Contextuality as a Diagnostic of Translation-Symmetry Breaking in Translation-Invariant 1D Hamiltonians

Abstract

Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we show that, in infinite one-dimensional translation-invariant chains, contextuality can acquire a genuinely thermodynamic meaning: within the witness families studied, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, producing strictly $p$-periodic ground states with $p>1$. Along natural continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking point where competing classical periodicities are lifted in favor of a unique quantum-selected period. At the quantum optimum, the studied families admit exact finite-size reductions: a translation-invariant contextuality witness induces a $p$-site periodic-boundary-condition inequality with identical classical and quantum bounds (hence no loss under reduction), and in several cases the resulting finite inequalities are tight. This reduction turns an infinite-chain contextuality certification into a compact, hardware-testable benchmark on a small ring, requiring only local energy measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using a translation-invariant adaptation of semidefinite-program hierarchies together with variational matrix-product-state algorithms.