On a class of orthogonal-invariant quantum spin systems on the complete graph
On a class of orthogonal-invariant quantum spin systems on the complete graph
复制标题
关于完全图上的一类正交不变量子自旋系统
DOI：
10.1093/imrn/rnac034
复制
发表时间：
2020
影响因子：
--
通讯作者：
Kieran Ryan
中科院分区：
文献类型：
--
作者：
Kieran Ryan
在复正交群下最一般的模型不变量完全图上研究了一类双参数量子自旋系统。在自旋S = 1 2时，它相当于XXZ模型，在自旋S = 1时，它相当于双线性双二次海森堡模型。这篇论文的动机是比约恩伯格的工作，他的模型在（更大的）复杂的一般线性群下是不变的。在自旋S = 1.2和S = 1时，我们给出了两个参数的所有值的自由能的显式公式，对于自旋S > 1，当其中一个参数为非负时，给出了自由能的显式公式。这使我们能够绘制相图，并确定临界温度。对于自旋S = 1 2和S = 1，我们给出了左和右导数的强度参数的某个磁化项趋于零，我们给出了一个公式，为一定的总自旋可观察的，和一组极端吉布斯状态在几个区域的相图，在最近的文件的风格比约恩贝格，Fröhlich和Ueltschi。其关键技术工具是用对称群的不可约特征标和Brauer代数表示配分函数。所考虑的参数包括，并超越，那些系统的概率表示为交换过程。
We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin S = 1 2 it is equivalent to the XXZ model, and in spin S = 1 to the bilinearbiquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin S = 1 2 and S = 1 we give an explicit formula for the free energy for all values of the two parameters, and for spin S > 1 for when one of the parameters is non-negative. This allows us to draw phase diagrams, and determine critical temperatures. For spin S = 1 2 and S = 1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.