Nonlinear Sigma model

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Sergei V. Ketov (2009), Scholarpedia, 4(1):8508. doi:10.4249/scholarpedia.8508 revision #91589 [link to/cite this article]
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A nonlinear sigma model is a scalar field theory whose (multi-component) scalar field defines a map from a `space-time' to a Riemann (target) manifold.

Basic facts

Consider a set of D real scalar fields \( \phi^a(x^{\mu}) \) mapping a d-dimensional (flat) space \(\Sigma\) into a D-dimensional target space M, with the action \[\tag{1} S[\phi]=\frac{1}{2}\int_{\Sigma}{\rm d}^dx\,g_{ab}(\phi)\partial^{\mu}\phi^a\partial_{\mu}\phi^b~,\]

where \( \partial^{\mu}=\eta^{\mu\nu}\partial_{\nu},\quad \partial_{\nu}=\partial/\partial x^{\nu},\quad \eta^{\mu\nu}\) is normally a Minkowski metric (though often \(\eta^{\mu\nu}\) is taken to be Euclidean). It is called a Non-Linear Sigma-Model (NLSM) with the metric \(g_{ab}(\phi).\) It is usually assumed that \(g_{ab}(\phi)\) is a positive-definite field-dependent matrix, in order to ensure the absence of negative norm states. The standard mechanisms for a construction of compact NLSM (without ghosts) are: (i) imposing constraints on the NLSM scalars so that they take their values in a compact (usually symmetric) space, and (ii) gauging internal symmetries of a NLSM.

The geometrical significance of NLSM lies in the invariance of the action (1) under the (infinitesimal) field reparametrizations \[\tag{2} \phi^a \to \phi^a + \xi^a(\phi) \]

that can be interpreted as diffeomorphisms of the target space M, provided that the metric \(g_{ab}(\phi)\) transforms as a second-rank tensor. The latter in the field theory context is a transformation of the fields and the coupling constants (defined as the coefficients in Taylor expansion of the metric components with respect to the fields). Hence, generically, there is no Noether current for the NLSM reparametrizational invariance. The two NLSM are physically equivalent when they are related by a field redefinition alone. It is not difficult to check that it happens only if \(\xi\) is a Killing vector in M. In this case the NLSM action (1) has an isometry (or an internal symmetry) leading to a conserved Noether current and the corresponding conserved charge. All the isometries of the target space M form the group G representing the global symmetry of M. The most important case is given by the target space that is a Lie group G itself. For example, the O(n) NLSM is defined by the action \[\tag{3} S[\vec{n}]=\frac{1}{2\lambda^2}\int {\rm d}^dx \,\partial^{\mu}\vec{n}\cdot \partial_{\mu}\vec{n}~,\]

where the real scalar fields \(\vec{n}(x^{\mu})\) are restricted by a condition \[\tag{4} \vec{n}\cdot \vec{n}=const.\]

After solving the condition in terms of some independent field variables (angles) \(\{\phi\},\) and substituting the result \(\vec{n}=\vec{n}(\phi)\) back into the action, one gets a NLSM action (1). The O(3) NLSM model can be interpreted as the continuum limit of an isotropic ferromagnet in statistical mechanics. The two-dimensional O(3) model is known to be integrable as a classical field theory. Some particular two-dimensional O(n) NLSM are used in connection to antiferromagnetic spin chains, quantum Hall effect and superfluid 3He.

To illustrate the gauging procedure, let's consider the simplest example of the CP(n) model with an action \[\tag{5} S[\vec{n}]=\frac{1}{2\lambda^2}\int {\rm d}^dx\,(D^{\mu}\vec{n})^{\dagger}\cdot D_{\mu}\vec{n}~,\]

in terms of the (n+1)-component complex scalar field \(\vec{n}\) subject to the constraint \[\tag{6} \vec{n}{}^{\dagger} \cdot \vec{n}=const~,\]

and the covariant derivatives \[\tag{7} D_{\mu}\vec{n}= (\partial_{\mu} +iA_{\mu})\vec{n}~.\]

The action (5) is invariant under Abelian gauge transformations, \[\tag{8} \vec{n} \to e^{i\Lambda(x)}\vec{n}~,\quad A_{\mu} \to A_{\mu}-\partial_{\mu}\Lambda ~,\]

with the gauge parameter \(\Lambda(x)\ .\) There are no kinetic terms for the gauge field \(A_{\mu}\) in the action (5), so that the gauge field can be integrated out from the action by the use of its algebraic equation of motion. Substituting the solution \(A_{\mu}=i\vec{n}^{\dagger} \cdot\partial_{\mu}\vec{n}\) back into the action (5) gives rise to a CP(n) NLSM.

