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In summary, the classification scheme for topological insulators shows that the existence of a TI state depends on how it transforms under symmetries. Time reversal symmetry is a key factor, with topological insulators being possible if TR symmetry is implemented by T = UK where U^2 = -1. This applies whether the transformation is on spin or another degree of freedom. Additionally, there are topological superconductors with full SU(2) spin symmetry. While there are known examples of TIs such as the Kane-Mele and Fu-Kane states, there are also other possibilities beyond the Z2 classification.

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The key point in applying the classification scheme to these particular models is that they have time-reversal (TR) symmetry implemented by [itex]T = i \sigma^y K[/itex] where [itex]K[/itex] is complex conjugation and [itex]\sigma^y[/itex] is the spin matrix. If the spin does not appear explicitly in the Hamiltonian but you still have TR symmetry, then you would have invariance under [itex]T = K[/itex] and you end up in the AI class, and there are no topological insulators. (Also, without TR symmetry, you cannot get a TI in three dimensions, but you can in two: the Haldane model.)

But more abstractly, the existence of a possible TI state "just" boils down to how it transforms under symmetries as outlined in the linked paper. So if you have time reversal symmetry which acts on your Hamiltonian as [itex]T = UK[/itex] where [itex]U^2 = -1[/itex], you can get a TI whether the matrix [itex]U[/itex] is acting on spin or some other degree of freedom.

Finally, if you're interested in topological insulators outside of the Kane-Mele/Fu-Kane Z2 classifications, the answer is emphatically yes! As shown in the linked paper, there are topological superconductors with full SU(2) spin symmetry.

I'm afraid I'm not familiar enough with TIs to discuss the specific microscopic models which have been proposed or realized experimentally, but the above generalities may be helpful.

A topological insulator is a material that can conduct electricity on its surface or edges, while the bulk of the material remains an insulator. This is due to the topological properties of the material, which protect the conducting surface or edge states from disorder or impurities.

Spin Orbit Coupling is a phenomenon in which the spin of an electron is coupled to its motion around the atomic nucleus, resulting in a splitting of energy levels. In the context of topological insulators, SOC is responsible for the formation of the conducting surface or edge states.

No, it is not possible for a topological insulator to exist without SOC. SOC is a key ingredient in the formation of the conducting surface or edge states in topological insulators. Without SOC, the material would not exhibit the unique properties of a topological insulator.

No, there are no known materials that exhibit topological insulator behavior without SOC. All current topological insulators require the presence of SOC to exhibit their unique conducting surface or edge states.

No, topological insulators cannot be engineered to have no SOC. The presence of SOC is intrinsic to the electronic band structure and cannot be removed or altered without significantly changing the material's properties.

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