Thinking in Waves Episode 37 ／ Part XIII · Frontier physics as waves

Breaking the assumption of a clean medium

Anderson localization ──
whether a wave stops depends on whether it comes back In a disordered medium a wave stops with no absorption at all. In one dimension any disorder, however weak, localizes ($\xi\propto W^{-2}$; measured slope −1.97).
★★ And the reason it depends on dimension came out ── a wave that never returns cannot interfere with itself.
★ The dimensional dependence of localization is Pólya's recurrence theorem.

Prerequisites: Episodes 5, 6, 8, 27, 28, 30Computation: kenshou/kyokuzai.js

For 36 episodes the medium has been clean and uniform. Now we break that.
The naive expectation is "it scatters, so it diffuses" ── but past a threshold it does not even diffuse (Anderson localization, 1958).
★ This is not restricted to quantum systems ── light, sound, microwaves, water waves, cold atoms: it is a pure wave phenomenon.

01Measuring the localization length in one dimension

The simplest model (tight binding) $$\psi_{n+1}=(E-\varepsilon_n)\psi_n-\psi_{n-1},\qquad \varepsilon_n\ \text{uniform on }[-W/2,\,+W/2]$$

Measure the growth rate of the transfer-matrix product (the Lyapunov exponent $\gamma$); the localization length is $\xi=1/\gamma$

Disorder $W$	$\gamma=1/\xi$	$\xi$ [sites]	$\xi W^2$
0.25	7.2262e−4	1383.86	86.49
0.50	2.8142e−3	355.34	88.84
1.00	1.1115e−2	89.97	89.97
2.00	4.3764e−2	22.85	91.40
4.00	1.6989e−1	5.89	94.18

The heart of §01

Regressing $\ln\xi$ on $\ln W$ gives a slope of −1.97 (weak-disorder prediction −2).
★★ However small $W$ is, $\xi$ stays finite ── it always localizes.
★ But $\xi\propto W^{-2}$ makes it enormous at weak disorder ── $\xi\approx8.6\times10^5$ sites at $W=0.01$, $8.6\times10^7$ at $W=0.001$.

★ So experiments look "not localized" ── the sample is simply too short. The same structure as Episode 5's "a period longer than the age of the universe is indistinguishable from a constant".

02The same exponential decay, from a different cause

Phenomenon	Form	Cause	Does it fluctuate?
Evanescent (Episode 6)	$e^{-\kappa x}$	$k^2<0$ (deterministic)	no
Tunnelling (Episode 28)	$e^{-2\kappa a}$	$k^2<0$ (deterministic)	no
★ Anderson localization	$e^{-x/\xi}$	★ multiple-scattering interference (statistical)	★★ yes

Numerically: at the same $W=1$, changing only the random seed scatters $\xi$ over 84.98–91.86, a relative standard deviation of 2.46 %.

The heart of §02

Every sample gives a different value ── the signature of a statistical exponent. (Increasing $N$ shrinks the spread as $1/\sqrt N$ ── self-averaging.)
★ An evanescent wave has no spread at all.

03★★ Why it depends on dimension ── does the wave come back?

Behaviour changes with dimension ── in 1D and 2D any disorder localizes; in 3D weak disorder still diffuses (there is a mobility edge).
★ Why is 2 the boundary? Localization happens because the wave returns and interferes with itself, so what matters is how often it returns.

Expected number of returns $$\int\frac{d^dk}{(2\pi)^d}\,\frac{1}{1-\lambda(k)},\qquad \lambda(k)=\frac1d\sum_i\cos k_i$$

Grid resolution	$d=1$	$d=2$	★ $d=3$
20	7.00	3.18	1.743111
40	13.67	3.64	1.655221
80	27.00	4.08	1.598384
160	53.67	4.53	1.563634

The heart of §03 ── ★★★ Pólya's theorem (1921)

$d=1$ and $d=2$ keep growing as the grid is refined (divergent). Only $d=3$ settles (Watson's integral 1.516386…; it approaches only as $1/N$, so convergence is slow).

