Exact transport and spectral dimensions for the almost Mathieu operator

Douglas, Seth; Mghirbi, Nidhal

doi:10.5281/zenodo.23864352

Results · Transport in quasi-periodic systems

Exact transport and spectral dimensions for the almost Mathieu operator

Seth Douglas and Nidhal Mghirbi · version 2.0.1 · Zenodo, 1 November 2026 · 50 pages · doi:10.5281/zenodo.23864352 · PDF

In plain wordsIn a standard model of a quasi-periodic crystal, usually called localized, whether a wave spreads, and how fast at its farthest, is decided by how well one number is approximated by fractions; for some computable choices of that number, no algorithm can compute that rate.

The resultat three depths

In one sentenceIn the Aubry–André model of a quasi-periodic crystal, even where it is usually called localized, whether a wave spreads, and how fast it spreads in its farthest bursts, is decided by how well one number, the frequency, can be approximated by fractions, weighed against the strength of the potential; the paper gives that upper rate exactly, for almost every phase, and shows that for some computable frequencies no algorithm can compute it.

As the paper states it, in briefFor the almost Mathieu operator with coupling $λ > 1$ , every irrational frequency $α$ and almost every phase, every moment of a wave packet started at one site has the same upper transport exponent $\max (0, 1 - \log λ / β (α))$ , where $β (α) = lim sup \log q_{n + 1} / q_{n}$ ; the lower bound holds at every phase, and when $β (α) > \log λ$ at the resonant times almost all of the packet makes the trip, uniformly in the phase. For almost every phase, both packing dimensions of every nonzero spectral measure equal the same number; transport switches on exactly at $λ = e^{β}$ ; and an explicit computable almost Mathieu operator has a fastest upper transport exponent of Turing degree $𝟎^{″}$ .

For a scientist
One paragraph

The almost Mathieu operator on $ℓ^{2} (ℤ)$ , $(H u) (n) = u (n + 1) + u (n - 1)$ $+ 2 λ \cos (2 π (θ + n α)) u (n),$ is the Aubry–André model, and above the self-dual point, $λ > 1$ , it is often summarized as localized. For every irrational $α$ and almost every phase $θ$ , the paper shows that every moment of a wave packet started at one site has the same upper transport exponent, $\max (0, 1 - \log λ / β (α))$ , where $β (α) = lim sup \log q_{n + 1} / q_{n}$ measures how well $α$ is approximated by its continued-fraction convergents $p_{n} / q_{n}$ . The lower bound holds at every phase, and when $β (α) > \log λ$ it is carried, at the resonant times, by almost all of the packet. The lower transport exponents vanish, by a theorem of Damanik and Tcheremchantsev, so in the window $1 < λ < e^{β}$ the spreading is strongly intermittent. For almost every phase, both packing dimensions of every nonzero spectral measure equal the same number; for $0 < \log λ < β < \infty$ this makes the 2025 bound $2 (1 - \log λ / β)$ of Jitomirskaya, Liu and Tcheremchantsev, who noted that its factor 2 might not be sharp, sharp with factor 1. Transport therefore switches on exactly at the arithmetic spectral transition $λ = e^{β (α)}$ of Avila, You and Zhou. Because $β (α)$ can be noncomputable when $α$ is computable, an explicit computable almost Mathieu operator has a fastest upper transport exponent of Turing degree $𝟎^{″}$ . A companion paper extends the law, with the least Lyapunov exponent on the spectrum in place of $\log λ$ , to every supercritical real-analytic quasi-periodic operator of type I. A third paper, Worst-case transport and light cones for quasi-periodic operators, treats the remaining phases: the worst phase has exponent $1 - \log λ / (2 β)$ when $β_{e} = β$ is finite and at least $\log λ$ , and it carries both laws to the light cones of the XY spin chain.

