From this chapter onwards, we turn to applications of the 3D structures. The tetrahedron equation $$RLLL=LLLR$$ can be composed n times in various directions. By tracing out a part of the spaces appropriately, it reduces to the Yang–Baxter equation among the remaining objects. The prescription, which we call the trace reduction, generates infinitely many solutions to the Yang–Baxter equation labeled with n in a matrix product form. In this chapter we demonstrate the method using the 3D R and 3D L in Sect. 3.5.2 . The resulting solutions to the Yang–Baxter equation are trigonometric. They are identified with the quantum R matrices for the symmetric and the anti-symmetric tensor representations of $$U_{\pm q^{-1}}(A^{(1)}_{n-1})$$ . The cyclicity of the trace is reflected in the Dynkin diagram of the affine Lie algebra $$A^{(1)}_{n-1}$$ . The matrix product formula of the quantum R matrices naturally leads to an interpretation of the commuting transfer matrix of $$m \times n$$ layer either as the one for a $$U_{q^{-1}}(A^{(1)}_{n-1})$$ vertex model with size m or a $$U_{q^{-1}}(A^{(1)}_{m-1})$$ vertex model with size n.
From this chapter onwards, we turn to applications of the 3D structures. The tetrahedron equation RLLL=LLLR can be composed n times in various directions. By tracing out a part of the spaces appropriately, it reduces to the Yang–Baxter equation among the remaining objects. The prescription, which we call the trace reduction, generates infinitely many solutions to the Yang–Baxter equation labeled with n in a matrix product form. In this chapter we demonstrate the method using the 3D R and 3D L in Sect. 3.5.2 . The resulting solutions to the Yang–Baxter equation are trigonometric. They are identified with the quantum R matrices for the symmetric and the anti-symmetric tensor representations of U±q−1(An−1(1)) . The cyclicity of the trace is reflected in the Dynkin diagram of the affine Lie algebra An−1(1) . The matrix product formula of the quantum R matrices naturally leads to an interpretation of the commuting transfer matrix of m×n layer either as the one for a Uq−1(An−1(1)) vertex model with size m or a Uq−1(Am−1(1)) vertex model with size n.
This work concerns studies of boundary integrability of the vertex models from representations of the Temperley–Lieb algebra associated with the quantum group 𝒰 q [ X n ] for the affine Lie algebras X n = , , and . A systematic computation method is used to construct solutions of the boundary Yang–Baxter equations. We find a 2 n 2 + 1 free parameter solution for spin-( n − 1/2) and vertex models. It turns out that for spin- n , and vertex models, the solution has 2 n 2 + 2 n + 1 free parameters.
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.
A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Gröbner-basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well-known C, P, and T reflections, and rotations and scalings R̃=κ(Q⊗Q)R(Q⊗Q)−1.
We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.
A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid G together with a map π:G→End(V⊗V) for some vector space V such that the Yang-Baxter commutator [[π(u),π(w),π(v)]]=0 if u,v∈G are such that the groupoid composition w=u⋆v is defined. An important role is played by an object map Δ:G→M for some set M such that Δ(u)=Δ(v′), Δ(w)=Δ(v) and Δ(w′)=Δ(u′), where v↦v′ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group GL(2)×GL(1). For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.