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

It is noted that the pair correlation matrix $\widehat{\chi }$ 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 $1\mathrm{/}3N+1$ out of $N$ eigenvalues of $\widehat{\chi }$ are degenerate at all temperatures $T$, and correspond to an eigenspace ${\mathbb{L}}_{-}$ of $\widehat{\chi }$, independent of $T$. Degeneracy of the eigenvalues, and ${\mathbb{L}}_{-}$ are an exact result for a complex $d=3$ statistical physical model. It is further noted that the eigenvalue degeneracy describing the same ${\mathbb{L}}_{-}$ 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 $m\to \mathrm{\infty }$ limits can be interpreted that the $m\to \mathrm{\infty }$ 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 ${\mathbb{L}}_{-}$ in the opposite limits of $m$ can imply their quasi-degeneracy for intermediate $1 \leqslant m < \infty$. For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from ${\mathbb{L}}_{-}$ that for $m⩾2$ contribute to $\widehat{\chi }$ at low $T$. The $m\to \mathrm{\infty }$ formulae can be thus quantitatively correct in description of $\widehat{\chi }$ 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 ${\mathbb{L}}_{-}$.