Despite extensive study, the phase structure of the wavefunctions in frustrated Heisenberg antiferromagnets (HAF) is not yet systematically characterized. In this work, we represent the Hilbert space of an HAF as a weighted graph, which we term the Hilbert graph (HG), whose vertices are spin configurations and whose edges are generated by off-diagonal spin-flip terms of the Heisenberg Hamiltonian, with weights set by products of wavefunction amplitudes. Holding the amplitudes fixed and restricting phases to $\mathbb{Z}_2$ values, the phase-dependent variational energy can be recast as a classical Ising antiferromagnet on the HG, so that phase reconstruction of the ground state reduces to a weighted Max-Cut instance. This shows that phase reconstruction HAF is worst-case NP-hard and provides a direct link between wavefunction sign structure and combinatorial optimization.

翻译：尽管已有广泛研究，受挫海森堡反铁磁体（HAF）波函数的相位结构尚未得到系统表征。本工作将HAF的希尔伯特空间表示为加权图，称为希尔伯特图（HG），其顶点为自旋构型，边由海森堡哈密顿量的非对角自旋翻转项生成，权重由波函数振幅的乘积设定。在固定振幅并将相位限制为$\mathbb{Z}_2$值的情况下，相位相关的变分能量可重构为HG上的经典伊辛反铁磁体模型，从而使基态的相位重构问题约化为加权最大割问题实例。这证明HAF相位重构在最坏情况下是NP难问题，并建立了波函数符号结构与组合优化之间的直接联系。