★ A random walk in $d\le2$ always returns to the origin (recurrent).
In $d\ge3$ it may never return ── in 3D the return probability is $1-1/1.516386=0.340537$.

★★ That is where the dimensional dependence of localization comes from.
A wave that never returns cannot interfere with itself, so it does not localize. In $d\le2$ it always returns, so the interference piles up and localization is inevitable. In 3D only 34 % return ── not enough unless the disorder is strong.

04Counting the weak-localization correction dimension by dimension

The return probability density of a diffusing wave, and its integral $$P(t)=\frac{1}{(4\pi Dt)^{d/2}}\qquad\Longrightarrow\qquad \Delta\sigma\propto-\int_\tau^{\tau_\phi}\frac{dt}{(4\pi Dt)^{d/2}}$$

$\tau_\phi/\tau$	$d=1$ ($\propto\sqrt t$)	★ $d=2$ ($\propto\ln t$)	$d=3$ (converges)
$10^{2}$	5.078	0.366468	0.04040710
$10^{4}$	55.85	0.732936	0.04444781
$10^{6}$	563.6	1.099403	0.04485188
$10^{8}$	5641.3	1.465871	0.04489229

The heart of §04

$d=1$ grows as $\sqrt{\tau_\phi}$, $d=2$ as $\ln\tau_\phi$ (0.3665 per two decades), and $d=3$ stops.
★ In $d\le2$ the correction grows without bound and perturbation theory fails ── the picture of diffusion itself breaks down ＝ localization. In 3D the correction stays finite, so weak disorder still diffuses.
★★★ $d=2$ is where it becomes logarithmic ── the signature of a marginal dimension.

05★ Is this $d=2$ the same as Episode 30's?

Episode	What happens at $d=2$	Where it comes from
30	$1/r^2$ sits exactly at critical $\lambda=1/4$	the geometry of the radial reduction (the minimum of $l(l+1)$)
★ Here	the return count starts to diverge	the integral of $P(t)\propto t^{-d/2}$

The heart of §05 ── ★ checking (the habit of Episodes 29–30)

Verdict: different origins, but a common root.
Both come from "a logarithm appears at $d=2$" ── in Episode 30 because $l=(d-3)/2$ hits $-1/2$ and $l(l+1)$ is minimal, here because $\int dt\,t^{-d/2}$ diverges logarithmically at $d=2$.
★★★ The dimension where a logarithm appears is, in both cases, "the dimension where the power becomes $-1$". The same mathematics as Episode 27's "1/f comes from log-uniform" ── when the exponent is $-1$, the integral is a logarithm. That is all.

★ But not overstating it: they do not follow from one theorem; the same integral diverges in both. The physics (geometry versus interference) is different.

06Any wave will do

System	Wave	Dimension	What was observed
Disordered photonic lattice	light	2 (transverse)	★ the beam stops spreading
Microwave waveguide	microwaves	1	★ transmission falls exponentially
Ultrasound	sound	3	★ a mobility edge was observed
Cold atoms	matter waves	1 and 3	★ expansion halts

The heart of §06

It looks like a quantum phenomenon and is explained by interference alone.
★ The same structure as Episode 8's "uncertainty is the bandwidth theorem times $\hbar$" ── $\hbar$ is stage machinery; being a wave is what matters.

