Coarse Grained

Mathematics and physics through TCM, in a non-representative stream

Topological insulators II: disorder

Last time we saw how a one particle, pure crystal theory of electrons might give rise to quantised transverse (i.e. Hall) conductance. The theory is simple, but rather leaves out crucial aspects of the phenomenon. Important aspects missing are interactions and disorder. Interactions lead to fractional quantum hall effects, and we do not wish to worry about those here. It's the usual post hoc justification where we see that a single particle theory seems to work, even though precisely why is mysterious — we ask for deliverance from the god of Landau, etc. As far as disorder goes however, things are not so simple. Loss of (lattice) translational invariance leads to a loss of the Brillouin zone, and its attendant topological structures. The classification of states by de Rham cohomology (the TKNN invariant) doesn't work if we don't have a manifold and a differential structure.

Disorder itself is a vast topic. One approach is to simply write down a Hamiltonian with a disorder potential written in, i.e. quenched, and work to compute averages over the disorder realisations. Powerful methods include using supersymmetric field theory and renormalisation. However, in keeping with the idea of simple physics, hard maths, we're going to hit the problem head on with noncommutative geometry.

Heuristically, we can build a picture via Anderson localisation. Although we can no longer use momentum to label states, we can still talk about the density of states for single particles. The disorder first causes a broadening of the Landau levels. Second, we know that in 2D arbitrarily weak disorder can cause localisation, albeit with a divergent localisation length. In a magnetic field the Landau levels are mobility edges, and so on approaching the level, the states start spanning the whole sample. Thus for a finite sample, around the Landau level there are extended states. It seems reasonable to hope that as long as the Fermi level is moving through the localised states, we still get a quantised transverse conductance. To show this, we just need to generalise the TKNN invariant to apply to a system which doesn't have momentum operators commute with the Hamiltonian. The full details can be found in arxiv:cond-mat/9411052.

First, we should say a few words about how noncommutative geometry works. On a usual space $X$ (a manifold say), we can consider the functions $C(X)$ of continuous functions from $X$ to (say) the reals. These functions from a commutative $C^*$-algebra. But we can also go backwards: start with an abstractly defined commutative algebra, ask for the space on which these elements have representation as functions on that space. It turns out that these two views are exactly equivalent, and we can therefore calculate topological quantities (such as the Chern classes) of $X$ by computations on the algebra $C(X)$.

Noncommutative geometry is what you get if you remove the restriction to commutative algebras. It turns out that many of the geometric concepts can be pushed through, and we get new, novel spaces in which to play. In our case, the strategy is to take the TKNN invariant, replace anything which mentions geometry by its algebraic equivalent, and then remove the commutativity requirement, and hope that it remains well-defined and integer valued. The paper above by Bellissard et al. also go through a first principles derivation of the algebraicised Kubo conductance formula, and show that it is related to the Chern class the same way as the TKNN invariant and consider the corrections to the quantum hall regime (it's an excellent paper).

To give a flavour of how it works, recall that the TKNN invariant is basically a suitably integrated derivative of some operator by momentum. Integration can be replaced by a suitable trace; the fact that it is over the Brillouin zone can be considered a normalisation, i.e. it is actually an integral divided by the area. Differentiation can be treated algebraically by treating it as a derivation: $$ \partial_{k} A \rightarrow -i[x,A], $$ relying on the commutator obeying a Leibniz rule and $[x,k] = i$. Thus, we have again turned the problem into a mathematical one, which means we can find clever people to solve it for us. In this case, it can be shown that subject to states at the Fermi level being localised, the algebraicised TKNN invariant is still integer valued. Thus we have a very direct way to show that the topological order is robust in the face of disorder.


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.