A scalar potential term \(V(\phi)\) may be added into the action (1). Given an isometry symmetry G of the NLSM terms, the scalar potential should be invariant under the same symmetry G. Reparametrizational invariance does not apply in the presence of a scalar potential. Nevertheless, the scalar field theory with free kinetic terms, \(g_{ab}(\phi)=\delta_{ab}~,\) and a G-invariant scalar potential is called a linear Sigma-model.

A d=2 NLSM is special since its fields \(\phi\ ,\) its metric \(g_{ab}\) and, hence, all of its coupling constants are dimensionless. In quantum theory it implies that the 2d NLSM (1) is renormalizable by index of divergence, i.e. its Ultra-Violet (UV) on-shell counterterms are of the same (mass) dimension two, as the NLSM Lagrangian itself. The NLSM action (1) has the most general (parity conserving) 2d kinetic terms, so that all the counterterms can be absorbed into the NLSM metric as its `quantum' deformations (geometrical renormalization). It is known as the on-shell renormalizability. Being non-linear, it should be distinguished from the usual (multiplicative) renormalizability that is only the case for a symmetric NLSM target space. In quantum theory, the renormalized NLSM metric is dependent upon the renormalization scale \(\mu\ ,\) while the NLSM renormalization group beta-functions are given by \[\tag{9} \mu \frac{d}{d \mu} g_{ab} = \beta_{ab}(\phi) \]

Another special feature in d=2 is the existence of the extension of the NLSM action (1) by the so-called generalized Wess-Zumino-Novikov-Witten (gWZNW) term, \[\tag{10} S_{\rm gWZNW}[\phi]=\frac{1}{2\lambda^2}\int {\rm d}^2x\,h_{ab}(\phi)\epsilon^{\mu\nu}\partial_{\mu}\phi^a\partial_{\nu}\phi^b~,\]

where \(h(\phi)\) is a two-form in M, and \(\epsilon^{\mu\nu}\) is a Levi-Civita antisymmetric symbol in d=2. The most important example is given by the standard WZNW model having a Lie group G as the target space, with the torsion \(T=dh\) to be constructed out of the Lie group structure constants. In quantum theory, the one-loop renormalization of the most general NLSM, described by a sum of the actions (1) and (10), is given by the one-loop beta-functions \[\tag{11} \beta^{1}_{ab} = \mu \frac{d}{d \mu} (g_{ab}+h_{ab}) = \frac{1}{2\pi}\hat{R}_{ab}~~, \]

where \(\hat{R}_{ab}\) is the Ricci curvature tensor in terms of the connection \((\Gamma_{abc} -T_{abc}),\ ,\) with the Christoffel symbols \(\Gamma\) and the NLSM torsion \(T=dh\ .\)

There exist an NLSM target space action \(S[g]\) in D dimensions, whose classical equations of motion can be identified with the NLSM renormalization group beta-functions, \[\tag{12} \frac{\delta S[g]}{ \delta g^{ab} } \propto \beta_{ab} \]

whereas the action \(S\) itself monotonically decreases under the renormalization group flow. It is known as the (Zamolodchikov) c-theorem. Equation (12) is valid on-shell only.

The one-loop renormalization group beta-function (11) vanishes, i.e. has its fixed point, when \(\hat{R}_{ab}=0\ .\) The one-loop fixed point is actually exact (i.e. valid to all loop orders of quantum perturbation theory) in the case of the parallelizing torsion when \(\hat{R}_{abcd}=0\ .\) It is only possible when the NLSM target space is either a group manifold or a seven-sphere. A quantized WZNW model (whose target space is a group manifold) at its fixed point is conformally invariant, being equivalent to a free quantum fermionic theory. This phenomenon is called a non-Abelian bosonization. There are infinitely many conserved currents in the quantized WZNW model at the fixed point. Those currents form an affine (or Kac-Moody) algebra, while the classical conformal algebra of the WZNW model is also extended to a quantum Virasoro algebra. Hence, the quantized NLSM on a group manifold G at its fixed point is the example of a conformal field theory with a chiral current algebra.

