We discuss systems which have some, but not all of the hallmarks of topological phases. These systems topological character is not fully captured by a local order parameter, but they are also not fully described at low energies by topological quantum
field theories. For such systems, we formulate the concepts of quasi-topological phases (to be contrasted with true topological phases), and symmetry-protected quasi-topological phases. We describe examples of systems in each class and discuss the implications for topological protection of information and operations. We explain why topological phases and quasi-topological phases have greater stability than is sometimes appreciated. In the examples that we discuss, we focus on Ising-type (a.k.a. Majorana) systems particularly relevant to recent theoretical advances and experimental efforts.
We present the first numerical computation of the neutral fermion gap, $Delta_psi$, in the $ u=5/2$ quantum Hall state, which is analogous to the energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We find $Delta_psi approx 0.027
frac{e^2}{epsilon ell_0}$, comparable to the charge gap, and discuss the implications for topological quantum information processing. We also deduce an effective Fermi velocity $v_F$ for neutral fermions from the low-energy spectra for odd numbers of electrons, and thereby obtain a correlation length $xi_{psi}={v_F}/Delta_{psi} approx 1.3, ell_0$. We comment on the implications of our results for electronic mechanisms of superconductivity more generally.
