Heisenberg coupling matrix for two spins ($C=2S-I$)

For two spin one-half systems, the dimensionless isotropic Heisenberg model coupling is $C=X\otimes X+Y\otimes Y+Z\otimes Z$, where $X,Y,Z$ are Pauli matrices. In the ordered computational basis $(|00⟩,|01⟩,|10⟩,|11⟩)$,where $S$ is the SWAP gate. This follows by acting on each basis state: the $XX+YY$ terms exchange the two opposite-spin states with coefficient two and annihilate the parallel-spin states. The symmetric three-dimensional sector has eigenvalue $1$ and the antisymmetric sector has eigenvalue $-3$. HenceThis identity explains how isotropic exchange interaction generates a SWAP gate at $a=\pi /2$ up to a global phase.