Sigma model

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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

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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.[1]

The sigma model was introduced by Gell-Mann & Lévy (1960, section 5); the name σ-model comes from a field in their model corresponding to a spinless meson called σ, a scalar meson introduced earlier by Julian Schwinger.[2] 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), or the vector subspace of quotient   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

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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. For   a field over a spacetime  , this may be written as

 

where the   is the metric tensor on the field space  , and the   are the derivatives on the underlying spacetime manifold.

This expression can be unpacked a bit. The field space   can be chosen to be any Riemannian manifold. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol   is avoided here to prevent clashes with many other common usages of   in geometry. Riemannian manifolds always come with a metric tensor  . Given an atlas of charts on  , the field space can always be locally trivialized, in that given   in the atlas, one may write a map   giving explicit local coordinates   on that patch. The metric tensor on that patch is a matrix having components  

The base manifold   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. The   is just the plain-old covariant derivative on the base spacetime manifold   When   is flat,   is just the ordinary gradient of a scalar function (as   is a scalar field, from the point of view of   itself.) In more precise language,   is a section of the jet bundle of  .

Example: O(N) non-linear sigma model

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Taking   the Kronecker delta, i.e. the scalar dot product in Euclidean space, one gets the   non-linear sigma model. That is, write   to be the unit vector in  , so that  , with   the ordinary Euclidean dot product. Then   the  -sphere, the isometries of which are the rotation group  . The Lagrangian can then be written as

 

For  , this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model. For  , 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 on neighboring lattice locations   Then   in the limit  , and   after dropping the constant terms   (the "bulk magnetization").

In geometric notation

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

  taking  

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

 

where the   is the wedge product, and the   is the Hodge star. This is an inner product in two different ways. In the first way, given any two differentiable forms   in  , 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

 

This makes it explicit and plainly evident that the sigma model is just the kinetic energy of a point particle. From the point of view of the manifold  , the field   is a scalar, and so   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 when integrating on curved spacetime. In the case that   is flat, it can be completely ignored, and so the action is

 

which is the Dirichlet energy of  . Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of  . Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function   one has   and so one may also write

 

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

That there is another, second inner product in play simply requires not forgetting that   is a vector from the point of view of   itself. That is, given any two vectors  , the Riemannian metric   defines an inner product

 

Since   is vector-valued   on local charts, one also takes the inner product there as well. More verbosely,

 

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

 

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

 

for   the metric on   Given this gluing-together, the   can be interpreted as a solder form; this is articulated more fully, below.

Motivations and basic interpretations

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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

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This follows directly from the expression

 

given above. Taking  , the function   can be interpreted as a wave function, and its Laplacian the kinetic energy of that wave function. The   is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is   In flat space, the Laplacian is conventionally written as  . Assembling all these pieces together, the sigma model action is equivalent to

 

which is just the grand-total kinetic energy of the wave-function  , up to a factor of  . To conclude, the classical sigma model on   can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of   to the Lagrangian results in the quantum mechanics of a wave-function in a potential. Taking   is not enough to describe the  -particle system, in that   particles require   distinct coordinates, which are not provided by the base manifold. This can be solved by taking   copies of the base manifold.

The solder form

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It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations.[3] In thumbnail form, the construction is as follows. Both   and   are Riemannian manifolds; the below is written for  , the same can be done for  . 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   on  , define the Hamiltonian function

 

where, as always, one is careful to note that the inverse of the metric is used in this definition:   Famously, the geodesic flow on   is given by the Hamilton–Jacobi equations

  and  

The geodesic flow is the Hamiltonian flow; the solutions to the above are the geodesics of the manifold. Note, incidentally, that   along geodesics; the time parameter   is the distance along the geodesic.

The sigma model takes the momenta in the two manifolds   and   and solders them together, in that   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 energy functionals. Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if   and   have the same real dimension. Furthermore, the conventional definition of a solder form takes   to be a Lie group. Both conditions are satisfied in various applications.

Results on various spaces

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The space   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, their tangent space (i.e. the place where   lives) naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.

On Lie groups

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For the special case of   being a Lie group, the   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

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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"

 

Such a quotient space is a symmetric space, and so one can generically take   where   is the maximal subgroup of   that is invariant under the Cartan involution. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on   to a metric on   or as a pullback of the Maurer–Cartan form.

Trace notation

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

 

Here, the   is the pullback of the Maurer–Cartan form, for  , onto the spacetime manifold. The   is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra   of  , the involution decomposes the space into odd and even parity components   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

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The equivalent metric form of this is to write a group element   as the geodesic   of an element   of the Lie algebra  . The   are the basis elements for the Lie algebra; the   are the structure constants of  .

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

 

where   is now obviously (proportional to) the Killing form, and the   are the vielbeins that express the "curved" metric   in terms of the "flat" metric  . The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written as

 

where   is a matrix whose matrix elements are  .

For the sigma model on a symmetric space, as opposed to a Lie group, the   are limited to span the subspace   instead of all of  . The Lie commutator on   will not be within  ; indeed, one has   and so a projection is still needed.

Extensions

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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 models. Here, one notes that the Maurer–Cartan form   looks like "pure gauge". In the construction above for symmetric spaces, one can also consider the other projection

 

where, as before, the symmetric space corresponded to the split  . This extra term can be interpreted as a connection on the fiber bundle   (it transforms as a gauge field). It is what is "left over" from the connection on  . It can be endowed with its own dynamics, by writing

 

with  . Note that the differential here is just "d", and not a covariant derivative; this is not the Yang–Mills stress-energy tensor. This term is not gauge invariant by itself; it must be taken together with the part of the connection that embeds into  , so that taken together, the  , 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

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  1. ^ page 114, David Tong: Lectures on Statistical Field Theory
  2. ^ Julian S. Schwinger, "A Theory of the Fundamental Interactions", Ann. Phys. 2(407), 1957.
  3. ^ Jurgen Jost (1991) Riemannian Geometry and Geometric Analysis, Springer
  • Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", Il Nuovo Cimento, 16 (4): 705–726, Bibcode:1960NCim...16..705G, doi:10.1007/BF02859738, S2CID 122945049