Research Project
Beyond Semilocality: Quantum-Critical Constraints on Density Functionals
量子临界普适性是否能够告诉我们，固定范围的局域/半局域泛函为什么必然失去正确的临界响应，以及如何构造一种能够跨越临界区、具备可预测性的改进泛函？
Research Log
研究日志
2026-09-25
确立课题方向
2026-10-18
自己尝试把目前的所有工作整理成对应的课题笔记文档，另外新增的这个研究日志就相当于一个备忘录吧，能记下自己一些灵感，或者单纯帮自己捋一下思路。
这个课题最开始的方向是去尝试思路强关联体系下半局域泛函的表达为什么失效，但后来发现局域性这一点在量子临界体系中有更清晰明确的对应概念。由此课题的主要侧重点逐渐偏向研究量子临界体系下泛函的表达能力。
首先当前课题的主要目的应该有两个
[1] 讨论量子临界体系泛函的一般理论，试着证明下面两个结论
Theorem Ⅰ ：固定有限范围、光滑且参考逆响应解析的泛函近似， 无法精确再现非高斯密度临界点的静态 Hessian。且误差无法控制在任意小（也即你甚至无法得到一个误差上界）。
Theorem Ⅱ ：即使体系不处在临界点，只要沿着合适的临界标度窗口逼近，固定范围、系数规则的半局域近似仍会产生渐近失配。
而在远离临界点的体系，局域/半局域泛函已经具有实用性，额外的泛函修复意义不大。
由此得到课题的第二个目标，也即修复临界处附近的泛函形式
[2] 从定理反推出泛函修正结构
这一部分的理论需要在具体的模型上验证。这个课题采用两个模型，一个是简单直接的transverse Ising model[8], [7]，一个则是临界性质更丰富的extended Hubbard model with bond dimerization[5], [4]。
2026-10-19
当前的最大难点在于理论上的合理性，自己希望得到的修正泛函形式是一种结构上的必然，而非强行拼凑的结果。
如果能够得到修正的唯一泛函形式，那我想整个课题的工作便可以字面意思上的宣告胜利。剩下的无非是在横场Ising和extend Hubbard上进行具体的计算验证。
我想一个研究最令人欣喜的时间点不是最终完成，而是当你意识到一切从不确定塌缩到确定的那一瞬。这无疑能给人以莫大的鼓舞。
2026-10-21
尽管泛函修复的思路是明确的，但形式上必定带有需要具体模型拟合的待定参数。这使得如果用横场Ising拟合出的修复泛函来验证横场Ising，就会很大程度上影响修复有效性的说服力。
因此测试的独立模型是必要的，且其必须与横场Ising为一个普适的临界类，对extend Hubbard，虽然文献证明其满足Ising临界，但泛函本身能否有效地迁移依然不能被保证，这一部分可能是之后工作的主要难点。
冬天也到了呢。
Research Notes
课题笔记 05
References
参考文献 18
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