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Toroidal lattices

Toroidal lattices

Introduction

In Lattice quivers, we examined lattice quivers generated by taking quotients of the path quiver of a fundamental quiver \(\quiver{F}\) by an affine path valuations induced by a cardinal valuation \(\groupoidFunction{ \phi }\). Our compact notation for this was \(\compactQuotient{\quiver{F}}{\vert{x}}{\groupoidFunction{ \phi }}\). Now, this is an algebraic way of expressing it of course, and perhaps obscure – the computational interpretation was that we were computing a multiway quiver from the action of a fundamental quiver with matrices associated with its cardinals. The fundamental quiver which was a bouquet quiver \(\bouquetQuiver{\sym{n}}\), where the groups associated with the cardinal matrices were (infinite) translation groups \(\translationCardinalValuation{1},\translationCardinalValuation{2},\translationCardinalValuation{3},\starTranslationCardinalValuation{3}\).

A straightforward generalization of this idea is to swap out these infinite translation groups for finite Abelian groups. These yield finite lattice quivers that we can interpreted as lattices defined on a torus.

Square torus

In the simple case of \(n\)-bouquet quiver with group being a product of cyclic groups, we obtain an \(n\)-dimensional discrete rectangular torus. We’ll only examine 2-dimensional tori for now, which we'll call the square tori.

Here is the square torus for group \(\mathbb{Z}_{10}\times \mathbb{Z}_4\):

Note the matrices use \(1_n\) to indicate the \(n^{th}\) root of unity. The \(\rform{\card{x}}\) cardinal is associated with traversal around the outer radius of the torus, and the \(\bform{\card{y}}\) cardinal with the inner radius. The outer radius spanned by \(\rform{\card{x}}\) has a width modulus of 10, and the inner radius spanned by \(\bform{\card{y}}\) has a height modulus of 4. We'll call these the modoli of the torus, and refer to the entire torus as a \(10 \times 4\) torus.

From a path-algebraic point of view, a square torus has a straightforward characterization. Recall that for the infinite square lattice, we have the single path relation \(\word{\rform{\card{x}}}{\bform{\card{y}}}\pathIso \word{\bform{\card{y}}}{\rform{\card{x}}}\). A square torus with moduli \(\tuple{\sym{w},\sym{h}}\) extends this to the set \(\list{\word{\rform{\card{x}}}{\bform{\card{y}}}\pathIso \word{\bform{\card{y}}}{\rform{\card{x}}},\repeatedPower{\word{\rform{\card{x}}}}{\sym{w}}\pathIso \word{1},\repeatedPower{\word{\bform{\card{y}}}}{\sym{h}}\pathIso \word{1}}\).

Other tori

This kind of construction gives us an finite analogs of the square, cubic, etc. lattices. How can we obtain triangular, hexagonal, etc. lattices?

This simplest way to accomplish this that I can find is to use similar quivers and cardinal matrices as for the infinite triangular and hexagonal tessellations, except over a finite field. Well, technically, we apply a per-component modulus to the elements of the matrix, since we want certain to apply different moduli to different axes. I believe this can be recast in terms of roots of unity, but this approach is easier to reason about.

We'll use the generic term tori to encompass both the cases where both axes are finite, yielding a torus, and where one radius can be infinite, yielding a cylinder.

Here’s such an equivalent of the triangular lattice, finite in one radius but infinite in the other:

Note that the modulus applied to the second axis is indicated by a blue subscript on that element in the cardinal matrices. Also note that we are using an artificial transparency effect to make the three dimensional structure easier to perceive.

And here’s a toroidal triangular lattice, rendered without arrowheads to avoid clutter:

A hexagonal torus lattice gives a familiar structure, corresponding to the bond connectivity of a carbon nanotube:

Axes

Let’s consider for a moment what axis-aligned geodesics look like for toroidal lattices. First, though, let’s consider the situation for plane lattices:

The geodesics are colored by which cardinal they are aligned with. In a hexagonal lattice we cannot form path words like \(\word{\card{a}}{\card{a}}{\card{a}}\), so our “axes” are composed of alternating pairs of cardinals, e.g. \(\word{\card{a}}{\ncard{b}}{\card{a}}{\ncard{b}}{\card{a}}{\ncard{b}}\); we color these paths by blending the colors of the alternated cardinals.

