Undecidability at finite temperature and under disorder/noise in many-body systems

Determine, for classical and quantum lattice spin systems described by local Hamiltonians, which decision problems previously shown undecidable in the zero‑temperature, noiseless setting remain undecidable when considering finite‑temperature states or when interactions include disorder and other natural sources of noise.

Background

The review surveys multiple undecidability results for many-body systems, including the spectral gap, ground-state energy density, and phase diagram properties in the thermodynamic limit. These results are typically established at zero temperature for noiseless, translation-invariant Hamiltonians.

The authors emphasize that realistic physical systems operate at finite temperature and often include disorder or noise in interactions. Understanding whether undecidability persists under these more realistic conditions is a significant unresolved issue, as it impacts both theory and potential experimental relevance.