It is well-known that some exotic phases in condensed matter physics are described by Schwarz-type TQFTs, such as Chern-Simons theory of quantum Hall states. My question is whether there are condensed matter systems that can realize Witten-type TQFTs?
[Physics] Realization of Witten-type topological quantum field theory in condensed matter physics
condensed-matterresearch-leveltopological-field-theorytopological-order
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That is simplification based on the Wannier theorem, later generalized by Luttinger and Kohn in their 1955 article "Motion of Electrons and Holes in Perturbed Periodic Fields". Basically, if you have energy band depending on the wave vector as $\varepsilon_{0} (\mathbf{k})$, and apply external potential $V(\mathbf{r})$, weak compared to crystal potential and slowly varying compared to unit cell size, then Schrödinger's equation can be written like that:
$$ \left(\varepsilon_{0} \left(- i \mathbf{\nabla}\right) + V(\mathbf{r})\right) \Psi(\mathbf{r}) = \varepsilon \Psi(\mathbf{r}) $$
Here $\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$. For details and proof go to the 1955 paper mentioned above or to Anselm's "Semiconductor theory", Ch. IV paragraph 3. The latter is simpler.
In many cases, especially in semiconductors, at low values of $k$ energy is parabolic: $\varepsilon_{0}(\mathbf{k}) \approx C * \left| \mathbf{k} \right| ^2$. Renaming $C = \frac{\hbar^2}{2 m^{\ast}}$, where $m^{\ast}$ is called effective mass, we have
$$ \left(- \frac{\hbar^2}{2 m^{\ast}} \nabla^{2} + V(\mathbf{r})\right) \Psi(r) = \varepsilon \Psi(\mathbf{r}), $$
which is very similar to the equation of actual free electron gas.
Actually, the question you asked puzzled scientists for a while, but they asked "why electrons in materials behave like free electron gas".
The crucial thing that QFT brings to the table is that it allows us to naturally model systems with an indefinite number of particles. For a relativistic system this is a necessity because particle number is not conserved at relativistic energies; for non-relativistic systems with a thermodynamically large number of particles, QFT is a very convenient tool through which we may implement the grand canonical ensemble, in which the particle number is indefinite not because of relativistic pair production and annihilation but rather because the system is in chemical contact with a particle reservoir.
Even for non-relativistic systems with definite particle number, QFT is a very convenient language for describing the dynamics of quasiparticles. When one considers the low-energy excitations of electrons in a solid, for example, the vast majority of the electrons are "frozen" below the Fermi level and do not participate in any dynamics. Near the Fermi level, electrons may be excited into higher energy states which do participate in dynamics (e.g. they can conduct current and scatter with phonons). Interaction with the surrounding lattice gives these excitations effective properties (mass, charge, etc) which differ from those of bare electrons in free space, and so the excitations are regarded as quasiparticles. Crucially, these quasiparticles are not conserved (since e.g. the absorption of a photon "creates" a quasiparticle excitation), and the formalism of QFT provides a convenient language in which we might describe quasiparticles as excitations above the ground state (filled) Fermi sea - in quite a precise analogy with how QFT allows us to describe actual elementary particles as excitations above the ground state vacuum in high energy physics.
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The TQFTs that Witten introduced are those obtained by a topological twist of a supersymmetric field theory. This includes notably the A-model and the B-model TQFTs.
Despite what seems to be suggested in the comments here and on Wikipedia, these are also "Schwarz type" (come from the Poisson sigma-model) and they do have a desciption in terms of functorial TQFT if only one allows what are called (infinity,1)-functors: they are "TCFTs" (i.e. non-compact 2d homotopy TQFTs).
Now, under homological Mirror symmetry these are related to other TCFTs known as Landau-Ginzburg models. And these do have applications in solid state physics.