For an expert
The main theorems, as stated in the paper (Section 1)

Notation from the paper: $H_{λ, α, θ}$ is the operator above, with coupling $λ > 0$ , irrational frequency $α$ and phase $θ \in 𝕋 = ℝ / ℤ$ ; $p_{n} / q_{n}$ are the continued-fraction convergents of $α$ , and $β (α) = \underset{n \to \infty}{lim sup} \log q_{n + 1} / q_{n} \in [0, \infty]$ . With $ψ_{t} = e^{- i t H} δ_{0}$ , the packet is averaged in time with the weight $(2 / T) e^{- 2 t / T}$ :

a (j, T) = \frac{2}{T} \int_{0}^{\infty} e^{- 2 t / T} | ⟨ δ_{j}, ψ_{t} ⟩ |^{2} d t,

P (N, T) = \sum_{| j | > N} a (j, T),

M_{p} (T) = \sum_{j \in ℤ} | j |^{p} a (j, T) .

The upper transport exponents are $β^{+} (p) = \underset{T \to \infty}{lim sup} \log M_{p} (T) / (p \log T)$ and $α_{u}^{+} = sup_{p > 0} β^{+} (p)$ . The phase set $G_{α}$ consists of the $θ \in 𝕋$ with $| \sin (2 π q_{n} θ) | \geq q_{n}^{- 2}$ for all sufficiently large $n$ ; it has full measure for every irrational $α$ . Further, $δ_{n} = q_{n} α - p_{n}$ and $b_{n} = q_{n}^{- 1} \log | δ_{n} |^{- 1}$ . For a finite Borel measure $μ$ , $d_{μ}^{+} (E) = \underset{r ↓ 0}{lim sup} \log μ ([E - r, E + r]) / \log r$ is its upper local dimension, and its lower and upper packing dimensions $\dim_{P}^{-} (μ)$ and $\dim_{P}^{+} (μ)$ are the $μ$ -essential infimum and supremum of $d_{μ}^{+}$ ; $μ_{φ}$ is the spectral measure of $φ \in ℓ^{2} (ℤ)$ . A real $x$ is right-c.e. relative to $𝟎^{'}$ if the set of rationals $r > x$ is computably enumerable relative to the halting problem. Throughout, $\max {0, 1 - \log λ / β (α)}$ is read as $0$ when $β (α) = 0$ and as $1$ when $β (α) = \infty$ .

Theorem A (The transport law)

Let $λ > 1$ and let $α$ be irrational.

If $β (α) < \infty$ , then for every $θ \in G_{α}$ and every $p > 0$ ,
$β^{+} (p) = α_{u}^{+}$ $= \max {0, 1 - \frac{\log λ}{β (α)}}$
for $H_{λ, α, θ}$ , where the right side is read as $0$ when $β (α) = 0$ . Moreover, for every $a > \max {0, 1 - \log λ / β (α)}$ and every $m > 0$ , $P (T^{a}, T) = O (T^{- m})$ as $T \to \infty$ .
If $β (α) = \infty$ , then $β^{+} (p) = α_{u}^{+} = 1$ for every phase and every $p > 0$ .

The lower bound $β^{+} (p) \geq \max {0, 1 - \log λ / β (α)}$ in part 1 holds for every phase and every $p > 0$ .

Theorem B (Almost all of the packet moves)

Let $λ > 1$ , let $α$ be irrational, let $h > \log λ$ and $c > h$ , and assume $b_{n} \geq c$ for infinitely many $n$ . Then

\lim_{n \to \infty, b_{n} \geq c} \inf_{θ} \inf_{e^{h q_{n}} \leq T \leq | δ_{n} |^{- 1}} P_{θ} (T e^{- h q_{n}}, T) = 1 .

The same holds, uniformly in the starting site, for packets started at any single site. Version 1.0.0 gave the fixed fraction $2^{- 32} / (1 + 4 π λ)$ .

Theorem C (Packing dimension)

Let $λ > 1$ and let $α$ be irrational.

