Yang–Baxter equation

In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix ${\displaystyle R}$, acting on two out of three objects, satisfies

$${\displaystyle ({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )=(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}),}$$

where ${\displaystyle {\check {R}}}$ is ${\displaystyle R}$ followed by a swap of the two objects. In one-dimensional quantum systems, ${\displaystyle R}$ is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. In quantum integrable models, the Yang–Baxter equation ensures that multi-particle scattering processes can be factorized into a sequence of two-body interactions, preserving exact solvability. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where ${\displaystyle R}$ corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation ensures that both paths are the same.