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Baxter Relations for Open Integrable Quantum Spin Chains

开放可积量子自旋链的巴克斯特关系

基本信息

批准号：
EP/R009465/1
负责人：
Robert Weston
金额：
$ 43.63万
依托单位：
Heriot-Watt University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR009465%2F1
关键词：
Baxter Relations Open Integrable Quantum

项目摘要

This proposal considers a class of one-dimensional quantum systems known as integrable quantum spin chains. The word integrable means that these systems possess enhanced symmetries - with the consequence that some of their properties can be computed exactly. In particular, it is in principle possible to compute their energy eigenvalues exactly. These eigenvalues are given in terms of the solution of a system of equations called 'Bethe ansatz equations', which in term come from a more fundamental system of equations called 'Baxter relations'. Baxter relations are difference equations for a polynominal Q(z).The modern construction and understanding of quantum spin chains relies on the representation theory of quantum groups, also know as quasi-triangular Hopf algebras. While this picture is well-developed for closed, periodic quantum spin chains, it is only very recently that Baxter relations have been fully understood in this language. A key tool in the derivation and proof of Baxter relations was the definition and use of 'q-characters' of representations of general quantum affine Lie algebras.The main goal of our proposal is to develop a parallel understanding of Baxter relations in 'open' quantum spin chains - that is, those with two independent integrable boundary conditions. We will start by producing an explicit construction of the Q-operator (whose eigenvalues give the polynomial Q(z)) for a simple open quantum spin chain known as the XXZ model (with arbitrary integrable boundary conditions). We will then define open analogues of q-characters, and use these objects in the formulation of a conjecture for the form of Baxter relations for a very general class of open systems. This conjecture will be proved.A secondary goal concerns an application of our Baxter relations for open chains. We will use these relations to derive Bethe ansatz equations for a wide class of open chains. These Bethe ansatz equations will in turn be used to identify sub-classes of these models that possess a lattice supersymmetry (SUSY) relating systems of different size (observed previously for some very simple open spin chains). Very similar lattice-size recursion relations have also been observed in certain non-equilibrium statistical-mechanical models known as ASEPs and ASAPs. We will use our systematic, algebraic understanding of lattice SUSY in order to clarify the relation of SUSY to ASEP and ASAP recursion relations.

这个提议考虑了一类被称为可积量子自旋链的一维量子系统。可积这个词意味着这些系统具有增强的对称性--其结果是它们的一些性质可以精确地计算出来。特别是，原则上可以精确地计算它们的能量本征值。这些本征值是以一个被称为‘Bethe ansatz方程’的方程组的解的形式给出的，该方程组的术语来自一个更基本的被称为‘Baxter关系’的方程组。Baxter关系是多项式q(Z)的差分方程组。量子自旋链的现代构造和理解依赖于量子群的表示理论，也称为准三角Hopf代数。虽然这幅图对于闭合的周期性量子自旋链来说已经很好地发展了，但直到最近才用这种语言完全理解了巴克斯特关系。推导和证明Baxter关系的一个关键工具是定义和使用一般量子仿射李代数表示的“q-特征标”。我们的主要目标是发展对“开的”量子自旋链中的Baxter关系的并行理解--即具有两个独立的可积边界条件的链。我们将首先给出一个简单的开放量子自旋链的Q算符的显式构造(其本征值给出多项式Q(Z))，称为XXZ模型(具有任意可积边界条件)。然后，我们将定义Q-特征标的开放类似物，并使用这些对象来对一类非常一般的开放系统的Baxter关系的形式提出一个猜想。这一猜想将得到证实。第二个目标涉及到我们的巴克斯特关系在开链中的应用。我们将利用这些关系推导出一大类开链的Bethe ansatz方程。这些Bethe ansatz方程将被用来识别这些模型的子类，这些模型具有不同大小的晶格超对称性(SUSY)相关系统(之前观察到的是一些非常简单的开式自旋链)。在某些称为ASEP和ASAPS的非平衡统计力学模型中，也观察到了非常相似的晶格尺寸递归关系。我们将使用我们对格SUSY的系统的、代数的理解来阐明SUSY与ASEP和ASAP递归关系的关系。