In fiber bundle picture of Berry connection, what is the vertical basis if the horizontal basis is the underlying parameter space?

Question

In Ref. [1], the authors show how

The geometric (Berry) phase is shown to have its origin in the nontrivial geometry of the fiber bundle: Hilbert space --—> space of states. The nontrivial geometry comes simply from the scalar product in Hilbert space.

Part of their derivation considers the tangent space of a point on a closed, adiabatic loop over some parameter space. They show that the tangent vectors in tangent space can be decomposed into vertical and horizontal parts. The state evolved by Schrodinger's equation is vertical, making the horizontal part zero (parallel transport?). Then, the covariant derivative that makes the basis of the horizontal part is used to show that the Berry connection $A_\mu$ can be written entirely in terms of the scalar product on Hilbert space. That is, the covariant derivative

$$ D_\mu = \frac{\partial}{\partial X^\mu} + A_\mu \frac{\partial}{\partial\theta}, $$

gives

$$ \langle m| D_\mu |m \rangle= \langle m|\frac{\partial}{\partial X^\mu} + A_\mu \frac{\partial}{\partial\theta}|m \rangle=0 \implies A_\mu = \langle m|\frac{\partial}{\partial X^\mu}|m \rangle. $$

This shows that the Berry phase is entirely geometric, as it can be defined entirely in terms of the horizontal parameter space coordinates $ X^\mu$ (such as $k$-space?). My question is, in a condensed matter system, if a function depending on $ X^\mu$ might make it depend on $k$ variables, what would a corresponding variable be for the $\theta$ variable? From the reference, it seems as if $\theta$ 'is defined by the fibers' (i.e. tangent to the fibers), and can correspond to the dynamical phase, or some arbitrary $U(1)$ phase of a state. The reference says:

Before choosing a horizontal subspace (connection) we will identify the vertical subspace (or vertical direction). The action of the group U(1) generates the fibers. Each element of a fiber points in the same direction (they just differ by a phase). This direction generated by the U(1) action is called the vertical direction.

However, I am still trying to understand this work, and so am not entirely sure what the $\theta$ variable for the vertical subspace means in less abstract notation. Is it just some arbitrary phase factor, or is it both the dynamical and geometric phase together? I am looking for familiar notation/ideas to help me understand this. Any clarification?

[1] Bohm, A., Boya, L. J., & Kendrick, B. (1991). Derivation of the geometrical phase. Physical Review A, 43(3), 1206–1210. doi:10.1103/physreva.43.1206

Answer 1

The $\theta$ variable is simply a representation of the ignorance that you have on determining a wave-function because what actually is physically observable is the state (or rather expectation values of the state).

When you have a Hilbert space, your (normalized) wave-function is a vector $|\psi\rangle$. Multiplying by a phase factor $|\psi\rangle \to e^{i\theta}|\psi\rangle $ does not change the state $|\psi\rangle\langle\psi|$.

If you have a curve of vectors $t\mapsto e^{i\theta(t)}|\psi\rangle$, this is equivalent to a constant curve of states. The velocity vector of the curve is

$i\frac{d\theta}{dt} (t)|\psi(t)\rangle$

Since what we have done is simply a gauge transformation, the associated tangent vector is "vertical". Meaning that the curve is moving along the fiber of the bundle of normalized vectors over the space of states.

For a general curve of states, the Berry connection gives you a decomposition of tangent vectors into vertical and horizontal tangent vectors. The horizontal projection gives you a unique lift of the tangent vector downstairs. If the tangent vector is always horizontal, this means that it's vertical projection is zero or, in other words, that the wave function is the result of parallel transport of the initial wave function with respect to the Berry connection along the curve of states downstairs.

The statement of the adiabatic theorem, ignoring the dynamical phase, is that the wave function at time $t$ is the result of parallel transportation of the wave-function at the initial time (initial vector, with respect to the Berry connection, along the curve of states induced by the curve in parameter space.