There exist yet another interesting connection between complex geometry and extended supersymmetry in d=2 NLSM. Though any action (1) can be supersymmetrized with respect to a simple (1,1) supersymmetry in d=2, requiring more supersymmetries implies geometrical restrictions on the target space M. For instance, in the case of (2,2) supersymmetry, the NLSM target space must be Kaehler (\(b=0\)) or bi-hermitean (\(b\neq 0\)), whereas in the case of the maximal (4,4) supersymmetry, the NLSM target space must be either hyper-Kaehler (\(b=0\)) or bi-hyper-complex (\(b\neq 0\)). In quantum theory, the one-loop NLSM renormalization group beta-functions of the two-dimensional supersymmetric NLSM are the same as their purely bosonic counterparts. However, unlike the bosonic NLSM, there are no two- and three-loop contributions to the supersymmetric NLSM renormalization group beta-functions. Any N=4 supersymmetric NLSM in d=2 is UV-finite to all orders of quantum perturbation theory, i.e. is free of UV divergences and has the vanishing beta-functions.

In string theory the d=2 space \(\Sigma\) represents a string world-sheet, the NLSM metric is identified with a D-dimensional space-time metric representing the gravitational background where the string propagates, and the antisymmetric field \(h_{ab}\) is called a B-field. The two-dimensional (world-sheet) conformal invariance of strings requires the string NLSM renormalization group beta-functions to vanish. Their vanishing yields the effective equations of motion of string modes in spacetime. The Zamoldchikov action of equation (12) in string theory represents the gravitational effective action. For example, the vanishing one-loop renormalization group beta-functions of the string NLSM with a B-field give rise to the effective action \[\tag{13} S[g,B] = \frac{1}{2\pi} \int d^D\phi \left( -R +\frac{1}{3}H^2\right)~~, \]

where we have introduced the notation \(H^2=H^{abc}H_{abc}\) and \(H=dB\ .\) Equation (13) contains the familiar Einstein-Hilbert term for gravity. The higher quantum loop-order terms in the string 2d NLSM renormalization group beta-functions represent string corrections to the Einstein gravity.

There exist many striking similarities between non-abelian gauge theories in four dimensions and NLSM in two dimensions, such as (i) geometrical significance, (ii) renormalizability, (iii) existence of topologically non-trivial field configurations (solitons and instantons), (iv) asymptotic freedom, (v) dynamical generation of vector bosons, (vi) mass gap in a non-perturbative spectrum, (v) 1/n quantum perturbation theory, etc.

Though a NLSM in d=4 is not renormalizable, it often arises in elementary particles physics and nuclear physics as the effective field theory of Goldstone bosons associated with spontaneous breaking of an internal symmetry. The most famous example is given by the chiral symmetry breaking in QCD with \(N_f\) flavors, where the phenomenological low-energy Lagrangian of pions (Goldstone pseudo-scalar mesons made out of quark-antiquark pairs) \( U_{ff'}\sim < \bar{q}^L_fq^R_{f'}> \) can be constructed as a NLSM, \[\tag{14} L[U] = -\frac{F^2_{\pi}}{16} tr (A_{\mu}A^{\mu}) -\frac{F^2_{\pi}}{2} tr [ \hat{M}^2(U^{\dagger}+U)] \]

in terms of the Cartan field \(A_{\mu}=U^{\dagger}\partial_{\mu}U\ ,\) the \(SU(N_f)\)-valued pion scalar field \(U\ ,\) the pion mass squared matrix \(\hat{M}^2\ ,\) and the pion decay constant \( F_{\pi} \) whose experimental value is 186 MeV. The effective action of pions is the stating point of chiral perturbation theory. Baryons can be indentified with solitons (classical exact solutions of finite energy) after adding to the NLSM Lagrangian (14) a Skyrme term \[\tag{15} L_{Skyrme}[U]= \frac{1}{32e^2} tr ([A_{\mu},A_{\nu}]^2) \]

