Hall viscosity of hierarchical quantum Hall states

Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter $\tau$, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor $\nu =2/5$ Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex $\tau$ plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.

Figure 1

Charge lattice of for $\nu =2/5$. Circles ($\circ$) denote ${\Gamma }^{☆}$ and dots ($•$) denote $\Gamma$. A unit cell of $\Gamma$ is spanned by ${\mathbf{q}}_{\alpha }$ (blue) and a unit cell of ${\Gamma }^{☆}$ is spanned by ${\mathbf{l}}_{\beta }$ (red). ${\Gamma }^{☆}/\Gamma$ is a finite subset of ${\Gamma }^{☆}$, constructed by equating all points in ${\Gamma }^{☆}$ related by a vector in $\Gamma$. In the figure, ${\Gamma }^{☆}/\Gamma$ is given by the points in the (blue) parallelogram. Note that ${\Gamma }^{☆}/\Gamma$ is one dimensional and spanned by ${\mathbf{h}}_0={\mathbf{l}}_1+{\mathbf{l}}_2$ (green). Note that ${\mathbf{l}}_{\alpha }={\mathbf{q}}_{\alpha }-2{\mathbf{h}}_0$.

Figure 2

Changes to a path in the $x$ direction (red) and $y$ direction (blue) under modular transformations for a rectangular torus. Under $\mathcal{S}$, the coordinate system is rotated and the $x$ and $y$ directions are effectively interchanged and so are the boundary conditions. Under $\mathcal{T}$, nothing happens with $y$, but $x\to x+y$, such that a path in the $y$ direction gets mapped on a path winding in the $x$ direction. Also the twisted path encloses $N_{\Phi }/2$ fluxes, compared to the original path.

Figure 3

Overlap of the terms $D_{1,0}$ (blue), $D_{0,1}$ (red), and $D_{1,0}+D_{0,1}$ (green) in Eq. (3.9) with the exact Coulomb state, for ${\tau }_1=0$ and $0.1<{\tau }_2<10$. Note the logarithmic scale of ${\tau }_2$ which makes the figure symmetric around ${\tau }_2=1$. Note that in the limit of a thin torus, ${\tau }_2\to \infty$ or ${\tau }_2\to 0$, the overlap goes to 1, and that the combination $D_{1,0}+D_{0,1}$ is good over the full range $0<{\tau }_2<\infty$. At ${\tau }_2\approx 5({\tau }_2\approx 0.2)$, there is a dip in the overlap, which is improved by adding the term $D_{2,0}$ ($D_{0,2}$).

Figure 4

(a) Overlap of the terms $D_{1,0}$ (blue), $D_{1,0}$ (red), and $D_{1,0}+D_{0,1}$ (green) (3.9) with the exact Coulomb state, for ${\tau }_2=1$ and $-1<{\tau }_1<1$. (b) The same as (a) except that the term $D_{-1,1}$ (purple) has been added to the linear combination $(D_{1,0}+D_{0,1}+D_{-1,1})$ (orange). For highly skew tori $\tau \approx 1+ı$, the term $D_{-1,1}$ plays a similar role as does $D_{0,1}$ at $\tau \approx ı$. At $\tau \approx \frac{1}{2}+ı$, the two terms have equal weight. Note that the overlap is boosted in the region around $\tau =1+ı$ when the term $D_{-1,1}$ is accounted for. We emphasize that keeping only two terms, the overlap with the Coulomb state is good for all ${\tau }_1$, but which are the terms that dominate depends on $\tau$.

Figure 5

Absolute value of the coefficients ${\lambda }_{m,n}$ as a function of increasing $|{\delta }_{m,n}|$, for $\tau =\frac{1}{2}+ı$ (green) and $\tau =\frac{1}{3}+\frac{ı}{10}$ (red). In order, the first six terms are $(m,n)=(1,0),(0,1),(-1,1)(-2,1),(1,1),(2,0)$ and $(m,n)=(-1,3),(0,1),(-1,2)(-2,4),(-2,6),(2,0)$. All terms $(-m,-n)$ have been omitted, as discussed in the text. The normalization is such that the weight of the first term is ${\lambda }_{m,n}=1$.

Figure 6

The cumulative overlap between (3.9) and the exact Coulomb result as terms are added, in order of increasing $|{\delta }_{m,n}|$ for $N_e=6$ at $\tau =\frac{1}{2}+ı$ (green) and $\tau =\frac{1}{3}+\frac{ı}{10}$ (red). The terms in the sum are ordered in the same way as in Fig. 5. For the first few terms added, the overlap is improved, but the curve is not monotonic. Note that since there are no adjustable parameters there is no reason to expect that monotonicity. The overlap is converging with increasing number of terms, but not at the maximum value of the overlap, so there is a point where adding more terms does not necessarily make the result better. In fact, for all values of $\tau$, it is sufficient to keep as few as only two terms, provided these are chosen appropriately.

