Quantum Graphs : Physics of Quantum Singularity (3-3)

Geometry of Parameter Space 3

We assume, for now, the torus parameters (\theta_+, \theta_-) are fixed. Since different choice of \mu and \nu doesn't affect the spectra of the system, the entire sphere spanned by (\mu, \nu) is a "equispectral sphere". On the other hand, if we fix the values of (\mu, \nu) and thus have a fixed V and consider a new spin operator \sigma_S given by

(3.7)

the entire collection of U spanned by all possible choice of (\theta_+, \theta_-) respects the relation

(3.8)

thus the torus (\theta_+, \theta_-) is mapped onto itself by \sigma_S. Namely, the set of point interaction with fixed (\mu, \nu) forms a "\sigma_S invariant torus" unaltered by the symmetry operation associated to \sigma_S.

Thus we have separated spectre-deciding two parameters (\theta_+, \theta_-) and shape-chaging, but spectral-invariant parameters (\mu, nu) from the entire four dimensional sphere SA^4. Moreover, the shapes of manifold for each subset of parameter space are determined as torus T^2 and sphere S^2.

Such unexpected appearence of nonflat manifold, or nontrivial geometryhas to have physical consequences. It suffices to recall the concept of Berry phase. The original example by Berry is a spin 1/2 particle in a constant magnetic field whose three-dimensional direction are to be varied. Two Euler angles, the parameter describing the direction of the magnetic field (unsurprisingly) form a two dimensional sphere which turned out to contain "monopole" at the center, which is the background for the appearence of Berry phase.

By a sheer chance, the same two dimensional sphere is found in our example as the equispectral sphere, and it does contain the same monopole at the center that results in the same Berry phase in our model.

Suppose a quantum state is defined on each point on the surface of two-dimensional parameter sphere. Let us further suppose that the variation of the quantum state while varing the location of the point is smooth. A consistent arrangement of states in such setting can be just regular unremarkable everywhere, but it can also involve nontrivial arrangement. If we consider a cyclic variation of parameters corresponding to the motion along a closed loop on the sphere. In any "regular" arrangement of the state wave function will change its shape along the loop and come backk to the original shape. But there is another possibility of having some nontrivial arrangements of the state such that after varying the parameter back to the original value, we of course obtain the original state back but with some extra phase. What Berry have shown is that that extra phase is given by the integer multiple of the solid angle the loop has spanned over the center of the sphere. That integer is what is referred to as the "monopole number" , and the situation is just like monopoles of that numbers are stting at the center causing that phase acquisittion.