In Kopec and Usadel's Phys. Rev. Lett. 78.1988, a spin glass Hamiltonian is introduced in the form:
$$ H = \frac{\Delta}{2}\sum_i \Pi^2_i - \sum_{i<j}J_{ij}\sigma_i \sigma_j, $$ where the variables $ \sigma_i (i = 1, \ldots, N) $ are associated with spin degrees of freedom [...] and canonically conjugated to the "momentum" operators $ \Pi_i $ such that $ [\sigma_i, \Pi_j] = i \delta_{ij} $.
Now, I am accustomed to writing the "kinetic" term in a transverse-field Ising-like Hamiltonian as $ \propto \sum_i \sigma^x_i $ (working in the standard basis of $ \{\sigma^z_i\} $), so this passage is raising some questions for me.
What are these $ \Pi_i $ operators? If $ \Pi_i^2 = \sigma^x_i $, like I initially believed, then they cannot be observables, for the square of a self-adjoint operator is positive semidefinite (which $ \sigma^x_i $ is not). In fact, if one restricts to the $ i $-th spin and takes $ i = j $, one can easily prove that $$ [\sigma^z, \Pi] = \sigma^z \Pi - \Pi^\dagger \sigma^z = i \mathbb 1 $$ is satisfied for $$ \Pi = \begin{pmatrix}i/2&b\\-\bar{b}&-i/2\end{pmatrix} $$ with $ b \in \mathbb C $. This squares to a multiple of the identity matrix, which seems like an odd choice for a kinetic term. I feel I am missing something here.
More generally speaking, can one even define a momentum "canonically conjugate" to $ \sigma^z $, or any other spin operator for that matter? As far as I understand, in classical mechanics the variables conjugate to physical rotations are angles, but this cannot be ported over to QM in any obvious way.