In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

Let's start with a simple but important example:

Let's consider the set of all the curves in the plane that don't pass thought the origin and that don't intersect themeseves. We can divide this set in two main subsets:

The origin in represented with a black dot. On the left the different equivalence classes are shown in different colors, and on the right you can see what happens when we apply noise

The set of curves that circle around the origin (orange)
The set of curves that don't circle around the origin (blue)

You cannot deform continously a orange curve in to a blue one without crossing the origin, but you can deform continously a curve to reach one of the same color

So we say that the two types of curves have different topologies, or alternatively that they form two different classes of equivalence.

As you can see from the image below, if we add noise, most of the time the curves will keep their topology. However, if the noise is strong enough, or the curve close enough to the origin, the curve sometimes changes its topology.

For a physicist, things that are inherently topological in nature are important because they are resistant to noise. For example, in quantum computing the main bottleneck comes from envorinment noise. Because of this reaserchers are trying to explore the possibility of using topological effects to execute noise-resistant computations.

The first example of topology in physics comes from Gauss's law $$ \oint \mathbf{E}\cdot d\mathbf s= \frac {1}{\epsilon_0} Q_{\textrm{inside}} $$ Lets say that we only have to work with one charge, then there are two types of closed surfaces:

The reason I bring this up is because with this we have a way to actually calculate the topology of something instead of relying on what you eyes see

There are a few other examples of things that link to topology in ElectroMagnetism, but we are going to skip them to jump straitgh into Quantum Mechanichs!

To start off we are goin to talk about a very specific case of a quantum particle in a ring around a infinitely long solenoid Wait! isn't it harder to start off with Quantum Mechanichs + Electromagnetism? Why don't just use QM?
Believe it or not, it's harder to use QM without EM for topological stuff, plus when we'll do that, we'll use tools that are better understood by doing analogies with the QM+EM case

The hamiltonian of a particle with charge $-e$ moving through a magnetic field $\mathbf B= \nabla \times \mathbf A$ is $$ H=\frac 1{2m}(\mathbf p + e\mathbf A)^2 $$

Since $$\mathbf p=p_\theta \hat\theta=-\frac{i\hbar\mathbf {\hat \theta}}{R}\frac\partial {\partial \theta}$$ $$ \oint \mathbf A\cdot d\mathbf r=\int \mathbf B\cdot d\mathbf s = \Phi $$ it means that $$ \mathbf A=\frac \Phi{2\pi R} \mathbf {\hat \theta} $$ Putting it back into the hamiltonian $$ H=\frac 1{2m}\bigg(-\frac{i\hbar}{R}\frac\partial {\partial \theta} + \frac{e\Phi}{2\pi R}\bigg)^2 $$

Here plotted the energies $E_n$ (eq. ), notice how the state "flow" as we change $\Phi$

The eigenstates of this hamiltonian are $$ \psi_n(\theta)=\frac {e^{in\theta}} {\sqrt{2\pi R}}; \quad n\in \mathbb Z $$ Interestingly the Energies of the eigenstates are influenced by the vector potential $$ E_n=\frac 1{2mR^2}\bigg(\hbar n+ \frac{e\Phi}{2\pi}\bigg)^2=\tilde E\bigg(n+\frac{\Phi}{\Phi_0}\bigg)^2 $$ where $$ \tilde E=\frac{\hbar^2}{2mR^2} \quad \textrm{and} \quad \Phi_0=\frac{2\pi \hbar}e $$

Suppose now that we start with the turned soleind off, and place the particle in the $n=0$ ground state. If we increase the flux then, by the time we have reached $\Phi=\Phi_0$ , the $n=0$ state has transformed into the state that we previously labelled $n = 1$. Similarly, each state $n$ is shifted to the next state, $n + 1$ It is tempting to invoke the adiabatic theorem here but, because of level crossing at $\Phi=\Phi_0/2$ it is not valid

This is an example of a phenomenon is called spectral flow: under a change of parameter the spectrum of the Hamiltonian changes, or “flows”. As we change increase the flux by one unit $\Phi_0$ the spectrum returns to itself, but individual states have morphed into each other.

The keen eyed among you might have noticed that Figure 3 is suspicially similar to a crystal band structure in the limit on which the periodic potential $V\to 0$ with periodicity $2\pi$, let's see if this analogy holds the test of math. Let's start by taking the single-particle free propagating Hamiltonian $$H=\frac 1{2m}p_\theta^2$$

