashed line, with no sign of the formation of a plateau.

Figure 7

The static structure factor $s\left(k\right)$ of the $\nu =1$ MR state for bosons, with $N=100$ particles. Measurements were taken over $2\times {10}^8$ MC steps. The inset shows $s\left(k\right)$ over the entire range of k.

Figure 8

The small $k$ behavior of $s_{{}_0}\left(k\right)$ (the lowest-LL–projected structure factor) of the $\nu =1$ MR state for 100 bosons, obtained from $s\left(k\right)$. Also shown is the expected small $k$ behavior $k^4/8$.

Figure 9

The coefficients of $k^4$ in $s\left(k\right)-k^2/2$ for the two smallest nonzero $k$ values plotted versus $\frac{1}{N}$ for various sizes. The expected value as $N\to \infty$ is shown as a horizontal line.

Figure 10

$s\left(k\right)$ at the smallest nonzero $k$ for systems of up to 150 anyons in the wave function in the text. The horizontal line is the expected value $s\left(0\right)=4/3$.