# Baxter Relations for Open Integrable Quantum Spin Chains

Lead Research Organisation:
Heriot-Watt University

### Abstract

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.

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.

### Planned Impact

We summarise the impacts in the four key areas identified by the EPSRC:

KNOWLEDGE:

* We will organise a school for young researchers in the final year of the project. There will be lectures from two high-profile invited speakers, as well as the PI and one CI.

* We will maintain a project website that we will also develop into a wider community portal for the field of quantum integrable systems.

* We will present the work at major international conferences such as the 2018 International Congress in Mathematical Physics.

PEOPLE:

* We will train the RA in this cutting edge area of mathematical physics.

* We will involve approximately 3 PhD students in aspects of the project.

* We will organize the concluding school in order to influence and train the next generation of research leaders.

* The PI will present a popular lecture at the Edinburgh International Science Festival on the general area of the project.

SOCIETY:

* The PI and the D. Johnston will tutor MSc projects on aspects of this proposal for students at the African Institute for Mathematical Sciences centres, where they both teach. These DFID funded centres train excellent students from across Africa. AIMS alumni are currently contributing enormously to developing the academic research base in the mathematical sciences across the African continent.

ECONOMY:

* Novel low-dimensional materials are being created in laboratories across the world. Quantum integrable systems have played an important part in the theoretical understanding of these materials over the last decade. Technology of the future will undoubtedly emerge from these experiments, and it is essential to the UK economy that it maintains a solid theoretical research base in this area in order to understand and successfully exploit these materials.

KNOWLEDGE:

* We will organise a school for young researchers in the final year of the project. There will be lectures from two high-profile invited speakers, as well as the PI and one CI.

* We will maintain a project website that we will also develop into a wider community portal for the field of quantum integrable systems.

* We will present the work at major international conferences such as the 2018 International Congress in Mathematical Physics.

PEOPLE:

* We will train the RA in this cutting edge area of mathematical physics.

* We will involve approximately 3 PhD students in aspects of the project.

* We will organize the concluding school in order to influence and train the next generation of research leaders.

* The PI will present a popular lecture at the Edinburgh International Science Festival on the general area of the project.

SOCIETY:

* The PI and the D. Johnston will tutor MSc projects on aspects of this proposal for students at the African Institute for Mathematical Sciences centres, where they both teach. These DFID funded centres train excellent students from across Africa. AIMS alumni are currently contributing enormously to developing the academic research base in the mathematical sciences across the African continent.

ECONOMY:

* Novel low-dimensional materials are being created in laboratories across the world. Quantum integrable systems have played an important part in the theoretical understanding of these materials over the last decade. Technology of the future will undoubtedly emerge from these experiments, and it is essential to the UK economy that it maintains a solid theoretical research base in this area in order to understand and successfully exploit these materials.

### Organisations

### Publications

Appel A
(2022)

*Universal K-matrices for quantum Kac-Moody algebras*in Representation Theory
Doikou A
(2021)

*Set-theoretic Yang-Baxter equation, braces and Drinfeld twists*in Journal of Physics A: Mathematical and Theoretical
Doikou A
(2021)

*Set-theoretic Yang-Baxter & reflection equations and quantum group symmetries*in Letters in Mathematical Physics
Doikou A
(2022)

*Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation*in Letters in Mathematical Physics
Doikou A
(2020)

*Discretizations of the generalized AKNS scheme*in Journal of Physics A: Mathematical and Theoretical
Doikou A
(2021)

*An algebraic approach to discrete time integrability*in Journal of Physics A: Mathematical and Theoretical
Doikou A
(2022)

*From braces to Hecke algebras and quantum groups*in Journal of Algebra and Its Applications
Johnston D
(2019)

*Lattice SUSY for the DiSSEP at ? 2 = 1 (and ? 2 = -3)*in Journal of Physics Communications
Regelskis V
(2018)

*Solutions of the $ \newcommand{\mfsl}{\mf{sl}} \newcommand{\mf}{\mathfrak} \newcommand{\wh}{\widehat} {U_q(\wh\mfsl_N)}$ reflection equations*in Journal of Physics A: Mathematical and TheoreticalDescription | In this project, a new algebraic picture of Baxter's Q-operator was constructed for open quantum systems. The existance and properties of this operator play a key role in 'solving' the system: namely finding the energy eigenstates and correlation functions of the system. The new Q-operator was constructed in terms of the underlying representation theory understanding of open quantum integrable systems. |

Exploitation Route | The new Q-operator allows for a systematic, and algebraic understanding of the Bethe Equation for open quantum integrable systems. Potential applications include condensed matter physics and the integrable systems arising in the holographic correspondence between gravity and CFTs. |

Sectors | Other |

Description | Quantum integrability from set theoretic Yang-Baxter & reflection equations |

Amount | £428,321 (GBP) |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 03/2021 |

End | 03/2024 |