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

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

In one dimensional quantum systems, ${\displaystyle R}$ is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. 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 two different ways, the Yang–Baxter equation enforces that both paths are the same.

Illustration of the Yang–Baxter equation

It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971.

General form of the parameter-dependent Yang–Baxter equation

Let ${\displaystyle A}$ be a unital associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for ${\displaystyle R(u,u')}$, a parameter-dependent element of the tensor product ${\displaystyle A\otimes A}$ (here, ${\displaystyle u}$ and ${\displaystyle u'}$ are the parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers+ in the case of a multiplicative parameter).

Let ${\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))}$ for ${\displaystyle i,j=1,...,3}$, with algebra homomorphisms ${\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A}$ determined by

${\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}$
${\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}$
${\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}$

The general form of the Yang–Baxter equation is

${\displaystyle R_{12}(u_{1},u_{2})\ R_{13}(u_{1},u_{3})\ R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})\ R_{13}(u_{1},u_{3})\ R_{12}(u_{1},u_{2}),}$

for all values of ${\displaystyle u_{1}}$ , ${\displaystyle u_{2}}$ and ${\displaystyle u_{3}}$ .

Parameter-independent form

Let ${\displaystyle A}$ be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for ${\displaystyle R}$, an invertible element of the tensor product ${\displaystyle A\otimes A}$. The Yang–Baxter equation is

${\displaystyle R_{12}\ R_{13}\ R_{23}=R_{23}\ R_{13}\ R_{12},}$

where ${\displaystyle R_{12}=\phi _{12}(R)}$, ${\displaystyle R_{13}=\phi _{13}(R)}$, and ${\displaystyle R_{23}=\phi _{23}(R)}$.

With respect to a basis

Often the unital associative algebra is the algebra of endomorphisms of a vector space ${\displaystyle V}$ over a field ${\displaystyle k}$, that is, ${\displaystyle A={\text{End}}(V)}$. With respect to a basis ${\displaystyle \{e_{i}\}}$ of ${\displaystyle V}$, the components of the matrices ${\displaystyle R\in {\text{End}}(V)\otimes {\text{End}}(V)\cong {\text{End}}(V\otimes V)}$ are written ${\displaystyle R_{ij}^{kl}}$, which is the component associated to the map ${\displaystyle e_{i}\otimes e_{j}\mapsto e_{k}\otimes e_{l}}$. Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map ${\displaystyle e_{a}\otimes e_{b}\otimes e_{c}\mapsto e_{d}\otimes e_{e}\otimes e_{f}}$ reads

${\displaystyle (R_{12})_{ij}^{de}(R_{13})_{ak}^{if}(R_{23})_{bc}^{jk}=(R_{23})_{jk}^{ef}(R_{13})_{ic}^{dk}(R_{12})_{ab}^{ij}.}$

Alternate form and representations of the braid group

Let ${\displaystyle V}$ be a module of ${\displaystyle A}$, and ${\displaystyle P_{ij}=\phi _{ij}(P)}$ . Let ${\displaystyle P:V\otimes V\to V\otimes V}$ be the linear map satisfying ${\displaystyle P(x\otimes y)=y\otimes x}$ for all ${\displaystyle x,y\in V}$. The Yang–Baxter equation then has the following alternate form in terms of ${\displaystyle {\check {R}}(u,u')=P\circ R(u,u')}$ on ${\displaystyle V\otimes V}$.

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u_{1},u_{2}))({\check {R}}(u_{1},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{2},u_{3}))=({\check {R}}(u_{2},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{1},u_{3}))({\check {R}}(u_{1},u_{2})\otimes \mathbf {1} )}$.

Alternatively, we can express it in the same notation as above, defining ${\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))}$ , in which case the alternate form is

${\displaystyle {\check {R}}_{23}(u_{1},u_{2})\ {\check {R}}_{12}(u_{1},u_{3})\ {\check {R}}_{23}(u_{2},u_{3})={\check {R}}_{12}(u_{2},u_{3})\ {\check {R}}_{23}(u_{1},u_{3})\ {\check {R}}_{12}(u_{1},u_{2}).}$

In the parameter-independent special case where ${\displaystyle {\check {R}}}$ does not depend on parameters, the equation reduces to

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

and (if ${\displaystyle R}$ is invertible) a representation of the braid group, ${\displaystyle B_{n}}$, can be constructed on ${\displaystyle V^{\otimes n}}$ by ${\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}}$ for ${\displaystyle i=1,\dots ,n-1}$. This representation can be used to determine quasi-invariants of braids, knots and links.

