  We construct a Hamiltonian lattice regularisation of the $N$-flavour
Gross-Neveu model that manifestly respects the full $\mathsf{O}(2N)$ symmetry,
preventing the appearance of any unwanted marginal perturbations to the quantum
field theory. In the context of this lattice model, the dynamical mass
generation is intimately related to the Coleman-Mermin-Wagner and
Lieb-Schultz-Mattis theorem. In particular, the model can be interpreted as
lying at the first order phase transition line between a trivial and
symmetry-protected topological (SPT) phase, which explains the degeneracy of
the elementary kink excitations. We show that our Hamiltonian model can be
solved analytically in the large $N$ limit, producing the correct expression
for the mass gap. Furthermore, we perform extensive numerical matrix product
state simulations for $N=2$, thereby recovering the emergent Lorentz symmetry
and the proper non-perturbative mass gap scaling in the continuum limit.
Finally, our simulations also reveal how the continuum limit manifests itself
in the entanglement spectrum. As expected from conformal field theory we find
two conformal towers, one tower spanned by the linear representations of
$\mathsf{O}(4)$, corresponding to the trivial phase, and the other by the
projective (i.e. spinor) representations, corresponding to the SPT phase.
