The large-m limit, and spin liquid correlations in kagome-like spin models

Abstract

It is noted that the pair correlation matrix χ 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 χ are degenerate at all temperatures T, and correspond to an eigenspace L_ of χ, 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≤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≥2 contribute to χ at low T. The m→ ∞ formulae can be thus quantitatively correct in description of χ 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_ .

Keywords

• kagome lattice
• frustration
• spin correlations
• exact result