research31 min studyLesson 541 in Physics

Berry curvature in crystal momentum

Define band geometry with explicit phase and orientation conventions and separate local curvature from a quantized global invariant.

Before you begin

On this page

The question for today

What you’ll understand

Track Berry connection and curvature under a Bloch-state phase change.
Calculate a geometric phase with a declared loop orientation.
State the band-isolation and adiabatic assumptions behind a geometric interpretation.

The periodic part of a Bloch eigenstate changes as crystal momentum moves through a band. Its energy alone does not describe that change: overlaps between nearby eigenvectors define a geometry. Berry curvature measures the local geometric twist while remaining independent of a smooth local phase convention. Its sign requires a declared orientation and connection convention. A large curvature at one momentum is not, by itself, a proof that an entire occupied band has nonzero topology.

Declare the connection before assigning a sign

Write a Bloch state as a plane wave times its cell-periodic normalized eigenvector. For an isolated nondegenerate band, define the connection with a positive i multiplying the overlap with its momentum derivative. Rephasing that eigenvector changes the connection by minus the phase gradient. The curl removes this gradient locally, leaving a gauge-invariant curvature. Different texts choose the opposite connection sign, so comparing a curvature formula without its convention can reverse every later topological sign.

Crystal momentum components have inverse-length units, so connection components have length units and planar curvature has length-squared units. A curvature integral over momentum area is dimensionless. Smooth eigenvectors may require several patches over the Brillouin zone; nontrivial topology is precisely an obstruction to choosing one globally smooth periodic phase. Gauge singularities in a selected eigenvector are therefore not necessarily physical band degeneracies, although a true degeneracy invalidates an isolated-band description at that point.

\mathcal A_n=i\langle u_n|\nabla_{\mathbf k}u_n\rangle,\qquad |u_n\rangle\mapsto e^{i\chi}|u_n\rangle:\ \mathcal A_n\mapsto\mathcal A_n-\nabla_{\mathbf k}\chi,\qquad \Omega_{n,z}=\partial_{k_x}\mathcal A_{n,y}-\partial_{k_y}\mathcal A_{n,x}

The cell-periodic eigenvector is unit-normalized; χ is real and dimensionless. The ordered axes are (kx,ky). With dimensional k, A has units of length and Ω has units of length squared. Formulas apply within smooth patches of an isolated band.↔ Scroll wide equations horizontally.

A two-level eigenvector makes the curvature visible

For a two-level Hamiltonian, the direction of its Pauli-vector coefficient determines its eigenvectors. An eigenvector pointing along spherical angles theta and phi has connection in the azimuthal direction equal to minus the squared sine of half theta. Its curvature on the sphere is minus half the sphere's area form. The other eigenstate has the opposite curvature. Keeping track of which band is occupied is as important as keeping track of the orientation.

A closed-loop geometric phase is the oriented connection integral modulo two pi, with patch transitions included when necessary. It is separate from the dynamical phase obtained by integrating energy over time. Interpreting a slowly driven state through this phase requires it to remain in the selected band, with a nonzero gap and drive rates small enough to suppress transitions. Merely drawing a closed path of Bloch vectors does not establish that an actual time-dependent Hamiltonian satisfies those assumptions.

|u_+\rangle=\begin{pmatrix}\cos(\theta/2)\\e^{i\phi}\sin(\theta/2)\end{pmatrix},\qquad \mathcal A_\phi=-\frac{1-\cos\theta}{2},\qquad \Omega_{\theta\phi}=-\frac12\sin\theta,\qquad \gamma=\oint\mathcal A

θ and φ are dimensionless angular coordinates, so these connection and curvature components use the angular measure rather than dimensional momentum. The spinor is the + eigenstate of n·σ. γ is geometric phase in radians modulo 2π, with the path orientation explicitly specified.↔ Scroll wide equations horizontally.

Band geometry is a local ingredient of transport

Berry curvature enters the semiclassical motion of a wavepacket as an anomalous velocity perpendicular to its force. This treatment assumes a packet sufficiently localized in momentum to sample a band smoothly, with interband transitions negligible. Curvature can vary strongly within a band while its total integral remains unchanged under a smooth gap-preserving deformation. A local geometry measurement and a global topological classification therefore answer related but different questions.

When occupied bands cross one another but remain separated as a group from empty bands, assigning each one an individual smooth curvature can become inappropriate. The occupied projector and its matrix-valued connection provide the reliable description. Strong interactions or disorder can require many-body or real-space formulations instead of a crystal-momentum band map. These extensions preserve the need to identify a gap and a physical response, rather than making topology a label attached to any visually curved eigenvector plot.

Work it through

A latitude loop with a definite orientation

Use the displayed + spinor at fixed θ=60°. Let φ increase from 0 to 2π once while a gapped Hamiltonian follows adiabatically. Find its geometric phase, then reverse the traversal. Exclude the dynamical phase.

At θ=60°, cosθ=1/2, so the azimuthal connection is $\mathcal A_\phi=-(1-1/2)/2=-1/4$ . This sign follows from the stated positive-i connection and the + eigenvector.
Integrate with increasing φ: $\gamma=\int_0^{2\pi}(-1/4)d\phi=-\pi/2$ modulo 2π. Equivalently, the enclosed oriented solid angle is π and this spinor contributes minus half that angle.
Reversing the same path reverses the connection integral, giving +π/2 modulo 2π. No change of the band energy was used; adding an energy-time integral would mix a separate dynamical phase into this calculation.

The result

The forward loop gives −π/2 and the reversed loop +π/2 under the declared convention. A global phase redefinition changes neither closed-loop result modulo 2π.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Inspect local eigenvector geometry

Reset and set both controls to θ=60° and φ=0°; predict Bloch (√3/2,0,1/2). For a fresh comparison, Reset and set both controls to θ=60° and φ=90°; predict (0,√3/2,1/2). These points lie on the latitude used in the example. Playback adds coherent relative-phase precession, rather than verifying an adiabatic band protocol.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe lab displays pure two-level geometry and basis probabilities. It contains no crystal momentum, Hamiltonian gap, Berry-phase integral, or Chern-number calculation.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Differentiating the phase factor produces the minus-gradient term in this connection convention. Curl removes a smooth gradient, and a single-valued smooth gauge changes a closed-loop phase only by an allowed multiple of 2π.

Your turn

Try it yourself

A two-band massive Dirac patch has $H=v(k_x\sigma_x+k_y\sigma_y)+m\sigma_z$ , with m=2 meV and v=4 meV nm. For its lower band in the stated connection convention, use $\Omega_-(0)=v^2\operatorname{sgn}(m)/(2m^2)$ . Estimate the Berry phase of a small positively oriented patch of momentum area 0.010 nm⁻² centred at zero.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First determine the curvature's length-squared units. Approximate it as constant only when v|k| is much smaller than |m| throughout the patch.

Compare with the solution

The lower-band curvature at the centre is $16/(2\times4)=2.0\,\mathrm{nm^2}$ . Thus the small-patch phase is approximately Ω times oriented area, or +0.020 rad. The estimate assumes a sufficiently small patch that curvature variation is negligible. It is a local nonquantized flux, not a Chern number; integrating an unregularized Dirac patch over an infinite momentum plane is not the same as integrating a complete lattice Brillouin zone.

Keep the idea

Take it with you

Berry connection is gauge dependent, while its local curvature is gauge invariant.
Band choice, orientation, and connection sign must accompany geometric-phase formulas.
Local curvature becomes a topological invariant only after a suitable complete-domain integral.

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