The Abelian Bosonization Dictionary

Abelian bosonization is an operator dictionary between one-dimensional fermions and a compact scalar field. Its power comes from translating strongly interacting fermion bilinears into Gaussian bosonic currents; its danger comes from mixing conventions. This page fixes the commutator, vertex normalization, cutoff, and Klein factors together so that charges and scaling dimensions can be checked rather than guessed.

Required background. Free Bosons and Vertex Operators supplies normal ordering and vertex-operator correlators; Second-Quantized Bosons and Fermions supplies canonical fermion fields; Fourier Series, Transforms, and Plancherel supplies the mode expansions and distributional Fourier identities used below.

Canonical fields and chiral fermions

Introduce real fields $\phi$ and $\theta$ with equal-time algebra

$$[\phi(x),\partial_y\theta(y)]=i\pi\delta(x-y), \qquad [\phi(x),\theta(y)]=-\frac{i\pi}{2}\operatorname{sgn}(x-y).$$

The second relation fixes the additive zero-mode convention implicit in the first. For chirality $r=+1$ ($R$) or $-1$ ($L$), define

$$\psi_r(x)=\frac{\eta_r}{\sqrt{2\pi a_0}} \exp[-i(r\phi(x)-\theta(x))],$$

where $a_0$ is a short-distance regulator and vertex products are normal ordered. The Hermitian Klein factors obey $\{\eta_r,\eta_{r'}\}=2\delta_{rr'}$ and commute with oscillator modes. The field commutator supplies fermionic exchange within one chirality; the Klein factors supply it between distinct species or chiral sectors. Equivalent conventions often exchange $\phi$ and $\theta$ or insert $\sqrt\pi$ factors. A formula can be transported only by translating the entire row of definitions.

Point splitting gives the smooth densities

$$\rho_r(x)=:\!\psi_r^\dagger\psi_r\!: =-\frac{r}{2\pi}\partial_x[r\phi(x)-\theta(x)],$$

and therefore

$$\rho_R+\rho_L=-\frac1\pi\partial_x\phi, \qquad \rho_R-\rho_L=\frac1\pi\partial_x\theta.$$

These normalizations reproduce the level-one U(1) current algebra. A direct point-splitting derivation and its regulator dependence are given by von Delft and Schoeller 1998, §§ 2–4.

Operator dictionary

Restoring the Fermi phases, $\Psi=e^{ik_Fx}\psi_R+e^{-ik_Fx}\psi_L$. The leading operators are

$$\begin{aligned} \rho(x)&=\rho_0-\frac1\pi\partial_x\phi +A_1\cos(2k_Fx-2\phi)+\cdots,\\ \psi_R^\dagger\psi_L&\propto e^{+2i\phi},\\ \psi_R\psi_L&\propto \eta_R\eta_L e^{2i\theta}. \end{aligned}$$

$A_1$ is nonuniversal; the harmonic and exponent are universal within the infrared fixed point. The first bilinear carries momentum $-2k_F$ and no charge. The pair field carries charge two and no $2k_F$ momentum. Those quantum numbers offer checks independent of correlation functions.

For the Gaussian Hamiltonian

$$H=\frac{u}{2\pi}\int dx\left[K(\partial_x\theta)^2 +K^{-1}(\partial_x\phi)^2\right],$$

the vertex $V_{m,n}=e^{i(m\phi+n\theta)}$ has bulk scaling dimension

$$\Delta_{m,n}=\frac14\left(m^2K+\frac{n^2}{K}\right).$$

Consequently $2k_F$ density correlations decay as $|x|^{-2K}$, pair correlations as $|x|^{-2/K}$, and a chiral fermion Green function as $|x|^{-(K+K^{-1})/2}$. At $K=1$ these reduce to free-fermion powers. The Gaussian-field derivation and finite-temperature conformal map are reviewed in Cazalilla 2004, §§ 2–4.

Deriving the dimension

In Euclidean coordinates, integrating out the dual field gives

$$S_\phi=\frac1{2\pi K}\int d\tau\,dx \left[u^{-1}(\partial_\tau\phi)^2+u(\partial_x\phi)^2\right].$$

Thus $\langle[\phi(x)-\phi(0)]^2\rangle=K\ln(|x|/a_0)+O(1)$ at equal time, and Gaussian averaging yields

$$\langle e^{im\phi(x)}e^{-im\phi(0)}\rangle \propto |x|^{-m^2K/2}.$$

Since a two-point function of a primary scales as $|x|^{-2\Delta}$, its dimension is $m^2K/4$. Repeating the calculation for $\theta$ gives $n^2/(4K)$. The absence of a real cross term in the scaling dimension reflects duality; the cross correlator instead fixes conformal spin and exchange phase.

Currents, dynamics, and limits

Hamilton’s equation gives $\partial_t\phi=uK\partial_x\theta$. With charge density $\delta\rho=-\partial_x\phi/\pi$, the continuity equation fixes

$$j=\frac1\pi\partial_t\phi=\frac{uK}{\pi}\partial_x\theta.$$

This is the particle current; multiply by the particle charge for electrical current. The dictionary describes the scaling limit below the cutoff $u/a_0$. It does not determine amplitudes such as $A_1$, band-curvature thresholds, or the global zero-mode sectors. Those are supplied by microscopic matching and by Zero Modes, Klein Factors, and Compactification.

Exercises

1. Find the scaling dimensions of the $2k_F$ density and pairing operators.

Solution

The density harmonic is $e^{\pm2i\phi}$, so $(m,n)=(\pm2,0)$ and $\Delta_{\mathrm{CDW}}=K$. The pair field is $e^{2i\theta}$, so $(m,n)=(0,2)$ and $\Delta_{\mathrm{pair}}=1/K$. Their equal-time correlations decay with twice these dimensions.

2. Check the free-fermion propagator exponent.

Solution

For $\psi_r$, $(m,n)=(-r,1)$ up to an overall sign. Hence $\Delta_\psi=(K+K^{-1})/4$. At $K=1$, $2\Delta_\psi=1$, so $\langle\psi_r(x)\psi_r^\dagger(0)\rangle\propto1/x$, as required for a free chiral fermion.

References

• Cazalilla, M. A. “Bosonizing One-Dimensional Cold Atomic Gases.” Journal of Physics B 37 (2004): S1–S47. DOI.
• von Delft, J., and H. Schoeller. “Bosonization for Beginners—Refermionization for Experts.” Annalen der Physik 7 (1998): 225–305. DOI.