The Complexity of Approximating Critical Points of Quantum Phase Transitions

Phase diagrams chart material properties with respect to one or more external or internal parameters such as pressure or magnetisation; as such, they play a fundamental role in many theoretical and applied fields of science. In this work, we prove that provided the phase of the Hamiltonian at a finite size reflects the phase in the thermodynamic limit, approximating the critical boundary in its phase diagram to constant precision is $P^{QMA_{EXP}}$-complete. This holds even for translationally-invariant nearest neighbour couplings, and even if the system's phase diagram is promised to have a single critical boundary delineating two phases. For the simpler case of a single parameter, the same problem remains $QMA_{EXP}$-hard. Our results extend the study of quantum phases to systems with more realistic phase diagrams than previously studied. Furthermore, our findings place complexity-theoretic constraints on the effectiveness of (computational or analytic) methods based on finite size criteria, similar in spirit to the Knabe bound, for the task of extrapolating the properties (e.g. gapped/gapless) of a system from finite-size observations to the thermodynamic limit.