Thinking in Waves Episode 41 / Part XIII — The Frontier, as Waves
With no road back, there is nothing to bounce into
Topological Waves —
They survive disorder not because they are strong
Episode 37 found that in one dimension any disorder, however weak, stops a wave.
★★ Yet some waves do not stop — not from strength, but because there is nowhere to backscatter into.
★ Checked numerically: along one edge branch the group velocity has the same sign at all 42 points. A counter-propagating state does not exist.
Episode 35 showed that light's orbital angular momentum carries a winding number no continuous deformation can change.
★ This episode is about that unchangeable integer counting how many roads a wave has.
01First, build a quantity that can only be an integer
| \(v\) | \(w\) | winding number (numerical) | gap \(|v-w|\) | Phase |
|---|---|---|---|---|
| 1 | 0.2 | 0.000000000 | 8.0e−1 | trivial |
| 1 | 0.999 | 0.000000000 | 1.0e−3 | trivial |
| 1 | 1.001 | ★ 1.000000000 | 1.0e−3 | ★ topological |
| 1 | 5 | 1.000000000 | 4.0e+0 | ★ topological |
| 1 | 1 | — | 0 | ★ gap closes (undefined) |
The point of §01
Move \(w\) from 0.999 to 1.001 — a change of 0.2 % — and the winding number jumps from 0 to 1.
★ And it never takes a value in between. No 0.5, no 0.9 — zero or one.
★★ The only place it can jump is \(v=w\), that is, where the gap closes. You cannot go from 0 to 1 without closing the gap — that is what "cannot be changed by continuous deformation" means.
★ It is the same kind of quantity as the winding number of an optical vortex in Episode 35 (that one in real space, this one in momentum space).
02Winding number 1 means states at the ends
Diagonalize a finite chain (40 unit cells = 80 sites) and look near \(E=0\).
| \(v\) | \(w\) | the four smallest \(|E|\) | edge states |
|---|---|---|---|
| 1 | 0.5 | 5.0e−1 5.0e−1 5.1e−1 5.1e−1 | none |
| 1 | 0.9 | 1.2e−1 1.2e−1 1.6e−1 1.6e−1 | none |
| 1 | 1.1 | 4.2e−3 4.2e−3 1.4e−1 1.4e−1 | ★ two |
| 1 | 2.0 | 1.4e−12 1.4e−12 1.0e+0 1.0e+0 | ★ two |
Their localization length is measurable too — theory says \(\xi=1/\ln(w/v)\).
| \(w/v\) | measured \(\xi\) | \(1/\ln(w/v)\) | ratio |
|---|---|---|---|
| 1.5 | 2.466303 | 2.466303 | 1.00000 |
| 2.0 | 1.442695 | 1.442695 | 1.00000 |
| 5.0 | 0.621335 | 0.621335 | 1.00000 |
The point of §02 — bulk–boundary correspondence
★ An integer computed from the interior tells you how many states sit at the edge.
★★ That is remarkable: the winding number was computed for an infinite system with no boundary, yet it knows what happens when you make one.
★ At \(w=1.1\) the energy is \(4.2\times10^{-3}\) rather than zero. With \(\xi=10.5\) and a chain of only 40 cells, the two edge states overlap — at \(w=2\) it falls to \(10^{-12}\).
03★ Protected against what, exactly?
People say "topological, therefore robust against disorder." Robust against everything? We try two kinds (\(v=1,\ w=2\)).
| Disorder | the two smallest \(|E|\) | Verdict |
|---|---|---|
| none | 1.36e−12 1.36e−12 | stays at \(E=0\) |
| 30 % on the bonds | 6.69e−12 6.69e−12 | ★ stays at \(E=0\) |
| 60 % on the bonds | 3.25e−11 3.25e−11 | ★ stays at \(E=0\) |
| just 0.05 on the sites | 1.21e−2 2.80e−2 | ★ moves off |
| 0.3 on the sites | 7.23e−2 1.68e−1 | ★ moves far off |
The point of §03 — ★★★ the caveat that matters
60 % disorder on the bonds does not move \(E=0\).
0.05 on the sites does.
★ What protects is a symmetry — bond disorder preserves sublattice (chiral) symmetry; site disorder breaks it.
★★ So it is not "topological, hence robust to anything," but robust to a specified thing. Against disorder that breaks that symmetry, it is not robust at all.
★ On Episode 15's scorecard this reads "◎, conditional on a stated assumption."
