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.
We recently posted a paper (https://arxiv.org/abs/1808.09394) to address this issue systematically.
We may use disordered symmetry
breaking states (which are described by non-linear $\sigma$-models) to realize a large class of topological orders. ... ... In this paper, we show that the phase transitions driven by
fluctuations with all possible topological defects produce disordered states
that have no topological order, and correspond to non-topological phase
transitions. While transitions driven by fluctuations without any topological defects usually produce disordered states that have non-trivial topological orders, and correspond to topological phase transitions.
Thus, it may be confusing to refer the transition driven by topological defects as a topological phase transitions, since the appearance of topological defects decrease the chance to produce topological phases of matter.
More precisely, if the fluctuating order parameter in a disordered state has no topological defects, then the corresponding disordered state will usually have a non-trivial topological order. The type of the topological order depends on the topology of the degenerate manifold $K$ of the order parameter (ie the
target space of the non-linear $\sigma$-model). For example, if $\pi_1(K)$ is a
finite group and $\pi_{n>1}(K)=0$, then the disordered phase may have a
topological order described by a gauge theory of gauge group $G=\pi_1(K)$. If
$\pi_1(K),\pi_2(K)$ are finite groups and $\pi_{n>2}(K)=0$, then the disordered
phase may have a topological order described by a 2-gauge theory of
2-gauge-group $B(\pi_1(K),\pi_2(K))$.
It is the absence of topological defects that enable the symmetric disordered state to have a non-trivial topological order. When there are a lot of
topological defects, they will destroy the topology of the degenerate manifold
of the order parameter (ie the degenerate manifold effectively becomes a
discrete set with trivial topology). In this case the symmetric disordered
state becomes a product state with no topological order. Certainly, if the
fluctuating order parameter contains only a subclass of topological defects,
then only part of the topological structure of the degenerate manifold is
destroyed by the defects. The corresponding symmetric disordered state may
still have a topological order.
Best Answer
Let us consider a quantum phase transition (at T=0) from an ordered phase to a disordered phase, driven by the quantum fluctuations of the order parameter. We like to ask if the disordered phase has topological order or not.
The importance of the topological defects in phase transitions have been emphasized by Kosterlitz and Thouless, who shared 2016 Nobel prize (with Haldane) ``for theoretical discoveries of topological phase transitions and topological phases of matter''.
In this paper, we show that the phase transitions driven by fluctuations with all possible topological defects produce disordered states that have no topological order, and correspond to non-topological phase transitions. While transitions driven by fluctuations without any topological defects usually produce disordered states that have non-trivial topological orders, and correspond to topological phase transitions. If we refer to phase transitions induced by topological defects as topological phase transitions, and refer to phase transitions between different topological orders as topological phase transitions, then our result can be restated as:
Thus, it may be confusing to refer to the transition driven by topological defects as a topological phase transitions, since the appearance of topological defects decrease the chance to produce topological phases of matter.