Lattice matter

Answered Are the critical exponents of lattice matter computable from its Hamiltonian?

Not in general.

1. Uncomputable critical exponents

   Seth Douglas and Nidhal Mghirbi · version 2.1.0 · 29 October 2026 · 64 pages

   Translation-invariant nearest-neighbour quantum Hamiltonians whose critical gap exponents are exactly the nonnegative ${\mathrm{\Delta }}_3^0$ reals, already in two dimensions; and robust laws of noisy media that no effective inquiry can learn beyond ${\mathrm{\Delta }}_2^0$.

   • Answered Are the critical exponents of lattice matter computable from its Hamiltonian?

   The paperPDFDOICode

2. Matter that pays the price of perfection

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 29 October 2026 · 27 pages

   Gapped translation-invariant lattices whose entanglement has any prescribed limit-computable logarithmic coefficient, and a referee cell that converts low energy into strategies for a nonlocal game. The game-driven material theorem is stated conditionally.

   • Answered Can gapped lattice matter carry any limit-computable amount of entanglement?

   The paperPDFDOICode

3. Spontaneous magnetization and thermal decay are undecidable at positive temperature

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 3 November 2026 · 36 pages

   Spontaneous magnetization is undecidable at a fixed positive temperature for translation-invariant finite-range classical lattice models with strictly positive rational Boltzmann weights. A computable family on 2304 local states has maximal visible nine-colour Potts magnetization exactly zero or a fixed positive constant, attained in a translation-invariant Gibbs state. Its pressure in a uniform field is uniformly computable, but differentiability at zero field is undecidable with a fixed derivative-jump gap. Rational weights are asserted at zero field. We also prove undecidability of exponential decay of one specified thermal spin correlation, in every Gibbs state, on 512 local states. Both constructions couple critical visible spins to cellular-automaton replicas. Surjectivity cancels the collision corrections; nonsurjectivity gives strict subcritical bond domination and exponential decay of the specified correlation uniformly over states. The interaction range depends on the input, and the branches are fine-tuned at their critical or coexistence points. The results transfer to diagonal quantum interactions and survive a prescribed output-reset noise family, the same construction with reduced retention. Nonsurjective full-state uniqueness and exponential clustering of arbitrary local observables remain open.

   • Partly answered Given a translation-invariant Hamiltonian, is it decidable whether the correlations in its thermal state decay exponentially with distance?

   The paperPDFDOICode

4. Worst-case transport and light cones for quasi-periodic operators

   Seth Douglas and Nidhal Mghirbi · version 2.0.1 · 2 November 2026 · 72 pages

   At the worst phase rather than a typical one. For every real-analytic quasi-periodic potential with positive Lyapunov exponent on the spectrum and every frequency with $0<\beta (\alpha )<\infty$, uniformly in the phase, the fastest upper transport exponent is at most the maximum over the spectrum of $1-\ell /(2\omega \beta )$ when $\beta \ge \ell$, where $\ell$ is the Lyapunov exponent and $\omega$ the acceleration, with an explicit formula when $\beta <\ell$; so a finite $\beta$ keeps transport below ballistic at every phase, and the factor $2\omega$ is attained. For the almost Mathieu operator, when ${\beta }_e=\beta$ is finite and greater than $\log \lambda /2$, where ${\beta }_e$ is the limsup along even denominators, the worst exponent is exactly $1-\log \lambda /(2\beta )$: it is attained at the phase $1/4$ and on a dense $G_{\delta }$ set of phases, while almost every phase has $\max (0,1-\log \lambda /\beta )$. In the Aubry–André XY spin chain, for the specified observables, the light cone at a fixed site follows the typical law at almost every phase, and the light cone uniform over sites follows the worst-case law; an explicit computable chain has two different light-cone exponents, both of Turing degree ${\mathrm{𝟎}}^{″}$. The Lyapunov exponent of a computable analytic potential is computable at every computable irrational frequency, and at one explicit frequency the transport exponent has degree exactly ${\mathrm{𝟎}}^{″}$ for every computable even potential in an explicit open neighbourhood of $4\cos 2\pi x$.

   • Proved How fast do signals travel in the quasi-periodic XY spin chain?

   The paperPDFDOICode

• Partly answered Given a translation-invariant Hamiltonian, is it decidable whether the correlations in its thermal state decay exponentially with distance?