I have a very naive question about the notion of time-reversal symmetry applied to topological insulators that are studied in experiments. If I understand correctly, the exsistance of time-reversal symmetry is crucial for the occurence of topologically protected surface states in topological insulators. At least this is what the theory of topological insulators tells us. But I have a problem with applying this theory to real materials. In real world there is always dissipation and therefore there is an "arrow of time" and time-reversal symmetry does not hold (right?). So how can we talk about time-reversal symmetry in real materials that we call topological insulators? Please help me find where I went wrong in my argument. I realize that there must be something wrong but can't figure out what exactly.
In real world there is always dissipation and therefore there is an "arrow of time" and time-reversal symmetry does not hold (right?).
There are many phenomena in physics that we understand by invoking the time reversal symmetry of microscopic interactions. This is distinct from the "arrow of time" and the fact that an out-of-equilibrium system increases in entropy.
In terms of microscopic interactions, time reversal symmetry may be broken by an external magnetic field. This is the context in which we think of Quantum Hall states. For (most?) topological insulators, it is important that the microscopic interactions obey time reversal symmetry. This is very generally true: even if though you can see the "arrow of time" when a hot cup of coffee cools and increases the entropy of the room, the individual interactions (atoms bouncing off of each other) are time reversal symmetric. For topological insulator, the time reversal symmetry prevents edge spin currents from scattering.
If you have a solid foundation in undergraduate or graduate physics, the Review of Modern Physics articles by Hasan and Kane or Qi and Zhang are very good. For a more general audience, the Physics Today article by Qi and Zhang is intriguing.
You are just wrong.
1) The time reversal symmetry you are speaking of is not the time reversal symmetry which is considered when topological isulators are discussed. In the latter case just no magnetic field (both external and internal) is enough. In that case effective Hamiltonian of the system allows for the time inversion symmetry. Formally you may inverse the time (which has rather definite meaning you may find in textbooks) and end up with equivalent system.
2) You misinterpret the statement "the time-reversal symmetry is crucial for the occurrence of topologically protected states". There are different cases. Some of them rely on time inversion, some do not.