= Heisenberg coupling matrix for two spins
{c}
{title2=$C=2S-I$}
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> $(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$,
$$
C=\begin{pmatrix}1&0&0&0\\0&-1&2&0\\0&2&-1&0\\0&0&0&1\end{pmatrix}=2S-I,
$$
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> $-3$. Hence
$$
e^{-iaC/2}=e^{ia/2}\bigl(\cos a\,I-i\sin a\,S\bigr).
$$
This identity explains how isotropic <exchange interaction> generates a <SWAP gate> at $a=\pi/2$ up to a <global phase>.
Back to article page
