Dependence of ground-state energy of classical $n$-vector spins on $n$

We study the ground state energy $E_G\left(n\right)$ of $N$ classical $O\left(n\right)$ vector spins with the Hamiltonian $\mathcal{H}=-{\sum }_{i>j}{J}_{ij}{\vec{S}}_i\bullet {\vec{S}}_j$ where the coupling constants $\left\{J_{ij}\right\}$ are arbitrary. We prove that $E_G\left(n\right)$ is independent of $n$ for all $n>n_{\max }\left(N\right)=\lfloor (\sqrt{8N+1}-1)∕2\rfloor$. We show that this bound is the best possible. We also derive an upper bound for $E_G\left(m\right)$ in terms of $E_G\left(n\right)$, for $m<n$. We obtain an upper bound on the frustration in the system, as measured by $F\left(n\right)\equiv [{\sum }_{i>j}\mid J_{ij}\mid +E_G(n\left)\right]∕{\sum }_{i>j}\mid J_{ij}\mid$. We describe a procedure for constructing a set of $J_{ij}$’s such that an arbitrary given state, $\left\{{\vec{S}}_i\right\}$, is the ground state. We show that the problem of finding the ground state for the special case $n=N$ is equivalent to finding the ground state of a corresponding soft-spin problem.

Figure 1

Sequence of examples for which $E_G\mathbf{\left(}{n}_{\max }\right(N\left)\mathbf{\right)}<E_G\mathbf{\left(}{n}_{\max }\right(N)-1\mathbf{)}$. Shown are three of those members of the sequence for which $(\sqrt{8N+1}-1)∕2$ is an integer [such $N$’s are $3,6,10,\dots =\frac{1}{2}k(k+1)$]. The rest of the members are obtained by adding appropriate number of free spins to the example of the last $N$ for which $(\sqrt{8N+1}-1)∕2$ is an integer, e.g., $N=8$ example has two more free spins added to the $N=6$ example.

Figure 2

Functions $f_{34}\left(\theta \right)$ and $q_{34}\left(\theta \right)=[1-f_{34}(\theta \left)\right]∕(1-\cos \theta )$.