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

TL;DR

This paper investigates how the ground state energy of classical n-vector spins depends on n, proving it becomes independent of n beyond a certain threshold, and provides bounds and construction methods related to the system's frustration.

Contribution

It establishes the independence of ground state energy from n for large n, derives bounds relating energies for different n, and offers a method to construct coupling constants for desired ground states.

Findings

E_G(n) is independent of n for n > n_{max}(N).

Derived bounds for E_G(m) in terms of E_G(n).

Provided a procedure to construct J_ij for specific ground states.

Abstract

We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - \sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n > n_{max}(N) = floor((sqrt(8N+1)-1) / 2) . We show that this bound is the best possible. We also derive an upper bound for E_G(m) in terms of E_G(n), for m<n. We obtain an upper bound on the frustration in the system, as measured by F(n), which is defined to be (\sum_{i>j} |J_ij| + E_G(n)) / (\sum_{i>j} |J_ij|). We describe a procedure for constructing a set of J_ij's such that an arbitrary given state, {S_i}, is the ground state.