Musings

Spin Chains

Ever since I failed to mention it in my coverage of Strings 2004, I’ve been meaning to say something about spin chains and anomalous dimensions in $\mathcal{N}=4$ $SU(N)$ gauge theory. It’s really pretty stuff and seemingly very deep.

The story starts with a paper by Minahan and Zarembo and a followup by Beisert, Kristjansen and Staudacher. We’ll work in the large-$N$ limit, and study an expansion in the 't Hooft parameter, $\lambda=g^2 N$. We are going to compute the anomalous dimensions of the spinless superconformal primary fields transforming as some representation of the $SO(6)$ R-symmetry.

In general, because of operator mixing, this is a messy business. But the observation of Minahan and Zarembo is that operators consisting of a trace of a product of scalars, which transforms in the representation $[p,q,p]$ of $SO(6)$ form a closed set under renormalization. If we write the six scalars in the $\mathcal{N}=4$ multiplet as

then a trace of a product of $p+q$ $x$’s and $p$ $y$’s (in any order) transforms in representation $[p,q,p]$ and has engineering dimension $\Delta_0= 2p+q$ and so cannot mix with any operators involving gauginos or field strengths or covariant derivatives. Moreover, each such operator

or

can be thought of as a state of an $SU(2)$ spin chain, where we identify $x\sim\uparrow$ and $y\sim\downarrow$. Minahan and Zarembo noticed that matrix of one-loop anomalous dimensions,

$H$ is none other than the Hamiltonian for the Heisenberg spin chain. This Hamiltonian is integrable. Beisert et al showed that dilation operator up to 2 loops is, again, an integrable spin-chain Hamiltonian, and this structure seems to persist to higher loops as well.

The interpretation of the higher conserved charges (which can be constructed) remains mysterious, as does the origin of this integrable structure. There’s been a whole flurry of work on this subject. For a flavour of what’s been going on, you can check out the recent, somewhat telegraphic review by Beisert.