Ok, now we can compare with square torus (top) and triangular torus (bottom):

As you probably expected, the square torus has predictable “orthogonal” axes.

But the situation with triangular torus appears to be more interesting: the \(\bform{\card{b}}\) and \(\gform{\card{c}}\) axes twist around and meet at an point opposite the origin. This behavior turns out to be sensitive to the moduli of the torus.

If the moduli are coprime, then either of the axes \(\bform{\card{b}}\) and \(\gform{\card{c}}\) will reach every vertex:

Modular plane

This is of course a straightforward consequence of basic number theory, but is best explained by plotting these lattices on a modular plane instead of a torus. We can think of this plane as “wrapping around” on its border – but we’ll number the edges that cross this border so that it is easy to trace how they connect up. First let’s look at the square lattice to familiarize ourselves with this visualization:

Now we can consider the modular plane for the triangular and hexagonal tori:

Let’s plot the \(\rform{\card{a}}\), \(\bform{\card{b}}\), \(\gform{\card{c}}\) axes on the plane for a \(8 \times 4\) torus:

Plotting the \(\rform{\card{a}}\) and \(\bform{\card{b}}\) axes for a range of moduli, we can see how the greatest common divisor of the two moduli determines how many orbits of the \(\sym{w}\) dimension the \(\bform{\card{b}}\) axis will make before returning to the origin:

Notice that if \(\sym{w} / 2\) and \(\sym{h} / 2\) are coprime, the axis \(\bform{\card{b}}\) intersects every vertex. In general the number of times \(\rform{\card{a}}\) will intersect \(\bform{\card{b}}\) for a \(\sym{w}\times \sym{h}\) torus is \(2 \, \lcm(\sym{h} / 2,\sym{w}) / \sym{h}\).

A similar situation applies for the hexagonal torus, when we use alternating axes:

Due to an arbitrary choice of orientation of these hexagons, our previously vertical axis is now horizontal, but a similar relation holds: the number of vertices in the intersection of the axis-aligned geodesics is given by \(3 \, \lcm(\sym{w},\sym{h} / 3) / \sym{h}\), where it is important to notice the \(\tuple{\sym{w} = 6,\sym{h} = 9}\) case actually has two such intersecting vertices, not one as it first appears. Similarly the \(\tuple{\sym{w} = 12,\sym{h} = 9}\) case has 4 intersecting vertices.

We are certainly not limited to these square, triangular, and hexagonal toroidal lattices. Any fundamental quiver employing the same translation groups has a toroidal version. For example, the rhombille lattice:

Sheared tori

There is a straightforward construction we can use to introduce an intuitive kind of shear or torsion into our lattice quivers.

We'll start with the square torus (of size \(5 \times 3\)), where the torsion is easiest to see. On the left is the sheared lattice, and on the right is the normal square lattice for comparison.

Notice that in the sheared lattice, the top left vertex connects to the second vertex on the bottom left; in the normal lattice it connects to the first. We’ll say that this lattice has a shear parameter of \(\sym{z} = 1\).

Here are square lattices with shear parameters \(-2 \le \sym{z} \le 2\):

Notice similar "orbital mechanics" for the sheared square lattice as we saw for the triangular lattice: for \(\sym{z} = \pm 1\), a \(\bform{\card{y}}\) geodesic will orbit through all vertices, but for \(\sym{z} = \pm 2\), a \(\bform{\card{y}}\) geodesic effectively skips the neighboring geodesic on completing a circuit, so if \(\sym{w}\) is even we will have exactly two distinct \(\bform{\card{y}}\)-orbits.

From a path-algebraic point of view, a \(\sym{z}\)-sheared square lattice can be expressed as the path relation set:

\[ \list{\word{\rform{\card{x}}}{\bform{\card{y}}}\pathIso \word{\bform{\card{y}}}{\rform{\card{x}}},\repeatedPower{\word{\rform{\card{x}}}}{\sym{w}}\pathIso \repeatedPower{\word{\bform{\card{y}}}}{\sym{z}},\repeatedPower{\word{\bform{\card{y}}}}{\sym{h}}\pathIso \word{1}} \]

It's clear then that if \(\sym{z} = \sym{h}\), we have no shear at all, and hence the shear parameter lives in the integers modulo \(\sym{h}\).

When plotted in three dimensions, the twist of a sheared square torus resembles the twisted magnetic field used in a tokamak.