with a dimensionless coupling constant \(e\) whose experimental value is 5,45. From the NLSM point of view, the Skyrme term (15) represents the 4th-order derivative contribution to the NLSM (14) that is of the 2nd-order in the derivatives of its scalar field \(U\ .\) From the phenomenological viewpoint, the Skyrme term (15) is needed for stability of solitons (skyrmions), which is ensured by their conserved topological winding number identified with the baryon number. For instance, numerical computations of the skyrmion mass with baryonic number \(B=1\) yield 850 MeV that is pretty close to the nucleon mass.


A sigma-model was first introduced in

  • Gell-Mann M, Lévy M (1960), The axial vector current in beta decay, Nuovo Cimento 16, 705

The non-linear realizations of symmetries in field theory were introduced in

  • Coleman S, Wess J, Zumino, B (1969), Structure of phenomenological Lagrangians, Phys. Rev. 177, 2239

The NLSM quantization was initiated in

  • Honerkamp J and Meetz K (1971), Chiral-invariant perturbation theory, Phys.Rev. D3, 1996
  • Gerstein I S, Jackiw R, Lee B W, Weinberg S (1971), Chiral loops, Phys. Rev. D3, 2486
  • Faddeev L D, Slavnov A A (1971), Invariant perturbation theory for nonlinear chiral Lagrangians. Theor. Math. Phys. 8, 843

The two-dimensional NLSM renormalizability was proved in

  • Brézin E, Le Guillou J C, Zinn-Justin J (1976), Renormalization of the nonlinear σ model in 2+ε dimensions, Phys. Rev. D14, 2615
  • Becchi C, Piguet O (1989), On the renormalization of two-dimensional chiral models, Nucl. Phys. B315, 153

The NLSM renormalization group properties and their applications to ferromagnets were considered in

  • Polyakov A M (1975), Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields, Phys. Lett. B59, 79
  • Brézin E, Zinn-Justin J (1976), Renormalization of the nonlinear σ model in 2+ε dimensions — application to the Heisenberg ferromagnets, Phys. Rev. Lett. 36, 691
  • Brézin E, Zinn-Justin J (1976), Spontaneous breakdown of continuous symmetries near two dimensions, Phys. Rev. B14, 3110
  • Bardeen W A, Lee B W, Shrock R E (1976), Phase transition in the nonlinear σ model in a (2+ε)-dimensional continuum, Phys. Rev. D14, 985

The NLSM on symmetric spaces and their multiplicative renormalizability were derived in

  • Pisarski R D (1979), Nonlinear σ models on symmetric spaces, Phys. Rev. D20, 3358
  • Brézin E, Hikami S, Zinn-Justin J (1980), Generalized non-linear σ-models with gauge invariance, Nucl. Phys. B165, 528
  • Voronov B L, Tyutin I V (1981), Renormalization of two-dimensional chiral theories, Yadern. Phys. 33, 1137

The perturbative (pure metric) NLSM renormalization group beta-functions (up to four loops) in two dimensions were calculated in

  • Friedan D H (1980), Nonlinear models in 2+ε dimensions. Phys. Rev. Lett. 45, 1057
  • Hikami S (1981), Three-loop ß-functions of non-linear σ models on symmetric spaces, Phys. Lett. B98, 208
  • Bernreuther W, Wegner F J (1986), Four-loop-order β Function for two-dimensional non-linear sigma models. Phys. Rev. Lett. 57, 1383
  • Graham S J (1987), Three-loop beta-function for the bosonic non-linear sigma-model, Phys. Lett. B197, 543
  • Jack I, Jones D R T (1989), The four-loop metric beta-function for the bosonic sigma-model, Phys. Lett. B220, 171

The Wess-Zumino-Novikov-Witten term was introduced in

  • Wess J, Zumino B (1971), Consequences of anomalous Ward identities, Phys. Lett. B37, 95
  • Novikov S P (1981), Multi-valued functions and functionals: an analogue of Morse theory, Sov. Math. Dokl. 24, 222
  • Witten E (1983), Non-abelian bosonization in two dimensions, Nucl. Phys. B223, 422

