Non-linear sigma model: Difference between revisions

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{{Short description|Class of quantum field theory models}}

In [[quantum field theory]], a '''nonlinear σ model''' describes a [[scalar field]] Σ which takes on values in a nonlinear manifold called the [[target manifold]] ''T''. The non-linear ''σ''-model was introduced by {{harvtxt|Gell-Mann|Lévy|1960|loc=section 6}}, who named it after a field corresponding to a spin 0 meson called ''σ'' in their model.

In [[quantum field theory]], a '''nonlinear ''σ'' model''' describes a field {{mvar|Σ}} that takes on values in a nonlinear manifold called the '''target manifold''' &nbsp;''T''. The non-linear ''σ''-model was introduced by {{harvtxt|Gell-Mann|Lévy|1960|loc=§6}}, who named it after a field corresponding to a [[spin (physics)|sp]] [[meson]] called ''σ'' in their model.<ref>{{Citation | last2=Lévy | first1=M. | last1=Gell-Mann | first2=M. | s2cid=122945049 | title=The axial vector current in beta decay | publisher=Italian Physical Society | doi=10.1007/BF02859738 | year=1960 | journal=Il Nuovo Cimento | issn=1827-6121 | volume=16 | issue=4 | pages=705–726| bibcode=1960NCim...16..705G }}</ref> This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the [[sigma model]] for general definitions and classical (non-quantum) formulations and results.

==Description==

The target manifold ''T'' is equipped with a [[Riemannian metric]]&nbsp;''g''. {{mvar|Σ}} is a differentiable map from [[Minkowski space]] ''M'' (or some other space) to&nbsp;''T''.

The [[Lagrangian density]] in contemporary chiral form<ref>{{citeCite journal doi| last1 = Gürsey | first1 = F. | s2cid = 122270607 | title = On the symmetries of strong and weak interactions | doi = 10.1007/BF02860276 |noedit journal = Il Nuovo Cimento | volume = 16 | issue = 2 | pages = 230–240 | year = 1960 | bibcode = 1960NCim...16..230G }}</ref> is given by:

:<math>\mathcal{L}={1\over 2}g(\partial^\mu\Sigma,\partial_\mu\Sigma)-V(\Sigma)</math>

where here, we have used a +&nbsp;−&nbsp;−&nbsp;− [[metric signature]] and the [[partial derivative]] <{{math>\partial\Sigma</math>| ''∂Σ''}} is given by a section of the [[jet bundle]] of ''T''&times;''M'' and ''{{mvar|V''}} is the potential.

In the coordinate notation, with the coordinates {{math|''Σ^''a''''}}, ''a''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'' where ''n'' is the dimension of&nbsp;''T'',

where here, we have used a +&nbsp;−&nbsp;−&nbsp;− [[metric signature]] and the [[partial derivative]] <math>\partial\Sigma</math> is given by a section of the [[jet bundle]] of ''T''&times;''M'' and ''V'' is the potential.

:<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 nonrenormalizablerenormalizable.

In the coordinate notation, with the coordinates Σ^''a'', ''a''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'' where ''n'' is the dimension of&nbsp;''T'',

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 |author2year= Zinn-Justin, Jean1976| title=Renormalization of the nonlinear sigma model in in 2 + epsilon dimensions | journal=Physical Review Letters| yearvolume= 1976 36| volumeissue=36 13|pages=691–693|bibcode=1976PhRvL..36..691B|doi=10.1103/PhysRevLett.36.691 |bibcode author2=Zinn-Justin, 1976PhRvL..36..691B Jean}}</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.

In both approaches, the non-trivial renormalization-group fixed point found for the [[n-vector model|''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.

