In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled "non-linear sigma model".

Overview

The name has roots in particle physics, where a sigma model describes the interactions of pions. Unfortunately, the "sigma meson" is not described by the sigma-model, but only a component of it.

The sigma model was introduced by ; the name σ-model comes from a field in their model corresponding to a spinless meson called , a scalar meson introduced earlier by Julian Schwinger. The σ-meson was eventually discovered and identified to thef_0(500) meson. The model served as the dominant prototype of spontaneous symmetry breaking of O(4) down to O(3): the three axial generators broken are the simplest manifestation of chiral symmetry breaking, the surviving unbroken O(3) representing isospin.

In conventional particle physics settings, the field is generally taken to be SU(N) of the product of left and right chiral fields. In condensed matter theories, the field is taken to be O(N). For the rotation group O(3), the sigma model describes the isotropic ferromagnet; more generally, the O(N) model shows up in the quantum Hall effect, superfluid Helium-3 and spin chains.

In supergravity models, the field is taken to be a symmetric space. Since symmetric spaces are defined in terms of their involution, their tangent space naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.

In its most basic form, the sigma model can be taken as being purely the kinetic energy of a point particle; as a field, this is just the Dirichlet energy in Euclidean space.

In two spatial dimensions, the O(3) model is completely integrable.

Definition

The Lagrangian density of the sigma model can be written in a variety of different ways, each suitable to a particular type of application. The simplest, most generic definition writes the Lagrangian as the metric trace of the pullback of the metric tensor on a Riemannian manifold a field over a spacetime, this may be written as

is the metric tensor are the derivatives on the underlying spacetime manifold.

can be chosen to be any Riemannian manifold is avoided here to prevent clashes. Given an atlas of charts, the field space can always be locally trivialized

must be a differentiable manifold; by convention, it is either Minkowski space in particle physics applications, flat two-dimensional Euclidean space for condensed matter applications, or a Riemann surface, the worldsheet in string theory is just the plain-old covariant derivative is just the ordinary gradient is a section of the jet bundle.

Example: O(n) non-linear sigma model

the Kronecker delta, i.e. the scalar dot product to be the unit vector-sphere, the isometries of which are the rotation group. The Lagrangian can then be written as

, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model, this is the continuum limit of the classical XY model. See also the n-vector model and the Potts model for reviews of the lattice model equivalents. The continuum limit is taken by writing

as the finite difference (the "bulk magnetization").

In geometric notation

The sigma model can also be written in a more fully geometric notation, as a fiber bundle over a differentiable manifold. Given a section The pushforward is a map of tangent bundles

is taken to be a local orthonormal vector space basis is a differential form. The sigma model action is then just the conventional inner product on vector-valued k-forms

is the wedge product is the Hodge star. This is an inner product in two different ways. In the first way, given any, the Hodge dual defines an invariant inner product on the space of differential forms, commonly written as

The above is an inner product on the space of square-integrable forms, conventionally taken to be the Sobolev space In this way, one may write

1. .

This makes it explicit and plainly evident that the sigma model is just the kinetic energy can be recognized just the ordinary gradient of a scalar function. The Hodge star is merely a fancy device for keeping track of the volume form is flat, it can be completely ignored, and so the action is

1. ,

which is the Dirichlet energy. Classical extrema of the action (the solutions to the Lagrange equations and so one may also write

is the Laplace–Beltrami operator, i.e., the ordinary Laplacian is flat.

That there is another itself. That is, given any defines an inner product

on local charts, one also takes the inner product there as well. More verbosely,

1. .

The tension between these two inner products can be made even more explicit by noting that

is a bilinear form; it is a pullback can be taken as vielbeins. The Lagrangian density of the sigma model is then

can be interpreted as a solder form; this is articulated more fully below.

Motivations and basic interpretations

Several interpretational and foundational remarks can be made about the classical (non-quantized) sigma model. The first of these is that the classical sigma model can be interpreted as a model of non-interacting quantum mechanics. The second concerns the interpretation of energy.

Interpretation as quantum mechanics

This follows directly from the expression

can be interpreted as a wave function. Assembling all these pieces together, the sigma model action is equivalent to

copies of the base manifold.

The solder form

It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations. In thumbnail form, the construction is as follows. Both. The cotangent bundle, supplied with coordinate charts, can always be locally trivialized, i.e.

The trivialization supplies canonical coordinates on the cotangent bundle. Given the metric tensor, define the Hamiltonian function

Famously, the geodesic flow is given by the Hamilton–Jacobi equations

The geodesic flow is the Hamiltonian flow is the distance along the geodesic.

is a solder form. In this sense, the interpretation of the sigma model as an energy functional is not surprising; it is in fact the gluing together of two to be a Lie group. Both conditions are satisfied in various applications.

Results on various spaces

is often taken to be a Lie group, usually SU(N), in the conventional particle physics models, O(N) in condensed matter theories, or as a symmetric space in supergravity models. Since symmetric spaces are defined in terms of their involution lives) naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.

On Lie groups

being a Lie group is the metric tensor on the Lie group, formally called the Cartan tensor or the Killing form. The Lagrangian can then be written as the pullback of the Killing form. Note that the Killing form can be written as a trace over two matrices from the corresponding Lie algebra; thus, the Lagrangian can also be written in a form involving the trace. With slight re-arrangements, it can also be written as the pullback of the Maurer–Cartan form.

On symmetric spaces

A common variation of the sigma model is to present it on a symmetric space. The prototypical example is the chiral model, which takes the product

of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal"

that is invariant under the Cartan involution or as a pullback of the Maurer–Cartan form.

Trace notation

In physics, the most common and conventional statement of the sigma model begins with the definition

is the pullback of the Maurer–Cartan form corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as

This is instantly recognizable as the first term of the Skyrme model.

Metric form

are the structure constants.

Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression

are the vielbeins. The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written as

.

will not and so a projection is still needed.

Extensions

The model can be extended in a variety of ways. Besides the aforementioned Skyrme model, which introduces quartic terms, the model may be augmented by a torsion term to yield the Wess–Zumino–Witten model.

Another possibility is frequently seen in supergravity looks like "pure gauge". In the construction above for symmetric spaces, one can also consider the other projection

. This extra term can be interpreted as a connection. It can be endowed with its own dynamics, by writing

. Note that the differential here is just "d", and not a covariant derivative; this is not, now with the connection as a part of it, together with this term, forms a complete gauge invariant Lagrangian (which does have the Yang–Mills terms in it, when expanded out).

References