For every phase $θ$ and every nonzero $φ \in ℓ^{2} (ℤ)$ , $d_{μ_{φ}}^{+} (E) \geq \max {0, 1 - \log λ / β (α)}$ for $μ_{φ}$ -almost every $E$ , with the right side read as $0$ when $β (α) = 0$ and as $1$ when $β (α) = \infty$ . In particular $\dim_{P}^{-} (μ_{φ})$ is at least this number.
If $β (α) < \infty$ and $θ \in G_{α} \cap (G_{α} - α)$ , then for every nonzero $φ \in ℓ^{2} (ℤ)$ ,
$d_{μ_{φ}}^{+} (E) = \max {0, 1 - \frac{\log λ}{β (α)}}$ $for μ_{φ} -a.e. E,$
so that
$\dim_{P}^{-} (μ_{φ})$ $= \dim_{P}^{+} (μ_{φ})$ $= \max {0, 1 - \log λ / β (α)},$
read as $0$ when $β (α) = 0$ .

Theorem D (Computability)

Let $λ > 1$ be computable, let $α$ be a computable irrational with $β (α) < \infty$ , and let $θ \in G_{α}$ be computable. Then $α_{u}^{+} (H_{λ, α, θ})$ is computable from $β (α)$ . If moreover $β (α) > \log λ$ , then $α_{u}^{+}$ and $β (α)$ have the same Turing degree.
Take $λ = 2$ and $θ = 1 / 4$ . As $α$ ranges over the computable irrationals whose convergent denominators are eventually odd, $α_{u}^{+} (H_{2, α, 1 / 4})$ ranges over exactly the reals in $[0, 1]$ that are right-c.e. relative to $𝟎^{'}$ .
There is an explicit computable irrational $α$ such that $α_{u}^{+} (H_{2, α, 1 / 4}) \in [11 / 23, 1 / 2]$ has Turing degree $𝟎^{″}$ .

As the paper says, part 2 of Theorem A is due to Jitomirskaya and Zhang, and the case $β (α) = 0$ of part 1 to Damanik and Tcheremchantsev; they are included so that the statement covers every frequency. The new content is part 1 for $0 < β (α) < \infty$ . The set $G_{α} \cap (G_{α} - α)$ has full measure, so for almost every phase the fastest upper exponent, every upper moment exponent and both packing dimensions coincide.

The question

Put a quantum particle on one site of a chain whose potential is quasi-periodic: never repeating, yet built from a single cosine sampled at an irrational step $α$ . Does its wave packet spread, and if so, how fast? For the Aubry–André model the usual summary is a single threshold: below $λ = 1$ the packet spreads, above it the particle stays put.

That summary is incomplete. Above $λ = 1$ , whether a packet spreads, and how fast it spreads at its farthest, depends on the arithmetic of the frequency, and the paper gives the exact upper exponent.

The answerone upper exponent, for every moment

Average the packet over time, with the standard exponential weight. For almost every phase, every moment of the packet, its average distance, its root-mean-square spread and every higher moment, then has the same upper growth exponent, $\max (0, 1 - \log λ / β (α))$ : along some sequence of times the spread grows like that power of time, and it never outgrows it by any fixed power. The lower bound holds at every phase. When $β (α) > \log λ$ it is not carried by a vanishing tail: at the resonant times almost all of the packet makes the trip, uniformly in the phase.

The lower exponents vanish, as Damanik and Tcheremchantsev proved; they reflect times at which the packet is confined. So for $1 < λ < e^{β}$ the spreading comes in bursts near resonant times, and the upper exponents record the largest of them. The paper proves nothing about the proportion of time spent in each regime.

The same number is, for almost every phase, the exact packing dimension of every nonzero spectral measure: its upper and lower packing dimensions both equal it, and at every phase they are at least it. For $0 < \log λ < β < \infty$ this makes the 2025 bound $2 (1 - \log λ / β)$ of Jitomirskaya, Liu and Tcheremchantsev sharp with factor 1; they had noted that the factor 2 might not be sharp.

Why arithmetic decides it

Write $α$ as a continued fraction. Its convergents $p_{n} / q_{n}$ are the best fractions near it, and $β (α)$ measures how fast their denominators can jump: it is positive when, infinitely often, the next denominator is exponentially larger than the one before. For the golden mean, whose digits are all 1, $β = 0$ ; $β$ can also be infinite.

