# Aharonov-Bohm Effect in Torus

I had a very brief introduction to the Aharonov-Bohm effect in class. The lecturer introduced the notion that $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ gives identical energy spectrum and that the Hamiltonians are related by a the large gauge unitary transformation.

I did a quick Google on the large gauge transformation, but I did not quite understand much about it aside from the fact that it is a topological related gauge transformation. Can someone explain a bit more about what that gauge is about and how it is performed?

Also, in a many-body system with the following Hamiltonian (1), in a 3D Torus with flux, $\Phi$ piercing through the hole on the torus, how do show that the energy eigenvalues of $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ are indeed identical? $\Phi_0 = \frac{h}{e}$ in this case.

$$H(\Phi) = \Sigma_{j} \frac{1}{2M}\biggl(\vec{p_j}+e\frac{\Phi}{L}\hat{x}\biggr)^2 +\Sigma_j \ U(\vec{r_j}) + \Sigma_{j<k} \ V\Bigl(\vec{r_j}-\vec{r_k}\Bigr)\tag{1}$$

The general idea of how I will go about solving this is by perhaps acting the hamiltonian on a wavefunction to determine the energy eigenvalue though I am not sure how to do it explicitly.

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A large transformation is a transformation that can't be continuously connected with the identity transformation (with the transformation that does nothing) though other allowed transformations. So large transformations are grouped into "sectors" that are discretely separated from each other.

In electromagnetism, the gauge transformations are $U(1)$ transformations parameterized by the number $\lambda(x,y,z,t)$ that is defined modulo $2\pi$. Charged fields of integral charge $Q$ transform as $$\Psi\to e^{iQ\lambda} \Psi$$ so you can see that only $\exp(i\lambda)$ matters: shifts of $\lambda$ by $2\pi N$ where $N\in{\mathbb Z}$ are unphysical.

In the case of the Aharonov-Bohm effect, there is a closed loop $C$ around the solenoid (where the magnetic field is localized) and the relevant "large gauge transformation" is given by $$\lambda = \phi$$ where $0\leq \phi\leq 2\pi$ is a periodic, angular variable parameterizing the closed loop $C$ (in the simplest parameterization, some angle in the axial or spherical coordinates).

Note that even though this $\lambda$ isn't a single-valued function of the spacetime coordinates $x,y,z$ because it jumps when $\phi$ is increased by $2\pi$ (which corresponds to the return to the original point in space), it is an allowed gauge transformation because $\lambda\mod 2\pi$ or, equivalently, $\exp(i\lambda)$ is a single-valued function of space, and that's everything that is needed. Such a gauge transformation may be ill-defined inside the contour $C$ i.e. inside the solenoid, however.

When this minimal large (topologically nontrivial) transformation is performed, the gauge potential $\vec A$ is charged by $\nabla\cdot \lambda$. The contour integral changes by $2\pi$ $$\oint_C \vec{d\ell}\cdot \vec A \to \oint_C \vec{d\ell}\cdot \vec A + 2\pi$$ because the integral is an integral of a gradient of $\lambda$, so it's just the difference of $\lambda$ between the initial and final points which is $2\pi$, up to a sign. By Stokes' theorem, $\oint_C \vec{d\ell}\cdot \vec A$ is the same thing as the integral $$\int_\Sigma \vec{B}\cdot \vec{dS}$$ over the interior $\Sigma$ of the contour $C$, i.e. inside the solenoid, so this magnetic flux jumps by $2\pi$ as well.

I was neglecting the factors $e,c,\hbar$ above. With the correct factors included in the sentences above, the jump of the magnetic flux is $2\pi\hbar / e$ in your units and conventions. So the two physical configurations may differ in their values of the magnetic flux through $\Sigma$ but they're physically equivalent i.e. indistinguishable because they're related by a large gauge transformation (although one that is only well-defined outside the solenoid).

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I did bit revising and your explanation helped me understand some concepts. Thank you very much. I guess the main thing that I was missing is the transformation of the wavefunction under a gauge transformation. Can you explain the terminology 'mod $2\pi$'? I have seen it a few times but I can't quite grasp its meaning. –  Shiki Jun 11 '13 at 14:01
Dear Shiki, $A$ is equal to $B$ mod $2\pi$ (or "modulo $2\pi$") if the difference between $A$ and $B$, i.e. $A-B$, is an integer multiple of $2\pi$. So if $A,B$ are angles showing a direction in plane, they correspond to the same direction because if you turn around by $2\pi$, you return to the same direction. –  Luboš Motl Jun 12 '13 at 5:51
Alright, I understand now. Just to confirm one more things, the gauge transformations performed in electromagnetism are actually large gauge transformation? –  Shiki Jun 13 '13 at 3:54