condensed matter - Simple models that exhibit topological phase transitions

There are a number of physical systems with phases described by topologically protected invariants (fractional quantum Hall, topological insulators) but what are the simplest mathematical models that exhibit topological phases? Is the toric code as simple as we can go?

Edit: Just to be clear, I'm talking about phases meaning states of matter, and not just the geometric phase that a wavefunction will pick up under parallel transport in a nontrivial configuration space. I'm looking for simple models where one can make a phase diagram, and as a function of the available couplings there is a change in some topological property of the system.

(For example, in the XY magnet, neglecting bound vortex-antivortex pair formation, there is an instability at finite temperature to creation a single vortex, which is topologically distinct from the vortex-free state.)

Answer

I think you need to define what you mean by a "topological state of matter", since the term is used in several inequivalent ways. For example the toric code that you mention, is a very different kind of topological phase than topological insulators. Actually one might argue that all topological insulators (maybe except the Integer Quantum Hall, class A in the general classification) are only topological effects rather than true topological phases, since they are protected by discreet symmetries (time reversal, particle-hole or chiral). If these symmetries are explicitly or spontaneously broken then the system might turn into a trivial insulator.

But one of the simplest lattice models (much simpler that the toric code, but also not as rich) I know of is the following two band model (written in k-space)

$H(\mathbf k) = \mathbf d(\mathbf k)\cdot\mathbf{\sigma},$

with $\mathbf d(\mathbf k) = (\sin k_x, \sin k_y, m + \cos k_x + \cos k_y)$ and $\mathbf{\sigma} = (\sigma_x,\sigma_y,\sigma_z)$ are the Pauli matrices. This model belongs to the same topological class as the IQHE, meaning that it has no time-reversal, particle-hole or chiral symmetry. The spectrum is given by $E(\mathbf k) = \sqrt{\mathbf d(\mathbf k)\cdot\mathbf d(\mathbf k)}$ and the model is classified by the first Chern number

$C_1 = \frac 1{4\pi}\int_{T^2}d\mathbf k\;\hat{\mathbf d}\cdot\frac{\partial \hat{\mathbf d}}{\partial k_x}\times\frac{\partial \hat{\mathbf d}}{\partial k_y},$

where $T^2$ is the torus (which is the topology of the Brillouin zone) and $\hat{\mathbf d} = \frac{\mathbf d}{|\mathbf d|}$. By changing the parameter $m$ the system can go through a quantum critical point, but this can only happen if the bulk gap closes. So solving the equation $E(\mathbf k) = 0$ for $m$, one can see where there is phase transitions. One can then calculate the Chern number in the intervals between these critical points and find

$C_1 = 1$ for $0 < m < 2$, $C_1 = -1$ for $-2 < m < 0$ and $C_1 = 0$ otherwise.

Thus there are three different phases, one trivial and two non-trivial. In the non-trivial phases the system has quantized Hall response and protected chiral edge states (which can easily be seen by putting edges along one axes and diagonalizing the Hamiltonian on a computer).

If one takes the continuum limit, the model reduces to a 2+1 dimensional massive Dirac Hamiltonian and I think the same conclusions can be reached in this continuum limit but the topology enters as a parity anomaly.

More information can be found here: http://arxiv.org/abs/0802.3537 (the model is introduced in section IIB).

Hope you find this useful.