Non-Abelian Bosonization and Spin Sectors

Non-Abelian bosonization packages the spin or flavor sector of multicomponent one-dimensional fermions into affine currents and a Wess–Zumino–Witten (WZW) field. It makes the symmetry representation, anomaly level, operator spectrum, and marginal interactions explicit in a way that separate Abelian fields can obscure.

Required background. The Abelian Bosonization Dictionary supplies chiral fermions and charge vertices; Affine Currents and WZW Models supplies current algebras and primary representations. Helpful background. Wess–Zumino and WZW Terms supplies the quantization and extension independence of the Wess–Zumino term.

Currents and the level

For chiral fermions $\psi_{r\alpha}$ in a representation with generators $t^a$, define normal-ordered currents

$$J_r^a=:\!\psi_{r\alpha}^\dagger(t^a)_{\alpha\beta}\psi_{r\beta}\!:.$$

In complex coordinates the SU(2) current operator product is normalized here as

$$J^a(z)J^b(0)\sim \frac{k\delta^{ab}}{2z^2}+\frac{i\epsilon^{abc}J^c(0)}{z}.$$

The integer $k$ is the affine level. It counts the anomaly coefficient, not an adjustable interaction. One spinful channel realizes SU(2)$_1$; $k$ equivalent channels can furnish a diagonal SU(2)$_k$ spin current. The Sugawara stress tensor gives

$$c=\frac{3k}{k+2}, \qquad h_j=\frac{j(j+1)}{k+2}, \quad j=0,\frac12,\ldots,\frac k2.$$

These formulas follow from the affine algebra and integrable representation condition Knizhnik and Zamolodchikov 1984, pp. 83–103.

The WZW field and fermion operators

The nonchiral matrix field $g(x,\tau)\in\mathrm{SU}(2)$ has action

$$S_k[g]=\frac{k}{8\pi}\int_\Sigma d^2x\, \operatorname{tr}(\partial_\mu g^{-1}\partial_\mu g) +ik\,\Gamma[g],$$

where $\Gamma[g]$ is defined using a three-dimensional extension. Changing the extension shifts $\Gamma$ by $2\pi$ times an integer in a compatible trace convention, so the path-integral phase is well defined for integer $k$. Witten established the equivalence between this current theory and non-Abelian fermion bosonization Witten 1984, §§ 2–4.

For one spinful channel the low-energy theory factorizes schematically as

$$\mathrm{U}(2)_1\simeq \mathrm{U}(1)_{\mathrm{charge}} \times \mathrm{SU}(2)_{1,\mathrm{spin}},$$

with a discrete identification needed for the complete operator spectrum. A fermion is a product of a charge vertex and a chiral spin-$1/2$ primary. The $2k_F$ spin density and dimerization contain the nonchiral matrix $g$ multiplied by a charge vertex; uniform spin density is $\mathbf J_R+\mathbf J_L$. Thus uniform and staggered responses are distinct operators, not different limits of one smooth current.

For a pure spin-$1/2$ Heisenberg chain, the leading continuum expansion is

$$\frac{\mathbf S_j}{a}\sim \mathbf J_R(x)+\mathbf J_L(x)+(-1)^j C\,\mathbf n(x),$$

where $\mathbf n\propto\operatorname{tr}(g\boldsymbol\sigma)$ and $C$ is nonuniversal. The SU(2)$_1$ WZW fixed point explains the half-integer-chain critical exponents Affleck 1986, pp. 409–447.

Perturbations and spin gaps

Spin-rotation symmetry permits the current interaction

$$\delta\mathcal H=g_s\,\mathbf J_R\cdot\mathbf J_L.$$

It is classically marginal. For the antiferromagnetic spin-$1/2$ chain its sign is marginally irrelevant, producing slow logarithmic corrections to correlations. The opposite sign flows strong and can generate a spin gap. A relevant dimerization couples to $\operatorname{tr}g$; staggered magnetic fields couple to components of $\mathbf n$. Their scaling dimensions follow from the $j=1/2$ primary, but whether the perturbation is allowed follows from lattice translation and internal symmetries.

At $k>1$, additional relevant primaries generally exist, so the WZW fixed point often requires tuning or protection. Conversely, a current algebra inferred from a central charge alone need not determine the microscopic realization; representation content and symmetry action must also match.

Limits of the description

Non-Abelian bosonization is an infrared operator equivalence. The level and fusion rules are universal, while velocities, amplitudes, irrelevant couplings, and the ultraviolet cutoff are not. Charge and spin sectors can couple through irrelevant operators, and lattice symmetries can impose discrete quotients on their product. Keeping only the Lie algebra while discarding these global constraints can introduce operators absent from the microscopic chain.

Exercises

1. Compute the central charge and spin-$1/2$ chiral weight for SU(2)$_1$.

Solution

For $k=1$, $c=3/(1+2)=1$. For $j=1/2$, $h_{1/2}=(1/2)(3/2)/(1+2)=1/4$. The nonchiral matrix field has scaling dimension $h+\bar h=1/2$.

2. Why can a one-site translation forbid $\operatorname{tr}g$ in the uniform Heisenberg chain?

Solution

$\operatorname{tr}g$ represents the staggered dimerization and changes sign under a one-site translation. A translation-invariant Hamiltonian therefore cannot contain it linearly. Explicit bond alternation breaks that symmetry and supplies precisely such a coupling.

References

• Affleck, I. “Exact Critical Exponents for Quantum Spin Chains, Nonlinear Sigma Models at $\theta=\pi$ and the Quantum Hall Effect.” Nuclear Physics B 265 (1986): 409–447. DOI.
• Knizhnik, V. G., and A. B. Zamolodchikov. “Current Algebra and Wess–Zumino Model in Two Dimensions.” Nuclear Physics B 247 (1984): 83–103. DOI.
• Witten, E. “Non-Abelian Bosonization in Two Dimensions.” Communications in Mathematical Physics 92 (1984): 455–472. DOI.