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.
There is no proof of bulk-boundary correspondence for topological phases in general. In fact, topological phases like toric code model does not have gapless excitations on the boundary.
For non-interacting fermion systems protected by internal symmetries (as in the "periodic table" classification), bulk-boundary correspondence holds. For non-interacting fermion systems protected by a spatial symmetry, gapless surface states also exist on those crystal surfaces preserving the symmetry.
In some sense, the existence of some kind of boundary states is all there is about topology in non-interacting fermions.
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To my knowledge there is no simple proof of the bulk-boundary correspondence. That's why it's so surprising and interesting. It is really weird that there is a correspondence because they are different setups - the edges are a property of the finite chain and topological invariant is a property of the bulk/ infinite chain. At the same time, it is really at the crux of topological phases, and this correspondence has been verified over and over experimentally and numerically, even if we don't have a simple answer of why the correspondence exists. I think there is some heavy maths like K-theory that have tried to explain this correspondence but I don't understand it.
One of my supervisors sent me this paper 'Edge states and the bulk-boundary correspondence in Dirac Hamiltonians' https://arxiv.org/abs/1010.2778 which proves why the topological invariants of the bulk lead to edge states but I haven't looked at in depth myself yet.
On a rather random note, there is a really nice set of notes on this website: https://topologicalphases.wordpress.com/lecture-notes/. At the end of Lecture 6 (on the SSH model!), they show why the winding number is equal to the number of edge states for 1D gapped chiral Hamiltonians. It is password protected though. I got access by cold-emailing them and Tomoki Ozawa was nice enough to give me the password (I am a random student and never met them) so maybe you can email them too if you like. They are really nice and clear notes. They consider a special case that has edge states and use the fact that other Hamiltonians with the same winding number can be continuously transformed into the special case (but this also uses homotopy theory which they don't explain in the notes - so the proof of this seemingly simple correspondence is not so simple).