Topology and the 2016 Nobel Prize in Physics
Roughly speaking: When a system consists of particles interacting strongly, you won't have large movements for any particle unless all of them move together. There is still "quantization", so you'll have jumps in the type of movement (example: the quantum Hall effect). So the topology of the system plays a role. For example, given a collection of electrons with their spins and an external magnetic field, there are alignment issues with how the spins interact with each other and the magnetic field (and that's where your vector fields and winding numbers pop up). But you're not going to really understand the use of the topology without the terminology coming from Solid State physics (alignments of particles in crystals, spin chains, band gaps, etc).
Granted that, I highly recommend Witten's "Three Lectures On Topological Phases Of Matter". He ends up giving a description of our new Nobelist Haldane's model for the quantum Hall effect. He also explains the relation to Chern-Simons theory, which I think is the last remaining "topological concept" that your question is looking for. In a sense, this is analogous to the gauge-theoretic formulation of electrodynamics. To make "in a sense" rigorous, I recommend Simons' "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase". He gives the relation to the work of Thouless on the quantum Hall effect, and the key here is Berry's geometrical phase factor (which for example appears in Yang-Mills theory that describes electrodynamics).
To better bring in the relation to "topological phase", consider the aforementioned collection of particles with spins (a "spin configuration"). You can imagine a region where these spins rotate 360 degrees (a "vortex"). Traversing this path of rotated spins gives you a phase -- the analog is the Aharanov-Bohm effect (the attainment of a "topological phase" to the electron wavefunction as you traverse a non-simply connected region that contains a magnetic field). An example of a "topological phase transition" is that it may be energetically favorible for more or less vortices to appear, as temperature of the system is varied (an analog here is superconductivity where you transition from penetrating magnetic fields to expulsions of the fields, with appearances of vortices inbetween). This is the work of our new Nobelists Kosterlitz and Thouless.
More references for topologists, related to the quantum Hall effect:
The Fractional Quantum Hall Effect,
Chern-Simons Theory, and Integral Lattices (Frohlich's 1994 ICM talk)
Mathematical Aspects of the Quantum Hall Effect (by Frohlich)
The theory of topological phase is a beautiful new theory, but is lacking any consensus. Here are places one might look:
- Edward Witten Three Lectures On Topological Phases Of Matter
- Anton Kapustin Topological Field Theory, Higher Categories, and Their Applications
- Xiao Gang Wen Theory of Topological Order
- David Tong Resources on Fractional Quantum Hall Effect
Being an established physical theory is a far cry from being mathematically proven. We are still figuring out the details of old theories like GR, Statistical Mechanics and parts of Electromagnetism...
Physicists seem to move back and forth between Chern-Simons ⇋ FQHE ⇋ Topological Order, they are likely not the same. How to they reliably know which theory to use?
The "advanced" Nobel prize summary gives an excellent orientation: we get that the topological phase theory has its origins in the work of Kousterlitz and Thouless
1972
The XY Hamiltonian -- seems to be an $O(1)$ version of the Ising model allowing all spin angles $\theta$, not just $| \uparrow \rangle$ and $ | \downarrow \rangle$ -- is the sum over neighboring spins:
$$ H_{XY} = - J \sum_{\langle ij\rangle} \cos (\theta_i - \theta_j) $$
if we take the continuum limit (anyone who has taken a decent analysis may realize how hard this is to define): $$ H_{XY} = \frac{J}{2} \int d^2r \; \big( \nabla \theta(\,\vec{r}\,)\big)^2$$
so now $\theta \in [0, 2\pi]$ is a section of the trivial line bundle $\theta: \mathbb{R}^2 \to U(1)$ or whatever. Page 8 is of the notes is priceless
A simplification is to extend the range of the angular variable to $−\infty < \theta < \infty$ to get a free field Hamiltonian and thus Gaussian fluctuations Nobel, p8
In fact, expanding the range of angles from $\theta \in [0, 2\pi]$ to $\theta \in \mathbb{R}$ does not give the correct experimental answer. They are led to study the vorticity
$$ v = \frac{1}{2\pi} \oint_C d\vec{r} \cdot \nabla \theta(\vec{r})$$
and this leads to the Kosterlitz-Thouless phase transition:
$$ \lim_{r \to \infty} \langle e^{i(\theta_c(\vec{r}) - \theta(\vec{0}))}\rangle = \left\{ \begin{array}{lc} c_1 & T < T_c \\ c_2 e^{r/\zeta} & T > T_c\end{array}\right. $$
Over time, we learn this theory matched with experiment. Who knows why this back-of-the-envelope calculation worked and others did not.
