I consider the quantum Ising model in a transverse field. Take the Hamiltonian, $$H = -\sum_i\left[Z_i Z_{i+1}+g X_i\right] $$ where $Z$ and $X$ are Pauli operators and $g\geq 0$ is a ratio of the external field to the coupling strength. The ground state spontaneously breaks the $\mathbb{Z}_2$ symmetry as $g$ is tuned from $>1$ to $<1$.
I can rewrite the Ising model in terms of Majorana fermions. I take the Jordan-Wigner transformation $X_i = i \gamma_{Ai}\gamma_{Bi}$ and $ Z_iZ_{i+1}= -i \gamma_{Ai}\gamma_{B,i+1}$ and find the Hamiltonian, $$H=i\sum_i\left[\gamma_{Ai}\gamma_{B,i+1}-g\gamma_{Ai}\gamma_{Bi}\right] $$ where for each spin at site i, I have taken two Majorana fermions labeled by $A$ and $B$.
I want to interpret the phases and phase transition of the spins in the Majorana language. I take periodic boundary conditions and in momentum space, I can write the Majorana Hamiltonian as, $$H=\frac{1}{2}\sum_k \Psi^\dagger_k H_k\Psi_k $$ where $H_k=(\sin k )X+(\cos k -g) Y=\mathbf{h}(k)\cdot \sigma$ where again $X$ and $Y$ are Pauli matrices, $\sigma=(X\: Y\: Z)^T$ is the pseudo spin for the two bands and $\Psi_k=(\gamma_{Ak}\: \gamma_{Bk})^T$. Using this I compute the winding number given by, $$ w=\frac{1}{2\pi}\int_{-\pi}^{\pi}d k\: \left(\hat{h}(k)\times \frac{d}{d k}\hat{h}(k)\right)_z$$ where $\hat{h}$ is the unit vector along the effective magnetic field $\mathbf{h}$. I find that $w=1$ for $g<1$ and $w=0$ for $g>1$ indicating that the ferromagnetic phase is topological while the paramagnetic phase is trivial. In this language, the two gapped bands close at $g=1$ (the quantum critical point) and mark a topological phase transition.
I don't understand why the the same phase transition can be topological in terms of one set of degrees of freedom while "Landau-type" (spontaneous symmetry breaking) in terms of a different set of degrees of freedom, when the mapping between them is exact. Is there a good way to understand why this may be the case? More generally, are there more examples where a topological phase transition can actually be spontaneous symmetry breaking when seen through a different set of variables?
