Yang–Baxter equation explained
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
, acting on two out of three objects, satisfies
(\check{R} ⊗ 1)(1 ⊗ \check{R})(\check{R} ⊗ 1)=(1 ⊗ \check{R})(\check{R} ⊗ 1)(1 ⊗ \check{R}),
where
is
followed by a swap of the two objects. In one-dimensional quantum systems,
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
corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation enforces that both paths are the same.
History
According to Jimbo,[1] the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire[2] in 1964 and C. N. Yang[3] in 1967. They considered a quantum mechanical many-body problem on a line having
as the potential. Using
Bethe's Ansatz techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly. Here YBE arises as the consistency condition for the factorization.
In statistical mechanics, the source of YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model[4] in 1944. Hunt for solvable lattice models has been actively pursued since then, culminating in Baxter's solution of the eight vertex model[5] in 1972.
Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory.[6] Zamolodchikov pointed out[7] that the algebraic mechanics working here is the same as that in the Baxter's and others' works.
of the
symmetric group in the work of
A. A. Jucys[8] in 1966.
General form of the parameter-dependent Yang–Baxter equation
Let
be a unital
associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for
, a parameter-dependent element of the
tensor product
(here,
and
are the parameters, which usually range over the
real numbers ℝ in the case of an additive parameter, or over
positive real numbers ℝ
+ in the case of a multiplicative parameter).
Let
Rij(u,u')=\phiij(R(u,u'))
for
, with algebra
homomorphisms
determined by
The general form of the Yang–Baxter equation is
R12(u1,u2) R13(u1,u3) R23(u2,u3)=R23(u2,u3) R13(u1,u3) R12(u1,u2),
for all values of
,
and
.
Parameter-independent form
Let
be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for
, an invertible element of the tensor product
. The Yang–Baxter equation is
where
,
, and
.
With respect to a basis
over a
field
, that is,
. With respect to a
basis
of
, the components of the matrices
R\inEnd(V) ⊗ End(V)\congEnd(V ⊗ V)
are written
, which is the component associated to the map
. Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map
ea ⊗ eb ⊗ ec\mapstoed ⊗ ee ⊗ ef
reads
(R12
(R13
(R23
=(R23
(R13
(R12
.
Alternate form and representations of the braid group
Let
be a
module of
, and
. Let
be the linear map satisfying
for all
. The Yang–Baxter equation then has the following alternate form in terms of
\check{R}(u,u')=P\circR(u,u')
on
.
(1 ⊗ \check{R}(u1,u2))(\check{R}(u1,u3) ⊗ 1)(1 ⊗ \check{R}(u2,u3))=(\check{R}(u2,u3) ⊗ 1)(1 ⊗ \check{R}(u1,u3))(\check{R}(u1,u2) ⊗ 1)
.
Alternatively, we can express it in the same notation as above, defining
\check{R}ij(u,u')=\phiij(\check{R}(u,u'))
, in which case the alternate form is
\check{R}23(u1,u2) \check{R}12(u1,u3) \check{R}23(u2,u3)=\check{R}12(u2,u3) \check{R}23(u1,u3) \check{R}12(u1,u2).
In the parameter-independent special case where
does not depend on parameters, the equation reduces to
(1 ⊗ \check{R})(\check{R} ⊗ 1)(1 ⊗ \check{R})=(\check{R} ⊗ 1)(1 ⊗ \check{R})(\check{R} ⊗ 1)
,
and (if
is invertible) a
representation of the
braid group,
, can be constructed on
by
\sigmai=1 ⊗ ⊗ \check{R} ⊗ 1 ⊗
for
. 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
matrix to be invariant under the action of a
Lie group
. For example, in the case
and
, the only
-invariant maps in
are the identity
and the permutation map
. The general form of the
-matrix is then
R(u,u')=A(u,u')I+B(u,u')P
for scalar functions
.
The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines
R'(ui,uj)=f(ui,uj)R(ui,uj)
, where
is a scalar function, then
also satisfies the Yang–Baxter equation.
The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments
must be dependent only on the translation-invariant difference
, while scale invariance enforces that
is a function of the scale-invariant ratio
.
Parametrizations and example solutions
A common ansatz for computing solutions is the difference property,
, where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization
, 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:
R12(u) R13(u+v) R23(v)=R23(v) R13(u+v) R12(u),
for all values of
and
. For a multiplicative parameter, the Yang–Baxter equation is
R12(u) R13(uv) R23(v)=R23(v) R13(uv) R12(u),
for all values of
and
.
The braided forms read as:
(1 ⊗ \check{R}(u))(\check{R}(u+v) ⊗ 1)(1 ⊗ \check{R}(v))=(\check{R}(v) ⊗ 1)(1 ⊗ \check{R}(u+v))(\check{R}(u) ⊗ 1)
(1 ⊗ \check{R}(u))(\check{R}(uv) ⊗ 1)(1 ⊗ \check{R}(v))=(\check{R}(v) ⊗ 1)(1 ⊗ \check{R}(uv))(\check{R}(u) ⊗ 1)
In some cases, the determinant of
can vanish at specific values of the spectral parameter
. Some
matrices turn into a one dimensional projector at
. In this case a quantum determinant can be defined .
