Nonlinear sigma model

A nonlinear sigma model has fields valued in a curved target space such as a vacuum manifold $G/H$. Its leading action is quadratic in spacetime derivatives but nonlinear in any unconstrained coordinates used for the target.

The $O(N)$ nonlinear sigma model has an $N$-component field constrained to a sphere, commonly ${\mathbf{n}}^2=N$ or $1$. An auxiliary Lagrange-multiplier field turns the constraint into a self-consistent mass gap in the Large-N expansion.

For the unit-sphere nonlinear sigma model with action $\int d^{D}x|\nabla \sigma {|}^2/(2g)$, parameterize a slowly rotating background by $n_0=(\cos \theta ,\sin \theta ,0,\dots )$ and tangent vectors $e_1=(-\sin \theta ,\cos \theta ,0,\dots )$, $e_a$ for $a\ge 2$. Write $\sigma =n_0\sqrt{1-{\pi }^2}+{\sum }_{a=1}^{n-1}{e}_a{\pi }_a$. For constant slow $\nabla \theta$, the coefficient of $|\nabla \theta {|}^2$ in $|\nabla \sigma {|}^2$ is $1-{\sum }_{a=2}^{n-1}{\pi }_a^2$. The fast Gaussian field theory gives $⟨{\pi }_a^2⟩=g{I}_D$, where $I_D={\int }_{\Lambda /b<|p|<\Lambda }{d}^{D}p/[(2\pi {)}^D{p}^2]$. The omitted linear term integrates to zero, and terms cubic in the fast variables have zero one-loop average. Thus precisely $n-2$ tangent components reduce the phase stiffness $1/g$. Restoring the cutoff in $D=2+ϵ$ gives $dg/d\log b=-ϵg+(n-2)S_D{\Lambda }^{ϵ}{g}^2/(2\pi {)}^D+O(g^3)$. This weak-coupling expansion requires $n\ge 2$; the discrete target $S^0$ has no tangent expansion.

With $A_D=S_D{\Lambda }^{ϵ}/(2\pi {)}^D$, the one-loop renormalization-group beta function is ${\beta }_g=-ϵg+(n-2)A_D{g}^2$. For $n>2$ and $ϵ>0$, its two nonnegative renormalization-group fixed points are zero and $g_{\ast }=ϵ/[(n-2)A_D]$. Differentiating the beta function gives the displayed thermal eigenvalues, so the nonzero point has correlation-length critical exponent $\nu =1/ϵ$ to leading order. At $ϵ=0$ the zero eigenvalue requires nonlinear analysis: $g$ is marginally relevant for $n>2$, but the $n=2$ perturbative spin-wave theory has a line of fixed points and omits vortices.

At $D=2$ and $n>2$, the weak-coupling renormalization-group flow obeys $dg/d\ell =(n-2)g^2/(2\pi )$. Integrating gives $1/g(\ell )=1/g_0-(n-2)\ell /(2\pi )$. The flow reaches order-one coupling at a length whose logarithm is $2\pi /[(n-2)g_0]+O(1)$ at this order. Restoring the microscopic scale ${\Lambda }^{-1}$ gives the displayed leading exponential correlation length. Higher loops affect its prefactor, and this conclusion does not apply to the compact $n=2$ model with its Berezinskii–Kosterlitz–Thouless transition.

In two spatial dimensions, a unit-vector field $\mathbf{n}:{\mathbb{R}}^2\to S^2$ has static energy $E=\int |\nabla \mathbf{n}{|}^2{d}^{2}x/2$. The sphere is also the complex projective line, so its stereographic projection gives the equivalent complex field description. A constant value at spatial infinity compactifies a regular configuration to a map $S^2\to S^2$, whose degree of a map between oriented manifolds is its topological charge.

In the oriented stereographic projection $\varphi =(2\mathrm{Re}w,2\mathrm{Im}w,1-|w{|}^2)/(1+|w{|}^2)$, the sphere's metric is $4|dw{|}^2/(1+|w{|}^2{)}^2$. For energy $V=\int |\nabla \varphi {|}^2$, its topological degree is $N={\pi }^{-1}\int \mathrm{Im}(\overline{w_{x_1}}{w}_{x_2})/(1+|w{|}^2{)}^2$. The inequality $|w_{x_1}{|}^2+|w_{x_2}{|}^2\ge 2|\mathrm{Im}(\overline{w_{x_1}}{w}_{x_2})|$ gives $V\ge 8\pi |N|$. Holomorphic maps and antiholomorphic maps that are rational maps saturate the respective signs. The more common energy with a prefactor $1/2$ has bound $4\pi |N|$ instead.

A spherical sigma model in polar fields $\varphi ,\alpha$ has kinetic metric $r(d{\varphi }^2+{\sin }^2\varphi d{\alpha }^2)/2$ in the usual one-half kinetic convention. The displayed positive potential selects the two polar vacua. For $r>0$ it supports static and rotating charged kinks, and a topological theta term shifts their canonical internal charge without changing the bulk equations.

