保持非相互作用费米气体一体密度矩阵的厄米性
Preserving the Hermiticity of the One-Body Density Matrix for a Non-Interacting Fermi Gas
AI总结：
针对非相互作用费米气体半经典 ODM 展开中不同方法厄米性和幂等性不一致的问题，本文通过引入对称坐标证明各方法等价，并首次显式推导 d 维 Grammaticos-Voros ODM。
摘要：
零温非相互作用费米气体的一体密度矩阵（ODM）可在半经典区域通过不同的 ℏ 展开技术近似得到。人们会期望，每种近似 ODM 的方法都应给出等价的密度矩阵，并且这些矩阵在 ℏ 的任意阶都同时具有厄米性和幂等性。然而，Kirzhnits 方法和 Wigner-Kirkwood 方法并不产生这些性质，而 Grammaticos-Voros 方法则可以。本文通过适当地变换到对称坐标，明确证明在任意 d 维中，每种方法实际上都是等价、厄米且幂等的。这一变量变换解决了各种方法之间的不一致，表明 Kirzhnits 和 Wigner-Kirkwood 方法的非厄米和非幂等行为，是对半经典 ℏ 展开执行非对称截断所造成的伪效应。我们的工作还首次显式推导了 d 维 Grammaticos-Voros ODM；该 ODM 最初由 Redjati 等人（2019 年《J. Phys. Chem. Solids》134 313-8）基于其 d=1、2、3、4 的表达式提出。
英文摘要：
The one-body density matrix (ODM) for a zero temperature non-interacting Fermi gas can be approximately obtained in the semiclassical regime through different $\hbar$-expansion techniques. One would expect that each method of approximating the ODM should yield equivalent density matrices which are both Hermitian and idempotent to any order in $\hbar$. However, the Kirzhnits and Wigner-Kirkwood methods do not yield these properties, while the Grammaticos-Voros method does. Here we show explicitly, for arbitrary $d$-dimensions through an appropriate change into symmetric coordinates, that each method is indeed identical, Hermitian, and idempotent. This change of variables resolves the inconsistencies between the various methods, showing that the non-Hermitian and non-idempotent behavior of the Kirzhnits and Wigner-Kirkwood methods is an artifact of performing a non-symmetric truncation to the semiclassical $\hbar$-expansions. Our work also provides the first explicit derivation of the $d$-dimensional Grammaticos-Voros ODM, originally proposed by Redjati et al (2019 $\textit{J. Phys. Chem. Solids}$ 134 313-8) based on their $d=1,2,3,4$ expressions.
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