In literature one sometimes find that spins are replaced by hardcore bosons. Formally one replaces spin operators $\sigma^- \leftrightarrow a$, $\sigma^+ \leftrightarrow a^\dagger$, $\sigma_z \leftrightarrow a^\dagger a - 1/2$ and appends a term $U a^\dagger a^\dagger aa$ to the spin Hamiltonian. The limit $U \to \infty$ in the end splits the energy of states with more than 2 bosons away from the $\{|0\rangle,|1\rangle\}$ manifold. Identifying $|0\rangle \leftrightarrow |\downarrow\rangle$, $|1\rangle \leftrightarrow |\uparrow\rangle$ then gives a mapping between the bosonic and spin system.
My question is: how rigorous is this mapping? Can there be systems (i.e. Hamiltonians) or states for which the spin treatment yields different results than the bosonic treatment?
A possible example I had in mind: Consider some Hamiltonian $H(\sigma^-, \sigma^+, \sigma_z)$ and do the mapping to $H_B = H(a, a^\dagger, a^\dagger a - 1/2) + U a^\dagger a^\dagger aa$. Maybe time evolution $H_B$ can be easily solved using coherent states, thus I decompose some initial state into the coherent state basis, do the time evolution there, project the results onto the $\{|0\rangle, |1\rangle\}$-manifold and then take the limit $U\to \infty$. However, the coherent states always have a non-zero overlap with the "bad manifold" $\{|n\rangle | n\ge 2\}$. Thus this procedure works only, if no probability flows from the $\{|n\rangle | n\ge 2\}$-manifold to the $\{|0\rangle, |1\rangle\}$-manifold.
No, it is easy to show, that all states in $\{|n\rangle | n\ge 2\}$ are approximative eigenstates of $H_B$ with eigenenergy $U$ and the full eigenstates with eigenenergy $\mathcal{O}(U)$ have an overlap $\mathcal{O}(1/U)$ with the states from $\{|0\rangle, |1\rangle\}$. Thus, in the limit $U\to\infty$ the states $\{|0\rangle, |1\rangle\}$ and $\{|n\rangle | n\ge 2\}$ belong to disjoint subspaces of $H_B$'s eigenspace. Formally, this would suffice for any time evolution for some time $t$.
However, for asymptotic states (i.e., $t\to\infty$) im not so sure. Comming from a pertubation theory perspective I can argue that for each infinitesimal time step states from the $\{|0\rangle, |1\rangle\}$-manifold can be mapped to the $\{|n\rangle | n\ge 2\}$-manifold with some rate $\mathcal{O}(1/U)$ and vice versa. Doing infinitely many of these infinitesimal time steps can allow that these transition add up to something $\mathcal{O}(1)$. This is, we don't necessarily know whether $\lim_{t\to\infty}\lim_{U\to\infty} e^{-iHt} = \lim_{U\to\infty}\lim_{t\to\infty} e^{-iHt}$.
