# How do physicists use solutions to the Yang-Baxter Equation?

As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions.

Often research grants in this area cite this as an "application" of their research. This being said, many mathematicians(definitely including myself) don't know why these solutions are important. So, I wonder;

What exactly do physicists do with solutions to the Yang-Baxter Equation once they have them?

Thanks.

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I do not really work in this part of the field, but my very vague impression is that they are primarily useful in finding exact solutions for lattice models for statistical systems in 2D. These "integrable models" may or may not have much direct relevance to the real world, but are theoretically interesting, as most realistic statistical mechanical models admit no analytical solutions. Hopefully someone else will come along and correct me or expand on this further. –  j.c. Nov 3 '10 at 13:55
@j.c. Well thanks for these initial thoughts. :) –  BBischof Nov 4 '10 at 3:34

Ah. Finally a topic I know something about !

There are many places in physics where the YB equation pops up. I can think of two at the moment.

a. Exactly solvable lattice models

b. Quantum Computation (QC)

It is the second application I find most exciting, so I'll focus on it.

The canonical reference (IMHO) on the link between the YB equation and QC is the wonderful paper by Lomonaco and Kauffmann (LK04) http://arxiv.org/abs/quant-ph/0401090

In topological quantum computation, the hope is to be able to perform unitary operations on qubits by moving them around each other. A typical arena is a 2D electron gas, where our qubits are the quasiparticles of the system. In 2D when we exchange two objects we get a richer symmetry group than in 3D, where we get the permutation group whose eigenvalues $\pm 1$ correspond to the case of bosons and fermions respectively. However, in 2D this symmetry group is enlarged to the braid group - one can exchange two objects by moving them around such that their worldlines "braid" around each other. This braiding cannot be eliminated by deforming the trajectories, because we don't have the third dimension to utilize.

Anyhow to cut a long story short, the YBE can be shown diagramatically as a relationship between three particles under exchange (see fig. 1 on pg. 8 of above ref.). What LK04 then show is that solutions of the YBE are unitary matrices which are universal for quantum computation. In much the same way that any classical binary circuit can be built out of NAND gates along, any quantum circuit can be built out of a set of universal quantum gates.

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Awesome, thanks. –  BBischof Nov 12 '10 at 5:28