# What are the implications that the Hamiltonian of a material lacks time reversal symmetry?

When reading about topological insulators and the quantum Hall effect, I've read that some Hamiltonians of the crystal structure representing the "materials" lack time reversal symmetry. My question is, what does this imply that the Hamiltonian representing those materials lacks the time reversal symmetry? Does that mean the entropy is always increasing (so there would be no equilibrium state possible, from a thermodynamics point of view)?

It usually just means that the material is magnetic, since magnetization ${\bf M}$ changes to $-{\bf M}$ under time reversal.
• Well, Maxwell equations are also modified, for instance $\vec J$ becomes $-\vec J$, $\vec B$ changes sign as well, and so on. Yet the Maxwell equations are invariant under time reversal symmetry. So I do not really understand the claim that lacking time reversal symmetry means that $\vec M$ becomes $-\vec M$. Could you comment or enhance your answer? – AccidentalBismuthTransform Feb 15 '18 at 21:22