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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来自数学物理，特别是量子可积性领域的想法，以及纯代数，特别是辫子群，大括号和环理论的领域。该建议考虑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在于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性质，如能量本征值，原则上可以精确计算，并可以用称为Bethe anomaly方程的方程组的解表示。用于构建可积量子自旋链及其光谱分辨率的主要方法是量子逆散射方法（QISM），这是一种优雅的代数技巧，自然地产生了贝特方程，从而产生了自旋链的能谱。QISM还直接导致了准三角霍普夫代数的发明，称为量子群或量子代数。杨-巴克斯特方程是量子可积性理论中的一个关键对象，因为方程的不同解产生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性，保证了关联系统的量子可积性。在这个项目中，我们专注于一类特殊的解决方案的YBE被称为集理论的解决方案，这也提供了某些表示的阿丁的辫子群。一个特殊的代数结构，推广幂零环，称为一个支撑，以描述所有有限的，对合的，集理论的解决方案的YBE。众所周知，每个支撑提供了一个集理论解决方案的YBE，和每个非退化，对合集理论解决方案的YBE可以得到从brackes. Central研究计划的主要目的是调查的代数和物理方面相关的量子可积系统构建从集理论解决方案的YBE。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一个关键目标。我们还旨在研究某些二次代数，如反射代数，并获得一个分类的可能的可积边界条件。这些发现将导致识别新的物理自旋链系统的周期性和开放的边界条件。另一个关键问题是检验我们是否可以将YBE的支撑型解表示为Drinfeld twists。一个霍普夫代数的“扭曲”是一个代数作用，它产生另一个霍普夫代数。对于某些特殊的集合论解类，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推广这些发现，包括更大的类集理论的解决方案，也调查的作用，这种扭曲的新兴量子群对称性。从物理学的角度来看，最终目标是识别的本征值和本征态的开放和周期性的可积量子自旋链构造的集合论解决方案。我们将系统地追求这个问题，通过实施广义的Bethe anonymous技术，将导致新的Bethe anonymous方程组和相关的量子自旋链的频谱。在我们的处置的频谱和相关的Bethe anomaly方程，我们将能够计算物理相关的量，如能量，散射振幅和运营商的期望值。