Matrix definition of the Ising XX gate

Question

On the English Wikipedia, $XX$ Ising gates are defined in matrix form as :

$$ XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}$$

However, on the French Wikipedia, the $XX$ Ising gates are defined as :

$$ XX(\phi) = \frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0& -i e^{i \phi}\\ 0&1&-i & 0 \\ 0 & -i & 1 & 0 \\ -i e^{i \phi} & 0 & 0 & 1 \\ \end{bmatrix}$$

From some sources, I believe the right one is the first one. I would like some confirmations about the second definition being an error, and I would be glad to take any explanation on it. In particular, does this matrix in the second definition corresponds to the definition of another gate?

Answer 1

Your first option is the correct one, being related to $e^{-i\phi X\otimes X}$, which is $$ XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}.$$

The second option doesn't make a whole lot of sense; it's not even unitary, so it's really not right. It cannot be the definition of a gate.