# A Is the Berry connection a Levi-Civita connection?

#### Joker93

Hello!
I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric).
When performing a parallel transport with the L-C connection, angles and lengths are preserved.
Now, I am trying to learn about the Berry phase which is full of differential geometry. What I want to learn is whether or not the Berry connection is a Levi-Civita connection.

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#### Orodruin

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No, the Berry connection is not a Levi-Civita connection. The Levi-Civita connection is a particular connection on the tangent bundle of a manifold given a metric. The Berry connection is a connection for the U(1) fiber bundle over the appropriate projective Hilbert space.

#### Joker93

No, the Berry connection is not a Levi-Civita connection. The Levi-Civita connection is a particular connection on the tangent bundle of a manifold given a metric. The Berry connection is a connection for the U(1) fiber bundle over the appropriate projective Hilbert space.
Now that I think about it, I think that this can also be seen through physical arguments. Since under adiabatic change each energy eigenstate might pick up its own non-trivial Berry phase, this can lead to interference effects, which shows that the parallely transported states(which pick up the geometric phase) do not preserve the angles between them.
Is this view right?

#### Orodruin

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What you are talking about here is the inner product of states in the Hilbert space itself, not its tangent vectors - which is what the metric, and therefore the Levi-Civita connection, is related to. The tangent bundle and the U(1) bundle are simply different objects, which means that connections on them will be different objects.

#### Joker93

What you are talking about here is the inner product of states in the Hilbert space itself, not its tangent vectors - which is what the metric, and therefore the Levi-Civita connection, is related to. The tangent bundle and the U(1) bundle are simply different objects, which means that connections on them will be different objects.
So, if Berry phase has to do with the U(1) bundle, that is why the various geometric quantities have to do with the states of the Hilbert state, |n>, and the tangent vectors,|dn>, rather than just the tangent vectors(which is the what the L-C connection is about)?

#### Jimster41

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Does U(1) here just refer generically to the circle bundle?

I am trying to understand if QM Berry Phase is only relevant to Electro-Magnetism. All the examples I’ve seen of it’s derivation are given w/respect to quantum spin. Does a similar real geometric phase shift emerge from the calculation of closed path integrals of position and momentum? I think this is a question similar to the OP.

Also the along same line I’m trying to understand the relationship, if any, between Berry Phase and non-zero commutators.

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"Is the Berry connection a Levi-Civita connection?"

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