Topological quantum phases

I will describe the problem of finding the homotopy type of the space of quantum Hamiltonians in dimension d under certain constraints. The common assumptions are that the interactions have finite range and that the ground state is separated from excited states by a constant energy gap; then the space of Hamiltonians is homotopy equivalent to the space of their ground states. The problem has been solved for free-fermion Hamiltonians, which are given by quadratic elements of a Clifford algebra. In this case, the answer is a shifted KO spectrum. I will also discuss a more general setting of "invertible systems", which should be described by a different homotopy spectrum. However, it is not known what spectrum it is.