Abelian Topological Orders and K-Matrix Data

An Abelian K matrix is a compact encoding of a multicomponent Chern–Simons theory. Once its integer basis, electromagnetic charge vector, spacetime orientation, and local-particle lattice are specified, it determines filling, quasiparticle charge and statistics, genus-dependent degeneracy, and the minimal edge anomaly. Different matrices related by an integral unimodular basis change describe the same data.

Required background. Fractional Hall fluids supplies the physical quasiparticles; Abelian Chern–Simons theory supplies line operators; level quantization supplies integrality and large-gauge constraints.

Action and quasiparticle lattice

For $N$ compact internal gauge fields, take

$$S=\int\left[ \frac{K_{IJ}}{4\pi}a_I\,da_J -\frac{e}{2\pi}t_I A\,da_I +l_Ia_I\,j_l \right],$$

where $K$ is symmetric, integral, and nondegenerate, $t\in\mathbb Z^N$, and $l\in\mathbb Z^N$. With the chapter’s orientation, the invariant data are

$$\nu=t^{\mathsf T}K^{-1}t, \quad Q_l=e\,t^{\mathsf T}K^{-1}l, \quad \theta_l=\pi l^{\mathsf T}K^{-1}l, \quad \theta_{ll'}=2\pi l^{\mathsf T}K^{-1}l'.$$

Overall Hall and braid signs reverse with spacetime orientation. Vectors $l$ and $l+K\Lambda$ differ by a local excitation and define the same anyon sector. Hence the anyon group is $\mathbb Z^N/K\mathbb Z^N$ and contains $|\det K|$ sectors Wen and Zee 1992.

On genus $g$, the ideal topological theory has $|\det K|^g$ ground states. The minimal edge has $N$ bosons and net chiral central charge

$$c_-=n_+-n_-=\operatorname{signature}K.$$

Adding an invertible neutral sector can change $c_-$ without changing the anyon group, so the local-particle and invertible-sector conventions matter.

Integral basis equivalence

Let $W\in GL(N,\mathbb Z)$ and $a=W a'$. Then

$$K'=W^{\mathsf T}KW, \qquad t'=W^{\mathsf T}t, \qquad l'=W^{\mathsf T}l.$$

Because $W^{-1}$ is integral, this relabels compact gauge fields and quasiparticles without changing $\nu$, charges, spins, mutual braiding, determinant magnitude, or signature. A real but nonintegral diagonalization is useful for propagation velocities but is not an allowed relabeling of the topological charge lattice.

Example: the 2/5 state

Take

$$K=\begin{pmatrix}3&2\\2&3\end{pmatrix}, \qquad t=\begin{pmatrix}1\\1\end{pmatrix}, \qquad K^{-1}=\frac15\begin{pmatrix}3&-2\\-2&3\end{pmatrix}.$$

Then $\nu=2/5$, $|\det K|=5$, and $\operatorname{signature}K=2$. The vector $l=(1,0)^{\mathsf T}$ has $Q_l=e/5$ and exchange angle $3\pi/5$. These values identify topological data more sharply than the filling alone.

The K-matrix description assumes a fully gapped Abelian bulk. It does not encode non-Abelian fusion spaces, microscopic energy gaps, edge velocities, disorder equilibration lengths, or which candidate is realized experimentally.

Exercise

For $K=(3)$ and $t=(1)$, show that $l=1$ and $l=4$ are the same anyon type.

Solution

$4=1+3\times1$, so the vectors differ by $K\Lambda$ with $\Lambda=1$. Their charge differs by $e$ and their exchange phase differs by an integer-local-particle contribution; their mutual braiding with every anyon is identical modulo $2\pi$.

References

• Xiao-Gang Wen and A. Zee, “Classification of Abelian Quantum Hall States and Matrix Formulation of Topological Fluids,” Physical Review B 46 (1992) 2290–2301, doi:10.1103/PhysRevB.46.2290.