[论文解读] Semi-classical limit of the Levy-Lieb functional in Density Functional Theory

Mathieu Lewin|arXiv (Cornell University)|Jun 1, 2017

Random Matrices and Applications参考文献 16被引用 50

一句话总结

该论文将 Bindini-De Pascale 正则化扩展到混合量子费米态，以证明 DFT 中 Levy-Lieb 泛函的半经典极限，并将其与带库仑代价的多边际最优传输联系起来。

ABSTRACT

In a recent work, Bindini and De Pascale have introduced a regularization of $N$-particle symmetric probabilities which preserves their one-particle marginals. In this short note, we extend their construction to mixed quantum fermionic states. This enables us to prove the convergence of the Levy-Lieb functional in Density Functional Theory , to the corresponding multi-marginal optimal transport in the semi-classical limit. Our result holds for mixed states of any particle number $N$, with or without spin.

研究动机与目标

• 在 DFT 中需要对 N-粒子费米态进行正则化，同时保持一粒子密度的动机。
• 将现有的玻色正则化扩展到混合费米态，以实现严格的半经典分析。
• 在半经典极限下建立 Levy-Lieb 泛函收敛到多边际库仑传输的结论。

提出的方法

• 引入一个量子扩展 Gamma_epsilon，在保持一粒子密度 rho_P 的同时，对对称的 N 粒子密度进行正则化。
• 将 Gamma_epsilon 构造为一个混合费米态，来自局部化函数 chi_epsilon 和 Slater 行列式以强制反对称性。
• 证明 Trace(-Δ) Gamma_epsilon 等于 N 乘以 [∫|∇√ρ_P|^2 dx + ε^{-2} ∫|∇χ|^2 dx]，从而确立式 (1.6)。
• 证明对于对称态 Φ 的期望收敛到在 P 下的经典期望，并给出明确误差界 (1.7)。
• 应用该构造导出 DFT 中 Levy-Lieb 泛函的半经典极限结果。

实验结果

研究问题

• RQ1 Can Bindini-De Pascale type regularization be extended from bosonic to mixed fermionic states while keeping the same one-particle density?
• RQ2 Does the extended regularization yield controlled kinetic energy and allow convergence of the Levy-Lieb functional in the semiclassical (eta -> 0) limit?
• RQ3 Can the semiclassical limit of the Levy-Lieb functional be characterized by a multimarginal Coulomb transport problem?
• RQ4 What are the quantitative error estimates when passing from quantum mixed states to the semiclassical transport limit?

主要发现

• The authors define a fermionic mixed-state extension Gamma_epsilon that preserves the one-particle density from P.
• They prove the exact kinetic energy identity Tr(-Delta) Gamma_epsilon = N(integral |∇√ρ_P|^2 + (1/ε^2) ∫|∇χ|^2).
• They establish convergence of Gamma_epsilon to the classical N-particle density P in the sense of reduced densities and expectations (via (1.7)).
• In the semiclassical scaling, the Levy-Lieb functional divided by eta is sandwiched between the multimarginal Coulomb transport energy and that energy plus O(√η) and O(η) terms.
• The main semiclassical result shows E_OT(ρ) ≤ E(η^3 ρ(η⋅))/η ≤ E_OT(ρ) + C(√η + η) for η → 0, linking DFT to multimarginal optimal transport.