advanced30 min studyLesson 410 in Physics

Aharonov–Bohm phase and holonomy

Calculate a magnetic flux phase with explicit charge and orientation conventions and identify the topology of the accessible region.

Before you begin

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The question for today

What you’ll understand

Derive the gauge-invariant phase difference between two paths around excluded magnetic flux.
Track charge signs, winding orientation, and flux units in SI.
Explain why a flat connection can have nontrivial holonomy in a multiply connected region.

A charged matter wave can acquire a measurable interference shift even when the magnetic field vanishes along both accessible paths. In the magnetic Aharonov–Bohm arrangement, flux is confined to an excluded region that the paths enclose. The observable depends on a loop integral of the vector potential and is gauge invariant. The lesson is a precise statement about coherence and global geometry; it does not make a locally gauge-dependent potential itself an independently measurable number.

A path phase and its gauge transformation

In SI units use minimal coupling $H=(\mathbf p-q\mathbf A)^2/(2m)+qV$ , where $q$ is the signed charge. The magnetic contribution to the action along a path is the charge times the line integral of the vector potential. Dividing by Planck's reduced constant gives a dimensionless phase. Under a gauge change adding the gradient of a scalar to the vector potential, the wavefunction also receives a position-dependent phase. Both changes must be included when comparing amplitudes.

A phase assigned to one open path depends on its endpoint convention. For two paths with the same initial and final points, subtracting their phases cancels the gauge endpoint terms. Define the closed loop by traversing path one forward and path two backward, and choose the flux orientation by the corresponding right-hand rule. Reversing this loop reverses the phase. Reversing the charge does too. These two sign choices are often hidden when a formula is written using the positive electron charge magnitude.

\Delta\varphi_{\rm AB}=\frac q\hbar\left(\int_{P_1}\mathbf A\cdot d\mathbf l-\int_{P_2}\mathbf A\cdot d\mathbf l\right)=\frac q\hbar\oint_C\mathbf A\cdot d\mathbf l=\frac{q\Phi_B}{\hbar}

The loop orientation is P1 followed by reversed P2. Phi_B is the signed enclosed flux in webers. The formula is in SI, with q the signed charge; Gaussian-unit formulas contain an additional c.↔ Scroll wide equations horizontally.

Zero local field need not mean trivial holonomy

Outside an ideal long flux tube, the magnetic field can vanish while the vector potential has a nonzero circulation around the excluded tube. Locally, on a sufficiently small simply connected patch, that potential can be removed by a gauge choice. A single-valued global gauge cannot generally remove its circulation throughout the accessible multiply connected region. The obstruction is visible through the phase factor accumulated around a closed loop, which is a holonomy of the electromagnetic connection.

Stokes' theorem is consistent with this result. A surface spanning the loop crosses the excluded flux region, so the full surface flux is not zero. If the calculation is restricted to the accessible exterior, one cannot simply fill the loop with a surface lying entirely inside that domain. The ideal arrangement assumes negligible field leakage and coherent paths with controlled ordinary dynamical phases. Real interferometers must assess those effects; the loop formula does not assert that every experimental fringe shift has a topological origin.

\mathbf A(r)=\frac{\Phi_B}{2\pi r}\,\hat{\boldsymbol\phi},\qquad \mathbf B=0\ \ (r>R),\qquad e^{i\Delta\varphi_{\rm AB}}=e^{iqw\Phi_B/\hbar}

The ideal flux tube has radius R and is excluded from the particle domain. The exterior cylindrical potential has SI units weber per metre. Integer w is the oriented winding number about the tube.↔ Scroll wide equations horizontally.

Periodicity and the comparison with Berry geometry

Increasing the enclosed flux by $h/|q|$ changes the loop phase by one full turn for unit winding, so ideal single-charge interference is periodic in that flux. This is a periodicity of a phase-sensitive observable, not a claim that the solenoid flux must take only discrete values. Superconducting flux quantization involves a distinct condensate setting and an appropriate effective charge; it should not be inferred from the isolated-particle interference formula without those extra assumptions.

