Yang-Baxter integrable open quantum systems

Abstract

The main result of this thesis is the use of the boost operator to develop a systematic method to construct new integrable spin chains with nearest-neighbour interaction and characterized by an R-matrix of non-difference form. This method has the advantage of being more feasible than directly solving the Yang-Baxter equation. We applied this approach to various contexts, in particular, in realm of open quantum systems, we achieved the first classification of integrable Lindbladians. These operators describe the dynamics of physical systems in contact with a Markovian environment. Within this classification, we discovered well-known models such as the Hubbard model, as well as a novel deformation of the Hubbard model spanning three sites of the spin chain. We extensively analyzed this range 3 model and uncovered the existence of multiple Non Equilibrium Steady States (NESS). The presence of multiple NESS is connected to the existence of certain "hidden strong symmetries" in the form of quasi-local charges. We computed the NESS analytically in the form of a Matrix Product Operator. Additionally, we applied our method to classify models with su(2)+su(2) symmetry. In this class of models, we recovered the matrix part of the S-matrix of AdS_5xS^5 derived by requiring centrally extended su(2|2) symmetry. We also obtained five models that, to the best of our knowledge, are new. Furthermore, we focus on spin 1/2 chain. We demonstrated that all Hermitian integrable Hamiltonians can be reconducted to models of 8-Vertex type. We classified models in this class and we found two difference form models that correspond to the known XXZ and XYZ spin chain. We also found two models of non-difference form type: the one of 6-vertex type is a known model, while the one of 8-vertex type is a newly discovered model. Furthermore, we showed that the two non difference form models satisfy the free fermion condition. This enables us to express the transfer matrix associated to some of the models in a diagonal form, simplifying the computation of the eigenvalues and the eigenvectors.