Solutions of the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models using a differential approach

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 their 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. Nevertheless, the remaining unknowns can be found by solving a few 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 consider the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models with a generalization based on the A n−1 symmetry. This differential approach allows us to solve the Yang–Baxter equation in a systematic way.

Integrability of two-species partially asymmetric exclusion processes

“…We remark that condition (6) comes from the form of the ansatz (5). However, a similar condition was derived from the algebraic BA for the 15-vertex model [31]. These two cases are linked, as with certain choices of parameters, the 15-vertex model can be mapped to a twospecies PASEP with overtaking.…”

“…An answer to this problem is through the Yang-Baxter equations (YBEs), which can be interpreted as consistency conditions for the BA; indeed, by checking whether the YBE hold on a general class of models, it is possible to select values of the parameters for which exact solutions exist. Though this approach was used for quantum [30] and vertex model [31], it has been applied less systematically to study classical interacting particles systems, see for example [27,[32][33][34].…”

Automorphic Symmetries, String integrable structures and Deformations

“…Zamolodchikov tetrahedral algebra, as well as constant YBE solutions were shown in [18,19]. More recently a classification of {4-8}-vertex via differential approach was given in [20,21]. Other crucial constraint that will appear more efficient for derivation of properties and algebra of models in the non-additive sector 5, 6, 7 -is the braiding algebra realised by intertwining the monodromy T (u) and the R-matrix, R 0 0(u − v)T 0 (u)T 0(v) = T 0(v)T 0 (u)R 0 0(u − v) (2.25) where the monodromy T 0,{1,...,L} (u) acts on L physical spaces and one extra space is auxiliary, which we can set through the sequence of the Lax matrices L as can noticed from Fig.…”

“…It defines its structure through the action in two-particle representation R e |φ ± φ ± = s e 1 |φ ± φ ± + q e 1 |ψ ± ψ ± R e |ψ ± ψ ± = s e 2 |ψ ± ψ ± + q e 2 |φ ± φ ± (D. 21)…”

“…vertex and 9 × 9 class below that obeys the ice-rule[20,102,21]. It is clear that there are objective limitations, since the more free parameters (functions in the non-additive case) the higher the complexity of problem becomes, and one needs to address with distinct techniques.…”