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

TL;DR

This paper demonstrates that finding the ground state energy of certain translationally-invariant spin models, like the Heisenberg model on a 2D lattice, is not NP-hard unless P=NP, suggesting potential tractability.

Contribution

It proves that specific translationally-invariant spin-model problems are unlikely to be NP-hard, using reductions from computational complexity theory.

Findings

Ground state energy problem is not NP-hard unless P=NP.

Results extend to Ising, t-J, and Fermi-Hubbard models.

Encourages further research into positive complexity results.

Abstract

Finding the ground state energy of the Heisenberg Hamiltonian is an important problem in the field of condensed matter physics. In some configurations, such as the antiferromagnetic translationally-invariant case on the 2D square lattice, its exact ground state energy is still unknown. We show that finding the ground state energy of the Heisenberg model cannot be an NP-Hard problem unless P=NP. We prove this result using a reduction to a sparse set and certain theorems from computational complexity theory. The result hints at the potential tractability of the problem and encourages further research towards a positive complexity result. In addition, we prove similar results for many similarly structured Hamiltonian problems, including certain forms of the Ising, t-J, and Fermi-Hubbard models.