= Aharonov-Bohm effect
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{title2=$\delta=q\Phi/\hbar\pmod{2\pi}$}
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= Aharonov-Bohm phase
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The Aharonov-Bohm effect is the phase $q\Phi/\hbar$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.
In the magnetic <Aharonov-Bohm effect>, a coherent <Electron> wave takes two routes around inaccessible <magnetic flux>. Recombination converts their relative <wave phase> into displaced <interference fringes>. In 1959, <Yakir Aharonov> and <David Bohm> proposed a shielded <solenoid> geometry and identified the obstruction to removing the <magnetic vector potential> throughout the accessible region.<Significance of Electromagnetic Potentials in the Quantum Theory by Yakir Aharonov and David Bohm>[1959, §2, pp. 486–487, Fig. 2 and reflecting-barrier discussion]{ref}
Use <International System of Units>, <electric charge> $q$, and <magnetic minimal coupling> $H=(-i\hbar\nabla-q\mathbf A)^2/(2m)$. Let $\gamma_1,\gamma_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows $\gamma_1$ and then $\gamma_2$ backwards. For a static <magnetic vector potential>, the magnetic contribution to the relative <wave phase> is
$$
\delta=\frac q\hbar\left(\int_{\gamma_1}\mathbf A\cdot d\mathbf l-\int_{\gamma_2}\mathbf A\cdot d\mathbf l\right)
=\frac q\hbar\oint_C\mathbf A\cdot d\mathbf l
=\frac{q\Phi_C}{\hbar}\pmod{2\pi}.
$$
Here $\Phi_C=\int_\Sigma\mathbf B\cdot d\mathbf S$ is the oriented <magnetic flux> through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an <Electron>, $q=-e$, where $e>0$ is the <elementary charge>; reversing $C$ reverses the sign. The observable is the <gauge-invariant> <holonomy> $e^{i\delta}$, rather than a unique local value of the <magnetic vector potential>.
The accessible region is a <multiply connected region>: $C$ cannot contract to a point without entering the excluded flux region. <Stokes theorem> does not set the <line integral> to zero, because a spanning surface must cross that excluded region, where the <magnetic field> need not vanish. There is real <magnetic flux> in the apparatus; only the accessible paths are idealized as free of the <magnetic field>. The exterior of a finite straight <solenoid> also permits routes around its ends, so its exact topology and leakage differ from those of an infinite <solenoid> or an excluded solid <torus>.
An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose <cylindrical coordinates> $(r,\theta,z)$ and
$$
\mathbf A=\frac{\Phi}{2\pi r}\,\widehat{\boldsymbol\theta},\qquad r>a.
$$
Its <curl> vanishes there, but a loop with <winding number> $n$ has $\oint_C\mathbf A\cdot d\mathbf l=n\Phi$. Locally $\mathbf A=\nabla(\Phi\theta/(2\pi))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi/\hbar$, that candidate gauge removal changes the <wave function>'s single-valued boundary condition. On a <simply connected> region, a smooth static <magnetic vector potential> with zero <curl> instead has a globally defined scalar primitive and is a <pure gauge potential>. Thus local zero <magnetic field> and globally removable <magnetic vector potential> are different conditions.
\Image[/aharonov-bohm-two-path-flux-geometry.png]
{title=Two accessible paths can link inaccessible <magnetic flux>}
{description=Original functional geometry. Left: a cross-section of an ideal infinite flux tube; $C$ traverses the blue route forwards and the orange route backwards. Right: a meridional section of an excluded solid <torus>; the <magnetic field> circulates inside its core, with opposite signs through the section. One route passes through the central hole and the other outside the <torus>. The drawn linking surface intersects one core section. Paths denote coherent alternatives, not measured trajectories; shielding thicknesses and beam optics are schematic.}
{source=aharonov-bohm-two-path-flux-geometry.py}
For an original <interference> example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in <off-axis holography>. Then
$$
I(y)=I_1+I_2+2\sqrt{I_1I_2}\cos(ky+\delta),\qquad
\Delta y=-\frac\delta k\pmod{2\pi/k}.
$$
The <interference fringe spacing> is $d=2\pi/k$. An <Electron> enclosing $\Phi=h/(4e)$ has $\delta=-\pi/2$, shifting maxima by $d/4$ for this orientation. Increasing the <magnetic flux> by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the <Electron> <Aharonov-Bohm effect> has <magnetic flux quantum> $h/e$. This uses $h=2\pi\hbar$, the <Planck constant>. It does not require the source's <magnetic flux> to vary continuously during a single exposure.
For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max}=9/4$, $I_{\min}=1/4$, and the <interferometric visibility> is $4/5$. A fixed <Aharonov-Bohm phase> translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed <magnetic flux>. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.
The appeal of the <Aharonov-Bohm effect> is that a local <Lorentz force> calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of <holonomy> and supplies the encircling phase used in <Chern--Simons flux attachment>. This alone does not establish <intrinsic topological order>. The static comparison also matters: changing <magnetic flux> can induce an <electric field>, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial <line integral>.
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