Sine–Gordon Model

The sine–Gordon model is a scalar theory with a periodic potential in $1+1$ dimensions. It contains small oscillations and topological solitons in the same field. Quantum mechanically it connects to the massive Thirring model, illustrating how apparently different elementary variables can describe corresponding observables.

Periodic potential and vacua

For the classical model, take $\beta>0$, $\mu>0$, and

$$\mathcal L=\frac12(\partial\varphi)^2 -\frac{\mu^2}{\beta^2}\bigl[1-\cos(\beta\varphi)\bigr].$$

The field and $\beta$ are classically dimensionless; $\mu$ has mass dimension one. Neighboring vacua differ by $2\pi/\beta$. Expanding about one minimum gives a small-oscillation mass $\mu$, because $1-\cos(\beta\varphi)=\beta^2\varphi^2/2+\cdots$.

Finite-energy static configurations approach vacua at spatial infinity. Their topological charge is

$$Q_{\mathrm{top}} =\frac{\beta}{2\pi} \bigl[\varphi(+\infty)-\varphi(-\infty)\bigr].$$

It depends on boundary values and is insensitive to a smooth local deformation that keeps those values fixed.

A kink and its classical energy

A charge-one kink centered at $x_0$ is

$$\varphi_K(x)=\frac4\beta \arctan e^{\mu(x-x_0)}.$$

It runs from $0$ to $2\pi/\beta$ and satisfies $\varphi'_K=(2\mu/\beta)\sin(\beta\varphi_K/2)$. Therefore its gradient and potential energies are equal. Changing the integration variable from $x$ to $\varphi_K$ gives

$$M_K^{\mathrm{classical}} =\int_0^{2\pi/\beta}\! \frac{2\mu}{\beta}\sin\frac{\beta\varphi}{2}\,\mathrm d\varphi =\frac{8\mu}{\beta^2}.$$

This is a classical soliton mass, not an exact quantum mass formula for arbitrary coupling.

Quantum coupling and the Thirring relation

The quantum cosine requires a normal-ordering or equivalent renormalization prescription. Around the canonically normalized free-boson fixed point, its scaling dimension is $\Delta=\beta^2/(4\pi)$. Its renormalized coefficient consequently has dimension $2-\Delta$; one should not identify it with a scheme-independent physical mass squared. The perturbation is relevant for $\beta^2<8\pi$.

With the Thirring interaction written as $-(g/2)j_\mu j^\mu$, the Coleman relation is

$$\frac{4\pi}{\beta^2}=1+\frac g\pi.$$

At $g=0$, $\beta^2=4\pi$: the fermionic description is free even though the bosonic potential is nonlinear. The correspondence requires matched operator definitions and sectors; the original result concerns the neutral sector and perturbative expansions in the mass interaction Coleman 1975, §§II–III.

Develop the calculation

Bosonization and Sine–Gordon–Thirring Duality develops the current map and coupling relation. Kinks and Domain Walls supplies the soliton framework. Compare the radius and vertex-operator data of the Compact Free Boson.

References

• Coleman, Sidney. “Quantum Sine-Gordon Equation as the Massive Thirring Model.” Physical Review D 11 (1975): 2088–2097. DOI.