In solving the Classical model for lattice dynamics [Rossler pag 38] we find that the lattice admits $$d\cdot N\cdot r = \#modes$$ where
$d=$dimension of the problem
$N=$ number of atoms
$r=$ number of atoms in the primitive cell
So, classically we have that only a finite number of modes (for a finite lattice) is possible.
When we quantize the Hamiltonian we get the hamiltonian of an HO
where $s$ is a branch index.
So, in the Quantum case it seems that we can have an infinite number of phonons.
Am I missing something? Since the creation of a phonon corresponds physically to excite a vibration mode in the lattice, it seem me strange that we can create an infinite number of phonons in QM while in classical mechanics we can have only a finite number of excitations.