One invariant for every chain that can be undone

A spectrum that is a Cantor set followed a wave through the Fibonacci chain by multiplying two-by-two transfer matrices, one per site, and found that the half-traces of the matrices at successive levels of the chain obey a recursion of their own. The recursion has a quantity it cannot change,

$$I = x^2 + y^2 + z^2 - 2xyz - 1,$$

built from the half-traces of three consecutive matrices. That quantity is zero exactly when the two kinds of site are the same, and positive otherwise. The essay closed on a guess about the rest of the family. Chains built from other inflation factors would have “a different trace map and a different invariant”, and comparing the invariants would show what the conserved quantity measures in general.

The comparison is made here, for three metallic chains and the golden chain written backwards, and against two chains that are not quasiperiodic in the same way. The first half of the guess is right and the second is wrong. The trace maps differ, but the invariant does not. Every chain built by a substitution that can be undone, from the same two tiles, conserves exactly the same number, at every level and every energy. The reason turns the conserved quantity from a property of the Fibonacci chain into a property of the two tiles.

Five rules, three that can be undone #

A substitution replaces each letter by a word. The golden rule sends A to AB and B to A. The silver rule sends A to AAB, the bronze rule A to AAAB, and in both B goes to A. Iterated, they build the Fibonacci chain and its silver and bronze relatives, with inflation factors $\varphi = 1.618$, $1 + \sqrt2 = 2.414$ and $(3 + \sqrt{13})/2 = 3.303$. Two more rules serve as controls. Period-doubling sends A to AB and B to AA; Thue–Morse sends A to AB and B to BA. Both build aperiodic chains that are not quasiperiodic in the cut-and-project sense.

The property that separates them is whether the rule can be undone. Treat A and B as symbols that can be multiplied and inverted, so that words are elements of the free group on two letters. The golden rule sends A to AB and B to A, and it can be run backwards. From the images AB and A, the original A is recovered as the second image and B as the first image with that A removed from the front. The images still generate everything the original letters did, so the rule is an automorphism of the free group. The silver and bronze rules can be undone in the same way. Period-doubling cannot. Its images AB and AA generate a subgroup in which B cannot be isolated, and the determinant of its letter-count matrix, minus two, says so at a glance, since a substitution that can be undone must have determinant plus or minus one. Thue–Morse’s matrix has determinant nought. The golden rule reversed, sending A to BA and B to A, builds the Fibonacci chain read from right to left. It can be undone as easily as the golden rule, and it is included because it gives exactly the same chain with different level words. It is the simplest test that the conservation belongs to the rule’s being reversible and not to the particular words it writes.

The invariant is a commutator #

The identity behind the conserved quantity is older than the Fibonacci chain. For any two matrices $X$ and $Y$ of determinant one, with half-traces $x$ and $y$ and the half-trace $z$ of their product $XY$, a computation Fricke made in the nineteenth century gives

$$x^2 + y^2 + z^2 - 2xyz - 1 \;=\; \tfrac14\big(\operatorname{tr}[X, Y] - 2\big),$$

where $[X, Y] = XYX^{-1}Y^{-1}$ is the commutator. So $I$ measures how far $X$ and $Y$ are from commuting.

Now take $X$ and $Y$ to be the transfer matrices of the two words the substitution produces at some level, $\sigma^n(A)$ and $\sigma^n(B)$. If the substitution can be undone, it is an automorphism of the free group. A theorem of Nielsen then says that it sends the commutator of A and B to a conjugate of that commutator or of its inverse. Conjugate matrices have equal traces, and a matrix and its inverse have equal traces when the determinant is one. So the trace of $[X, Y]$ equals the trace of the commutator of the two single-site matrices, at every level:

$$I \;=\; \tfrac14\big(\operatorname{tr}[M_A, M_B] - 2\big) \;=\; \frac{V^2}{4},$$

since the two site matrices $M_A$ and $M_B$ differ only by the site energy $V$ in one corner, and their commutator’s trace is $2 + V^2$ at every energy. The invariant is the same for every substitution that can be undone, and it is a property of the two tiles, not of the chain. It is zero when the tiles are the same, as the earlier essay found, because two equal matrices commute.

