O(N) Nonlinear Sigma Model

The O(N) nonlinear sigma model is a field theory whose values lie on a sphere. Its kinetic term looks quadratic, but the fixed-length constraint produces interactions. In $1+1$ dimensions with $N>2$, it is a particularly useful example of asymptotic freedom without a non-Abelian gauge field.

A field constrained to a sphere

For this profile, take a unit vector $n^a(x)$, $a=1,\ldots,N$, with no topological term:

$$n^an^a=1,\qquad S=\frac1{2t}\int\mathrm d^2x\, \partial_\mu n^a\partial^\mu n^a, \qquad t>0.$$

The target is $S^{N-1}$ and $t$ is dimensionless. The global O(N) symmetry rotates the target vector. There is no fluctuating radial mode.

In a patch with $n^N>0$, write $n=(\boldsymbol\pi,\sqrt{1-\boldsymbol\pi^2})$. Direct differentiation gives

$$\partial_\mu n\cdot\partial^\mu n =(\partial\boldsymbol\pi)^2 +\frac{(\boldsymbol\pi\cdot\partial_\mu\boldsymbol\pi) (\boldsymbol\pi\cdot\partial^\mu\boldsymbol\pi)} {1-\boldsymbol\pi^2}.$$

The denominator generates arbitrarily high interactions in the local coordinates. Its singularity at the patch edge is a coordinate failure, not automatically a physical singularity. This geometric construction is explained in Tong 2017, §4.3.

Running and the infrared problem

With the displayed $1/(2t)$ normalization, the one-loop ultraviolet beta function is

$$\mu\frac{\mathrm dt}{\mathrm d\mu} =-\frac{N-2}{2\pi}t^2+O(t^3).$$

For $N>2$, increasing the energy weakens the coupling. Integration identifies the scale

$$\Lambda_\sigma =\mu\exp\left[-\frac{2\pi}{(N-2)t(\mu)}\right]$$

at this order. The one-loop calculation predicts where perturbation theory loses control; it does not calculate the exact mass-gap coefficient.

The classical space of constant configurations suggests angular Goldstone modes. In a two-dimensional relativistic quantum theory at finite $N$, continuous spontaneous symmetry breaking is constrained by infrared fluctuations. One cannot simply retain the classical massless modes as the exact spectrum.

Dimensions and variants matter

The ordinary $N>2$ sphere model at theta zero is the intended asymptotically free laboratory. For $N=2$, the one-loop coefficient vanishes and the target is a circle; winding defects and their allowed weights require separate treatment. For O(3), a theta term can change the infrared theory. Neither case inherits an unqualified mass-gap claim from the displayed beta function.

In spacetime dimension $d>2$, the coefficient convention gives $[t]=2-d$. The same geometric action is commonly a derivative effective theory with a cutoff, not automatically the same ultraviolet construction as its two-dimensional counterpart.

Develop the calculation

Sigma-Model Dynamics: Target Geometry, Dimension, and Control develops the geometric description. Sigma-Model Beta Function and Asymptotic Freedom gives a calculation route. Compare the linear O(N) Vector Model when a radial mode remains active.

References

• Tong, David. Statistical Field Theory. University of Cambridge lecture notes, 2017, §4.3. Author’s text.