Ordering the topological order in the fractional quantum Hall effect

Abstract

We discuss the possible topological order/topological quantum field theory of different quantum Hall systems. Given the value of the Hall conductivity, we constrain the global symmetry of the low-energy theory and its anomaly. Specifically, the one-form global symmetry and its anomaly are presented as the organizing principle of these systems. This information is powerful enough to lead to a unique minimal topological order (or a small number of minimal topological orders). Almost all of the known experimentally discovered topological orders are these minimal theories. Since this work is interdisciplinary, we made a special effort to relate to researchers with different backgrounds by providing translations between different perspectives.

Figures (5)

• Figure 1: A cartoon illustrating the difference between our approach and the usual classification of SETs. (a) Illustration of the usual approach taken to classify SETs. Here, the topological order is already known, and this information, along with the symmetry, is put into the classification. The classification then provides the possible consistent ways of assigning symmetry charges to the anyons. If the symmetry is $\mathrm{U}(1)$, each symmetry assignment will come with a possibly distinct value of $\sigma_H$. In contrast, our approach, illustrated in (b), assumes the $\mathrm{U}(1)$ symmetry and the value of $\sigma_H$ and returns a list of consistent topological orders. This is relevant experimentally as in this setting the topological order of the system is not already known.
• Figure 2: Illustration of the flux-threading argument. (a) The setup of the argument; an annulus with inner radius $R_1$ and outer radius $R_2$. Flux $\Phi$ is threaded through the hole of the annulus, resulting in a radial electric field, $\vec{E}$. This electric field in turn induces a current flow from the outer edge to the inner edge, $\vec{J}$. When $2\pi$ flux is threaded through the hole, the system can be thought of as having returned to its ground state, albeit having accumulated a charge of $\sigma_H$ in the inner hole. (b) Display of the braiding of an anyon $a$ around this flux, which will detect its charge. The flux itself can be thought of as an anyon, the vison $v$, whose charge is $\sigma_H$ and whose topological spin is $\sigma_H/2$.
• Figure 3: A cartoon of the action of translation on a TQFT that respects translation and $\mathrm{U}(1)$. (a) Display of the setup in a toroidal geometry with $N_x$ unit cells along the $x$ direction and $N_y$ along the $y$ direction. The filling constraints ensure that there is a background anyon, $a_b$, at the center of each unit cell with a charge $Q(a_b) = \nu_l$. If the system is translated in the $x$ direction by the size of the cell, as in (b), each $a_b$ moves to its neighbor. This can be seen to produce $N_y$ loop defects of $a_b$. In the presence of $2\pi$ flux threading through the hole of the torus, translation thus leads to a phase of $2\pi \nu_l N_y$. We note that we cannot embed a flat torus in three-dimensional space. Hence, the torus in the figure is not flat. However, since we discuss only discrete translation, we can have interactions preserving them.
• Figure 4: Illustration of the descent of single-layer spin$^c$ theories with $\sigma_H = p/q$. Solid arrows indicate descent under gauging, while dashed arrows indicate removing neutral, decoupled theories. The vertical axes display theories with increasing total quantum dimension, $\mathcal{D}$. The larger $\mathcal{D}$, the larger the number of theories, illustrated by the darker color at larger values. In (a), the lower bound on $\mathcal{D}^2$ is $q$, illustrated by the gray line. It is seen to be saturated by $\mathcal{V}^{q,p}$, which is the unique endpoint of the descent discussed in the main text. In (b), we see that for even $q$ the descent no longer has a single endpoint. Some, but not all, are $\mathsf{Pf}^f_{q,p,n}$ and ${\cal U}_{q,p,6}$, discussed in the main text.
• Figure 5: For even $q$, we see that allowing ourselves to violate the spin/charge relation restores the unique endpoint. To do this, we gauge the one-form symmetry associated with $v^q$ and descend to $\ell=1$, indicated by solid arrows. We then remove the uncharged sectors, indicated by dashed arrows. As shown in the main text, the theories $\mathsf{Pf}^f_{q,p,n}$ are special because they are the only theories without an uncharged sector upon descending to $\ell=1$.

Theorems & Definitions (35)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Definition 3.3
• Remark 3.4
• Definition 3.5
• Theorem 3.6
• proof
• Theorem 3.7