The eigenstates this time have to respect the condition that $u_{n,q}(\theta)=e^{i2\pi q}u_{n,q}(\theta+2\pi)$.This come from the block's theorem that states that the eigenstates of the hamiltonian of a periodic potential with periodicity $a$ mush obey that $u_{n,q}(x)=e^{iaq}u_{n,q}(x+a)$. In our case the periodicity $a=2\pi$ and the variable of the function is $\theta$ instead of $x$ This also means that if we substitute $q\to q+1$ the spectrum doesn't change A brief proof of this is the following:
The problem of the eigenvalues can be written like so $$ He^{iq\theta}u_{n,q}(\theta)=E_n(q)e^{iq\theta}u_{n,q}(\theta) $$ if we now send $q\to q+1$ the equation above becomes $$ He^{iq\theta}e^{i\theta}u_{n,q+1}(\theta)=E_n(q+1)e^{iq\theta}e^{i\theta}u_{n,q+1}(\theta) $$ The terms $u'_{n,q}(\theta)\equiv e^{i\theta}u_{n,q+1}(\theta)$ are also periodic $u'_{n,q}(\theta)=u'_{n,q}(\theta+2\pi)$. If we put it back into the equation above we get $$ He^{iq\theta}u'_{n,q}(\theta)=E_n(q+1)e^{iq\theta}u'_{n,q}(\theta) $$ That is completely equivalent to the equation at the start of this footnote, this means that the eigenvalues are periodic with periodicity 1

We now make the following unitary transformation to obtain a $q$-depentant Hamiltonian. $$ H(q)=e^{-iq\theta}He^{iq\theta}=\frac 1{2m}\bigg (p_\theta + \frac {\hbar q}R \bigg)^2 $$ We can easly map this hamiltonian to the one in equation just by difining $\Phi$ such that $\hbar q= \frac {e\Phi}{2\pi}$. $$ H(\Phi)=\frac 1{2m}\bigg(p_\theta + \frac{e\Phi}{2\pi R}\bigg)^2 $$ Since before we said that the system doesn't change if we substitute $q\to q+1$, it means that now the system remains unchanged if we send $\Phi \to \Phi+\Phi_0$. This is exactly the result obtained in the previous section! ()

And The tranformed eigenstates $\psi_{n,q}=e^{-iq\theta}u_{n,q}$ is just the cell-periodic part of the Bloch function. It satified the stricter periodic boundary condition $$\psi_{n,q}(\theta)=\psi_{n,q}(\theta +2\pi)$$ As you can see equations and create a system that is mathematically equivalent to the particle moving in a ring around a flux tube (subsection )

The spectral flow is applicable in much more complex geometries than the one we have seen so far.
Suppose that now the particle can moove in a 3D potential $V(\mathbf r)$, the Hamiltonian is $$ H(\mathbf A)=\frac 1{2m}(\mathbf p + e\mathbf A)^2 + V(\mathbf r) $$ Since the solenoid is still the same, the formula for $\mathbf A$ remains unchanged (eq. ) $$ H(\Phi)=\frac 1{2m}\bigg(\mathbf p + \frac{e\Phi}{2\pi R}\hat \theta\bigg)^2 + V(\mathbf r) $$ and since it's expressed in cylindrical coordinates it's better to express also $\mathbf p$ in cylindrical coordinates. $$ \mathbf p =-i\hbar \mathbf \nabla=-i\hbar\bigg( \mathbf{\hat r}\frac{\partial}{\partial r}+ \frac{\mathbf {\hat \theta}}{r}\frac{\partial}{\partial \theta} + \mathbf{\hat z}\frac{\partial}{\partial z} \bigg) \equiv \mathbf{\hat r} p_r+ \mathbf {\hat \theta} p_\theta + \mathbf{\hat z} p_z $$ Of course if we send $\theta \to \theta + 2\pi$ the system should be unchanged. $$ \psi(r,\theta,z)=\psi(r,\theta+2\pi,z) $$ Following the inverse reasoning done in subsection we make the following unitary transformation. $$ H=e^{i\theta\Phi/\Phi_0}H(\Phi)e^{-i\theta\Phi/\Phi_0}=\frac {\mathbf p^2} {2m} + V(\mathbf r) $$ This means that the eigenvalue probelm is now written like so $$ He^{i\theta\Phi/\Phi_0}\psi(r,\theta,z)=E(\Phi)e^{i\theta\Phi/\Phi_0}\psi(r,\theta,z) $$ If we send $\Phi \to \Phi+\Phi_0$ we get an equivalent equation $$ He^{i\theta\Phi/\Phi_0}\psi(r,\theta,z)=E(\Phi+\Phi_0)e^{i\theta\Phi/\Phi_0}\psi(r,\theta,z) $$ this means that the energy spectrum is unchanged if we send $\Phi \to \Phi+\Phi_0$. This is true regardless of the shape or geometry of $V(\mathbf r)$.

We'll now see the effects of the spectral flow on physical properties of materials. Suppose we have a system like the one of the figure on the side. Now we slowly increase $\Phi$ from 0 to $\Phi_0$ in a total time $T$. This introduces a electromagnetice force arround the ring $\mathcal{E}=-\partial_t\Phi=-\Phi_0/T$.

Let's suppose that the disc has the property that due to spectral flow $n$ electrons are transferred form the inner cirlce to the outer cicle in this time $T$. This would result in a radial current $I_r=-ne/T$. This means that the resistance is $$R_{xy}=\frac{\mathcal {E}}{I_r}=\frac{2\pi\hbar}{e^2}\frac 1n$$

However, to be able to calculate $n$ we need to calculate how is the spectrum of the system as we change $\Phi$. This means that $n$ depends on the system, but equation is independent of the system. There is the caveaut here that the system has to be in a defined quantum state, so in real world system it means that $T\approx0$

What does Topology have to do with Physics?