The first thing to observe is that the complex classes only have \(\mathbb{Z}\) invariant. We already know what these invariants are in low dimensions: the Chern number, which we encountered in quantum Hall systems (class A, \(d=2\)), and the winding number of the reflection matrix, which we encountered when we studied reflection from the Thouless pump.
The higher dimensional invariants are simple generalizations of these two. Their mathematical expression can be found in several papers, for instance this one.
Another useful feature of the table is that in a given column, all \(\mathbb{Z}\) or \(2\mathbb{Z}\) entries, which are grouped by the color gradients below, have the same topological invariant.
We can check this statement for some cases we know. For instance, in \(d=0\) the \(\mathbb{Z}\) topological invariant is the number of filled energy levels, which applies to quantum dots with broken time-reversal symmetry (class A), spinless time-reversal symmetry (class AI) and spinful time-reversal symmetry (class AII, which has \(2\mathbb{Z}\) because of Kramers degeneracy). In \(d=2\), the \(\mathbb{Z}\) topological invariant is the Chern number, and we saw how it applies to both the Chern insulators in class A and the \(p\)-wave superconductor in class D.
An important pattern visible in the table is the descending sequence \(\mathbb{Z} \,\to\,\mathbb{Z}_2\,\to\,\mathbb{Z}_2\) that appears in every symmetry class. That is, starting from the \(\mathbb{Z}\) invariant, reducing the dimensionaility twice by one we encounter two \(\mathbb{Z}_2\) invariants in a row:
Again, this dimensional reduction can best be understood with an example we already know. Consider the symmetry class \(D\). In \(d=2\) it has a \(\mathbb{Z}\) topological phase, the \(p\)-wave superconductor.
If you take a finite ribbon of a \(p\)-wave superconductor, it will have an integer number of edge states, as determined by the Chern number. Let’s now imagine that you take a ribbon and roll it up into a thin, long cylinder, by pasting two of its edges together. The two remaining edges at this point form a circle.
You can now view this cylinder as a one-dimensional system whose ends are the two rolled-up edges, and ask: how many zero-energy Majorana modes can there be at the ends? We know the answer from last week’s material: the number of zero-modes can be zero or one, depending on whether the boundary conditions are periodic or anti-periodic. The topological invariant is thus reduced to \(\mathbb{Z}_2\). This is no surprise, since the system is topologically in the same class as the Kitaev chain.
We can proceed further with our dimensional reduction. If we take our one dimensional system and make it into a ring, we obtain a zero-dimensional system. Depending on how the two ends are coupled, the two Majorana modes can favour the even or odd fermion parity state, and this quantity cannot change without a Fermi level crossing. This is the \(\mathbb{Z}_2\) invariant of zero-dimensional systems in class D.