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 , acting on two out of three objects, satisfies

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.

Illustration of the Yang–Baxter equation

History edit

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.

The YBE has also manifested itself in a study of Young operators in the group algebra   of the symmetric group in the work of A. A. Jucys[8] in 1966.

General form of the parameter-dependent Yang–Baxter equation edit

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+ in the case of a multiplicative parameter).

Let   for  , with algebra homomorphisms   determined by

 
 
 

The general form of the Yang–Baxter equation is

 

for all values of  ,   and  .

Parameter-independent form edit

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 edit

Often the unital associative algebra is the algebra of endomorphisms of a vector space   over a field  , that is,  . With respect to a basis   of  , the components of the matrices   are written  , which is the component associated to the map  . Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map   reads

 

Alternate form and representations of the braid group edit

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   on  .

 .

Alternatively, we can express it in the same notation as above, defining   , in which case the alternate form is

 

In the parameter-independent special case where   does not depend on parameters, the equation reduces to

 ,

and (if   is invertible) a representation of the braid group,  , can be constructed on   by   for  . This representation can be used to determine quasi-invariants of braids, knots and links.

Symmetry edit

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   for scalar functions  .

The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines  , 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 edit

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:

 

for all values of   and  . For a multiplicative parameter, the Yang–Baxter equation is

 

for all values of   and  .

The braided forms read as:

 
 

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[clarification needed].

Example solutions of the parameter-dependent YBE edit

  • 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,   (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.
  • The  -matrices of the evaluation modules of the quantum group   are given explicitly by the matrix
 

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

 

Classification of solutions edit

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 edit

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 edit

  •  .
  •   where  , the transposition map.
  • If   is a (right) shelf, then   is a set-theoretic solution to the YBE.

Classical Yang–Baxter equation edit

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 edit

References edit

  • 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, ISBN 3-540-53503-9.
  • Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
  • Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), arXiv:math-ph/0606053.
  1. ^ Jimbo, M. (1989). "Introduction to the Yang-Baxter Equation". International Journal of Modern Physics A. 4 (15). World Scientific: 3759–3777. Bibcode:1989IJMPA...4.3759J. doi:10.1142/S0217751X89001503.
  2. ^ McGuire, J. B. (1964-05-01). "Study of Exactly Soluble One‐Dimensional N‐Body Problems". Journal of Mathematical Physics. 5 (5). The American Institute of Physics (AIP): 622–636. Bibcode:1964JMP.....5..622M. doi:10.1063/1.1704156. ISSN 0022-2488.
  3. ^ Yang, C. N. (1967-12-04). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23). American Physical Society (APS): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312. ISSN 0031-9007.
  4. ^ Onsager, L. (1944-02-01). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition". Physical Review. 65 (3–4). Americal Physical Society (APS): 117–149. Bibcode:1944PhRv...65..117O. doi:10.1103/PhysRev.65.117.
  5. ^ Baxter, R. J. (1972). "Partition function of the Eight-Vertex lattice model". Annals of Physics. 70 (1). Elsevier: 193–228. Bibcode:1972AnPhy..70..193B. doi:10.1016/0003-4916(72)90335-1. ISSN 0003-4916.
  6. ^ Zamolodchikov, Alexander B.; Zamolodchikov, Alexey B. (1979). "Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models". Annals of Physics. 120 (2). Elsevier: 253–291. Bibcode:1979AnPhy.120..253Z. doi:10.1016/0003-4916(79)90391-9. ISSN 0003-4916.
  7. ^ Zamolodchikov, Alexander B. (1979). "Z4-symmetric factorized S-matrix in two space-time dimensions". Comm. Math. Phys. 69 (2). Elsevier: 165–178. Bibcode:1979CMaPh..69..165Z. doi:10.1007/BF01221446. ISSN 0003-4916.
  8. ^ Jucys, A. A. (1966). "On the Young operators of the symmetric group" (PDF). Lietuvos Fizikos Rinkinys. 6. Gos. Izd-vo Polit. i Nauch. literatury.: 163–180.
  9. ^ Drinfeld, Vladimir (1992). Quantum groups : proceedings of workshops held in the Euler International Mathematical Institute, Leningrad, Fall 1990. Berlin: Springer-Verlag. doi:10.1007/BFb0101175. ISBN 978-3-540-55305-2. Retrieved 4 February 2023.
  10. ^ Belavin, A. A.; Drinfel'd, V. G. (1983). "Solutions of the classical Yang - Baxter equation for simple Lie algebras". Functional Analysis and Its Applications. 16 (3): 159–180. doi:10.1007/BF01081585. S2CID 123126711. Retrieved 4 February 2023.

External links edit