Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

Jon Links

doi:10.3390/axioms1020226

Abstract

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn).

Highlights

Integrable quantum systems which admit exact solutions are central in advancing understanding of many-body systems
With the development of the Quantum Inverse Scattering Method [6] as a systematic prescription for constructing integrable quantum systems through the Yang–Baxter equation [3,7,8], and solving them through the algebraic Bethe ansatz, it subsequently emerged that Hopf algebraic structures are fundamental in quantum integrability
In the remainder of this work it will be shown that for certain further restrictions on the coupling parameters there are additional Hopf algebraic symmetries of the system. These non-Abelian symmetries are not related to a quantum algebra Uq (sl(2)) structure, but are realised through the Drinfel’d doubles of dihedral group algebras

Summary

Introduction

Integrable quantum systems which admit exact solutions are central in advancing understanding of many-body systems. Leads to the six-vertex model solution of the Yang–Baxter equation, which establishes integrability of the anisotropic (XXZ) Heisenberg chain. It was found that two-dimensional representations of these algebras belong to the aforementioned six-vertex model solution in the symmetric case. The symmetric solution was employed in [17] to construct an integrable anyonic pairing Hamiltonian, which generalises the pairing Hamiltonian with uniform scattering interactions solved by Richardson [4]. This integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries given by D(Dn ).

The Integrable Hamiltonian for Anyonic Pairing

Drinfel’d Doubles of Dihedral Group Algebras

Symmetries of the Transfer Matrix and Hamiltonian

Conclusions