Berry Phase in the Aharonov–Bohm Effect

The Aharonov–Bohm phase is often discussed beside Berry phase because both are gauge-invariant loop phases. The shared structure is holonomy: a quantum amplitude is transported around a closed loop and returns with a phase.

The important caution is that the standard magnetic Aharonov–Bohm effect is not, in its basic form, the same construction as Berry’s adiabatic eigenstate phase. It is electromagnetic gauge holonomy in real configuration space. Berry phase is eigenstate-bundle holonomy in a parameter space under an adiabatic following assumption.

The value of comparing them is that the same geometric grammar appears in both:

$$\text{connection} \quad \longrightarrow \quad \text{closed-loop phase} \quad \longrightarrow \quad \text{interference}.$$

Aharonov–Bohm Phase

For a charged particle moving around an excluded magnetic flux region, the magnetic Aharonov–Bohm phase is

$$\gamma_{\mathrm{AB}}[C] = \frac{q}{\hbar} \oint_C \mathbf A(\mathbf r)\cdot d\mathbf r.$$

If the loop encloses magnetic flux $\Phi_B$, then

$$\gamma_{\mathrm{AB}}[C] = \frac{q\Phi_B}{\hbar} \quad \operatorname{mod}2\pi.$$

The full physical setup, gauge-invariance argument, and topology of the punctured plane are developed in Aharonov–Bohm Effect. This page uses that effect to sharpen the Berry-phase comparison.

Berry Phase

For a nondegenerate eigenstate transported adiabatically around a loop $C$ in parameter space,

$$H(R)\lvert n(R)\rangle = E_n(R)\lvert n(R)\rangle,$$

the Berry phase is

$$\gamma_n[C] = \oint_C A_n, \qquad A_n = i\langle n(R)|d n(R)\rangle.$$

The adiabatic state also accumulates a dynamical phase:

$$\alpha_n = -\frac{1}{\hbar} \int_0^T E_n(t)\,dt + \gamma_n[C].$$

Thus Berry phase depends on the path of the eigenspace, while the Aharonov–Bohm phase depends on electromagnetic gauge holonomy around flux in configuration space.

Shared Holonomy Structure

Both phases can be written as integrals of a one-form around a loop:

$$\gamma[C] = \oint_C \mathcal A.$$

For Berry phase,

$$\mathcal A=A_n = i\langle n|dn\rangle.$$

For the magnetic Aharonov–Bohm phase,

$$\mathcal A_{\mathrm{AB}} = \frac{q}{\hbar} A_i(\mathbf r)\,dx^i.$$

In both cases the observable object is the phase factor

$$e^{i\gamma[C]}.$$

In both cases an open-path integral is gauge dependent unless endpoint comparison data are supplied. Closed-loop interference is the clean gauge-invariant setting.

Important Differences

Feature	Berry phase	Aharonov–Bohm phase
Base space	Hamiltonian parameter space	particle configuration space
Connection	$i\langle n	dn\rangle$ from eigenvectors
Dynamical assumption	adiabatic following of an eigenspace	coherent propagation around flux
Local curvature	Berry curvature $F_n=dA_n$	magnetic field $dA$
Canonical loop	loop of control parameters	real-space loop around excluded flux
Typical invariant	Berry holonomy $e^{i\oint A_n}$	Wilson loop $e^{(iq/\hbar)\oint A\cdot dr}$

The table explains why the phrase “Aharonov–Bohm as Berry phase” must be used carefully. The Aharonov–Bohm effect is a geometric phase effect, but the canonical page for its physics remains the Aharonov–Bohm Effect.

Gauge Transformations

The two cases have parallel gauge laws.

For Berry phase, a phase choice of the eigenvector changes by

$$\lvert n(R)\rangle \mapsto e^{i\chi(R)}\lvert n(R)\rangle,$$

which gives

$$A_n \mapsto A_n-d\chi.$$

For the electromagnetic Aharonov–Bohm phase, with minimal coupling $p\mapsto p-qA$,

$$\mathbf A \mapsto \mathbf A+\nabla\lambda,$$

while the wavefunction changes by a compensating local phase

$$\psi \mapsto \exp \left( \frac{iq\lambda}{\hbar} \right) \psi.$$

In both cases, the connection itself is not the observable. The closed-loop phase factor is.

Curvature versus Topology

Berry phase is often introduced through curvature:

$$\gamma_n[C] = \int_\Sigma F_n$$

when $C=\partial\Sigma$ and a smooth gauge exists on $\Sigma$. For spin-$1/2$ in a rotating magnetic field, the Berry curvature resembles a monopole in parameter space, and the phase is proportional to solid angle.

The ideal magnetic Aharonov–Bohm effect emphasizes a different possibility. Outside the flux tube,

$$\mathbf B = \nabla\times\mathbf A = \mathbf 0,$$

but a loop winding around the excluded region can have

$$\oint_C \mathbf A\cdot d\mathbf r = \Phi_B.$$

This is flat local connection with nontrivial global holonomy. The loop cannot be shrunk to a point inside the accessible region without crossing the excluded flux. The topology of the accessible space matters.

The moral for Berry phase is broader than the Aharonov–Bohm effect itself: local curvature is not always the whole story. Global patching, singularities, degeneracies, and noncontractible loops can all affect holonomy.

Can the Aharonov–Bohm Phase Be Viewed as a Berry Phase?

There are extended formulations where Aharonov–Bohm phases appear in Berry-like language. For example, a slow parameter describing the location of a flux tube, a constrained particle coordinate, or an effective adiabatic subspace can produce a Berry connection whose holonomy reproduces an electromagnetic phase.