The intuition, not the proof, as the paper sketches it. When $α$ lies extraordinarily close to a fraction $p / q$ , the potential nearly repeats itself every $q$ sites, drifting by only $| q α - p | \approx e^{- β q}$ per period. On that scale the packet moves as in a periodic crystal whose band has width about $λ^{- q}$ , so its speed is about $λ^{- q}$ , and the slow drift lets this motion last for a time about $T = e^{β q}$ . In that time the packet covers a distance about $T λ^{- q} = T^{1 - \log λ / β}$ . The resonances win exactly when $β (α) > \log λ$ , which is the threshold $λ = e^{β}$ found by Avila, You and Zhou; the exponent records by how much they win. The theorem shows that no other mechanism moves any polynomial fraction of the packet faster.

So the Aubry–André model is not simply “localized above $λ = 1$ ”: whether a packet spreads, and how far, depends on the arithmetic of the frequency.

An exponent no algorithm computes

A frequency can be computable, with every digit producible by a program, while $β (α)$ , a limit superior over its whole continued fraction, is not. At coupling $λ = 2$ and phase $θ = 1 / 4$ , the paper gives an explicit computable frequency whose almost Mathieu operator has a fastest upper transport exponent of Turing degree $𝟎^{″}$ , the jump of the halting problem: no algorithm computes it, not even one allowed to consult the halting problem. As the frequency ranges over the computable irrationals whose convergent denominators are eventually odd, the exponents at that coupling and phase are exactly the reals in $[0, 1]$ that are right-computably enumerable relative to the halting problem.

The exponent is an infinite-time quantity of an infinite system; as the paper notes, probabilities at finite computable times can still be approximated to any prescribed accuracy. Exponents that no algorithm computes appear elsewhere in the lab’s work, for the price of near-perfect play and for the critical exponents of lattice materials.

What a lab would seea numerical illustration, not a theorem

These are numerical simulations by the authors, not results in the paper. Take an Aubry–André lattice at $λ = 1.25$ ( $V / J = 2.5$ ), phase $θ = 0.05$ , a particle started on one site, and time in tunnelling times ( $ℏ / J = 1$ ). Change one continued-fraction digit of the frequency, the fifth, from the golden mean’s 1 to 60, giving $α = [0; 1, 1, 1, 1, 60, 1, 1, \dots]$ : its convergent $3 / 5$ then sets a resonant time $T = 1 / | 5 α - 3 | \approx 306$ . With the paper’s exponential time average at $T$ , the root-mean-square spread is 2.9 sites for the golden mean against 14.7 sites, and the share of the packet beyond 10 sites is 0.3% against 39%. At single instants up to $t = 900$ , the spread stays below 4.1 sites for the golden mean and peaks at 23.4 sites near $t = 280$ , on a plateau from about $t = 250$ to $t = 330$ , for the changed frequency. Lattices of 601 and 901 sites agree to $4 \times 10^{- 12}$ .

One digit does not change $β (α)$ , which depends only on the tail of the continued fraction; the illustration shows finite-time behaviour of the mechanism, not the exponent itself.

Questionson the index

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Novelty and credit

Novelty: Extends named prior work: for $0 < \log λ < β < \infty$ and almost every phase, it makes the 2025 packing bound $2 (1 - \log λ / β)$ of Jitomirskaya, Liu and Tcheremchantsev sharp with factor 1, where they had noted that the factor 2 might not be sharp. It credits the cases $β (α) = 0$ and $β (α) = \infty$ to Damanik and Tcheremchantsev and to Jitomirskaya and Zhang, earlier lower bounds of the form $1 - C \log λ / β$ to Jitomirskaya and Zhang and to Haeming, and the arithmetic spectral transition to Avila, You and Zhou.

