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.

This post imported from StackExchange Physics at 2014-04-04 16:12 (UCT), posted by SE-user Heidar