Those reformulations are useful, but they should not erase the basic distinction. In the textbook magnetic Aharonov–Bohm interferometer, the phase is already present as electromagnetic gauge holonomy. No slow change of a Hamiltonian parameter is required for the effect to exist.

A safe phrasing is:

The Aharonov–Bohm phase is a quantum holonomy closely analogous to Berry phase; in special adiabatic reformulations it can be represented as a Berry phase, but its canonical physical origin is electromagnetic gauge connection.

Interference and Observability

Neither phase is observed as an absolute phase of one isolated state. Both become visible through comparison.

For Berry phase, an experiment may compare two internal states, two paths in parameter space, or a cyclic evolution against a reference. For the Aharonov–Bohm effect, two spatial paths enclose different flux and recombine:

$$\Delta\gamma_{\mathrm{AB}} = \frac{q}{\hbar} \left( \int_{\Gamma_1}\mathbf A\cdot d\mathbf r - \int_{\Gamma_2}\mathbf A\cdot d\mathbf r \right).$$

The difference is a closed-loop integral over $\Gamma_1-\Gamma_2$. Gauge-dependent endpoint pieces cancel in the interference phase.

Ring Example

A particle on a ring threaded by flux $\Phi_B$ is the simplest exactly solvable Aharonov–Bohm system. The vector potential can be locally gauged away, but the boundary condition changes by a phase:

$$\psi(\theta+2\pi) = \exp \left( \frac{iq\Phi_B}{\hbar} \right) \psi(\theta).$$

This is a useful bridge to Berry phase because the phase is global. Locally the Hamiltonian may look free after a gauge transformation, but the allowed wavefunctions remember the holonomy around the circle. See Particle on a Ring for the canonical system.

Common Mistakes

• Saying the Aharonov–Bohm phase is simply a Berry phase without specifying the reformulation.
• Forgetting that Berry phase requires an adiabatic eigenspace in its standard form.
• Treating either connection as directly observable rather than its holonomy or curvature.
• Using Stokes’ theorem over a surface that crosses an excluded flux region without accounting for the flux.
• Assuming every geometric phase is topological.
• Confusing parameter space with real configuration space.
• Ignoring the dynamical phase when measuring Berry phase, or ignoring ordinary propagation phases in an Aharonov–Bohm interferometer.

Cross-Links

• Aharonov–Bohm Effect
• Berry Phase
• Holonomy
• Parallel Transport
• Berry Connection
• Berry Curvature
• Global Phase and Physical States
• Local Phase Transformations
• Gauge Transformations in Quantum Mechanics
• Dirac Monopole Preview
• Aharonov–Bohm Effect: First Encounter
• Particle on a Ring
• Aharonov–Bohm Rings distinguishes electromagnetic, dynamical, and Berry phases in open-ring transport data.
• Holonomy in the Mathematical Toolkit
• U(1) Bundles and Quantum Phase
• Homotopy and Winding

References

• Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Physical Review 115, 485-491, 1959.
• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45-57, 1984.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167-2170, 1983.
• Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Physical Review Letters 58, 1593-1596, 1987.
• M. Peshkin and A. Tonomura, The Aharonov–Bohm Effect, Springer, 1989.
• A. Shapere and F. Wilczek, eds., Geometric Phases in Physics, World Scientific, 1989.

Exercises

1. Write the Aharonov–Bohm phase as a holonomy of a dimensionless connection.

Solution

Define the dimensionless one-form

$$\mathcal A_{\mathrm{AB}} = \frac{q}{\hbar} A_i(\mathbf r)\,dx^i.$$

Then

$$\gamma_{\mathrm{AB}}[C] = \oint_C\mathcal A_{\mathrm{AB}} = \frac{q}{\hbar} \oint_C A_i\,dx^i.$$

The phase factor is $e^{i\gamma_{\mathrm{AB}}[C]}$.

2. Compare gauge transformations in the two cases.

Show that both closed-loop phases are invariant under single-valued gauge transformations.

Solution

For Berry phase,

$$A_n\mapsto A_n-d\chi,$$

so

$$\oint_C A_n \mapsto \oint_C A_n-\oint_Cd\chi.$$

For single-valued $\chi$, $\oint_Cd\chi=0$.

For Aharonov–Bohm phase,

$$\mathbf A\mapsto\mathbf A+\nabla\lambda,$$

so

$$\oint_C\mathbf A\cdot d\mathbf r \mapsto \oint_C\mathbf A\cdot d\mathbf r + \oint_C\nabla\lambda\cdot d\mathbf r.$$

For single-valued $\lambda$, the last integral is zero. Thus both closed-loop phase factors are invariant.

3. A loop encloses one flux quantum $\Phi_0=h/\lvert q\rvert$. What is the Aharonov–Bohm phase factor?

Solution

The phase is

$$\gamma_{\mathrm{AB}} = \frac{q\Phi_0}{\hbar} = \frac{q}{\lvert q\rvert} \frac{h}{\hbar} = \pm 2\pi,$$

with the sign set by the charge. Therefore

$$e^{i\gamma_{\mathrm{AB}}}=1.$$

4. State one reason why the ideal Aharonov–Bohm exterior can have nontrivial phase even when $\mathbf B=0$ on the accessible region.

Solution

The accessible region is multiply connected because the flux tube is excluded. A loop that winds around the excluded region cannot be shrunk to a point while staying in the accessible region. The vector potential can be locally pure gauge outside the flux tube while still having nonzero circulation around a noncontractible loop.

5. Give a precise sentence distinguishing Berry phase from the Aharonov–Bohm phase.

Solution

One precise sentence is: Berry phase is holonomy of an eigenstate line bundle over Hamiltonian parameter space under adiabatic transport, while the magnetic Aharonov–Bohm phase is holonomy of the electromagnetic gauge connection over the charged particle’s configuration space around excluded magnetic flux.