Chern-Simons gauge field theory of two-dimensional ferromagnets

Martina, L.; Pashaev, Oktay K.; Soliani, G.

doi:10.1103/physrevb.48.15787

Cited by 21 publications

(19 citation statements)

References 24 publications

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“…It has been shown that the Chern-Simons gauged Landau-Ginsburg model plays the role of effective theory for the Fractional Quantum Hall Effect [2]. The Chern-Simons equation of motion can describe time evolving two-dimensional surfaces in such a way that the deformation is not only locally compatible with the Gauss-Codazzi equation, but completely integrable as well [3].…”

Section: Copyright C 2002 By P Bracken and A M Grundlandmentioning

confidence: 99%

On Complete Integrability of the Generalized Weierstrass System

Bracken

Grundland

2002

JNMP

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In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsingular 2 × 2 matrices based on an sl(2, C) representation. We derive the explicit form of the Darboux transformation for the Weierstrass system. New classes of multisoliton solutions of the Generalized Weierstrass system are obtained through the use of the Bäcklund transformation and some physical applications of these results in the area of classical string theory are presented.

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Section: Copyright C 2002 By P Bracken and A M Grundlandmentioning

confidence: 99%

On Complete Integrability of the Generalized Weierstrass System

Bracken

Grundland

2002

JNMP

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“…This idea has been recently applied to the classical Heisenberg model in [12]. Furthermore, the stationary magnetic vortices of the model have been related to the Chern-Simons solitons, while the topological charge of the former has been related to the electric charge of the latter.…”

Section: Introductionmentioning

confidence: 99%

“…But, as we will see below the mapping of the model (2.1) to zero curvature equations connects these fields. Finally, let us emphasize that if for the Heisenberg model it was sufficient to have SU (2) zero curvature mapping [12], for the model considered here an additional gauge field, related to the velocity field v is necessary. …”

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confidence: 99%

Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

Pashaev

1996

Mod. Phys. Lett. A

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The classical spin model in planar condensed media is represented as the U (1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge -invariant fields whose evolution is described by In the present paper I reduce the self-dual CS model from an integrable model in 2+1 dimensions. This model was first derived by Ishimori and has integrable dynamics of the magnetic vortices [7]. Its formulation in terms of the vorticity for continuous flow tends to clarify the physical meaning of the model and allows an extension for higher dimensions, admitting the bilinear Hirota representation [8,10]. The physical applications of the model are connected with the resolution of the anomalous behavior of the linear momentum in ferromagnets [9]. As it was shown in the latter paper by using the de-localized electron model of ferromagnets, the canonical momentum becomes well defined due to a fermionic background.The model considered in the present paper is a modification of the classical Heisenberg model for condensed media, having hydrodynamical flow with its vorticity related to a topological charge density. In this case a hydrodynamical vortex in the magnetic (anyon) fluid is also inducing a magnetic vorticity.The CS gauge field theory is then obtained by projecting spin variables on the tangent plane to the sphere (spin phase space). This idea has been recently applied to the classical Heisenberg model in [12]. Furthermore, the stationary magnetic vortices of the model have been related to the Chern-Simons solitons, while the topological charge of the former has been related to the electric charge of the latter. 2In the case of equality between hydrodynamical vorticity and the "statistical" magnetic field, the theory can be formulated for a zero curvature effective gauge field. This allows me to reformulate the model in terms of gauge invariant matter fields. As shown, their evolution is described by the DS equations and at the same time the fields satisfy the CS system. For the self-dual case, we reproduce CS solitons with an integrable evolution replacing the statistical gauge field with a velocity field.As a by-product we obtain: a) new reduction conditions for the DS-I model related to the Ishimori-I (IM-I) model, b) the 2+1 dimensional zero curvature formulation for the DS equation. Zero Curvature ConditionThe crucial moment of our construction is a zero curvature condition in 2+1 dimensional spacefor u(2)(u(1, 1)) Lie algebra valued connection J µ , (µ = 0, 1, 2). Decomposing the algebra for the diagonal and off-diagonal partsand parametrizingin terms of two U (1) gauge potentials W µ and V µ , and the q µ matter fields, we have the set of U (1) × U (1) gauge invariant equationsHere the covariant derivative is defined asFor the spatial part of the matter fields we introduceand denote D ± =...

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“…For instance, for Ψ + ≡ 0 we can find Ψ − in terms of solutions of the Liouville equation, to which our system of equations reduces. Such solutions of multivortex type are widely discussed in [9,10].…”

mentioning

confidence: 99%

“…The vorticity is determined by the density of the topological charge. These types of systems have been discussed in [9] - [12]. They can be considered as generalizations of the well-known Ishimori model [13].…”

mentioning

confidence: 99%

Chern-Simons field theory and completely integrable systems

1996

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We show that the classical non-abelian pure Chern-Simons action is related in a natural way to completely integrable systems of the Davey-Stewartson hyerarchy, via reductions of the gauge connection in Hermitian spaces and by performing certain gauge choices. The Bäcklund Transformations are interpreted in terms of Chern-Simons equations of motion or, on the other hand, as a consistency condition on the gauge. A mapping with a nonlinear σ-model is discussed.

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Chern-Simons gauge field theory of two-dimensional ferromagnets

Cited by 21 publications

References 24 publications

On Complete Integrability of the Generalized Weierstrass System

On Complete Integrability of the Generalized Weierstrass System

Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

Chern-Simons field theory and completely integrable systems

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