Topological insulators

Currently, topological insulators are very fashionable. However, the usual discussions we've been enjoying (enduring?) have gone pretty much over the heads of most of us little ones. I think the primary problem so far has been that there are plenty of obvious things, which really should have been repeated, but have not. Part of the problem is that people seem to think that the integer quantum hall effect is well-understood by the audience. Below, we hopefully introduce the necessary context.

We are dealing with pure electrons in a lattice. No phonons, no impurities, no interactions. This means that the many-body theory reduces in the canonical way to single particle theory — the many body wavefunction is just a suitably antisymmetrised product of single particle states. Further, invariance under translation by lattice vectors imply that the single particle space separates into sectors labelled by momentum $\mathbf{k}$, which are not mixed by the Hamiltonian. Thus for each sector $\mathbf{k}$ we have a separate Hamiltonian $H_\mathbf{k}$, the total Hamiltonian $H$ being the direct sum of these, over all $\mathbf{k}$. We may assume that there are no degenerate energy levels in any of the sub-Hamiltonians, since either they are protected by some symmetry, or they will not be generic, i.e. we are not in a stable phase. This is essentially just band theory.

The lattice induces the topology of a torus in reciprocal space. The sub-Hamiltonians $H_\mathbf{k}$ can be seen as a mapping from the torus $T^d$ to self-adjoint operators on a suitable Hilbert space. For physically reasonable suitations, the sub-Hamiltonians $H_\mathbf{k}$ should be continuous as functions of $\mathbf{k}$. We can now invoke some algebraic topology, and ask how many equivalence classes exist of such mappings, where equivalence is defined to be any continuous deformation without causing any of the $H_\mathbf{k}$ to become degenerate by having a level-crossing, i.e. how many phases exist. This is a well-defined mathematical question, and has a well-defined answer in terms of the Chern classes, which physically are called TKNN invariants. It unimportant for us what they are, but just that they exist and may be computed for real situations.

Note that in a magnetic field the above is not strictly true. A uniform magnetic field causes orthogonal translations to become non-commutative. We are therefore unable to simultaneously label the states with $k_x$ and $k_y$ (say in 2D). However, at particular values of $B$-field the commutator vanishes — exactly those at which the quantum hall plateaus exist. The TKNN invariant also turns out to be proportional to the transverse conductance. The different integer filling factors are thus proper phases, separated by some complex quantum phase transition. In practise, impurities actually produce most of the observed phenonmenon (i.e. the broad plateaus), but these can be thought of as simply fighting against the magnetic non-commutativity and maintaining the phase even for not exactly correct field strengths.

Now, since we are concerned with insulators, we may assume that our Fermi level lies in a gap; this is also true of the integer quantum hall states. The vacuum outside our sample is also such a state. It has, by assumption, a different value for the TKNN invariant. Thus somewhere on the edge, there will be a level crossing, and the gap will close forming a conducting edge state. Thus we see that topological phases will necessarily be accompanied by edge states.

This understanding is still deficient in a very crucial way. The construction above relies strongly on single particle behaviour and momentum (or at least pseudomomentum) being a good quantum number. As mentioned, magnetic fields which are not exactly integer quanta per unit cell cause problems --- in fact we get fractional quantum hall effects, where the interaction leads to intrinsically many-body effects. Furthermore, it is not clear that these effects are impervious to impurities — to say that it is pretected because it is topological is getting things mixed up; we can only declare it to be topological because it is insensitive to impurities. Indeed, in the case of the integer quantum hall effect impurities actually extend the "radius of convergence" of the phase.