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Symmetry Protected Topological Order by Folding a One-Dimensional Spin-$1/2$ Chain

We present a toy model with a Hamiltonian $H^{(2)}_T$ on a folded one-dimensional spin chain. The non-trivial ground states of $H^{(2)}_T$ are separated by a gap from the excited states. By analyzing the symmetries in the model, we find that the topological order is protected by a $\mathbb{Z}_2$ global symmetry. However, by using perturbation series and excluding thermal effects, we show that the $\mathbb{Z}_2$ symmetry is stable in comparison to a standard nearest-neighbor Ising model with a Hamiltonian $H_I$. We find that $H^{(2)}_T$ is a member of a family of Hamiltonians that are adiabatically connected to $H_I$. Furthermore, the generalizations of this class of Hamiltonians, their adiabatic connection to $H_I$, and the relation to quantum error-correcting codes are discussed. Finally, we show the correspondence between the two ground states of $H^{(2)}_T$ and the unpaired Majorana modes, and provide numerical examples.