The geometry

On a flat torus with a puncture, I've been playing with families of SL₂(C) local systems with the following form.

The transition operators A(E) and B(E) describe the parallel transport of vectors in C² across the sides of the square. If you follow a straight path on the torus, you get a sequence of transitions, like

... A B A A B A B A A B ...

There's often a line in C² that shrinks exponentially as you travel forward along the path, and another line that shrinks exponentially as you travel backward. You can think of these lines as points on the boundary of hyperbolic 3-space, H³. In this way, the local system maps straight paths on the torus to geodesics in H³. A family of parallel paths on the torus maps to a family of geodesics with ideal triangles stretched naturally between them, sweeping out a surface in H³ which is pleated along the geodesics.

The pleated surface shown above comes from a family of parallel paths whose slope is infinitesimally below 45°. The constants a and b are 0 and 2, and the parameter E is set to 1. Since A(E) and B(E) are real, the pleated surface lies flat in the equatorial hyperbolic plane.

The movie below shows how the pleated surface changes as E varies along the real axis.

The plots at the bottom show how the shear coordinates of the pleated surface, also known as the spectral coordinates of the local system, depend on E. Color describes phase. For reference, the first coordinate is zero-like and the second coordinate is pole-like at E ≈ 0.6.

At the singular points E ≈ 0.6 and E ≈ 0.0, you can see clearly that two pleating geodesics’ endpoints collide. Heuristic arguments, backed up by the pleated surface pictures, suggest that the singularities are precisely at E = 2 − √2 and E = 0.

The physics

For local systems of our special form, the sequence of transitions

... A B A A B A B A A B ...

is determined by the sequence of numbers

... a b a a b a b a a b ...

Let’s view that sequence of numbers as a function u on Z.

The parallel transport of the local system turns out to be related to the eigenvalue problem

(−T + u) ψ = E ψ

in L²(Z), where T is the sum of the left and right shift operators. Physically, the operator −T + u models an electron hopping along a chain of atoms. In our case, there are two kinds of atoms, and the numbers a and b say how much the electron likes each kind. (This model is a "tight-binding approximation" of a more detailed one.) In physics, the eigenvalue equation above is called the time-independent Schrödinger equation, so the kind of local system we're looking at is called a Schrödinger cocycle.

The exponentially shrinking lines that gave us our pleated surface can only exist if E falls outside the spectrum of the operator −T + u. (This isn't obvious. It's a consequence of the nuclear spectral theorem in the rigged Hilbert space with Schwartz test functions.) It should follow that the spectrum of −T + u must lie in the part of the real axis where the shear coordinates are singular.

For the movie, I used a slope infinitesimally below 45°, so almost every straight path gives the same reflection-symmetric function u:

... b a b a b a a b a b a b ...

The energy 2 − √2 is in the point spectrum of −T + u. Its eigenvector peaks at the reflection point, and decays exponentially in both directions. Intuitively, this describes a state where the electron is snagged on the reflection point.

More sophisticated examples

Same binding energies, different slope

We can see more interesting behavior by traveling across the torus at a more interesting slope. The phases of the shear coordinates at slope 2/(√5 − 1) are plotted below.

Smaller binding energy difference

Let's set a and b to 0 and −1, keeping the slope at 2/(√5 − 1). Here's a movie of how the pleated surface varies at the low end of the energy spectrum.

The next movie focuses on the approach to the lowest point in the spectrum.