Each row in the table corresponds to a certain symmetry class, that is to a given combination of the presence or absence of three fundamental discrete symmetries.
You already encountered these three symmetries all the way back in week one. They are time-reversal symmetry (\(\mathcal{T}\)), particle-hole symmetry (\(\mathcal{P}\)) and chiral symmetry (\(\mathcal{C}\)). We previously referred to chiral symmetry \(\mathcal{C}\) as sublattice symmetry. This is because in condensed matter physics, a natural realization of chiral symmetry is a system composed of two sublattices, such that sites in one lattice only couple to sites in the other.
Why do we consider these symmetries fundamental, and restrict the periodic table to them only?
As you may recall from week one, and as Shinsei Ryu explained in the introductory video, normal unitary symmetries of a Hamiltonian do not have particularly interesting consequences for topological classification. One may always make the Hamiltonian block-diagonal, reducing the problem to the study of a single block. This procedure can be repeated until one runs out of unitary symmetries and is left with an irreducible block of the Hamiltonian, i.e. one which cannot be block diagonalized.
Time-reversal, particle-hole and chiral symmetries act in a different way. They impose certain constraints on an irreducible Hamiltonian - for instance, by forcing it to be a real matrix, or to be block off-diagonal.
Note, however, that it is possible to extend the periodic table to study the interplay between these three discrete symmetries and other unitary symmetries. For instance, we have already touched upon crystalline symmetries in week seven, and we will return to them in week ten.
But for now, let’s focus on the three fundamental discrete symmetries: \(\mathcal{P}\), \(\mathcal{T}\) and \(\mathcal{C}\). Their basic properties are:
Recall that an anti-unitary operator can be written as the product of a unitary operator and the complex conjugation operator \(\mathcal{K}\). The next important thing to know is that time-reversal and particle-hole symmetry may come in two separate flavors, depending on whether they square to plus or minus one.
For instance, you will recall that for the time-reversal operator acting on electronic states, \(\mathcal{T}^2=-1\). This was the crux of Kramers theorem, which in turn was the key to topological insulators. If you go back to week one, you will also remember that we discussed real matrices, which were symmetric under a time-reversal operator \(\mathcal{T}=\mathcal{K}\). This operator satisfies \(\mathcal{T}^2=1\).
Thus, a system can behave in three ways under time-reversal symmetry \(\mathcal{T}\): (1) it does not have time-reversal symmetry, (2) it has it and \(\mathcal{T}\) squares to \(+1\), (3) it has it and \(\mathcal{T}\) squares to \(-1\). The same holds for particle-hole symmetry, which can also have \(\mathcal{P}^2=\pm 1\). On the other hand, the chiral symmetry only comes in one flavor, \(\mathcal{C}^2=1\).
How do we arrive to having ten symmetry classes? Let’s count all the possible cases carefully. By combining the three cases for \(\mathcal{P}\) and the three cases for \(\mathcal{T}\) we arrive at nine possible combinations.
The important thing to notice now is that \(\mathcal{C}\) is not completely independent from \(\mathcal{T}\) and \(\mathcal{P}\). Whenever a system has both \(\mathcal{T}\) and \(\mathcal{P}\), there is also a chiral symmetry \(\mathcal{C}=\mathcal{P\cdot T}\).
This also means that if a system only has either \(\mathcal{T}\) or \(\mathcal{P}\) but not both, it cannot have a chiral symmetry \(\mathcal{C}\). In other words, the presence of any two out of the three symmetries implies that the third is also present.
On the other hand, if both \(\mathcal{P}\) and \(\mathcal{T}\) are absent, then \(\mathcal{C}\) may or may not be present. This gives us two distinct cases.
Adding all the possibilities, we indeed find 10 symmetry classes:
\[(3\times 3 - 1) + 2 = 8 + 2 = 10\,.\]
The first term in the sum corresponds to the eight cases where there is at least one anti-unitary symmetry: either \(\mathcal{P}\), or \(\mathcal{T}\), or both. These eight symmetry classes are called real, because an anti-unitary symmetry always involves the complex conjugation operator. This does not necessarily mean that the Hamiltonian is a real matrix, but it is a reminder that there is a constraint between its real and imaginary parts.
The second term in the sum covers the two cases when there are no anti-unitary symmetries. These symmetry classes are called complex.
Let’s have another look at the 10 rows in the table, this time specifying which combination of the three fundamental symmetries each row has:
The somewhat cryptic notations in the leftmost column are just the names of the different symmetry classes. (Also the ‘I’s appearing there are Roman cardinal numbers, so for instance BDI is ’B D one’, and AIII is “A three”.)
Their names come from an elegant mathematical classification of symmetric spaces worked out by Elie Cartan in 1926. While it is definitely intriguing that a group theory result from 1926 reappears in a totally different context almost 80 years later, the origin of this nomenclature is not directly relevant to most of the theory done in the field. The two complex classes are A and AIII.
Finally, let us make an extra observation:
While the particle-hole symmetry appears in any superconductor, it must satisfy \(\mathcal{P}^2 = -\mathcal{T}^2\) due to the way the Bogoliubov quasiparticles are related to the original electrons. This makes the symmetry classes BDI, and CII rely on a fine-tuned Hamiltonian, just like the symmetry class AIII.