Non-linear sigma model: Difference between revisions

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:<math>\mathcal{L}={1\over 2}g_{ab}(\Sigma) (\partial^\mu \Sigma^a) (\partial_\mu \Sigma^b) - V(\Sigma).</math>

In more than two dimensions, nonlinear ''σ'' models contain a dimensionful coupling constant and are thus not perturbatively renormalizablenonrenormalizable.

Nevertheless, they exhibit a non-trivial ultraviolet fixed point of the renormalization group both in the lattice formulation<ref>{{cite book | last = Zinn-Justin | first= Jean | title= Quantum Field Theory and Critical Phenomena | publisher = Oxford University Press | date = 2002 }}</ref><ref>{{ cite book | last= Cardy | first= John L. | title = Scaling and the Renormalization Group in Statistical Physics | publisher = Cambridge University Press | date = 1997 }}</ref> and in the double expansion originally proposed by [[Kenneth G. Wilson]].<ref>{{cite journal | last= Brezin | first= Eduard |author2= Zinn-Justin, Jean| title=Renormalization of the nonlinear sigma model in in 2 + epsilon dimensions | journal=Physical Review Letters| year= 1976 | volume=36 |pages=691–693|doi=10.1103/PhysRevLett.36.691 |bibcode = 1976PhRvL..36..691B }}</ref> In both approaches the non-trivial renormalization group fixed point found for the O(n) symmetric model is seen to simply describe, in dimensions greater than two, the critical point separating the ordered from the disordered phase. In addition, the improved lattice or quantum field theory predictions can then be compared to laboratory experiments on critical phenomena, since the O(n) model describes physical Heisenberg ferromagnets and related systems. The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the O(n) symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation.

This means they can only arise as [[effective field theory|effective field theories]]. New physics is needed at around the distance scale where the two point [[connected correlation function]] is of the same order as the curvature of the target manifold. This is called the [[UV completion]] of the theory. There is a special class of nonlinear σ models with the [[internal symmetry]] group&nbsp;''G''&nbsp;*. If ''G'' is a [[Lie group]] and ''H'' is a [[Lie subgroup]], then the [[Quotient space (topology)|quotient space]] ''G''/''H'' is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a [[homogeneous space]] of ''G'' or in other words, a [[nonlinear realization]] of&nbsp;''G''. In many cases, ''G''/''H'' can be equipped with a [[Riemannian metric]] which is ''G''-invariant. This is always the case, for example, if ''G'' is [[compact group|compact]]. A nonlinear σ model with G/H as the target manifold with a ''G''-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear {{mvar|σ}} model.

In both approaches, the non-trivial renormalization-group fixed point found for the ''O(n)''-symmetric model is seen to simply describe, in dimensions greater than two, the critical point separating the ordered from the disordered phase. In addition, the improved lattice or quantum field theory predictions can then be compared to laboratory experiments on [[critical phenomena]], since the ''O(n)'' model describes physical [[Heisenberg ferromagnet]]s and related systems. The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the ''O(n)''-symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation.

This means they can only arise as [[effective field theory|effective field theories]]. New physics is needed at around the distance scale where the two point [[connected correlation function]] is of the same order as the curvature of the target manifold. This is called the [[UV completion]] of the theory. There is a special class of nonlinear σ models with the [[internal symmetry]] group&nbsp;''G''&nbsp;*. If ''G'' is a [[Lie group]] and ''H'' is a [[Lie subgroup]], then the [[Quotient space (topology)|quotient space]] ''G''/''H'' is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a [[homogeneous space]] of ''G'' or in other words, a [[nonlinear realization]] of&nbsp;''G''. In many cases, ''G''/''H'' can be equipped with a [[Riemannian metric]] which is ''G''-invariant. This is always the case, for example, if ''G'' is [[compact group|compact]]. A nonlinear σ model with G/H as the target manifold with a ''G''-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear {{mvar|σ}} model.

When computing [[functional integration|path integrals]], the functional measure needs to be "weighted" by the square root of the [[determinant]] of&nbsp;''g'',

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This model proved to be relevant in string theory where the two-dimensional manifold is named '''worldsheet'''. Proof of renormalizability was given by [[Daniel Friedan]].<ref name="Frie80">

{{cite journal|last=Friedan|first=D.|authorlink=Daniel Friedan|title=Nonlinear models in 2+ε dimensions | journal = PRL | volume = 45 | issue = 13| pages = 1057 |publisher=|___location=| year = 1980 | url = http://link.aps.org/doi/10.1103/PhysRevLett.45.1057 |doi= 10.1103/PhysRevLett.45.1057 | bibcode=1980PhRvL..45.1057F}}</ref> He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form

:<math>\lambda\frac{\partial g_{ab\mu\nu}}{\partial\lambda}=\beta_{ab\mu\nu}(T^{-1}g)=R_{ab\mu\nu}+O(T^2).</math>

<math>R_{ab\mu\nu}</math> being the [[Ricci tensor]] of the target manifold.

This represents a [[Ricci flow]], obeyinghaving [[Einstein field equations]] for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that [[conformal field theory|conformal invariance]] is not lost due to quantum corrections, so that the [[quantum field theory]] of this model is sensible (renormalizable).

Further adding nonlinear interactions representing flavor-chiral anomalies results in the [[Wess–Zumino–Witten model]],<ref>{{cite journal |first=E. |last=Witten |title=Non-abelian bosonization in two dimensions |journal=[[Communications in Mathematical Physics]] |volume= 92| issue= 4 |year=1984 | pages= 455–472 | doi= 10.1007/BF01215276|bibcode = 1984CMaPh..92..455W }}</ref> which

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==O(3) non-linear sigma model==

A celebrated example, of particular interest due to its topological properties, is the ''O(3)'' nonlinear {{mvar|σ}}-model in 1&nbsp;+&nbsp;1 dimensions, with the Lagrangian density

:<math>\mathcal L= \tfrac{1}{2}\ \partial^\mu \hat n \cdot\partial_\mu \hat n </math>

where <math>\hat n=(n_1,n_2,n_3)</math> with the constraint <math>\hat n\cdot \hat n=1</math> and {{mvar|μ}}=1,2.

AOne celebratedof the most famous exampleexamples, of particular interest due to its topological properties, is the ''O(3)'' nonlinear {{mvar|σ}}-sigma model in 1&nbsp;+&nbsp;1 dimensions, with the Lagrangian density

This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning <math>\hat n=\textrm{const.}</math> at infinity. Therefore, in the class of finite-action solutions, one may identify the points at infinity as a single point, i.e. that space-time can be identified with a [[Riemann sphere]].

:<math>\mathcal L= \tfrac{1}{2}\ \partial^\mu \hat n \cdot\partial_\mu \hat n </math>

where <math>\hat n=(n_1,n_2,n_3)</math> with the constraint <math>\hat n\cdot \hat n=1</math> and <math>\mu=1,2</math>. This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning <math>\hat n=\textrm{const.}</math> at infinity. Therefore in the class of finite-action solutions we may identify the points at infinity as a single point, i.e. that space-time can be identified with a [[Riemann sphere]]. Since the <math>\hat n</math>-field lives on a sphere as well, onewe seeshave a mapping <math>S^2\rightarrow S^2</math>, the solutions of which are classified by the second [[homotopy group]] of a 2-sphere. These solutions are called the O(3) [[Instantons]].

==See also==