Necessary and sufficient conditions for quantum Hamiltonians to be exactly solvable within mean-field (MF) theories have not been formulated so far. To resolve this problem, first, we define what MF theory is, independently of a Hamiltonian realization in a particular set of operators. Second, using a Lie-algebraic framework we formulate a criterion for a Hamiltonian to be MF solvable. The criterion is applicable for both distinguishable and indistinguishable particle cases. For the electronic Hamiltonians, our approach reveals the existence of MF solvable Hamiltonians of higher fermionic operator powers than quadratic. Some of the MF solvable Hamiltonians require different sets of quasi-particle rotations for different eigenstates, which reflects a more complicated structure of such Hamiltonians.

迄今为止，量子哈密顿量在平均场 (MF) 理论中完全可解的充分必要条件尚未制定。为了解决这个问题，首先，我们定义了什么是 MF 理论，独立于特定算子集合中的哈密顿实现。其次，使用李代数框架，我们制定了哈密顿量是 MF 可解的标准。该标准适用于可区分和不可区分的粒子情况。对于电子哈密顿量，我们的方法揭示了比二次方具有更高费米算子幂的 MF 可解哈密顿量的存在。对于不同的本征态，一些 MF 可解哈密顿量需要不同的准粒子旋转集，这反映了这种哈密顿量的更复杂的结构。