Chandra Samarth
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Feb;77(2 Pt 1):021125. doi: 10.1103/PhysRevE.77.021125. Epub 2008 Feb 26.
We study the ground state energy E(G)(n) of N classical O(n) vector spins with the Hamiltonian H=-Sigma(i>j)J(ij)S(i).S(j) where the coupling constants {J(ij)} are arbitrary. We prove that E(G)(n) is independent of n for all n>n(max)(N)= left floor(sq rt[8N+1]-1)/2 right floor. 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) triple bond [Sigma(i>j) J(ij) + E(G)(n)]/Sigma(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. 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.
我们研究具有哈密顿量(H = -\sum_{i>j}J_{ij}\vec{S}i\cdot\vec{S}j)的(N)个经典(O(n))向量自旋的基态能量(E_G(n))，其中耦合常数({J{ij}})是任意的。我们证明，对于所有(n > n{max}(N)=\lfloor\frac{\sqrt{8N + 1}-1}{2}\rfloor)，(E_G(n))与(n)无关。我们表明这个界限是最优的。我们还根据(E_G(n))推导出了(m < n)时(E_G(m))的一个上界。我们得到了用(F(n)\equiv[\sum_{i>j}J_{ij}+E_G(n)]/\sum_{i>j}J_{ij})衡量的系统中受挫程度的一个上界。我们描述了一种构造一组(J_{ij})的方法，使得任意给定的态({\vec{S}_i})是基态。我们表明，对于(n = N)的特殊情况，找到基态的问题等同于找到一个相应的软自旋问题的基态。