Measured in exact integers #

The picture at the head of this essay is the measurement, and it has to be made with care. Outside the spectrum the entries of a transfer matrix grow exponentially with the length of the word. At a few hundred sites the four terms of $I$ are numbers of hundreds of digits whose sum is 0.09, and floating-point arithmetic returns nonsense, as the first attempt here did. So every matrix is computed in exact integers. The energy and the site energy are given in tenths, each site matrix becomes ten times an integer matrix, and $I$ is compared with $V^2/4$ as a single integer equation. For the golden chain to 610 sites, the silver to 239, the bronze to 469, and the golden rule reversed, at five energies each, the equation holds exactly at every level.

The controls fail it, and by a great deal. At the same energy, period-doubling’s $I$ climbs from 0.2 to over ten thousand by its eighth level. Thue–Morse’s rises and falls irregularly, and at another energy it reaches $10^{19}$. Their rules cannot be undone, so nothing sends the commutator of their level words back to the commutator of the tiles, and the quantity records whatever those particular matrices happen to do.

The energy test makes the same point from another side. The golden chain’s $I$ is 0.09 at every energy from minus two to two, although the individual traces $x$, $y$ and $z$ change enormously across that range, from values between minus one and one inside the bands to large numbers in the gaps. Period-doubling’s $I$ at its fourth level varies with the energy by five orders of magnitude. A quantity that does not depend on the energy of the wave cannot be telling anything about the wave. It is telling about the tiles, and the tiles are the same at every energy.

Why a Fibonacci-only proof looked specific #

The earlier essay derived the conserved quantity from the Fibonacci recursion itself: the trace of the level-$(n+1)$ matrix is twice the product of the two before it, minus the one before that. It checked that the combination $I$ survives each step. That derivation is correct and it looks specific to the golden rule, because the recursion is. The silver chain’s trace map is different: its level word is the previous one twice followed by the one before, and its half-traces obey a longer recursion involving a Chebyshev polynomial of the level before. Written out, the two recursions share nothing obvious.

Written out, the silver map needs all three half-traces as its state. The silver chain’s level word is the previous level word twice followed by the one before, $W_{n+1} = W_n W_n W_{n-1}$. The Cayley–Hamilton identity for a two-by-two matrix of determinant one, $W^2 = \operatorname{tr}(W)\,W - 1$, turns the trace of a square into traces of first powers. Writing $x$ and $y$ for the half-traces of the two most recent level matrices and $z$ for the half-trace of their product, one level of the silver chain sends

$$(x,\ y,\ z) \;\longmapsto\; \big(2xz - y,\ \ x,\ \ 2x(2xz - y) - z\big).$$

The golden map, by contrast, sends the three most recent half-traces $(x, y, z)$ to $(2xy - z, x, y)$. It needs no product, because the Fibonacci level word is the concatenation of the two before it, so the product of consecutive level matrices is itself the next level matrix. The two maps have different degrees and different variables, and composing either with itself produces polynomials that grow quickly apart. Substituting either into $x^2 + y^2 + z^2 - 2xyz - 1$ gives back the same polynomial, which takes a line of algebra to check for the golden map and a few lines for the silver. Run numerically at an arbitrary energy, the silver map carries the value 0.09 unchanged through every step.

What they share is the reason both conserve $I$, and the reason is not in the recursion. Each is the shadow of an automorphism of the free group on the two traces, and the Fricke quantity is the one polynomial in the traces that every such automorphism preserves. Once that is seen, the invariant stops being a lucky feature of one recursion. It becomes a property of the group, and the individual trace maps become different ways of moving round a surface on which it is constant. That is the sense of the earlier remark, which read the family of inflation factors as a family of trace maps. The family is real, and all its members move on the same surface $I = V^2/4$.

A surface the traces cannot leave #

The invariant has a picture. Think of the three half-traces $(x, y, z)$ as a point in space. The equation $x^2 + y^2 + z^2 - 2xyz - 1 = V^2/4$ is a surface, and the conserved quantity says that every trace map, golden or silver or bronze, moves its point about on that one surface and never off it. The map is different for each chain, so each takes its point on a different journey, but the surface is the same.

