连续费米子的变分完备性可由固定通用行列式实现
A Fixed Universal Determinant is Variationally Complete for Continuum Fermions
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中文总结 AI 辅助
该研究证明，描述连续费米子的变分完备性最多仅需2个固定通用Slater行列式，无需更多，且广义Slater-Jastrow神经量子态具备变分完备性。
中文摘要 AI 辅助
准确描述相互作用费米子的变分描述需要多少个Slater行列式？有限基组中的精确展开需要组合数量级的Slater行列式，而最先进的费米子神经量子态则会堆叠越来越多的此类行列式。我们证明，在控制变分计算的范数下，最多仅需2个，且与粒子数和目标精度无关。一个预先指定的、与系统和状态均无关的通用Slater行列式，乘以一个平滑玻色子波函数，可在控制变分能量的一阶Sobolev范数下近似任意三维空间中的费米子波函数。对于库仑相互作用，若要达到二阶Sobolev范数（即哈密顿量的定义域，该范数限制了变分蒙特卡洛核心处局域能量的方差），最多仅需额外1个固定行列式，且仅在三维空间中需要。因此，反对称性最多仅需2个通用行列式，且不会损失表达能力：广义Slater-Jastrow神经量子态是变分完备的。
英文摘要
How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.
