Chaitin Phase Transition

We construct a family of Hamiltonians whose phase diagram is guaranteed to have a single phase transition, yet the location of this phase transition is uncomputable. The Hamiltonians $H(\phi)$ describe qudits on a two-dimensional square lattice with translationally invariant, nearest-neighbour interactions tuned by a continuous parameter $\phi\in(0,1]$. For all $\phi\in(0,1]$, $H(\phi)$ is in one of two phases, one a gapless phase, the other a gapped phase. The phase transition occurs when $\phi$ equals the Chaitin's constant $\Omega$, a well-defined real number that encodes the Halting problem, and hence is uncomputable for Turing machines and undecidable for any consistent recursive axiomatization of mathematics. Our result implies that no general algorithm exists to determine the phase diagrams even under the promise that the phase diagram is exceedingly simple, and illustrates how uncomputable numbers may manifest in physical systems.