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Finding the ground state energy of the Heisenberg Hamiltonian is an importantproblem in the field of condensed matter physics. In some configurations, suchas the antiferromagnetic translationally-invariant case on the 2D squarelattice, its exact ground state energy is still unknown. We show that findingthe ground state energy of the Heisenberg model cannot be an NP-Hard problemunless P=NP. We prove this result using a reduction to a sparse set and certaintheorems from computational complexity theory. The result hints at thepotential tractability of the problem and encourages further research towards apositive complexity result. In addition, we prove similar results for manysimilarly structured Hamiltonian problems, including certain forms of theIsing, t-J, and Fermi-Hubbard models.

Non-NP-Hardness of Translationally-Invariant Spin-Model Problems