Quantum integrability from set theoretic Yang-Baxter & reflection equations

集合论 Yang-Baxter 的量子可积性

基本信息

批准号：
EP/V008129/1
负责人：
Anastasia Doikou
金额：
$ 54.58万
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：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV008129%2F1
关键词：
Quantum integrability set theoretic Yang

项目摘要

The proposed research program aims at bringing together ideas from mathematical physics and in particular the domain of quantum integrability, and pure algebra specifically the areas of braid groups, braces and ring theory. The proposal regards a special class of one dimensional interacting N-body quantum systems known as integrable quantum spin chains. Integrable quantum systems are characterized by the existence of a set of mutually commuting algebraic objects, usually as many as the associated degrees of freedom. This set of commuting objects ensures the exact solvability of the quantum system. This means that some of the fundamental physical properties of the system, such as the energy eigenvalues can be in principle computed exactly and can be expressed in terms of solutions of a system of equations known as Bethe ansatz equations.The main methodology used for the construction of integrable quantum spin chains and the resolution of their spectra is the Quantum Inverse Scattering Method (QISM), an elegant algebraic technique that naturally yields the Bethe ansatz equations and consequently the energy spectrum of the spin chains. The QISM has also led directly to the invention of quasitriangular Hopf algebras known as quantum groups or quantum algebras. The Yang-Baxter equation is a key object in the theory of quantum integrability, given that distinct solutions of the equation generate different types of quantum spin chains and distinct sets of algebraic constraints, i.e. quantum algebras. The algebraic constraints guarantee the existence of mutually commuting algebraic objects, ensuring the quantum integrabiltiy of the associated system. In this project we focus on a particular class of solutions of the YBE known as set theoretic solutions, which also provide representations of certain quotients of Artin's braid group. A special algebraic structure that generalizes nilpotent rings, called a brace was developed in order to describe all finite, involutive, set-theoretic solutions of the YBE. It is well established that every brace provides a set theoretic solution of the YBE, and every non-degenerate, involutive set theoretic solution of the YBE can be obtained from a brace.The central aim of the proposed research program is to investigate both algebraic and physical aspects associated to quantum integrable systems constructed from set theoretic solutions of the YBE. From the algebraic point of view the study of the representation theory of the quantum groups emerging from braces is one of the key objectives. We also aim at investigating certain quadratic algebras, such as the refection algebra, and obtain a classification of possible integrable boundary conditions. These findings will lead to the identification of new classes of physical spin chain systems with periodic and open boundary conditions. Another key issue is to examine whether we can express brace type solutions of the YBE as Drinfeld twists. The 'twisting' of a Hopf algebra is an algebraic action that produces yet another Hopf algebra. Explicit expressions of such twists have been derived for some special classes of set theoretic solutions. One of our fundamental objectives is to generalize these findings to include larger classes of set theoretic solutions and also investigate the role of such twists on the emerging quantum group symmetries. From a physical viewpoint the ultimate goal is the identification of the eigenvalues and eigenstates of open and periodic integrable quantum spin chains constructed from set theoretic solutions. We will systematically pursue this problem by implementing generalized Bethe ansatz techniques that will lead to sets of novel Bethe ansatz equations and the spectrum of the associated quantum spin chains. Having at our disposal the spectrum and the associated Bethe ansatz equations we will be able to compute physically relevant quantities, such as energy, scattering amplitudes and operator expectation values.

拟议的研究计划旨在汇集数学物理学（特别是量子可积性领域）和纯代数（特别是辫群、花括号和环理论领域）的思想。该提案涉及一类特殊的一维相互作用 N 体量子系统，称为可积量子自旋链。可积量子系统的特征是存在一组相互交换的代数对象，通常与相关的自由度一样多。这组交换对象确保了量子系统的精确可解性。这意味着系统的一些基本物理特性（例如能量特征值）原则上可以精确计算，并且可以用称为 Bethe ansatz 方程的方程组的解来表示。用于构造的主要方法可积量子自旋链及其光谱的分辨率是量子逆散射法（QISM），这是一种优雅的代数技术，可以自然地产生 Bethe ansatz 方程，从而产生自旋的能谱链。 QISM 还直接导致了称为量子群或量子代数的准三角 Hopf 代数的发明。杨-巴克斯特方程是量子可积性理论中的一个关键对象，因为该方程的不同解会生成不同类型的量子自旋链和不同的代数约束集（即量子代数）。代数约束保证了相互交换的代数对象的存在，保证了相关系统的量子可积性。在这个项目中，我们重点关注 YBE 的一类特定解，称为集合论解，它还提供了 Artin 辫子群的某些商的表示。为了描述 YBE 的所有有限、对合、集合论解，开发了一种特殊的代数结构，它推广了幂零环（称为括号）。众所周知，每个括号都提供了 YBE 的集合论解，并且 YBE 的每个非退化、对合集合论解都可以从括号中获得。所提出的研究计划的中心目标是研究代数和与由 YBE 的集合论解构建的量子可积系统相关的物理方面。从代数的角度来看，研究从括号中出现的量子群的表示论是关键目标之一。我们还旨在研究某些二次代数，例如反射代数，并获得可能的可积边界条件的分类。这些发现将导致识别具有周期性和开放边界条件的新型物理自旋链系统。另一个关键问题是检验我们是否可以将 YBE 的支撑类型解表达为 Drinfeld 扭转。霍普夫代数的“扭曲”是产生另一个霍普夫代数的代数作用。对于某些特殊类别的集合论解，已经导出了这种扭曲的显式表达式。我们的基本目标之一是将这些发现推广到更大类别的集合论解决方案，并研究这种扭曲对新兴量子群对称性的作用。从物理角度来看，最终目标是识别由集合论解构造的开放且周期性可积量子自旋链的特征值和特征态。我们将通过实施广义的 Bethe ansatz 技术来系统地解决这个问题，该技术将产生一组新颖的 Bethe ansatz 方程组和相关量子自旋链的光谱。有了光谱和相关的 Bethe ansatz 方程，我们将能够计算物理相关量，例如能量、散射幅度和操作员期望值。