Collinear and Non-Collinear Configurations in Classical Geometrically Frustrated Ring-Shaped Systems

Abstract

Geometrically frustrated quantum spin systems, with competing antiferromagnetic couplings, show the Kahn degenerate frustration for some specific values of Heisenberg Hamiltonian parameters. It has been recently shown for rings with a defect bond and centered rings. In the case of classical counterparts of these systems, degenerated configurations with the lowest energy are present for the energy function parameter greater than a certain threshold. In these domains such configurations are planar but non-collinear with continuous changes of the net magnetic moment with respect to the Hamiltonian parameter. Outside these domains there is unique collinear ground state configuration (neglecting choice of the net magnetic moment direction). However, these collinear configurations are the same in both non-frustrated and geometrically frustrated domains. Numerically exact calculations for quantum systems strongly confirm that determined properties of their classical counterparts realize the classical limit s Ñ 8.

Highly Degenerated Ground States in Some Rings Modeled by the Ising Spins with Competing Interactions

“…Therefore, all of them exhibit the third type of frustration in this domain. Analogous behavior has been found in the classical and quantum counterparts of models discussed here [9,10,20]. However, in each of these cases systems behave in different ways when the ratio α is large enough.…”

The Lagrange variety approach applied to frustrated classical wheels

“…Such a system is another example of models in which the collinear LEC is 'retained' when competing interactions are 'turned on' by a relatively weak antiferromagnetic coupling(s) [13,14,21]. This feature can be considered as the classical counterpart of the third type frustration according to the classification scheme recently proposed [8].…”

Spectrum of some arrow-bordered circulant matrix

“…Note that higher degeneracy is not excluded and Schmidt [51] have provided an example of a classical spin system, when the lowest energy configuration can be realized in nonphysical four-dimensional space R 4 . For small systems some results have been obtained with simple calculus [55,59], but more general considerations need strictly proved properties, i.e., the results of this paper [see 60]. It is worth noting that for actually synthesized magnetic molecules [see, e.g., [61][62][63] there is needed analysis of some more general matrices (not regularly weighted graphs), e.g., alternating circulant matrices or arrow-bordered alternating circulant ones.…”

“…They showed that for traceless dressed matrix (i.e., for j c j = 0) some physical quantities are "gauge independent". For fixed parameters of the system the eigenvalues of the dressed matrix, including its minimum eigenvalue λ min (c), depend on the gauge vector c. They proved that there exists max c∈R N λ min (c) and it equals, up to a constant factor, the minimum of the system energy E. The considerations performed in this paper give us a general solution of this problem for spins placed at vertices and the center of a regular n-gon with an even number n [see, e.g., [55][56][57][58]. The further physical analysis take into account also eigenvectors of the dressed matrix J, but this problem is out of scope here.…”