Universal R-matrix and representation theory for quantum groups

Quantum Annealing and Thermalization: Insights from Integrability

“…So, integrability is not only compatible but it is needed naturally to find the Gibbs distribution after QA. In [19], we support this conclusion by showing that the model's symmetry reflects invariance of the evolution matrix under action of the Braid Group and the associated with it quantum group SU q (2) [29][30][31] where the deformation parameter q ≡ e −ε/2k B T defines the temperature scale.…”

“…By representing σ = pR, with p being just the spin permutation operator, the cubic relations of the BG adopt a form R j,j+1 R j,j+2 R j+1,j+2 = R j+1,j+2 R j,j+2 R j,j+1 (26) of the Yang-Baxter-Zamolodchikov (YBZ) or triangle equation, whose solutions have been classified, and in the 2 × 2 case of interest are represented by a one-parameter family R = (1/ √ q)(I ⊗ I + (q − 1)(X 11 ⊗ X 11 + X 22 ⊗ X 22 ) + (q − 1/q)X 12 ⊗ X 21 ), (27) with X ab ≡ |a b|, a = 1, 2, being the Hubbard operators, and |1 = | ↑ , |2 = | ↓ , and q = e ih being the quantum deformation parameter, which can be identified by comparing the eigenvalues of σ = pR, obtained to be −q −3/2 and √ q, with degeneracy 1 and 3, respectively, with the corresponding eigenvalues for the two-spin system in the correlated region, which can be computed, yielding −e 3πg/2 and e −πg/2 , respectively, resulting in q = e −πg . An R matrix that satisfies the YBZ equation creates a quantum group/algebra [29][30][31], by identifying the needed commutation relations, which in the case under considerations happens to be SL q (2, C). Upon introducing a natural involution (complex conjugation) operation, it is reduced to its real "compact" version SU q (2), which provides a representation theory with good properties, in particular, (i) decomposition of representations in irreducibles, (ii) adding two spins using (q-deformed) 3j-symbols, (iii) comparing the result of adding three spins in different order that gives rise to (q-deformed) 6j-symbols, (iv) there is a notion of unitary representations of SU q (2), which makes the q-deformed 3-j and 6-j symbols scalar product preserving, and, very importantly, (v) the actions of the quantum group and the BG commute.…”