Berry phase and Wilson loop

Question

According to the definition, the Wilson loop is \begin{equation} W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right] \end{equation} where $\mathcal{P} $ is the path ordering, $A_{\mu }$ is the gauge field.

And closed-path Berry phase is defined as \begin{equation*} \gamma _{n} =\oint _{c} A\cdot dR \end{equation*} where A is the Berry connection \begin{equation} A=i\ \langle n(R)| \nabla _{R}| n(R)\rangle \end{equation}

$| n(R)\rangle $ is the eigenstate of Hamiltonian $\hat{H}(R)$, which depends on parameter $R$.

Since they can both be viewed as holonomy , my question is under what condition could they be identical? Or when could Berry connection be interpreted as gauge field in Yang-Mills theory?

Answer 1

1. Both the usual gauge fields ($A_\text{QFT}$) and the Berry connection ($A_🍓$) are connections on a principal bundle.
2. In usual gauge theory in QFT you have a spacetime manifold, $S$ and a Lie algebra $\mathfrak{g}$. The gauge fields are $\mathfrak{g}$-valued one-forms on $S$, $A_\text{QFT}\in\Omega^1(S;\mathfrak{g})$.
3. In the Berry phase business you have a quantum mechanical theory defined on some space with a parameter space, $P$. Moreover, the underlying Lie algebra is $\mathfrak{u}(1)$. The Berry connection takes values in $A_🍓 \in\Omega^1(P;\mathfrak{u}(1))$.
4. They could be, therefore, formally identical, if the parameter space of your quantum mechanics is the spacetime manifold of an abelian gauge theory.
5. A difference between them is that the Berry connection is a background gauge field; namely fixed by the system, while in gauge theories you need to path-integrate over all possible connections. Hence the word formally, above.
6. You can always build an abelian gauge theory with the Berry connection as its gauge connection but you should not expect that it would be a very physical theory.
7. Finally, another thing you can do, if you have a map $f:M\to P$, and a QFT defined on $M$, is that you can pull back the Berry connection along $f$, so that $f^*A_🍓\in\Omega^1(M;\mathfrak{u}(1))$. Now $f^*A_🍓$ behaves like a usual abelian background gauge field, which you can now couple to your physical fields.