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arXiv:2312.00064

Yang-Baxter integrable open quantum systems

Chiara Paletta

Published Nov 29, 2023• 0 citations
arXivHTMLPDF

Abstract

This work is based on the author's PhD thesis. The main result of the 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 the 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 a novel deformation of the Hubbard model spanning three sites of the spin chain. Additionally, we applied our method to classify models with $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ symmetry and we recovered the matrix part of the S-matrix of $AdS_5 \times S^5$ derived by requiring centrally extended $\mathfrak{su}(2|2)$ symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type and we showed that the models of this class 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 eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231, 2011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.

DOI
10.48550/arxiv.2312.00064

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This work is based on the author's PhD thesis. The main result of the 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 the 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 a novel deformation of the Hubbard model spanning three sites of the spin chain. Additionally, we applied our method to classify models with su(2)⊕su(2)\mathfrak{su}(2)\oplus \mathfrak{su}(2)su(2)⊕su(2) symmetry and we recovered the matrix part of the S-matrix of AdS5×S5AdS_5 \times S^5AdS5​×S5 derived by requiring centrally extended su(2∣2)\mathfrak{su}(2|2)su(2∣2) symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type and we showed that the models of this class 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 eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231, 2011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times16 16 × 16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2) 𝔰 𝔲 ( 2 ) ⊕ 𝔰 𝔲 ( 2 ) symmetry, which include the one-dimensional Hubbard model and the S S -matrix of the {AdS}_5 \times {S}^5 A d S 5 × S 5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.
Abstract We classify all fundamental integrable spin chains with two-dimensional (2D) local Hilbert space which have regular R -matrices of difference form. This means that the R -matrix underlying the integrable structures is of the form and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, eight of which correspond to well-known eight or lower vertex models. The remaining six models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R -matrices, we find a bijection between solutions of the Yang–Baxter equation (YBE) and the graded YBE which extends our results to the graded 2D case.
Abstract In this paper we find new integrable one-dimensional lattice models of electrons. We describe all such nearest-neighbour integrable models with s u ( 2 ) × s u ( 2 ) symmetry by classifying solutions of the Yang–Baxter equation following the procedure first introduced in [1]. We find 12 R-matrices of difference form, some of which can be related to known models such as the XXX spin chain and the free Hubbard model, and some are new models. In addition, integrable generalizations of the Hubbard model are found by keeping the kinetic term of the Hamiltonian and adding all terms which preserve fermion number. We find that most of the new models cannot be diagonalized using the standard nested Bethe Ansatz.
We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the AdS_{2} and AdS_{3} integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.
Subject of this work are integrable spin chains with general boundary conditions.In the framework of the Quantum Inverse Scattering Method Sklyanin has shown how to construct a family of commuting operators (transfer matrix) containing the hamiltonian of the XXX or XXZ spin chain with general boundary fields.Key ingredient is the underlying algebraic structure which is a combination of the Yang-Baxter algebra, using the known R-matrix representations, and a so-called Reflection algebra.The latter includes fields of arbitrary strength and direction acting on the first and last position of the chain.This setup is solvable via algebraic Bethe ansatz in the case of diagonal boundaries, i.e. the fields are parallel to each other and in the case of the XXZ model parallel to the distinguished direction.Kitanine et al. have managed to express local operators in terms of the non-local elements of the underlying algebraic structure in the case of half-infinite chain length hence establishing a possible approach to evaluate expectation values of physical observables.Their results will be picked up in this work and generalized to spin chains of arbitrary (including finite) lengths using non-linear integral equations for the lowest lying state with zero magnetization.In the case of non-diagonal boundary fields the lack of a reference state or pseudo vacuum prohibits the solution by algebraic Bethe ansatz.The method of separation of variables proposed by Sklyanin is not constrained in that sense and will be applied to this situation.In this approach for the XXX spin chain no restrictions to the boundary parameters are needed.The result is a TQ-equation on a finite discrete set of points and the eigenvalues of the transfer matrix are obtainable from this finite difference equation.As the underlying algebraic structure is independent of the representation, the analysis for the XXX spin chain can be extended to a spin-boson model.Using a known representation of the algebra for the bosonic degree of freedom again a TQ-equation is obtained but in this case being defined on an infinite discrete set of points.Finally a different approach to the case of non-diagonal boundary conditions using non-linear integral equations is studied.By construction of higher dimensional representations of the Yang Baxter algebra a hierarchy of transfer matrices is obtained.In this construction two R-matrices are fused together yielding the name of fusion hierarchy.From such hierarchies it is possible to derive non-linear integral equations for a certain eigenvalue of the transfer matrix using Fourier transform techniques.The equations are valid for arbitrary system sizes and a range of boundary parameters.By careful analysis of the zero and pole structure it is possible to extract information about the eigenvalue of a specific state.