The model is Aubry and André’s; as the almost Mathieu operator it is the central model of quasi-periodic spectral theory, and it has been realized with cold atoms in bichromatic optical lattices (Roati and others) and with light in quasi-periodic waveguide arrays (Lahini and others). For $λ > 1$ its Lyapunov exponent on the spectrum is $\log λ$ (Bourgain and Jitomirskaya). The arithmetic spectral transition at $λ = e^{β}$ is Avila, You and Zhou’s.

The paper credits the two ends of its transport law to others: $β (α) = 0$ to Damanik and Tcheremchantsev, and $β (α) = \infty$ , with packing dimension 1, to Jitomirskaya and Zhang. Jitomirskaya and Zhang also proved lower bounds of the form $1 - C \log λ / β$ for every moment, which Haeming refined, and for packing dimensions. Jitomirskaya and Krüger proved exponential dynamical localization when $β (α)$ is sufficiently small, and Bradshaw, de Jong, Liu, Wang, Wang and Yang refined the growth rates for weakly Liouville frequencies. The packing bound $2 (1 - \log λ / β)$ that the paper makes sharp is Jitomirskaya, Liu and Tcheremchantsev’s (2025), who noted that its factor 2 may not be sharp; their phase set and the paper’s are both of full measure, and neither is shown to contain the other. That the Hausdorff dimension of these spectral measures is zero is Jitomirskaya and Last’s.

The proofs use Chambers’ formula, the periodic magic formula of Damanik, Killip and Simon, the complex-energy method of Damanik and Tcheremchantsev, the inequality of Guarneri and Schulz-Baldes in the form of Barbaroux, Germinet and Tcheremchantsev, and, for computability, a lemma of Becher, Bugeaud and Slaman. Undecidable physical quantities were known for families of Hamiltonians, most prominently the spectral gap of Cubitt, Perez-Garcia and Wolf; here the uncomputable exponent belongs to one explicit operator.

The companion paper, Universality of the transport law for supercritical quasi-periodic operators, carries the law to every supercritical real-analytic quasi-periodic operator of type I.

Versionsthree, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23830304, which always resolves to the newest.
Version	Date	DOI	Notes
2.0.1	1 Nov 2026	10.5281/zenodo.23864352	The phase set of Jitomirskaya, Liu and Tcheremchantsev is contained in the paper's, with proof; Remark 9.11's table corrected; precision fixes and two citation locators. PDF hosted here.
2.0.0	1 Nov 2026	10.5281/zenodo.23848977	Theorem B strengthened (almost all of the packet moves); a larger phase set; a correction to the outer-barrier lemma.
1.0.0	31 Oct 2026	10.5281/zenodo.23830305	Titled “Exact transport and packing dimension for the almost Mathieu operator”.

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DOI

10.5281/zenodo.23864352; all versions 10.5281/zenodo.23830304

Code

github.com/Apsiape/almost-mathieu-exact-transport

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284d0a9d55dc2391a27927b73b28aa580421d61e

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Page checked against the paper, the release repository and the Zenodo record on 1 November 2026.

Corrections

1 Nov 2026Version 2.0.1. Remark 9.11’s table of numerical illustrations had entries that did not match the quantity it states, the strict outside probability; version 2.0.1 corrects them and states how they were computed. Smaller precision fixes and two citation locators. No theorem or proof was affected. Found by a blind audit and its fix-check.doi:10.5281/zenodo.23864352
1 Nov 2026Version 2.0.0. The outer-barrier lemma (Lemma 4.1 of version 1.0.0) allowed the endpoint J = 0 with V₀ = W, where its bound reads 0/0; version 2.0.0 adds V₀ > W. A Lean blueprint of the paper found it. No application is affected: every use has J = 1 and a strict gap. Version 2.0.0 also strengthens Theorem B and enlarges the phase set.doi:10.5281/zenodo.23848977

All corrections, and what counts as one.

Cite

@misc{douglas2026almostmathieu,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Exact transport and spectral dimensions for the almost {M}athieu operator},
  year    = {2026},
  version = {2.0.1},
  doi     = {10.5281/zenodo.23864352},
  url     = {https://doi.org/10.5281/zenodo.23864352},
  note    = {Preprint}
}

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