1988
Some fairly classical invariants are defined which in turn match up with experiment leading up to the Fractional Quantum Hall Effect
$$ S = \frac{\theta}{4\pi} \int d^2x \;\vec{n} \cdot \big( \partial_1 \vec{n} \times \partial_2 \vec{n} \big) $$
This is the linking number + the main term (looks like wave equation)
$$ S = \frac{1}{2g} \int dt\, dx\, \big( \frac{1}{v} (\partial_t \vec{n})^2 - v ( \partial_x\vec{n})^2\big)$$
Again, why these particular predictions happened to work and not others is beyond me.
2015+
Also the starting point of the Witten notes is the search for dispersion relations of a similar kind:
We will start by asking under what conditions we should expect to find a relativistic dispersion relation for electrons in a crystal. Witten p3
Certainly this guy is an expert (but not the only valid perspective) in Chern-Simons theory. Mathematically these are all a disaster... I'd get kicked out of MO for writing equations like this :)
One thing that mildly confuses me about Witten's notes is that in 1D he writes - near $p = p_0$ the fermi energy
$$ H = v \int_\mathbb{R} dx \langle \psi | \frac{\partial }{\partial } | \psi \rangle $$
and in 3D he only writes a single $2 \times 2$ Hermitian matrix:
$$ H = a(p) + b(p) \cdot \vec{\sigma} + ... $$
by linear approximation any Hamiltonian should take this form. I mean in quantum mechanics $\vec{p} \leftrightarrow i\hbar \nabla$ so that's fine. We can put Dirac fermions in 3+1 dimensions.
The other problem is that Kousterliz-Thouless is in 2+1 dimensions... leading to vortices
Let me try to give a brief explanation: All this is in the context of Fermi liquid theory, the idea that you can describe the low energy physics of these kinds of systems by pretending they are generated by free fermions in an external potential. So, all you need to do is to solve the single particle problem for the external potential and then fill up the energy levels from the bottom until you reach the total particle number (or actually the density). It is tempting (and conventional) to call these particles electrons, and I will do so here, but of course actual electrons are not free but interacting. This "Fermi Liquid" explanation is just and effective description for long wavelength (the IR end of the renormalization group flow) where it turns out, that at those scales the interactions play no role (they are "irrelevant operators" in the language of the renormalization group).
The upshot is, we are dealing with free "electrons" and the previous paragraph was only essential if you want to connect to the physical world (but this is MATH overflow anyway).
Since the external potential comes from a lattice (crystal) it is invariant under lattice translations. So Bloch theory tells you, you can restrict your attention as far as solving the Schrödinger equation to wave functions living in the unit cell of the lattice. But you need to allow for quasi-periodic boundary conditions, i.e. when you go once around the unit cell you are allowed to pick up a phase. In fact, there is one phase for each generator of the first homotopy group of the unit cell. Each choice of these phases corresponds to one choice of boundary conditions for the wave function and you can compute the eigenvalues of the Hamiltonian for these given boundary conditions (the unit cell is compact so we expect discrete eigenvalues, bounded from below).
But these eigenvalues depend on the boundary conditions and you can think of the as a function of the phases. Each of the phases takes values in U(1) so the space of possible phases is a torus and you can think of the eigenvalues as functions on the torus. Actually, when going once around an irreducible cycle of the torus not all eigenvalues have to come back to themselves, you can end up with a permutation it this is not really a function but a section of a bundle but let's not worry too much about this as generally this "level crossing" does not happen in two dimensions and only at discrete points in 3D (this is Witten's argument with the 2x2 Hamiltonian above).
The torus of possible phases is called the "Brillouin zone" by physicists and its elements "inverse lattice vectors" (as you can think of the Brillouin zone as obtained from modding out the dual lattice of the lattice we started with).
