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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

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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.

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  • $\begingroup$ Thanks for the answer. However, now I wonder, what would it mean in the fiber bundle picture if the final result was somehow $\langle m | \partial_\theta m\rangle$ and not $\langle m | \partial_X m\rangle$? That is, only the quantity in the vertical direction survives. Does this mean that it is some form of gauge-dependent velocity/transport? Or can it depend on base manifold coordinates as well (because the covariant derivative depends on the base manifold too)? I know there won't be parallel transport in this case, but I am making sure I understand correctly. $\endgroup$ – TribalChief Nov 25 '20 at 21:43
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    $\begingroup$ You can have a curve which is purely vertical, but that essentially is something that doesn't change the state as I said before. Something like this occurs when you take a Hamiltonian depending on some parameters, say $x\in M$ where $M$ is some smooth manifold, but, for some reason, one eigenstate is $x$ independent but it's energy is $x$ dependent. Then, the adiabatic theorem would give you an evolution of the form $|\psi(t)\rangle =e^{-i\int_{0}^t E_0(x(s))ds} |\psi_0\rangle$, where $|\psi_0\rangle$, independent of $x$, satisfies $H(x)|\psi_0\rangle=E_0(x)|\psi_0\rangle$, for $x\in M$. $\endgroup$ – B. Mera Nov 25 '20 at 23:07
  • $\begingroup$ @B-Mera, Thanks, it somewhat makes sense, but I am still a bit confused. Is the phase in your comment essentially the dynamical phase (phase due to E)? Or is it separate from that? $\endgroup$ – TribalChief Nov 26 '20 at 4:34
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    $\begingroup$ You are welcome! Yes, it is the dynamical phase. I've considered this special case to illustrate the point. This is the kind of situation you expect when topology is not relevant and the dominant contribution comes from the energy of the configuration. $\endgroup$ – B. Mera Nov 26 '20 at 11:44
  • $\begingroup$ @B-Mera, thanks for clarifying. $\endgroup$ – TribalChief Nov 26 '20 at 23:48

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