全$(2,q)$-正性中的虚假零色散连续谱：来自一个精确可解计数算子的线性层级深度下界
Spurious zero-dispersion continua in full $(2,q)$-positivity: A linear hierarchy-depth lower bound from an exactly solvable counting operator
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中文总结 AI 辅助
研究变分两粒子约化密度矩阵方法的层级能否恢复基本费米子谱，通过精确可解计数算子证明除非层级深度随粒子数线性增长，否则会出现虚假非整数本征值区间，并给出精确阈值。
中文摘要 AI 辅助
变分两粒子约化密度矩阵方法施加了半定约束的层级。我们询问这种层级的固定水平能否通过色散检验恢复甚至基本的多费米子谱，在该检验中，当最小化色散为零时，该值被接受为本征值。对于一个精确可解的算子，该算子计数自旋轨道一半中的费米子，其谱由整数组成，我们在粒子-空穴半定约束的完整层级的每一水平上确定了所有被接受值的完整集合。除了整数之外，该集合还包含一个完整的非整数值区间，除非层级水平随粒子数线性增长，并且我们给出了精确阈值。证明构造了显式的伪约化密度矩阵，这些矩阵在占据数基上是对角的，通过将固定粒子数态的占据概率延续到非整数粒子数获得。对占据关联的经典半定约束，辅以非负占据模式概率，足以建立费米子正性。对于这个计数模型，两种松弛具有相同的色散零集，但不一定具有相同的可行态，并且一个逆论证表明没有其他虚假本征值出现。该限制特定于层级：编码整数谱的两体不等式将移除所有虚假值，但在固定水平上不会生成。我们还验证了在半定约束及其在二维粒子约化密度矩阵空间上的完全投影在有限维中是等价的，因此该结果同样适用于极形式。
英文摘要
Variational two-particle reduced-density-matrix methods impose hierarchies of semidefinite constraints. We ask whether a fixed level of such a hierarchy can recover even an elementary many-fermion spectrum, using the dispersion test, in which a value is accepted as an eigenvalue when the minimized dispersion vanishes. For an exactly solvable operator that counts fermions in one half of the spin orbitals, whose spectrum consists of integers, we determine the complete set of accepted values at every level of the full hierarchy of particle-hole semidefinite constraints. Besides the integers, this set contains an entire interval of noninteger values unless the hierarchy level grows linearly with the particle number, and we give the exact threshold. The proof constructs explicit pseudo reduced density matrices that are diagonal in the occupation basis, obtained by continuing the occupation probabilities of a fixed-particle-number state to a noninteger particle number. Classical semidefinite constraints on occupation correlations, supplemented by nonnegative occupation-pattern probabilities, suffice to establish fermionic positivity. For this counting model the two relaxations have the same dispersion zero set, not necessarily the same feasible states, and a converse argument shows that no other false eigenvalues occur. The limitation is specific to the hierarchy: two-body inequalities that encode the integer spectrum would remove every false value but are not generated at a fixed level. We also verify that the full semidefinite constraints and their complete projection onto two-particle reduced-density-matrix space are equivalent in finite dimensions, so that the result applies equally to the polar formulation.
发表机构
- RIKEN Pioneering Research Institute（理化学研究所先驱研究机构）
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