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Dynamics of the Heisenberg model and a theorem on stability
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Image of FIG. 1.
FIG. 1.

The phase space structure for the discrete classical HM. The 3Λ-dimensional phase space is foliated into 2Λ-dimensional, connected, compact, regular, invariant submanifolds . The 2Λ − 1 (Λ + 2)-dimensional submanifolds of longitudinal fixed points (LFPs), denoted by , are connected, unbounded, and intersect every leaf orthogonally along an S 2.

Image of FIG. 2.
FIG. 2.

Lemma 5.1.

Image of FIG. 3.
FIG. 3.

Theorem 5.1.

Image of FIG. 4.
FIG. 4.

The FR state for 3 spins (left) and the AF state for 6 spins (right). A FR state is stable if g jl ⩾ 0 for all j, l or if g j(A)l(A) = g j(B)l(B) = 0, g j(A)l(B) ⩽ 0. An AF state with non-zero total spin is stable if g j(A)l(A) = g j(B)l(B) = 0, g j(A)l(B) ⩾ 0 or if g j(A)l(A) ⩾ 0, g j(B)l(B) ⩾ 0, g j(A)l(B) ⩽ 0.

Image of FIG. 5.
FIG. 5.

Solutions for a 4-spin ring with the two spins in each sublattice equal. If the total spin v of the AF state is not zero (left), the AF state is stable. If v vanishes (right), the AF state is unstable.


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Scitation: Dynamics of the Heisenberg model and a theorem on stability