A Comment On Berry Connections

Understand

When families of quantum systems are equipped with a continuous family of Hamiltonians such that there is a gap in the common spectrum one can define a notion of a Berry connection.

• In this note we stress that, in general, since the Hilbert bundle defining the family of quantum systems does not come with a canonical trivialization there is in fact not a single Berry connection but rather a family of Berry connections.
• Two examples illustrate that this remark can have physical consequences.