Musings

Abelian Anyons

You’ll recall that the Fractional Quantized Hall Effect concerns strongly-interacting quasi-2-dimensional electron systems in a strong magnetic field at very low temperatures. The electrons form a partially-filled Landau level, and the bulk system has gap. (There are gapless excitations that propagate on the boundary of a droplet of Quantum Hall fluid.) For filling fraction $\nu = 1/m$, for $m$ an odd integer, Laughlin proposed that the multi-electron ground state wave function takes the form $$\Psi_0(z_1,\dots,z_N) = \prod_{i\gt j} (z_i-z_j)^m e^{-\sum |z_i|^2/4l_0^2}$$ where $l_0=\sqrt{hc/eB}$ is the magnetic length.

The quasiparticles are excitations above the gap, with fractional charges, $e/m$ and fractional statistics, $\theta = \pi/m$. The wave function in the presence of $n$ quasiparticles, at locations $w_1,\dots,w_n$, is $$\Psi_n = \prod_{i\lt j} (w_i-w_j)^{1/m} \prod_{i=1}^N\prod_{j=1}^n (z_i-w_j) \Psi_0$$ Aside from the Gaussian factor (which can be understood as introduction of a uniform background charge to cancel the net charge of the insertions), this looks like nothing so much as the holomorphic part of a correlation function of primary fields in a $c=1$ CFT, specifically, the one at radius $R^2=k$. Here, the electrons are the holomorphic primary fields, $e^{i\sqrt{m} \phi(z)}$, which generate the extended chiral algebra¹, and the quasiparticles are the operators $e^{i \phi(z)/\sqrt{m}}$

Nonabelian Anyons

The next step was taken by Greg Moore and Nick Read in 1991. They wrote down a Laughlinesque wave function of the form $$\Psi = \mathop{Pfaf}\left(\frac{1}{z_i-z_j}\right)\prod_{i\lt j} (z_i-z_j)^m e^{-\sum |z_i|^2/4l_0^2}$$ which, for $m$ even, should correspond to a filling fraction, $\nu$, with even denominator. This, again, has an interpretation as a CFT correlator, this time in the tensor product of an Ising model and a $c=1$ CFT at radius $R^2=m$. The electron operator is $\psi e^{i\sqrt{m}\phi(z)}$. Quasiparticles are then created by $\sigma e^{i\phi(z)/2\sqrt{m}}$. These quasiparticles have interesting nonabelian braiding relations

In recent years, considerable evidence has emerged that the $\nu=5/2$ FQHE state is described by Moore-Read.

The obvious generalization generalization of these constructions is to the product of a $\mathbb{Z}_k$ parafermion and a free scalar (the Moore-Read case is $k=2$), as worked out by Read and Rezayi.

There’s some fleeting evidence for a $\nu = 12/5$ FQHE state, described by the $\mathbb{Z}_3$ Reed-Rezayi system.

Chern-Simons

Wave functions are all very well and good, but we would like to have an effective field theory. Since the bulk theory has a gap, the low energy effective theory is a topological one. For the abelian anyons described above, the answer has been known since my graduate student days. It’s an abelian Chern-Simons Theory, $$S= \int \frac{k}{2\pi} a\wedge d a - \frac{1}{\pi} A \wedge d a - j_{\text{QP}} \wedge a$$ Here, $A$ is the external electromagnetic field, which couples to the electromagnetic current (written here as 2-form), $$j_{\text{EM}} = \frac{1}{\pi} d a$$ The action is Gaussian, so one can actually integrate out $a$, and obtain the Hall conductance $$\sigma_{x y} = \frac{1}{m} \frac{e^2}{h},\quad \sigma_{x x} =0$$ The quasiparticles are heavy, and so appear as classical sources, represented by the 2-form $j_{\text{QP}}$.

The obvious generalization to the Moore-Read/Read-Rezayi systems is the $SU(2)_k$ Chern Simons theory, $$S = \int \frac{k}{4\pi} Tr\left(a \wedge d a +\frac{2}{3} a\wedge a\wedge a\right) - \frac{1}{2\pi} A\wedge Tr\tau_3(d a + a\wedge a) - Tr (j_{\text{QP}}\wedge a)$$

And, so I’ve been watching with bemusement as the condensed matter theorists start drawing knots on the blackboard and speaking animatedly about the Jones Polynomial, pentagon and hexagon identities, etc.

Quantum Computing

So why are the condensed matter theorists so interested in this subject, all of a sudden? Quantum Computing. Recall that a quantum computer is, basically, a system with a finite-dimensional Hilbert Space. We perform a computation by preparing the system in a certain state, make some manipulations, and then measure the final state of the system.

To build such a quantum computer, one wants a system with a finite number of low-lying states, separated by a gap from the rest of the (invariably infinite-dimensional) Hilbert space. That’s exactly what nonabelian FQHE systems provide us. The abelian systems have a unique ground state, even in the presence of $n$ quasiparticles. But the state of the nonabelian system is degenerate, with degeneracy equal to the number of conformal blocks of the associated $\mathbb{Z}_k$ parafermion correlation function. Moreover, we can manipulate the state of the system (do computations) by moving the quasiparticles around.

The braiding of the spin field in the Ising model is a bit too simple. But for $k\neq 2,4$, the Read-Rezayi system has been shown to be universal for quantum computation.

You can read a lot more about this in a wonderful review by Das Sarma et al.