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?
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.
In mathematical physics you use solutions of the Yang Baxter equation in many contexts. In special in the quantum integrability solutions of the Yang Baxter equation are used for obtain the commutation relations from the Yang Baxter algebra $R_{12}(u,v)T^{1}(u)T^{2}(v) = T^{2}(v)T^{1}(u)R_{12}(u,v)$ can further include boundary terms. The $R$ in the context of statistical mechanics (two dimensional lattices) is related with the weights of Boltzmann, the $T's$ are associated with the monodromy matrix, which are the products of the locals $R$ for the fundamental models (fundamental models: which the lax operators are expressed in terms of $R$) this first part is the quantum inverse scattering method the second part we wish to diagonalize $[\tau(u),\tau(v)] = 0$ ($\tau$ is the transfer matrix) thus we use the first part for obtain the states of Bethe and Bethe equations this is the Algebraic Bethe Ansatz (The important thing here is the existence of something called the pseudo-vacuum), the most fundamental is the Yang Baxter equation it is the essence. You may also have 1+1 quantum integrable field models and apply such techniques. After the algebraic bethe ansatz you have to compute the inner product and correlation functions, this is the part that has more open problems. Another method different from algebraic bethe ansatz is the SOV-Sklyanin, which is a method applied to both the quantum and the classical case it arises from the separation of variables in classical mechanics, Sklyanin extends this by making the method most general, currently this method is proving more favorable to compute correlation functions (G. Nicolli work's), in SOV you also have the yang baxter equation.
With respect to the experiments, experimental techniques in trapping and cooling atoms in 1D have provided the realization of exactly solved models in the lab.
Lieb-Liniger Bose gas:
T. Kinoshita et al Science 2004, PRL 2005, Nature 2006; A. van Amerongen et al PRL2008; T. Kitagawa et al PRL 2010; J. Armijo et al PRL 2010
Super Tonks-Girardeau gas:
E. Haller et al Science 2009
Degenerate spin-1/2 Fermi gas:
Y. Liao et al Nature 2010; S. Jochim et al, Science 2011,PRL 2012: Deterministic preparation of few-fermion system; 2 fermions in a 1D harmonic trap
Two-component spinor Bose gas: J. van Druten et al arXiv:1010.4545
Quenches is a very current topic in this research follows the day July 26, this topic will enable further experiments (I don't know this). http://arxiv.org/abs/1407.7167
In the classical context solutions of the classical Yang Baxter equation, serve to compute the Poisson algebras $\lbrace T^{1}(u),T^{2}(v)\rbrace = [r_{12}(u,v),T^{1}(u)T^{2}(v)]$ (there are other depending on which model and boundary conditions you have) from this you have a connection with the Hamiltonian formalism.
