具有极化自旋反转算符的D_N型有理量子可积系统
Rational quantum integrable systems of D_N type with polarized spin reversal operators
摘要：
本文研究了具有极化自旋反转算符的D_N型自旋Calogero模型及其相应的Haldane-Shastry型自旋链，包括反铁磁和铁磁两种情况.我们以封闭形式计算了前一模型的谱和配分函数，由此导出了A型Polychronakos-Frahm自旋链的配分函数的乘积的精确公式。利用本文导出的后一种配分函数的递推关系，我们能够数值计算出自旋数N较大时D_N型自旋链的配分函数，从而计算出自旋链的谱。我们分析了链的谱的几个全局性质，如渐近水平密度，展开谱的连续间距的分布，和平均简并度。特别是，我们的结果表明，该链是不变的下一个合适的Yangian群，它的频谱符合的Yangian不变的顶点模型的线性能量函数和色散关系。
英文摘要：
We study the spin Calogero model of D_N type with polarized spin reversal operators, as well as its associated spin chain of Haldane-Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos-Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the D_N-type spin chain for relatively high values of the number of spins N. We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.
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