Exact solutions of a nonlinear sigma-model in a curved space and the theory of magnetically ordered media with variable saturation magnetization

Abstract

Constancy of the abso.lute value of magnetization is an essential assumption for the phenomenologtcal consideration of weakly excited states of magnets, because then the variation of the magnetization is limited to its rotation, which is described by the Landau-Lifshits equation [1]. This model works well far from the Curie point. At moderate temperatures, one can expect spatial variation of the saturation magnetization of the sample material. Such a conjecture enables us to study the magnet within the framework of the phenomenological approach in the vicinity of the critical point (but outside the fluctuation region) and to describe a smooth spatially inhomogeneous* transition to the paramagnetic phase.1. Consider the stationary Landau-Lifshits equation for an isotropic two-dimensional ferromagnet:In (1), a > 0 is the square of the saturation magnetization of the sample and ~" = (x, y). Introduce the matrix M = 2~rY, where as are the Pauli matrices. Standard transformations of (1) implyHere I is the identity matrix and z = x + iy. The key point for further consideration is the assumption that c~ is harmonic:Thus, a two-dimensional isotropic magnet whose square of saturation magnetization is a harmonic function of the coordinates is described by the nonlinear sigma-model in a curved space. Equation (2), in view of (3), formally belongs to the same type as the gravity equations with two Killing vectors for the part of the metric g containing off-diagonal terms [3][4][5][6]. In the situation considered, however, M = M* (* denotes the Hermitian conjugation) and det M = -o~, while in gravity theory we have g = gT and det g = -cr 2. Let S = M/v~, S 2 = I, and S = ffrT, where S is the normalized magnetization vector, if2 = 1. Then (2), (3) implyi.e., we obtain the equation of the 0(3) sigma-model in a curved space. Note that similar equations occur also in other applications interesting from the physical point of view. In particular, the above-mentioned case of gravity equations is reduced to the 0(2, 1)-sigma-model in a curved space [5,6]. The dimensional reduction of axially symmetric** self-duality equations in the Yang-Mills field theory also leads to a similar model with the analogue of the matrix S equal to g'g, where g E SL(N, R) [7].*For possible integrable models of the nonstationary one-dimensional Heisenberg magnet, see [2] and the references therein.

The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

“…Besides that, other reductions of the non-autonomous chiral model (for some m) are of interest, see e.g. [20,21], and the set of solutions that we obtained in this work will typically be reducible to solutions of them.…”

“…[9,10,11,12,13,14,15,16,17,18,19]). A version of the above equation also arises as the cylindrically symmetric case of the (2 + 1)-dimensional principal chiral model [20] and as a special case of the stationary Landau-Lifshitz equation for an isotropic two-dimensional ferromagnet [21]. The first construction of "multi-soliton" solutions of (1.1) has been carried out by Belinski and Zakharov [5,6] (also see [7]) using the "dressing method" 1 .…”

“…[9,10,11,12,13,14,15,16,17,18,19]). A version of the above equation also arises as the cylindrically symmetric case of the (2 + 1)-dimensional principal chiral model [20] and as a special case of the stationary Landau-Lifshitz equation for an isotropic two-dimensional ferromagnet [21].…”