Locally this density is a total derivative, so it changes neither the bulk field equations nor the energy. Its contribution to the angular canonical momentum is a spatial boundary derivative. With ${ϵ}^{01}=+1$, it produces the theta-angle shift of a sigma-model kink charge.

With ${ϵ}^{01}=+1$, the Noether density for the angular shift gains $\theta {\partial }_x\cos \varphi /(4\pi )$. Its integral gives the displayed topological shift. Mechanical charge $Q_0$ determines the classical rotating-kink energy, while canonical symmetry charge $Q_{\theta }$ is conjugate to the periodic angle and is integral after quantization. Reversing the epsilon convention reverses the theta sign.

For $\alpha =\omega t+{\alpha }_0$ and $|\omega |<m$, the kink is $\varphi =2\arctan e^{\kappa (x-x_0)}$ with $\kappa =\sqrt{m^2-{\omega }^2}$. Its rest energy and mechanical internal charge are $M=r{m}^2/\kappa$ and $Q_0=r\omega /\kappa$. Eliminating the frequency gives the displayed mass relation. The orientation rotates while the energy profile remains stationary.

In units $\hslash =1$, periodic-angle quantization gives canonical charge $Q_{\theta }=\ell \in \mathbb{Z}$. The mechanical charge is $Q_0=\ell +\theta T/(2\pi )$ and the rotating-kink mass gives the displayed leading semiclassical spectrum. The $T=1$ and $T=-1$ families are related by reversing the kink. Theta periodicity relabels the integer; fluctuation corrections are not excluded by this leading calculation.

For $T=-[\cos \varphi {]}_{-\infty }^{+\infty }/2$, complete the static energy into squares $({\varphi }_x\mp m\sin \varphi {)}^2$ and ${\sin }^2\varphi {\alpha }_x^2$. The boundary term is $\pm rmT$. A $T=1$ saturating kink is $\varphi =2\arctan e^{m(x-x_0)}$, with constant $\alpha$ and mass $rm$. This is a model-specific Bogomolny bound.

A sigma-model lump is a finite-energy classical field-theory soliton of the two-dimensional O3 nonlinear sigma model. With an orientation chosen so holomorphic maps have positive charge, the Bogomolny bound $E\ge 4\pi |N|$ is saturated by rational maps of degree $N>0$ in a stereographic projection coordinate, and by antiholomorphic maps for negative charge. The field remains smooth at coordinate poles, which are points at the other pole of the target sphere. Scale invariance permits lumps of arbitrary size; topology alone does not prevent concentration to a singular limiting configuration.

Fix the value $w(\infty )=0$ for positive-degree sigma-model lumps. Their holomorphic rational maps can be written $p/q$, where $q$ is monic of degree $k$, $p$ has degree at most $k-1$, and the polynomials have no common zero. The coefficients of $p$ and the nonleading coefficients of $q$ give $2k$ complex parameters; coprimality is an open condition. The topological degree is $k$, obtained by counting generic preimages with multiplicity. Allowing a common factor lowers the degree after cancellation and can describe a degeneration rather than another smooth point of this degree-$k$ parameter space.

For the O3 nonlinear sigma model with energy $\int |\nabla \mathbf{n}{|}^2/2$, the rational map $w(z)=\rho /(z-z_0)$, $\rho \ne 0$, is a degree-one sigma-model lump. Its energy density is $4|\rho {|}^2/(|z-z_0{|}^2+|\rho {|}^2{)}^2$, whose integral is $4\pi$. The complex center $z_0$ gives its position, while $|\rho |$ gives its size and $\arg \rho$ its internal phase. Varying the size preserves energy. The limit $\rho \to 0$ is singular at the center, so conserved degree under smooth deformations does not prevent concentration.

A unit-vector field $\mathbf{n}:{\mathbb{R}}^2\to S^2$ approaching a fixed value at infinity extends to $S^2\to S^2$. Its topological charge is the topological degree $Q=(4\pi {)}^{-1}\int \mathbf{n}\cdot ({\partial }_1\mathbf{n}\times {\partial }_2\mathbf{n})d^{2}x$. This normalized target area pullback counts the number of oriented coverings.

For $E=\frac{1}{2}\int (|{\partial }_1\mathbf{n}{|}^2+|{\partial }_2\mathbf{n}{|}^2)$, completing the square gives $E=\frac{1}{2}\int |{\partial }_1\mathbf{n}\pm \mathbf{n}\times {\partial }_2\mathbf{n}{|}^2\pm 4\pi Q$, hence $E\ge 4\pi |Q|$. Saturation yields the Bogomolny equations of the sigma-model lump. With the oriented stereographic projection $\mathbf{n}=(2\mathrm{Re}w,2\mathrm{Im}w,1-|w{|}^2)/(1+|w{|}^2)$, holomorphic rational maps have positive degree and saturate the bound; antiholomorphic maps have negative degree.