I think the difference between the representation of the spin operator in the <PERSON> vs. the <PERSON> models is due to the fact that in the <PERSON> model, one tries to describe the situation of spin-1/2 fermions sitting in a lattice, whereas, in the <PERSON> model, the spins are distinguishable and hence have neither fermionic nor bosonic symmetry. 469
In general, the second-quantized representation of one-body operators follows the same structure for either fermions or bosons, e.g.,
$O_1 = \sum_{pq} \langle p | o_1 | q \rangle a_p^+ a_q,$
where $|p\rangle$ and $|q\rangle$ represent the single-particle states with which we are describing the problem and $a_p^+ (a_p)$ is the creation (annihilation) operator that obeys either the fermionic or bosonic (anti)commutation relations.
For both the <PERSON> and <PERSON> models, the single-particle states are the spin-1/2 states, $|\alpha\rangle \equiv |1/2,1/2\rangle$ and $|\beta\rangle \equiv |1/2,-1/2\rangle$. If we follow the above recipe for calculating a one-body operator, we can evaluate the form of each component of $\vec{S}$ using the usual raising and lowering operators $S_+$ and $S_-$.
$S_{x} = \langle \alpha | s_x |\alpha\rangle a_\alpha^+ a_\alpha + \langle \alpha | s_x |\beta\rangle a_\alpha^+ a_\beta + \langle \beta | s_x |\alpha\rangle a_\beta^+ a_\alpha + \langle \beta | s_x |\beta\rangle a_\beta^+ a_\beta = \frac{\hbar}{2} a_\alpha^+ a_\beta + \frac{\hbar}{2} a_\beta^+ a_\alpha$
$S_{y} = \langle \alpha | s_y |\alpha\rangle a_\alpha^+ a_\alpha + \langle \alpha | s_y |\beta\rangle a_\alpha^+ a_\beta + \langle \beta | s_y |\alpha\rangle a_\beta^+ a_\alpha + \langle \beta | s_y |\beta\rangle a_\beta^+ a_\beta = -i\frac{\hbar}{2} a_\alpha^+ a_\beta + i\frac{\hbar}{2} a_\beta^+ a_\alpha$
$S_{z} = \langle \alpha | s_z |\alpha\rangle a_\alpha^+ a_\alpha + \langle \alpha | s_z |\beta\rangle a_\alpha^+ a_\beta + \langle \beta | s_z |\alpha\rangle a_\beta^+ a_\alpha + \langle \beta | s_z |\beta\rangle a_\beta^+ a_\beta = \frac{\hbar}{2} a_\alpha^+ a_\alpha - \frac{\hbar}{2} a_\beta^+ a_\beta$
The above expressions coincide with how it is usually written in terms of the Pauli spin matrices, namely, $\vec{S} = \frac{\hbar}{2} a^+ \vec{\sigma} a$. In fact, in the <PERSON> model (and other fermionic/bosonic models), we likely would have other quantum numbers related to the particle's spatial wave function (such as momentum $\vec{k}$ of a plane wave or an orbital label $p$), thus the more complete form of the spin operator for these situations would be $\vec{S} = \sum_{pq} \delta_{pq} \frac{\hbar}{2} a_p^+ \vec{\sigma} a_q$, where the $\delta_{pq}$ is explicitly included to show that the spin is diagonal in the spatial degrees of freedom. 66