A question on the Chern number and the winding number?

Question

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). The Chern number of $\mid \psi(x,y) \rangle$ then reads $$C=\frac{1}{2\pi i}\int \text{Tr}(P[\partial_xP,\partial_yP])$$ where $P=\mid\psi \rangle \langle \psi \mid$ is the projector.

When $d=2$, the projector can be written as $P=\frac{1}{2}(1+\mathbf{n}\cdot \mathbf{\tau})$, where the unit vector $\mathbf{n}(x,y)$ maps the closed surface to $S^2$ and $\mathbf{\tau}=(\tau_x,\tau_y,\tau_z)$ the $2\times2$ Pauli matrices. Now the above Chern number can be rewritten as $$W=\frac{1}{4\pi}\int \mathbf{n}\cdot(\partial_x\mathbf{n}\times\partial_y\mathbf{n})$$ which is the winding number that counts the times of wrapping $S^2$.

My question is: What about $d>2$, can the Chern number still be interpreted as some kind of winding number similar to the above $d=2$ case?

This post imported from StackExchange Physics at 2016-06-26 09:53 (UTC), posted by SE-user Kai Li

Answer 1

The Chern number you mention is the thing you get when you integrate a particular two-form over a surface. It turns out that this two form represents the first Chern class of the system (the system, in this case, consists of the parameter space and a line bundle describing the relative Berry phase along paths in the parameter space). The most important things about the first Chern class are that 1) it is a topological invariant of the system, and 2) if the parameter space is 2-dimensional you can integrate it over the parameter space to obtain a number which will also be a topological invariant of the system.

If your parameter space has dimension $d>2$, then you can still define the first Chern class, but now you will only be able to integrate it over 2 dimensional subspaces of the parameter space. It will still measure a winding number, as you mentioned, but now this winding number doesn't depend only on the system itself, but also on your choice of subspace. For this reason, it is harder to think of the numbers we get by doing these sorts of integrals as topological invariants of the system, though they can have other useful interpretations.

There are generalizations of this setup where, instead of measuring an abelian Berry phase that takes values in $U(1)$, we can measure non-abelian "phases" (holonomy is a better word) which take values in more complicated Lie groups like $SU(n)$. This sort of thing happens when, in the adiabatic theorem, you remove the assumption that the ground state is non-degenerate. In these cases you can define the higher Chern classes associated to the system (in the case of abelian phases all these higher classes vanish). Whereas the first Chern class was something you could integrate over a surface, the second Chern class can be integrated over a 4 dimensional manifold, the third Chern class over a 6 dimensional manifold, and so on. Moreover, just as with the first Chern class, there are nice formulas for forms representing these classes in terms of the analogue of the Berry curvature. If your parameter space is $2d$ dimensional, thenntegrating the $d$-th Chern class over the parameter space will give you a topological invariant of the system (for the abelian case, when $d > 1$ this Chern class vanishes, so the topological invariant doesn't tell you anything useful).

Another example where the second Chern class appears is in $SU(2)$ Yang-Mills theory on $\mathbb{R}^4$, where the value of the action on an (anti) self-dual connection measures the "winding number" of a given decay condition on the fields. More precisely, if we require the connection to decay (up to gauge transformation) at infinity, then this is the winding number of a map from a sphere $S^3\subset \mathbb{R}^4$ of very large radius to the gauge group $SU(2) \cong S^3$.

This post imported from StackExchange Physics at 2016-06-26 09:53 (UTC), posted by SE-user Sean Pohorence