一个新的具有近邻相互作用的准精确可解自旋链
A novel quasi-exactly solvable spin chain with nearest-neighbors interactions
摘要：
本文研究了一种新的自旋链，它的最近邻相互作用依赖于格点坐标，在某种意义上它介于Heisenberg链和Haldane-Shastry型自旋链之间.我们表明，当自旋的数量是足够大的网站的密度和连续的自旋之间的相互作用的强度遵循高斯定律。我们开发了一个扩展的标准冻结技巧参数，使我们能够准确地计算一定数量的本征值和相应的本征函数。这样计算的特征值都是整数，事实上，我们的数值研究证明，这些是唯一的整数特征值的链正在考虑。这一事实表明，这条链可以被视为一类准精确可解的薛定谔算子的有限维模拟，这在过去的二十年中得到了广泛的研究。我们应用矩量法研究了链的谱的一些统计性质，特别表明本征值的密度遵循Wigner定律。最后，我们强调，不同于原来的冻结技巧，在本文中开发的扩展可以应用到自旋链，其相关的动力学自旋模型是只准精确可解的。
英文摘要：
In this paper we study a novel spin chain with nearest-neighbors interactions depending on the sites coordinates, which in some sense is intermediate between the Heisenberg chain and the spin chains of Haldane-Shastry type. We show that when the number of spins is sufficiently large both the density of sites and the strength of the interaction between consecutive spins follow the Gaussian law. We develop an extension of the standard freezing trick argument that enables us to exactly compute a certain number of eigenvalues and their corresponding eigenfunctions. The eigenvalues thus computed are all integers, and in fact our numerical studies evidence that these are the only integer eigenvalues of the chain under consideration. This fact suggests that this chain can be regarded as a finite-dimensional analog of the class of quasi-exactly solvable Schroedinger operators, which has been extensively studied in the last two decades. We have applied the method of moments to study some statistical properties of the chain's spectrum, showing in particular that the density of eigenvalues follows a Wigner-like law. Finally, we emphasize that, unlike the original freezing trick, the extension thereof developed in this paper can be applied to spin chains whose associated dynamical spin model is only quasi-exactly solvable.
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