O(N) Vector Model

The linear O(N) vector model generalizes a real scalar to $N$ components with a rotation symmetry in field space. It connects spontaneous symmetry breaking, universal critical behavior, and the large- $N$ expansion. Keeping the radial degree of freedom distinguishes it from an O(N) nonlinear sigma model.

An O(N)-invariant multiplet

Let $\boldsymbol\phi=(\phi^1,\ldots,\phi^N)$ , and define

\mathcal L=\frac12\partial_\mu\phi^a\partial^\mu\phi^a -\frac r2\phi^a\phi^a -\frac{u}{4N}(\phi^a\phi^a)^2, \qquad u>0.

The repeated component label $a$ is summed. The $1/N$ convention makes the large- $N$ limit convenient; it differs from a $1/4!$ quartic convention. The symmetry acts as $\boldsymbol\phi\mapsto R\boldsymbol\phi$ with $R^TR=1$ . In $d$ spacetime dimensions, $[r]=2$ and $[u]=4-d$ .

The term “O(N) model” can also denote a fixed-length lattice spin system or a nonlinear sigma model. Here the field length fluctuates, and $r,u$ are parameters of a linear continuum theory.

Radial and transverse fluctuations

At the classical level, $r<0$ gives minima at $\boldsymbol\phi^2=Nv^2$ , with $v^2=-r/u$ . Choose a minimum along the last axis. The potential Hessian is

\frac{\partial^2V}{\partial\phi^a\partial\phi^b} =\left(r+\frac uN\boldsymbol\phi^2\right)\delta_{ab} +\frac{2u}{N}\phi^a\phi^b.

At the minimum the first term vanishes. There is one radial eigenvalue $-2r$ and $N-1$ zero transverse eigenvalues. Those flat directions are the classical Goldstone directions for $O(N)\to O(N-1)$ . The symmetry argument and quantum qualifications are developed in Srednicki 2007, §32.

This calculation does not guarantee a quantum broken phase in every dimension. Long-wavelength fluctuations can invalidate a naive ordered saddle; temperature and infinite-volume limits matter.

Criticality and controlled limits

A two-point function measures correlation length and field scaling. Susceptibility and the response to a small symmetry-breaking source distinguish phases. The large- $N$ limit reorganizes diagrams around an auxiliary-field saddle, with corrections in $1/N$ ; it is not the same approximation as weak coupling. Near four dimensions, the epsilon expansion gives another route to the critical fixed point.

If radial excitations are much heavier than the momenta of interest, a fixed-length description can approximate the low-energy angular dynamics. Integrating out a heavy radial mode is the reason for that relation; deleting it without a scale hierarchy is not.

Develop the calculation

The O(N) Model as a Strong-Coupling Laboratory develops its nonperturbative uses. Vector Models, Auxiliary Fields, and Large-N Saddles develops the expansion. Compare the O(N) Nonlinear Sigma Model before identifying the two theories.

References

Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§24 and 32. DOI.