When the tiles are equal, $V = 0$, the surface is the Cayley cubic, a surface with four singular points. The part of it inside the cube $|x|, |y|, |z| \le 1$ is exactly the set of half-traces of pairs of commuting rotations. That is the periodic crystal, whose transfer matrices all commute because every site is the same, and a point stays in that cube precisely at the energies inside the single band a uniform chain has. As $V$ grows the surface pulls away from the cube, the four singular points open into narrow necks, and a trace point can start inside the cube and leave it after some number of levels. That is how gaps open, and it is the geometric content of the earlier essay’s remark that the size of the gaps is controlled by $I$. The Fibonacci spectrum is the set of energies whose starting point stays on a bounded orbit for ever, and Sütő’s theorem that it has measure nought is a statement about how few such orbits there are.

Seen this way, the result here is that every metallic chain uses the same surface, and differs from the others only in the dynamics on it. The spectra below show what different dynamics on one surface look like.

Same invariant, different spectra #

If the invariant were all that mattered, the three metallic chains with the same two tiles would have the same spectrum, and they do not.

Each spectrum is a set of many thin bands, the approximant’s version of the Cantor set. The big gaps sit in different places for the three chains, because the positions of the gaps are labelled by the frequencies of the chain’s letters and patches. How often each patch occurs found those frequencies as an eigenvector, and they are different numbers for the golden, silver and bronze rules. By the gap-labelling theorem, each gap is labelled by the fraction of states below it, and for a chain of this kind those fractions are whole-number multiples, modulo one, of the frequency of one letter. That frequency is 0.618 for the golden chain, 0.707 for the silver and 0.768 for the bronze, the long letter’s share in each, read off the eigenvector of its rule. Different frequencies give different labels, so the gaps sit in different places. The invariant fixes how far the chain is from a crystal, and the substitution fixes where the gaps open.

The division of labour has a physical reading for a real quasicrystal. What the invariant knows is the difference between the local environments of the two kinds of site, a property of the chemistry of the tiles. What the substitution knows is how those environments are arranged, a property of the structure. The measurements here say the first sets how much of the energy axis survives and the second sets where the holes are. Two quasicrystals built from the same pair of tiles in different arrangements would have spectra that thin at the same rate with holes in different places. Two built from different tiles in the same arrangement would have holes carrying the same labels, since the labels depend on the arrangement alone, thinning at different rates. Which labelled gaps are actually open, and how wide each is, depends on both, and the chains here are too few to separate the two contributions gap by gap.

How much of the energy axis survives is another matter, and here the invariant seems to decide nearly everything. The total width of the allowed energies falls with the approximant’s length as a power, and at $V = \tfrac12$ the powers are $-0.128$, $-0.127$ and $-0.126$ for the golden, silver and bronze chains, the same to within one per cent. At $V = 1$ they are $-0.287$, $-0.283$ and $-0.277$, within four per cent. The chains have approximants of quite different lengths, and their gaps fall in different places, yet the rate at which the spectrum thins is almost entirely set by the strength of the difference between the tiles. The bands are found on a fine grid of energies, and the grid misses the narrowest: it finds 579 of the 610 bands the golden approximant has at $V = \tfrac12$. The widths are therefore slight overestimates, the same way for all three chains.

The agreement is a measurement, not a theorem, and it should be read as one. That a spectrum’s fractal dimension depends on the modulation strength and, at weak modulation, on little else is known for the Fibonacci chain in the limit of small $V$. That three different chains agree so closely at moderate $V$ is what these approximants show, and nothing here proves it continues at longer lengths or stronger modulation.

What the comparison has to refuse #

The refusal is the guess itself. A silver-chain invariant different from the golden chain’s would have been evidence that the conserved quantity measures something about the chain. It is refused by an exact integer equation at every level computed. The controls are the other half of the evidence. A quantity conserved by every substitution in sight would be a property of matrix multiplication, not of reversible rules, and period-doubling and Thue–Morse show that it is not conserved in general.

Still open: two tiles, three tiles, and the surface #

The Fricke quantity belongs to two matrices. A chain of three kinds of tile has three site matrices, and the free group on three letters has no single commutator whose trace every automorphism preserves: its automorphisms act on a space of traces of dimension more than three, and the invariants of that action are more than one polynomial. Whether the chains built from cubic inflation factors, with their three letters and their fractal windows, conserve a set of quantities that likewise measures the tiles and not the chain is the natural extension. The algebra for it is classical, and the measurement would be the same integer computation as here, run on three matrices.