能级的铁磁有序
Ferromagnetic Ordering of Energy Levels
摘要：
我们研究了海森堡模型中关于能级铁磁有序性的一个自然猜想，它补充了1962年关于反铁磁体的Lieb-Mattis定理：对于铁磁海森堡模型，固定总自旋的每个子空间中的最低能量严格地根据总自旋排序，最低的，即，基态，属于最大总自旋子空间。我们的主要结果证明了一维自旋为1/2的海森堡铁磁体XXX和XXZ的猜想。我们的证据有两个主要成分。第一个结果是Koma和Nachtergaele的结果的延伸，该结果表明，作为总自旋的函数的单调性如下从每个总自旋子空间中的基态能量的单调性作为链的长度的函数。对于证明的第二部分，我们使用Temperley-Lieb代数计算，在适当的基础上，限制到具有给定总自旋的最高权向量的每个子空间的哈密顿量的矩阵元素。然后，我们表明，这些矩阵元素的积极性意味着必要的单调性的体积。我们的方法还表明，XXX铁磁体的第一个激发态的任何有限树有一个小于最大总自旋。
英文摘要：
We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.
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