04In two dimensions — how many times do you cover the sphere?
| \(m\) | Chern number (numerical) | minimum gap | Phase |
|---|---|---|---|
| −3.0 | 0.000000188 | 2.000000 | trivial |
| −2.5 | 0.000001419 | 1.000000 | trivial |
| −2.0 | — | 0.000000 | ★ gap closes |
| −1.0 | 0.999994618 | 2.000000 | ★ \(C=+1\) |
| 0.0 | — | 0.000000 | ★ gap closes |
| +1.0 | −0.999994618 | 2.000000 | ★ \(C=-1\) |
| +3.0 | −0.000000188 | 2.000000 | trivial |
The point of §04
★ Again, only integers (the \(10^{-7}\) residue is the grid spacing).
★★ It jumps only at \(m=-2,\ 0,\ +2\) — and those three points are exactly where the gap closes.
★ "An integer cannot change unless the gap closes" — the same rule as §01, now in a different dimension. Episode 30 found that dimension sets the order; this integer, by contrast, stays an integer in any dimension.
05Edge bands — the branch that crosses the gap
Cut \(y\) to a finite 24 rows and keep \(k_x\) as a momentum. At \(m=-1\) (\(C=+1\)) the bulk bands sit at \(|E|\ge1\), and a branch appears in the gap between them.
| \(k_x/\pi\) | \(E\) of the left-edge state | \(E\) of the right-edge state |
|---|---|---|
| −0.333 | +0.866025 | −0.866025 |
| −0.200 | +0.587785 | −0.587785 |
| −0.067 | +0.207912 | −0.207912 |
| +0.067 | −0.207912 | +0.207912 |
| +0.200 | −0.587785 | +0.587785 |
| +0.333 | −0.866025 | +0.866025 |
The point of §05
The numbers are exactly \(E=-\sin k_x\) — a single branch running all the way through the gap from below to above.
★ There is one branch because \(C=1\) (bulk–boundary correspondence in 2D).
★★ And the two edges have opposite signs — at any given energy there is one state on the left edge and one on the right.
06★★ One edge, one direction
Count the sign of the branch's group velocity \(dE/dk_x\) at every point inside the gap.
| Edge | positive velocity (rightward) | negative velocity (leftward) |
|---|---|---|
| left | 0 | ★ 42 |
| right | ★ 42 | 0 |
The point of §06 — ★★★ the core of this episode
On the left edge all 42 points are negative. A counter-propagating state does not exist.
★ To backscatter you need a state on the same edge, at the same energy, going the other way. If there is none, no amount of disorder gives the wave anywhere to bounce into.
★★ The wave is not strong. There is simply no destination.
★ The counter-propagating state is on the other edge — reaching it means crossing the whole width of the system. The wider the sample, the exponentially smaller that chance.
07Squaring this with Episode 37
| Claim | Assumption | |
|---|---|---|
| Episode 37 | in 1D any disorder localizes | ★ you can go both forward and back |
| Here | an edge branch survives disorder | ★ there is no backward state |
The point of §07
★ No contradiction. Episode 37's localization comes from interference between forward and backward waves, so with no backward wave it cannot happen.
★★ Episode 37 put it as Pólya's recurrence theorem — a wave that never returns cannot interfere with itself. Here the road back does not exist at all, so the same reason applies even more strongly.
★ But §03's caveat still bites — disorder strong enough to connect the two edges will stop it after all. It is not unconditional.
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Winding number is an integer | ◎ numerics | 0 or 1; changes only at \(v=w\) |
| Winding 1 → two \(E=0\) edge states | ◎ numerics | bulk–boundary correspondence |
| Edge length \(1/\ln(w/v)\) | ◎ numerics | ratio 1.00000 |
| ★ What protects is a symmetry | ◎ numerics | ★ bonds yes, sites no |
| Chern number is an integer | ◎ numerics | \(0,\pm1\); jumps at \(m=-2,0,2\) |
| ★ The edge branch is one-way | ◎ numerics | ★ all 42 points the same sign |
| Hence no backscattering | ◎ logic | Episode 37's premise fails |
| ★ What real materials do | × no | ★ minimal models only |
| ★ What interactions do | × no | ★ single-particle problems only |
Exercises
- §01 found the winding number never takes 0.5, yet Episode 13 found the order varies continuously. Which is right?
Show answer
Both — they are different quantities. Episode 13's order \(s=\frac12\pm\sqrt{\frac14-\lambda}\) is a continuous function of a continuous \(\lambda\). The winding number here is the degree of a map, an integer by definition.