Figure 7

The overlap with the exact Coulomb state as a function of increasing $|{\delta }_{m,n}|$. The setup is the same as in Figs. 5 and 6. Larger values of $|{\delta }_{m,n}|$ have smaller overlap, which is in agreement with ${\delta }_{m,n}$ representing a derivative. From this image it is clear that when truncating the sum in Eq. (3.9) to only a few terms, the most important ones have small ${\delta }_{m,n}$.

Figure 8

The overlap with the exact Coulomb ground state in a region $-1<{\tau }_1<1$ and $0.37<{\tau }_2<2.72$ of the $\tau$ plane, for $N_e=8$ particles. The thick black lines mark the boundaries of regions with different minimal translation steps ${\delta }_{m,n}$. Note the logarithmic scale of ${\tau }_2$ for a more symmetric plot. (a) Only the dominant term is included in the sum (3.9). The overlap with the exact Coulomb state is good everywhere. (b) The eight most dominant terms at $\tau =ı((m,n)=(1,0),(1,0),(2,0),(0,2),(-1,1),(1,1),(-2,2),(2,2))$ are included in the sum (3.9). Overlap with the Coulomb state is better or equal in most parts of the $\tau$ plane; at the edges, other terms than the eight used here are dominant (see, e.g., Fig. 4). This shows that our ansatz is valid in the entire $\tau$ plane, and that the graphs in Figs. 3 to 7 are indeed representative.

Figure 9

Viscosity for the exact Coulomb ground state extracted from the Berry phase around a circle with radius $r=0.005$ in the $\tau$ plane, which is discretized into $n=200$ points. The calculation is for $N_e=6$ (green), $N_e=8$ (blue), and $N_e=10$ (cyan) particles. (a) Scan over $-1<{\tilde{\tau }}_1<1$ with ${\tilde{\tau }}_2=1$ fixed. (b) Scan over $0.368<{\tilde{\tau }}_2<2.72$ with ${\tilde{\tau }}_1=0$ fixed. Note the logarithmic scale of ${\tau }_2$. The viscosity is not constant over the $\tau$ plane, but seems to converge on $\overline{s}=2$ for larger system sizes. Note that the plateau with $\overline{s}\approx 2$ becomes wider as the system size grows, which indicates that the deviations from $\overline{s}=2$ are finite-size effects.

Figure 10

Viscosity for CFT trial wave functions at $\tilde{\tau }=ı$. Evaluation is for $D_{1,0}{\Psi }_s$ (blue), $D_{0,1}{\Psi }_s$ (red), and $(D_{1,0}+D_{0,1}){\Psi }_s$ (green) for $N_e=4,6,8,10$. As $N_e\to \infty$, the viscosity approaches $\overline{s}=2$. Note the logarithmic scale of ${\tau }_2$. The circle size and number of steps are the same as in Fig. 9.

Figure 11

Viscosity for CFT trial wave functions calculated for $N_e=8$ at ${\tilde{\tau }}_1=0$ and $-0.22<{\tilde{\tau }}_2<4.5$. Evaluation is for $D_{1,0}{\Psi }_s$ (blue), $D_{0,1}{\Psi }_s$ (red), and $D_{1,0}{\Psi }_s+D_{0,1}{\Psi }_s$ (green). At extremal geometries ${\tilde{\tau }}_2\ne 1$ the viscosity differs substantially from $\overline{s}=2$ but in the rectangular geometry it fits rather well. The discretization of the path in the $\tau$ plane is the same as in Fig. 9.

Figure 12

Charge lattice of (a) $K=\left(\begin{array}{cc}\hfill 3& \hfill 2\\ \hfill 2& \hfill 3\end{array}\right)$ for $\nu =2/5$, which satisfies (B1), and (b) $K=\left(\begin{array}{cc}\hfill 2& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ for $\nu =1$, which does not. In both panels open circles ($\circ$) denote ${\Gamma }^{☆}$ and closed circles ($•$) denote $\Gamma$. A unit cell of $\Gamma$ is spanned by ${\mathbf{q}}_{\alpha }$ (blue) and a unit cell of ${\Gamma }^{☆}$ is spanned by ${\mathbf{l}}_{\beta }$ (red). ${\mathbf{h}}_0={\mathbf{l}}_1+{\mathbf{l}}_2$ (green) is also marked in the figure. In (a) it is clearly seen that ${\mathbf{l}}_{\alpha }={\mathbf{q}}_{\alpha }-2{\mathbf{h}}_0$ whereas no such relation exists for (b). Also in (a) ${\mathbf{q}}_1+{\mathbf{q}}_2=detK·{\mathbf{h}}_0$, whereas for (b) ${\mathbf{q}}_1+{\mathbf{q}}_2\ne detK·{\mathbf{h}}_0$. The left panel is the same as in Fig. 1.