The c-theorem in two dimensions was formulated and proved in

  • Zamolodchikov A B (1986), Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 32, 730

As regards the two-dimensional NLSM in the context of string theory, see e.g.,

  • Callan C G, Friedan D, Martinec E J, Perry M J (1985), Strings in background fields, Nucl. Phys. B262, 593
  • Osborn H (1988), String theory effective actions from bosonic sigma models, Nucl. Phys. B308, 629
  • Curci G, Paffuti G (1989), Infrared problems in two-dimensional generalized sigma models, Nucl. Phys. B312, 227

The supersymmetric non-linear sigma-models were introduced in

  • Zumino B (1979), Supersymmetry and Kaehler manifolds, Phys. Lett. B87, 203
  • Alvarez-Gaume L, Freedman D Z (1981), The background-field method and the ultraviolet structure of the supersymmetric non-linear sigma-model, Annals Phys. 134, 85
  • Alvarez-Gaume L, Freedman D Z (1981), Geometrical structure and ultraviolet finiteness of the supersymmetric sigma model, Commun. Math. Phys. 80, 443
  • Gates, Jr. S J, Hull C M, Rocek M (1984), Twisted multiplets and new supersymmetric non-linear sigma models, Nucl. Phys. B248, 157

The geometrical structure and the perturbative renormalization (up to three loops) of a bosonic NLSM with a WZNW term were derived in

  • Braaten E, Curtright T L, Zachos C K (1985), Torsion and geometrostasis in non-linear sigma models, Nucl. Phys. B260, 630
  • Ketov S V (1987), Two-loop calculations in the non-linear sigma-model with torsion, Nucl. Phys. B294, 813
  • Ketov S V, Deriglazov A A, Prager Y S (1990), Three-loop beta-function for the two-dimensional non-linear sigma-model with a Wess-Zumino term, Nucl. Phys. B332, 447

The renormalization group beta-functions of a supersymmetric non-linear sigma-model with a WZNW term in two dimensions were calculated (up to four loops) in

  • Alvarez-Gaumé L, Freedman D Z, Mukhi S (1981), The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model, Annals Phys. 134, 85
  • Grisaru, M T, van de Ven A E M, Zanon D (1986), Four-loop divergences for the N=1 supersymmetric non-linear sigma-model in two dimensions, Nucl. Phys. B277, 409
  • Deriglazov A A, Ketov S V (1990), Four-loop divergences of the two-dimensional (1,1) supersymmetric non-linear sigma-model with a Wess-Zumino-Witten term, Nucl. Phys. B359, 498

The Skyrme model was introduced in

  • Skyrme T H R (1962), A unified field theory of mesons and baryons, Nucl. Phys. 31, 556

Many more references about the NLSM and their applications in quantum field theory, statistical field theory, hadron physics, conformal field theory and string theory, can be found in the reviews and books cited below (see Further Reading)

Internal references

  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Further Reading

  • Olshanetsky M A, Perelomov A M (1981), Classical integrable systems related to Lie algebras, Physics Reports 71, 313
  • Novikov V A, Shifman M A, Vainstein A I, Zakharov V I (1984), Two-dimensional sigma models: modeling non-perturbative effects of Quantum Chromodynamics, Phys. Reports 116, 103
  • Zakrzewski W J (1989), Low Dimensional Sigma Models, IOP Publishers, Bristol, ISBN 9780852742310
  • Makhankov V G, Rybakov Y P, Sanyuk V I (1993), Skyrme Model, Springer-Verlag, Berlin-Heidelberg, ISBN 9789810207052
  • Ketov S V (1995), Conformal Field Theory, World Scientific, Singapore, ISBN 9810216084
  • Ketov S V (2000), Quantum Non-Linear Sigma-Models, Springer-Verlag, Berlin-Heidelberg, ISBN 3540674616
  • Zinn-Justin J (2002), Quantum Field Theory and Critical Phenomena (4th edition), Oxford University Press, Oxford, ISBN 0198509235
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