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An uncomputable phase diagram at positive temperature for one classical lattice model

Year: 2026
Type: preprint
Abstract: Can the phase diagram of one fixed classical lattice model be uncomputable? For quantum spin systems at zero temperature it can: there are families of Hamiltonians whose spectral gap or ground-state phase cannot be computed from the parameter. We prove a classical counterpart at positive temperature. There is one translation-invariant nearest-neighbour interaction on ℤ², with local energies 0, 1 and +∞ (hard constraints) and a fixed universal Turing machine U built in, and there are disjoint intervals J_n with rational endpoints computable from n, such that at every inverse temperature β in J_n the model has exactly one translation-invariant Gibbs measure if U does not halt on input n, and at least two if U halts. So uniqueness of the translation-invariant Gibbs measure at rational inverse temperature is undecidable for this single model. Uniqueness among all Gibbs measures fails at every temperature, because the hard constraints freeze a non-periodic background.The background is a hierarchical tiling that contains arbitrarily large square grids. On each grid a Turing machine reads a label n from the size of the grid and runs U on n. Where U halts, small sensors made of free bits and unit energies couple the corner spins of all grids of that size into an ordinary Ising model, whose coupling is large only when β is close to n log 2. Assuming the exact solution of the two-dimensional Ising model, the inverse temperatures with at least two translation-invariant Gibbs measures form exactly a union of explicit intervals around n log 2 over the halting inputs n; restricted to rational β, this set is Σ⁰₁-complete.
DOI: 10.5281/zenodo.24115744
Language: English
Open Access status: green
OpenAlex ID: W7225756338
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