Column 5 — The Berry Phase

Every quantum state carries a phase. Most of them vanish when you take a measurement, but some cannot be extinguished if you close the right loop. This column is about that specific kind of phase; a phase that depends only on the shape of a path.

The Phase That We Didn’t Know Was There

Let’s start with a general setup. Suppose you have a Hamiltonian that depends on some external parameter R. It could be anything such as nuclear separation in the molecule or the direction of a magnetic field. We can write down the Schrödinger’s equation:

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For a specific time t, we can define the energy eigenvalue as:

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As the parameters change slowly, a system starting in the n-th eigenstate stays in the corresponding n-th eigenstate, according to the adiabatic approximation. So the time-evolved state has the form:

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Applying Hamiltonian operator:

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Projecting this to the bra-vector:

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If we solve this differential equation of the coefficient C:

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The first exponential factor is the dynamical phase. This is trivial result that we often have seen when solving the simple Schrödinger’s equation. What we should focus on is the second one. If we define:

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then

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If we convert the time-derivative to a parameter-derivative using the chain rule,

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Here, the A is called the Berry connection, and γ is the geometric phase — the Berry phase. The important thing is that the phase depends only on the path C that R evolves through parameter space; no other dependencies such as time, energy or speed of the parameter changes. This is the key difference from the dynamical phase.

The Geometric Phase Cannot Be Gauged Away

When physicists first encountered the geometric phase, they did not much care about this. Because for an open path in parameter space, we can always choose the phase of the energy eigenstate through gauge transformation:

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Here the Berry connection shifts by:

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and the accumulated phase changes as:

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So for the open path such that

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we can shift γ by any amount we want by choosing appropriate ζ. So the phase is gauge-dependent, thus is not a physical observable.

However, if we consider a closed loop where the parameter always satisfies

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the phase shift from the gauge transformation vanishes, and

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Here, no matter what ζ we choose, the closed-loop integral of Berry connection is unchanged. What left here is a gauge-invariant physical quantity. This is Berry’s discovery: the geometric phase accumulated through a closed loop in parameter space is a genuine physical observable, independent of any phase convention.

In addition, if we use the Stokes’ theorem on the Berry phase formula:

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where Ω is the Berry curvature. The Berry phase can thus be interpreted as the flux of the Berry curvature through the enclosed surface S.

Spin-1/2 particle in a Rotating Field

To make this physically straightforward, let’s consider the simplest possible system that exhibits a Berry phase: a spin-1/2 particle in a magnetic field which slowly rotates. The Hamiltonian of this system is:

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where:

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If we solve for the eigenvalues and eigenvectors:

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We can see that the eigenstates depend on the orientation of the external magnetic field. Thus, as the field direction traces a closed path on the unit sphere, these eigenstates rotate continuously. We can compute Berry connection explicitly for the spin eigenstates. For the lower eigenstate:

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Integrating around a closed loop gives the Berry phase:

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where F is a Berry curvature tensor component and Ωs is the solid angle subtended by the path C from the origin. The same result holds for spin-up state, with the opposite sign.

Thus, the Berry phase is half the solid angle enclosed by the trajectory on the unit sphere. There is no energy, no time, no speed; only the area of the loop on S. A path that encircles the hemisphere (Ωs = 2π) accumulates a phase of π, regardless of whether it takes a second or a year.

3-Dimensional parameter sphere

There is one more thing worth mentioning. F is the Berry curvature component on the unit sphere of field directions, ignoring the magnitude of the field h. If we consider Berry phase in the full parameter space:

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where b is the curvature seen as a field in 3-dimensional parameter space and dS is the area element of the sphere of radius h. Comparing with the value we already obtained:

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This is exactly the field of a magnetic monopole sitting at the origin, where the two eigenstates become degenerate:

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Therefore, we can say that the degeneracy point radiates the Berry curvature.

Berry Phase in Crystal Structure

Discussion so far has been about a spin in a rotating magnetic field. But the Berry phase formalism assumed no specific system or parameter. It needs only two things: a Hamiltonian parameterized by some external parameter, and a closed loop in the parameter space.

According to the Bloch’s theorem, the wavefunction of an electron in a periodic potential is:

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and the Hamiltonian gives the energy eigenvalue:

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if we massage the Hamiltonian equation a bit, we can get a parameterized Hamiltonian:

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Here the crystal momentum k plays the role of external parameter.

Now the topology of the Brillouin zone becomes crucial. Periodic boundary condition in reciprocal space makes the opposite endpoints of the first Brillouin zone equal:

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producing the torus-shaped Brillouin zone in 2 dimension. As k traverses the torus, it is always travelling through a closed surface.

In consequence, for any closed path in the Brillouin zone, the Bloch wavefunction accumulates a Berry phase:

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The Berry curvature — the local field whose flux gives the Berry phase — is defined at every point in the Brillouin zone:

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given that:

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we can conclude:

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and thus, the Berry curvature:

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In analogy with electrodynamics, the Berry connection A is a vector potential in k-space, and the Berry curvature is a magnetic field in the Brillouin zone. It is not a handy interpretation; it has a physical consequences. For instance, when an electric field is applied to a crystal, the Berry curvature generates an anomalous velocity:

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The first term is group velocity, and the second term deflects electrons sideways, perpendicular to both the E-field and the Berry curvature, without presence of an external magnetic field. This is called the anomalous Hall velocity.

Let’s wrap up. We started from the adiabatic approximation to derive the Berry phase, and defined key physical quantities along the way, such as the Berry connection and Berry curvature. We then got a more intuitive understanding of the Berry phase formalism through the example of the spin-1/2 particle. Finally, we applied this concept to a crystal structure. In the next column, we will further dive into various physical consequences that stem from the Berry curvature, including the definition of the Chern number, quantized Hall conductance, and more.

References

• Girvin & Yang, Modern Condensed Matter Physics (2019)
• Sakurai & Napolitano, Modern Quantum Mechanics (2nd ed.)