Symmetry

Solutions to the Yang–Baxter equation are often constrained by requiring the ${\displaystyle R}$ matrix to be invariant under the action of a Lie group ${\displaystyle G}$. For example, in the case ${\displaystyle G=GL(V)}$ and ${\displaystyle R(u,u')\in {\text{End}}(V\otimes V)}$, the only ${\displaystyle G}$-invariant maps in ${\displaystyle {\text{End}}(V\otimes V)}$ are the identity ${\displaystyle I}$ and the permutation map ${\displaystyle P}$. The general form of the ${\displaystyle R}$-matrix is then ${\displaystyle R(u,u')=A(u,u')I+B(u,u')P}$ for scalar functions ${\displaystyle A,B}$.

The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines ${\displaystyle R'(u_{i},u_{j})=f(u_{i},u_{j})R(u_{i},u_{j})}$, where ${\displaystyle f}$ is a scalar function, then ${\displaystyle R'}$ also satisfies the Yang–Baxter equation.

The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments ${\displaystyle (u,u')}$ must be dependence only on the translation-invariant difference ${\displaystyle u-u'}$, while scale invariance enforces that ${\displaystyle R}$ is a function of the scale-invariant ratio ${\displaystyle u/u'}$.

Parametrizations and example solutions

A common ansatz for computing solutions is the difference property, ${\displaystyle R(u,u')=R(u-u')}$ , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization ${\displaystyle R(u,u')=R(u/u')}$ , in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations:

${\displaystyle R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),}$

for all values of ${\displaystyle u}$ and ${\displaystyle v}$. For a multiplicative parameter, the Yang–Baxter equation is

${\displaystyle R_{12}(u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),}$

for all values of ${\displaystyle u}$ and ${\displaystyle v}$.

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(u+v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u+v))({\check {R}}(u)\otimes \mathbf {1} )}$
${\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(uv)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(uv))({\check {R}}(u)\otimes \mathbf {1} )}$

In some cases, the determinant of ${\displaystyle R(u)}$ can vanish at specific values of the spectral parameter ${\displaystyle u=u_{0}}$. Some ${\displaystyle R}$ matrices turn into a one dimensional projector at ${\displaystyle u=u_{0}}$. In this case a quantum determinant can be defined[clarification needed].

Example solutions of the parameter-dependent YBE

• A particularly simple class of parameter-dependent solutions can be obtained from solutions of the parameter-independent YBE satisfying ${\displaystyle {\check {R}}^{2}=\mathbf {1} }$, where the corresponding braid group representation is a permutation group representation. In this case, ${\displaystyle {\check {R}}(u)=\mathbf {1} +u{\check {R}}}$ (equivalently, ${\displaystyle R(u)=P+uP\circ {\check {R}}}$ ) is a solution of the (additive) parameter-dependent YBE. In the case where ${\displaystyle {\check {R}}=P}$ and ${\displaystyle R(u)=P+u\mathbf {1} }$ , this gives the scattering matrix of the Heisenberg XXX spin chain.
• The ${\displaystyle R}$-matrices of the evaluation modules of the quantum group ${\displaystyle U_{q}({\widehat {sl}}(2))}$ are given explicitly by the matrix
${\displaystyle {\check {R}}(z)={\begin{pmatrix}qz-q^{-1}z^{-1}&&&\\&q-q^{-1}&z-z^{-1}&\\&z-z^{-1}&q-q^{-1}&\\&&&qz-q^{-1}z^{-1}\end{pmatrix}}.}$

Then the parametrized Yang-Baxter equation with the multiplicative parameter is satisfied:

${\displaystyle ({\check {R}}(z)\otimes \mathbf {1} )({\check {R}}(zz')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(z'))=({\check {R}}(z')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(zz'))({\check {R}}(z)\otimes \mathbf {1} )}$

Classification of solutions

There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to quantum groups known as the Yangian, affine quantum groups and elliptic algebras respectively.