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. Under $l\mapsto l+K\Lambda$,

$$Q_{l+K\Lambda}-Q_l=e\,t^{\mathsf T}\Lambda, \qquad \theta_{l+K\Lambda,l'}-\theta_{ll'} =2\pi\Lambda^{\mathsf T}l',$$

so the charge changes by that of a local excitation and the mutual braiding with every integer vector $l'$ is unchanged. The self-exchange angle instead changes by

$$\theta_{l+K\Lambda}-\theta_l =2\pi\Lambda^{\mathsf T}l +\pi\Lambda^{\mathsf T}K\Lambda.$$

For an even integral lattice, meaning $\Lambda^{\mathsf T}K\Lambda$ is even for every integer $\Lambda$, every local excitation is bosonic. Then $l$ and $l+K\Lambda$ define the same anyon including its topological spin, and the anyon group is $\mathbb Z^N/K\mathbb Z^N$ with $\lvert\det K\rvert$ sectors Wen and Zee 1992.

For an odd integral lattice, meaning $\Lambda^{\mathsf T}K\Lambda$ is odd for at least one integer $\Lambda$, the theory is a spin TQFT and a local excitation can be a fermion. If all local particles are quotiented, $\mathbb Z^N/K\mathbb Z^N$ still labels topological-charge sectors and mutual braiding, but self-statistics requires a spin-dependent quadratic refinement—or, equivalently, is defined only up to the sign of the local fermion Belov and Moore 2005, § 1, eqs. (1.3)–(1.7), Open PDF. Alternatively, one may retain the transparent fermion as a line; then the line set is enlarged and $l$ and $l+K\Lambda$ need not be identified. For one-component odd-level $U(1)_k$, the retained-line labels are $\mathbb Z_{2\lvert k\rvert}$, and the line $k$ is transparent with fermionic spin Okuda, Saito, and Yokoyama 2021, § 4.1, pp. 12–13, Open PDF.

On a closed genus-$g$ surface, an even-$K$ bosonic theory has $\lvert\det K\rvert^g$ states. For an odd integral lattice, each Hilbert-space block at fixed spin structure has that dimension; the spin-structure dependence remains part of the theory Belov and Moore 2005, § 5.3, after eq. (5.17), Open PDF. 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 $\nu=2/5$, $\lvert\det K\rvert=5$, 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 $t=(1)$, compare $l=1$ and $l=4$ in both conventions: (i) quotient all local electrons, and (ii) retain the transparent local fermion as a line. Compute their charge difference, mutual-braiding difference with an arbitrary integer label $l'$, and exchange-angle difference.

Solution

Because $4=1+3\times1$, the labels differ by $K\Lambda$ with $\Lambda=1$. Their charges differ by

$$Q_4-Q_1=e,$$

and for every integer $l'$ their mutual-braiding angles differ by

$$\theta_{4l'}-\theta_{1l'} =2\pi\frac{(4-1)l'}{3} =2\pi l',$$

so all mutual-braiding phases agree. Their self-exchange angles differ by

$$\theta_4-\theta_1 =\pi\frac{4^2-1^2}{3} =5\pi \equiv\pi\pmod{2\pi},$$

hence $e^{i\theta_4}=-e^{i\theta_1}$. If the local electron is quotiented, $l=1$ and $l=4$ represent the same class in $\mathbb Z_3$, but its spin is defined only up to the electron sign. If the transparent electron is retained, the lines are labeled modulo $6$: $f=3$ has $\theta_f=3\pi\equiv\pi\pmod{2\pi}$, and $l=4$ is the distinct line obtained by fusing $l=1$ with $f$.

References

• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235 (2005). arXiv.
• Okuda, Takuya, Koichi Saito, and Shuichi Yokoyama. “$U(1)$ Spin Chern–Simons Theory and Arf Invariants in Two Dimensions.” Nuclear Physics B 962 (2021): 115272. DOI. Open PDF: arXiv:2005.03203v2.
• 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.