Hall viscosity of composite fermions

Hall viscosity, also known as the Lorentz shear modulus, has been proposed as a topological property of a quantum Hall fluid. Using a recent formulation of the composite fermion theory on the torus, we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form $\nu =n/(2pn\pm 1)$, where $n$ and $p$ are integers, from the explicit wave functions for these states. The calculated Hall viscosities ${\eta }^A$ agree with the expression ${\eta }^A=(\hslash /4)\mathcal{S}\rho$, where $\rho$ is the density and $\mathcal{S}=2p\pm n$ is the “shift” in the spherical geometry. We discuss the role of modular covariance of the wave functions, projection of the center-of-mass momentum, and also of the lowest-Landau-level projection. Finally, we show that the Hall viscosity for $\nu =\frac{n}{2pn+1}$ may be derived analytically from the microscopic wave functions, provided that the overall normalization factor satisfies a certain behavior in the thermodynamic limit. This derivation should be applicable to a class of states in the parton construction, which are products of integer quantum Hall states with magnetic fields pointing in the same direction.

Figure 1

Hall viscosities as a function of the system size at fillings $2/5, 3/7, 2/9, 2/3, 3/5$, and $2/7$. The calculations are performed for a square torus with $\tau =i$. For $\nu =\frac{n}{2pn+1}$, the Hall viscosities are calculated for both the LLL-projected and -unprojected wave functions, given in Eq. (32) and Eq. (27), depicted by circles and stars, respectively. For $\nu =\frac{n}{2pn-1}$, the Hall viscosities are calculated only for the unprojected wave functions given in Eq. (28). The expected quantized Hall viscosities are given by $\frac{\hslash n\mathcal{S}}{4}$, marked by dashed horizontal lines, where $\mathcal{S}$ is the shift, and $n=\frac{N}{V}$. The Hall viscosities calculated from both the projected and the unprojected wave functions approach the expected value with increasing $N$, the number of particles. The inset shows the difference between the Hall viscosities of the projected and the unprojected wave functions; the difference vanishes with increasing $N$.

Figure 2

Hall viscosities for $\tau =e^{i\varphi }$ along the unit circle. We choose $N=20$ for $2/5, 2/9, 2/3$, and $2/7$ while $N=21$ for $3/7$ and $3/5$. The dashed lines represent the quantized values given by Eq. (7).

Figure 3

The Hall viscosity for the 16-particle $\nu =2/5$ state at several positions in the $\tau$ plane. The different symbols represent different values of ${\tau }_1$, whereas the horizontal axis plots $ln\left({\tau }_2\right)$. For a fairly large region around $\tau =i$ the Hall viscosity is seen to be quantized at the value given by Eq. (7) with $\mathcal{S}=4$. This figure illustrates that the results are consistent with the modular covariance of our wave functions. First, Hall viscosities for points related by ${\tau }_1\to {\tau }_1+1$ coincide; there are two groups of such points: ${\tau }_1=-1,0,1$ plotted in red while ${\tau }_1=-0.5,0.5$ plotted in blue. Furthermore, the ${\tau }_1=0$ curve is symmetric between $-ln\left({\tau }_2\right)$ and $ln\left({\tau }_2\right)$, consistent with the symmetry under $\tau \to -1/\tau$.

Figure 4

Hall viscosity of the $\nu =2/5$ state as a function of $L_1$, the shorter side of the torus. Results are given for two systems with $N=10$ and $N=20$, for a rectangle torus with $\tau =i\frac{V}{L_1^2}$, where $V$ is the area of torus. The length $L_1$ is quoted in units of $l_B$. When $L_1$ becomes comparable to the magnetic length $l_B$, there are obvious fluctuations in the Hall viscosity. In the thin-torus limit $L_1/l_B\to 0$, the Hall viscosity approaches the value consistent with $\mathcal{S}=1$.

Figure 5

The Hall viscosity of ${\mathrm{\Psi }}_1{\left|{\mathrm{\Psi }}_1\right|}^{\alpha }$ with varying $\alpha$ for different system sizes. The Hall viscosity for nonzero $\alpha$ appears, by visual inspection, to approach ${\eta }^A=\frac{\hslash }{4}\frac{N}{V}$ with increasing system size.

Figure 6

The absolute values of ${\alpha }_0^2$ for the CF Laughlin state in the whole $\tau$ plane for (a) $q=2$ and (c) $q=3$. The middle panel (b) shows the components $|{\alpha }_k{|}^2$ along the line $\tau =i{\tau }_2$. As a guide to the eye, panels (a) and (c) contain a dashed line satisfying the equation ${\tau }_2^2+{({\tau }_1\pm 1/2)}^3=1/4$ and ${(6{\tau }_2/7)}^2+{({\tau }_1\pm 1/2)}^2=1/4$, respectively.

Figure 7

$\frac{1}{N}{\left(\frac{\partial lnZ}{\partial {\tau }_1}\right)}_{{\tau }_2}$ (red dots) and $\frac{1}{N}{\left(\frac{\partial lnZ}{\partial {\tau }_2}\right)}_{{\tau }_1}$ (blue stars) are depicted for the unprojected $\nu =2/5$ wave function as a function of $1/N$. Both clearly vanish in the thermodynamic limit. The calculations are performed for $\tau =i$.

Figure 8

The numerical results of the correction $\mathrm{\Delta }$ at $\tau =i$ for $\nu =2/5$, for momentum-projected and -unprojected wave functions. As a cross-check the actual difference $\mathrm{\Delta }=\frac{4\eta }{\hslash n}-4$, where $n=\frac{N}{V}$ with $\eta$ taken from Fig. 1, is also included.