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

TL;DR

This work develops a geometric, adiabatic-curvature framework that links Hall viscosity in gapped quantum fluids to the mean orbital spin per particle and to the topological shift. By combining analytical constructions (including homogeneous bundles and SL(d,R) geometry) with exact results in 2D quantum Hall systems and comprehensive numerical tests on Laughlin and Moore–Read states, it shows that the Hall viscosity is robust to perturbations within a phase and is quantized (rational) under rotational invariance. It also connects the Hall viscosity to the static structure factor and the isothermal compressibility, providing exact k^4 coefficients and highlighting a deep link to the shift S. The results offer a practical diagnostic tool for identifying topological phases and motivate experimental probes of Hall viscosity in quantum Hall and paired-superfluid systems.

Abstract

The Hall viscosity, a non-dissipative transport coefficient analogous to Hall conductivity, is considered for quantum fluids in gapped or topological phases. The relation to mean orbital spin per particle discovered in previous work by one of us is elucidated with the help of examples, using the geometry of shear transformations and rotations. For non-interacting particles in a magnetic field, there are several ways to derive the result (even at non-zero temperature), including standard linear response theory. Arguments for the quantization, and the robustness of Hall viscosity 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 also.

Figures (10)

• Figure 1: (color online) $\bar{s}$ 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 which is the corresponding expected result.
• Figure 2: (color online) Same as Fig. \ref{['fig:sLJ']}, but dependence on $\tau$ at the center of the circular path is shown. Writing $\tau=|\tau|\exp{i\theta}$, the horizontal axis is $\theta$, and the corresponding $|\tau|$ is shown for each point. The square geometry is at $\theta=90^\circ$.
• Figure 3: (color online) Same as Fig. \ref{['fig:sLJ']}, but for $\nu=1$ (boson) MR state for various sizes.
• Figure 4: (color online) Same as Fig. \ref{['fig:sLJ']}, but for $\nu=1/2$ (fermion) MR state for various sizes. Convergence here is slower than previous cases.
• Figure 5: (color online) $\bar{s}$ 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.