The Aharonov–Bohm connection lives over real-space paths, whereas the Berry connection of a transported eigenstate lives over Hamiltonian parameter space. Both use gauge choices and closed-loop holonomies, but their physical inputs differ. In the ideal exterior Aharonov–Bohm domain, curvature vanishes locally and winding controls the flux contribution. A generic Berry loop can sample nonzero curvature and change continuously under deformation. Topology constrains possible global structures; it does not make every geometric phase an integer or every phase immune to experimental perturbations.

\Phi_0=\frac{h}{|q|},\qquad \Delta\varphi_{\rm AB}=2\pi w\frac{q}{|q|}\frac{\Phi_B}{\Phi_0},\qquad P_0=\frac{1+\cos(\Delta\varphi_{\rm AB}+\varphi_{\rm ref})}{2}

Phi0 is the flux period, in webers, for a coherent particle of charge magnitude |q|. The port formula assumes a balanced two-path interferometer with unit visibility and a specified additional reference phase.↔ Scroll wide equations horizontally.

Work it through

An electron encloses a quarter flux period

For an electron $q=-e$ , a positively oriented unit-winding loop encloses $\Phi_B=\Phi_0/4$ , where $\Phi_0=h/e$ . Find the Aharonov–Bohm phase and the balanced-port probability for additional reference phase $\varphi_{\rm ref}=\pi/2$ . Then reverse the loop orientation.

Insert the signed charge and positive flux. The magnetic phase is $2\pi(-1)(1/4)=-\pi/2$ . Using a positive e in place of q without its minus sign would reverse this result.
Adding the reference phase gives zero, so $P_0=(1+\cos0)/2=1$ . The second output has probability zero in the ideal balanced coherent model.
Reversing the loop changes its winding to minus one and the magnetic phase to $+\pi/2$ . The same reference then gives total phase $\pi$ , hence $P_0=0$ . The reference makes the phase sign detectable in this comparison.

The result

The original orientation gives phase −pi/2 and port-zero probability one; the reversed orientation gives probability zero with the same reference.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Interference exposes a supplied flux phase

In fresh comparisons Reset and set phase difference and amplitude ratio to $(0^\circ,1)$ and $(180^\circ,1)$ . Predict resultant amplitudes 2 and 0, representing the worked example's total phases after reference addition. The two plotted waves have equal fixed frequency. This is coherent addition only: the lab has no magnetic flux, charge, path winding, vector potential, or Aharonov–Bohm dynamics.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). 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 forA gauge-invariant loop phase changes coherent interference after the paths recombine.

Pause & reason

Check your understanding

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

Read the checkpoint explanation

The excluded flux prevents contracting the loop within the accessible domain. Local zero curvature therefore does not imply that a single-valued gauge removes the potential globally, and no discrete solenoid-flux requirement follows.

Your turn

Try it yourself

A positively charged particle with charge $q=e$ makes relative winding $w=2$ around flux $\Phi_B=\Phi_0/6$ . With zero reference phase, calculate the magnetic phase and both balanced-port probabilities. Then increase the flux by one full period and check the probabilities.

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

Give me a hint

Keep the winding factor when computing the phase. Adding a flux period changes this two-winding phase by four pi.

Compare with the solution

The phase is $2\pi(2)(1/6)=2\pi/3$ . Thus $P_0=(1-1/2)/2=1/4$ and $P_1=3/4$ . Adding $\Phi_0$ changes the phase to $2\pi/3+4\pi$ , leaving its cosine and both probabilities unchanged. This verifies the expected period without claiming that the magnetic flux itself is quantized.

Keep the idea

Take it with you

The Aharonov–Bohm loop phase uses signed charge and oriented flux, with unit conventions stated.
A multiply connected accessible region can support nontrivial holonomy despite zero local magnetic field.
Flux periodicity of interference is distinct from flux quantization in a condensate.

Read further

Original sources and deeper explanations to follow at your own pace.

Robert G. Littlejohn, Berkeley Physics 221: Charged Particles in Magnetic Fields, sections on the Aharonov–Bohm effect

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