Baxter Relations for Open Integrable Quantum Spin Chains

这个建议考虑一类一维量子系统称为可积量子自旋链。“可积”一词意味着这些系统具有增强的对称性——其结果是它们的某些性质可以精确计算。特别是，原则上可以精确地计算出它们的能量特征值。这些特征值是用Bethe ansatz方程组的解来表示的，而Bethe ansatz方程组的解来自于一个更基本的方程组，叫做Baxter关系。巴克斯特关系是多标称Q(z)的差分方程。量子自旋链的现代构造和理解依赖于量子群的表示理论，也被称为准三角霍普夫代数。虽然这幅图在封闭的、周期性的量子自旋链上得到了很好的发展，但直到最近，巴克斯特关系才在这种语言中得到充分的理解。推导和证明巴克斯特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字符”。我们建议的主要目标是发展对“开放”量子自旋链中的Baxter关系的平行理解-即具有两个独立可积边界条件的自旋链。我们将首先为一个被称为XXZ模型（具有任意可积边界条件）的简单开放量子自旋链生成Q算子的显式构造（其特征值给出多项式Q(z)）。然后，我们将定义q字符的开放类似物，并将这些对象用于非常一般的一类开放系统的Baxter关系形式的猜想的公式中。这个猜想将被证明。第二个目标是将我们的Baxter关系应用于开放链。我们将利用这些关系推导出一类开链的Bethe ansatz方程。这些Bethe ansatz方程将被用来识别这些模型的子类，这些模型具有晶格超对称（SUSY），与不同大小的系统相关（之前在一些非常简单的开自旋链中观察到）。在某些非平衡统计力学模型（asep和asap）中也观察到非常相似的格大小递推关系。我们将用系统的，代数的理解晶格超对称性，以澄清超对称性与ASEP和ASAP递归关系的关系。

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.