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
, where the corresponding braid group representation is a permutation group representation. In this case,
\check{R}(u)=1+u\check{R}
(equivalently,
) is a solution of the (additive) parameter-dependent YBE. In the case where
and
, this gives the scattering matrix of the
Heisenberg XXX spin chain.
-matrices of the evaluation modules of the quantum group
are given explicitly by the matrix
\check{R}(z)=\begin{pmatrix}
qz-q-1z-1&&&\\
&q-q-1&z-z-1&\\
&z-z-1&q-q-1&\\
&&&qz-q-1z-1\end{pmatrix}.
Then the parametrized Yang-Baxter equation with the multiplicative parameter is satisfied:
(\check{R}(z) ⊗ 1)(\check{R}(zz') ⊗ 1)(1 ⊗ \check{R}(z'))=(\check{R}(z') ⊗ 1)(1 ⊗ \check{R}(zz'))(\check{R}(z) ⊗ 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.
Set-theoretic Yang–Baxter equation
Set-theoretic solutions were studied by Drinfeld.[9] In this case, there is an
-matrix invariant basis
for the vector space
in the sense that the
-matrix maps the induced basis on
to itself. This then induces a map
given by restriction of the
-matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting
as maps on
. The equation can then be considered purely as an equation in the
category of sets.
Examples
.
where
, the transposition map.
is a (right) shelf, then
r(x,y)=(y,x\trianglelefty)
is a set-theoretic solution to the YBE.
Classical Yang–Baxter equation
Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld.[10] Given a 'classical
-matrix'
, which may also depend on a pair of arguments
, the classical YBE is (suppressing parameters)
This is quadratic in the
-matrix, unlike the usual quantum YBE which is cubic in
.
This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the
-matrix admits an asymptotic expansion in terms of an expansion parameter
The classical YBE then comes from reading off the
coefficient of the quantum YBE (and the equation trivially holds at orders
).
See also
References
- H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989, Springer-Verlag Berlin, .
- Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .
- Jacques H.H. Perk and Helen Au-Yang, "Yang - Baxter Equations", (2006), .
Notes and References
- Jimbo . M. . Introduction to the Yang-Baxter Equation . International Journal of Modern Physics A . World Scientific . 4 . 15 . 1989 . 3759-3777. 10.1142/S0217751X89001503 . 1989IJMPA...4.3759J.
- McGuire . J. B. . Study of Exactly Soluble One‐Dimensional N‐Body Problems . Journal of Mathematical Physics . The American Institute of Physics (AIP) . 5 . 5 . 1964-05-01 . 0022-2488 . 10.1063/1.1704156 . 622-636 . 1964JMP.....5..622M .
- Yang . C. N. . Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction . Physical Review Letters . American Physical Society (APS) . 19 . 23 . 1967-12-04 . 0031-9007 . 10.1103/PhysRevLett.19.1312 . 1312-1315 . 1967PhRvL..19.1312Y .
- Onsager . L. . Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Physical Review. Americal Physical Society (APS) . 65 . 3-4 . 1944-02-01 . 117-149. 10.1103/PhysRev.65.117. 1944PhRv...65..117O.
- Baxter . R. J. . Partition function of the Eight-Vertex lattice model. Annals of Physics. Elsevier. 70 . 1 . 1972 . 193-228. 0003-4916. 10.1016/0003-4916(72)90335-1. 1972AnPhy..70..193B.
- Zamolodchikov . Alexander B.. Zamolodchikov. Alexey B.. Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Annals of Physics. Elsevier. 120 . 2 . 1979 . 253-291. 0003-4916. 10.1016/0003-4916(79)90391-9. 1979AnPhy.120..253Z.
- Zamolodchikov . Alexander B.. Z4-symmetric factorized S-matrix in two space-time dimensions. Comm. Math. Phys.. Elsevier. 69 . 2. 1979 . 165-178. 0003-4916. 10.1007/BF01221446. 1979CMaPh..69..165Z.
- Jucys . A. A. . On the Young operators of the symmetric group . Lietuvos Fizikos Rinkinys . Gos. Izd-vo Polit. i Nauch. literatury. . 6 . 1966 . 163-180 .
- Book: Drinfeld . Vladimir . Quantum groups : proceedings of workshops held in the Euler International Mathematical Institute, Leningrad, Fall 1990 . 1992 . Springer-Verlag . Berlin . 10.1007/BFb0101175 . 978-3-540-55305-2 . 4 February 2023.
- Belavin . A. A. . Drinfel'd . V. G. . Solutions of the classical Yang - Baxter equation for simple Lie algebras . Functional Analysis and Its Applications . 1983 . 16 . 3 . 159–180 . 10.1007/BF01081585 . 123126711 . 4 February 2023.