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

TL;DR

This paper investigates the eigenvalue degeneracy of the pair correlation matrix in kagome-like spin models, showing exact results in certain limits and suggesting implications for spin liquid correlations and frustration effects.

Contribution

It demonstrates that eigenvalue degeneracy in the correlation matrix is exact in the infinite spin dimensionality limit and explores its implications for finite spin models.

Findings

Eigenvalues degenerate at all temperatures in the infinite $m$ limit.

Eigenvalue degeneracy is partially exact for the 3D kagome-like lattice.

Results suggest quasi-degeneracy and relevance to spin liquid behavior.

Abstract

It is noted that the pair correlation matrix $\hat{\chi}$ 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 $1/3 \, N+1$ out of $N$ eigenvalues of $\hat{\chi}$ are degenerate at all temperatures $T$, and correspond to an eigenspace $\mathbb{L}_{-}$ of $\hat{\chi}$, 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 $m\rightarrow \infty$ limits can be interpreted that the $m\rightarrow\infty$ 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 $m\geqslant 2$ contribute to $\hat{\chi}$ at low $T$. The $m\rightarrow\infty$ formulae can be thus quantitatively correct in description of $\hat{\chi}$ 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 $\mathbb{L}_{-}$.