Now if your electron density is N electrons per unit cell of the lattice Fermi Liquid theory asks you to think of the lowest N energy levels as occupied. This is the "Fermi level" or more precisely the graph of the N-th eigenvalue over the Brillouin zone. This graph (views as a hyper-surface) can have non-trivial topology and the idea is that by doing small perturbations to the system (like changing the doping of the physical probe or changing the pressure or external magnetic field or whatever) stuff behaves continuously and thus the homotopy class cannot change and is thus robust (or "topological" as the physicist would say).
If we want to inquire about the quantum Hall effect, this picture is also useful: The Hall conductivity can be computed to leading order by linear response theory. This allows us to employ the Kubo formula to compute it as a certain two-point function or retarded Green's function. The relevant operators turn out to be related to the N-th level wave function and how it changes when we move around in the Brillouin zone: If we denote by u the coordinates of the Brillouin zone and by $\psi_u(x)$ the N-th eigenfunction for the boundary conditions implied by u, we can define a 1-form $$ A = \sum_i \langle \psi_u|\partial_{u_i}|\psi_u\rangle\, du^i = \langle\psi_u|d_u|\psi\rangle.$$ This 1-form is actually the connection of a U(1) bundle and the expression the Kubo-formula asks us to compute turns out to be the first Chern number of that bundle (over the Brillouin zone).
Again that, as in integer, cannot change upon small perturbations of the physical system and this is the explanation of the levels in the QHE.
In modern applications, an important role is played by the (N-dimensional and thus finite dimensional) projector the subspace of Hilbert space spanned by the eigenfunctions corresponding to he N lowest eigenvalues, again fibered over the Brillouin zone. Then one can use K-theory (and KO-theory in fact) related to this projector to classify the possible classes of Fermi surfaces (these are the "topological phases of matter", as eventually, when the perturbation becomes too strong even the discrete invariants can jump which then physically corresponds to a phase transition).
Ah, I forgot to explain the Hamiltonian above attributed to Witten. The expression $$H(u) = a(u) + \vec b(u)\cdot\vec\sigma +\cdots$$ as nothing to do with spin but is just the most general form of a Hamiltonian in a two dimensional Hilbert space that depends on a parameter u (in the application the u as in my post above is a point in the Brillouin zone): The Pauli matrices $\sigma_i$ form a basis of traceless hermitian 2x2 matrices, and the a adds the trace. So any hermitian 2x2 matrix can be written in that form.
What Witten worries about is when the two eigenvalues (as function of u) can cross. The two dimensional Hilbert space is actually the two dimensional subspace of the physical Hilbert space spanned by the two eigenfunctions whose eigenvalues' crossing we want to study.
Naively, one might expect the condition $\lambda_1 = \lambda_2$ to determine a co-dimension 1 manifold in u-space (since it is a single real equation). But as always "naive" means actually "wrong": A 2x2 matrix with degenerate eigenvalues is not only similar (in the linear algebra sense) to a multiple of the unit matrix, it is a multiple of the unit matrix (since the unit matrix is diagonal in all bases)! So the level crossing only happens at points u where $\vec b(u)=0$. And these are three real equations, so generically, this happens on a co-dimension 3 manifold in u-space, i.e. it typically does not happen at all if u-space (and thus also the physical space relevant to the system) is two-dimensional and it only happens at discrete points in three dimensional u-space (around which $\vec b(u)$ can be approximated by a linear function $\sum_j B_{ij} u^j$ which then gives rise to "Dirac cones" with "metric" $B_{ij}$ as you hear about in in discussions of graphene, but that is a story for another day.
The upshot of this level-crossing analysis is that the picture of the Fermi surface as a eigenvalue graph over the Brillouin zone is actually valid and not complicated by eigenvalues changing occupancy as a function of u.
The story is slightly different if there is a symmetry (call "time reversal" by physicists) that tells you that (in an appropriate) basis, all matrix elements are real which means $b_y$ is always 0 and we end up with co-dimension 2. This "reality" leads to the appearance of KO-theory (as the real version of K-theory) and physically corresponds to the absence of magnetic fields (as those show up as the imaginary components of H-matrix elements).