Exercises

If 1D always localizes, why does current flow in a copper wire?
Show the answer
Three reasons. ① A wire is not one-dimensional (tens of millions of transverse channels ＝ effectively 3D). ② The localization length exceeds the sample ── by §01, $\xi\propto W^{-2}$, astronomically long in a clean metal. ③ ★ Phase is not preserved ── localization is an interference effect, so phonon and electron–electron scattering that destroys phase destroys it.
★ ③ dominates in practice, which is why localization experiments are done cold ── to stop the phase-breaking scattering.
★★ This is Episode 22's decoherence exactly: if the phase-coherence length $L_\phi$ is shorter than $\xi$, localization is invisible.
§02 showed the exponent fluctuates. What does that mean for measurement?
Show the answer
One sample does not give a representative value. The right thing to average is $\ln$ of the conductance, not the conductance (the distribution is log-normal).
★ Turned around ── the size of the fluctuation itself carries information. It is called universal conductance fluctuation, and is of order $e^2/h$ regardless of sample details.
★★ "Look at the distribution, not the mean" is the practical difference between a statistical and a deterministic exponent (the same habit as the scatter of the 1/f exponent $\beta$ in Episode 27).
Is §03's "does it return" related to Episode 30's "does it leave a tail"?
Show the answer
Not directly ── they are different questions. Episode 30 asked what remains behind the front in a clean medium (Huygens). This episode asks whether a wave keeps spreading in a disordered one.
★ They do share something ── in both, the boundary is where a logarithm appears at $d=2$ (§05).
★★ The caution this series keeps repeating: when you see the same number (here $d=2$), first check which integral diverged. Same number, different integral, different phenomenon.

Summary　Whether it stops depends on whether it comes back

Disorder a medium and the wave stops with no absorption at all (Anderson localization).

★ In one dimension any disorder localizes. Measuring the Lyapunov exponent by transfer matrix gives $\xi W^2$ roughly constant (86–94) and a slope of −1.97 for $\ln\xi$ against $\ln W$ (prediction −2). But $\xi\approx8.6\times10^7$ sites at $W=0.001$ ── experiments miss it because samples are short (Episode 5's structure again).

The same exponential decay has a different cause ── an evanescent wave's $k^2<0$ exponent is deterministic and does not fluctuate, while localization is a statistical exponent from multiple-scattering interference and varies by 2.5 % between samples.

★★★ And the reason for the dimensional dependence came out ── Pólya's recurrence theorem.
A random walk in $d\le2$ always returns; in 3D the return probability is 0.340537.
A wave that never returns cannot interfere with itself, so $d\le2$ always localizes while 3D needs strong disorder.

The weak-localization correction says the same thing ── $d=1$ grows as $\sqrt{\tau_\phi}$, $d=2$ as $\ln\tau_\phi$, $d=3$ converges. In $d\le2$ the correction diverges and the diffusion picture itself breaks.

★ Episode 30 also singled out $d=2$, but on checking it is a different origin (geometry versus interference) ── though with a common root: in both, it is the dimension where the power becomes $-1$ and the integral turns logarithmic. The same mathematics as Episode 27's 1/f.

Finally ── none of this needs quantum mechanics. It happens with light, sound and microwaves. $\hbar$ is stage machinery; being a wave is what matters (Episode 8's conclusion again).

Next episode So far spacetime has been a given stage. ★ Next time we look at the stage itself as waves ── quantum gravity.
★ A striking number of approaches say the same strange thing ── at short distances the "dimension" runs from 4 to 2 (dimensional reduction of the spectral dimension).
★★ Since Episode 30 established $\alpha=(d-3)/2$, that means ── ★ the order itself runs. And $d=2$ is the dimension Episode 30 found to be critical.

This document is Episode 37 of the "Thinking in Waves" series. The computations are in kenshou/kyokuzai.js. On the numbers here: the 1D localization lengths, the sample-to-sample spread, the lattice return-count integrals and the weak-localization corrections were all computed for this article. Anderson localization, complete localization in 1D with $\xi\propto W^{-2}$, the scaling theory of the dimensional dependence, Pólya's recurrence theorem and Watson's integral 1.516386, and the dimensional dependence of weak localization are all well-known results and not discoveries of this article ── what this article did was line them up under the single phrase "does the wave come back", and show the correspondence numerically. The return counts in §03 are for a free random walk, not a calculation of localization itself (they correspond to the quantity weak-localization perturbation theory actually uses). "Always localized for $d\le2$" is the conclusion of scaling theory; only the 1D case is directly verified here. $\xi W^2$ is not exactly constant because the weak-disorder approximation degrades at larger $W$. The experiments listed in §06 are reported results, not verified here. ── To print, use your browser's Print → Save as PDF.