The large-m limit, and spin liquid correlations in kagome-like spin models - Yavors'kii, T. (Condensed Matter Physics, 2017)
It is noted that the pair correlation matrix χˆ of the nearest neighbor Ising model on periodic three-dimensional (d = 3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly. Specifically, a macroscopic number 1/3N + 1 out of N eigenvalues of χˆ are degenerate at all temperatures T , and correspond to an eigenspace L− of χˆ, independent of T . Degeneracy of the eigenvalues, and L− are an exact result for a complex d = 3 statistical physical model. It is further noted that the eigenvalue degeneracy describing the same L− is exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models. A peculiar match of the opposite m = 1 and m → ∞ limits can be interpreted that the m → ∞ considerations are exact for m = 1. It is not clear whether the match is coincidental. It is then speculated that the exact eigenvalues degeneracy in L− in the opposite limits of m can imply their quasi-degeneracy for intermediate 1 É m < ∞. For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from L− that for m Ê 2 contribute to χˆ at low T . The m → ∞ formulae can be thus quantitatively correct in description of χˆ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime. An exception may be an interval of T , where the order-by-disorder mechanisms select sub-manifolds of L−.