Yang-Baxter equation (YBE)

introduction

roles in the following fields
- Integrable systems and solvable models
exact solvability of many models is explained by commuting transfer matrices
- quantum groups
- Conformal field theory (CFT)
- Topological quantum field theory(TQFT)
- braid groups
manifestations of Yang-Baxter equation
- Exact S-matrices in ATFT
\(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
for vertex models, YBE becomes the star-triangle relation
see [Baxter1995] for a historical account

[Yang1967] interacting particles with potential
- Bethe ansatz gave rise to an equation
[Baxter1972] considered the problem of eight-vertex model and quantum XYZ model
- commutation of transfer matrices

in the space of couplings a submanifold exists, such as that the transfer matrices corresponding to any two points P and P' on it commute
characterized by a set of equations on the Boltzmann weights
- this set of equations is called the Yang-Baxter equation
solutions to Yang-Baxter equation can lead to a construction of integrable models

borrowed from transfer matrix in statistical mechanics
transfer matrix is builtup from matrices of Boltzmann weights
we need the transfer matrices coming from different set of Boltzman weights commute
partition function = trace of power of transfer matrices
so the problem of solving the model is reduced to the computation of this trace

we make a matrix from the Boltzmann weights
if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models
that is why we care about the quantum groups
spectral parameters
anistropy parameters
with an R-matrix satisfying the YBE, we obtain a representation of the Braid group, which then gives a link invariant in Knot theory
R-matrix

Introduction to the Yang-Baxter equation
- Jimbo
[Baxter1995]Solvable models in statistical mechanics, from Onsager onward
- Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
[Baxter1972]Partition Function of the Eight-Vertex Lattice Model
- Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
[Yang1967]Some exact results for the many-body problem in one dimension with repulsive delta-function interaction
- C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315