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

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

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 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'',

:<math>\sqrt{\det g}\mathcal{D}\Sigma.</math>

==Renormalization==

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

{{cite journal|last=Friedan|first=D.|authorlinkauthor-link=Daniel Friedan|title=Nonlinear models in 2+ε dimensions | journal = PRL Physical Review Letters| volume = 45 | issue = 13| pages = 10571057–1060 |publisher=|___location=| year = 1980 | url doi= http://link.aps.org/doi/10.1103/PhysRevLett.45.1057 |doi bibcode= 101980PhRvL.1103/PhysRevLett.45.1057 1057F| bibcodeurl=1980PhRvLhttps://digital.library.45unt.1057Fedu/ark:/67531/metadc841801/}}</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}}{\partial\lambda}=\beta_{ab}(T^{-1}g)=R_{ab}+O(T^2).~,</math>

<{{math|''R<sub>R_{ab}</mathsub>''}} being the [[Ricci tensor]] of the target manifold.

This represents a [[Ricci flow]], havingobeying [[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 |s2cid=122018499 |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 |url=http://projecteuclid.org/euclid.cmp/1103940923 }}</ref> which

augments the geometry of the flow to include [[Torsion tensor|torsion]], preserving renormalizability and leading to an [[infrared fixed point]] as well, on account of [[teleparallelism]] ("geometrostasis").<ref>{{citeCite journal doi| last1 = Braaten | first1 = E. | last2 = Curtright | first2 = T. L. | last3 = Zachos | first3 = C. K. | doi = 10.1016/0550-3213(85)90053-7 |noedit title = Torsion and geometrostasis in nonlinear sigma models | journal = Nuclear Physics B | volume = 260 | issue = 3–4 | pages = 630 | year = 1985 |bibcode = 1985NuPhB.260..630B }}</ref>

{{further|Ricci flow}}

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

OneA of the most famouscelebrated examplesexample, of particular interest due to its topological properties, is the ''O(3)'' nonlinear sigma {{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 ''n&#770;''=(''n₁, n₂, n₃'') with the constraint ''n&#770;''⋅''n&#770;''=1 and {{mvar|μ}}=1,2.

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&#770;'' =\textrm{const.}</math> constant at infinity. Therefore, in the class of finite-action solutions, weone 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, we have 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]].

One of the most famous examples, of particular interest due to its topological properties, is the O(3) nonlinear sigma model in 1&nbsp;+&nbsp;1 dimensions, with the Lagrangian density

Since the ''n&#770;''-field lives on a sphere as well, the mapping {{math|''S²→ S²''}} is in evidence, the solutions of which are classified by the second [[homotopy group]] of a 2-sphere: These solutions are called the O(3) [[Instantons]].

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

This model can also be considered in 1+2 dimensions, where the topology now comes only from the spatial slices. These are modelled as R^2 with a point at infinity, and hence have the same topology as the O(3) instantons in 1+1 dimensions. They are called sigma model lumps.

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, we have 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==

Line 46 ⟶ 51:

* [[Little Higgs]]

* [[Skyrmion]], a soliton in non-linear sigma models

* [[Polyakov action]]

* [[WZW model]]

* [[Fubini–Study metric]], a metric often used with non-linear sigma models.

* [[Ricci flow]]

* [[Scale invariance]]

==References==

{{Reflist|12}}

==External links==

* [http://www{{cite journal |doi=10.4249/scholarpedia.org/article/Nonlinear_Sigma_model '8508|title=Nonlinear Sigma model' on |year=2009|last1=Ketov|first1=Sergei|journal=Scholarpedia]|volume=4|issue=1|page=8508|bibcode=2009SchpJ...4.8508K|doi-access=free}}

* {{cite journal |doi=10.1023/A:1021009008129|year=2002|last1=Kulshreshtha|first1=U.|last2=Kulshreshtha|first2=D. S.|s2cid=115710780|title=Front-Form Hamiltonian, Path Integral, and BRST Formulations of the Nonlinear Sigma Model|journal=International Journal of Theoretical Physics|volume=41|issue=10|pages=1941–1956}}

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{{DEFAULTSORT:Non-Linear Sigma Model}}

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