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.
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.
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) symmetry and we recovered the matrix part of the S-matrix of AdS5×S5 derived by requiring centrally extended 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.
All regular 4×4 solutions of the Yang-Baxter equation
Multiple equations in math, physics, quantum information, and elsewhere are referred to as "the" Yang-Baxter equation, in spite of being a broad family of equations. Most of the equations are nonlinear matrix equations, where the unknown variable is a matrix. This is the case for the so called braided, algebraic, and generalized forms of "the" equation, which are the primary focus of this dissertation. Finding solutions to the various forms of these equations has been the subject of much research. The equations in all their forms are largely considered intractable in high dimensions, and only in dimension 2 have the solutions been fully classified. We begin with an introduction to quantum computation, with a focus on the topological model and its connection to the braid group. Next, we introduce the braided, algebraic, and generalized forms of "the" Yang-Baxter equation. We provide a full classification of diagonal solutions to each form. In particular, we show that any diagonal matrix is a solution to the algebraic form in any dimension, and each instance of the braided and generalized forms only have diagonal solutions that are scalar multiples of the identity. We exploit the relationship between the algebraic and braided forms to construct a solution in any dimension that is applicable to topological quantum computation as a universal gate. The generalized form of the equation is parameterized by three natural numbers, (d,m,l), and we show that the only invertible solutions when l ≥ m are scalar multiples of the identity. We completely classify all solutions arising from an X-shaped ansatz for five different choices of (d,m,l), and provide a complete classification of X-shaped solutions to every odd dimensional braided equation, where there are no X-shaped solutions in any dimension. We fully classify permutation solutions to each instance of the braided and algebraic equations that can be written as a product of 3 or fewer transpositions. We show that the problem of classifying all invertible upper triangular solutions to the 4-dimensional algebraic Yang-Baxter equation can be split into 48 cases, and fully classify one of the cases.
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) 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) 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.
Abstract The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.
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.