★ The difference is local versus once-around: the order describes behaviour at a point, the winding number describes the result of going all the way round. Since you come back to where you started, the total angle can only be an integer multiple of \(2\pi\).
★★ They are not unrelated — as §04 showed, the integer jumps where the gap closes, i.e. at a critical point, and Episode 13's order went complex at a critical point too. "Something only happens at criticality" is common to both. - §06 says there is no counter-propagating state. Doesn't that violate conservation of momentum?
Show answer
No. Once there is disorder there is no translation symmetry, so momentum is not conserved anyway.
★ When a wave bounces, the missing momentum is absorbed by the disorder (the medium). The statement here is not "momentum cannot be transferred" but "there is no state to land in."
★★ With no final state the transition rate is zero — not through the matrix element of Fermi's golden rule but through the final density of states. ★ So supply a final state — the other edge — and it can bounce: that is §07's caveat. - §03 found it weak against site disorder. Why is the SSH model called "topological" at all?
Show answer
Because the protecting symmetry is declared up front. Topological phases are always classified together with which symmetries are assumed, and SSH belongs to the chiral-symmetric class.
★ Within that class the winding number is an integer and cannot be changed continuously — that much is true. Step outside the class and the guarantee is gone.
★★ In this series' language: not unconditional robustness but robustness under a stated premise. ★ Just as Episodes 7 and 15 insisted on asking "what did you divide by," one must ask each time which symmetry was assumed.
Summary — it survives not from strength but from having nowhere to go
First we built a quantity that can only be an integer — the SSH winding number. Move \(w\) from 0.999 to 1.001, a change of 0.2 %, and it jumps from 0 to 1, never taking a value in between. It can only jump at \(v=w\), where the gap closes.
And that integer predicts how many states sit at the edges (bulk–boundary correspondence) — two \(E=0\) states, and only when \(w>v\). Their localization length matched \(\xi=1/\ln(w/v)\) with a ratio of 1.00000.
★ But there is a caveat that matters — 60 % disorder on the bonds leaves \(E=0\) untouched, while 0.05 on the sites moves it. What protects is a symmetry (chiral symmetry), not robustness against anything whatsoever.
Two dimensions behaved the same way — the Chern number takes only \(0,\pm1\), and jumps only at \(m=-2,0,+2\): exactly the three points where the gap closes.
★★★ The core — at \(C=1\) a single branch crosses the gap.
Counting the sign of its group velocity: all 42 points negative on the left edge, all positive on the right.
On one edge, a counter-propagating state does not exist.
★ Backscattering needs a state on the same edge, at the same energy, going the other way. With none, disorder has nowhere to send the wave — the wave is not strong; there is simply no destination.
★ This does not contradict Episode 37 — that localization comes from interference between forward and backward waves, and with no backward wave it cannot occur. It restates "a wave that never returns cannot interfere with itself" (Pólya) in a stronger form. ★ Yet disorder strong enough to link the two edges will stop it after all.
★ Next time we go where the order stops being a single number at all — intermittency in turbulence.
★★ Kolmogorov's \(-5/3\) is one order, but measurements depart from it systematically. The departure is the order becoming a distribution — multifractality. To Episode 14's three, a fourth is added: the order is not one number.
This document is Episode 41 of the "Thinking in Waves" series. Computations are in kenshou/topo.js. On the numbers here: the numerical winding integral, the finite-chain diagonalization and edge-state localization lengths, the shift of \(E=0\) under each kind of disorder, the Chern-number integral, and the strip band structure with its group-velocity sign counts were all computed here. The SSH model and its winding number, bulk–boundary correspondence, the edge length \(1/\ln(w/v)\), the QWZ model's Chern numbers and phase diagram, protection by chiral symmetry, and the suppression of backscattering by chiral edge states are all well-known results and none are discoveries of this article — what this article did was place them between Episode 37 (localization) and Episode 35 (winding), and check "protected against what" by varying the kind of disorder inside one and the same model. §03's conclusion (robust to bond disorder, fragile to site disorder) follows from SSH belonging to the chiral-symmetric class and does not generalize to every topological phase. §05–06 are single-particle calculations (no interactions) based on a width of 24 rows and 121 values of \(k_x\). The \(10^{-7}\) residue in §04's Chern numbers comes from the grid; the true values are integers. — To print, use your browser's Print and "Save as PDF."