Aharonov–Bohm phase
A charged quantum wavefunction responds to enclosed magnetic flux even along paths with zero local magnetic field.
$$\Delta\varphi=\frac{q}{\hbar}\oint\mathbf A\cdot d\mathbf l=\frac{q\Phi}{\hbar}$$
What this means
A charged quantum wavefunction responds to enclosed magnetic flux even along paths with zero local magnetic field.
Deep dive: mathematical and physical mechanism
Minimal coupling changes the phase along a path by $(q/\hbar)\int\mathbf A\cdot d\mathbf l$. Interference between two paths depends on their closed-loop difference, which equals enclosed flux by Stokes theorem. This is a gauge holonomy, not a claim of nonlocal force.
How it appears in quantum phenomena
Electron interferometers and mesoscopic rings display flux-periodic interference.
Assumptions, caveats and limitations
Requires coherent paths; the effect is distinct from Berry phase generated by parameter-space eigenstate transport.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.