# 量子材料的物理概念与数学结构

**Repository Path**: jzh12/QuMaMathPhys

## Basic Information

- **Project Name**: 量子材料的物理概念与数学结构
- **Description**: Quantum Materials: Mathematical & Physical Foundations
- **Primary Language**: Julia
- **License**: Apache-2.0
- **Default Branch**: master
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2025-04-13
- **Last Updated**: 2025-12-22

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

量子材料（Quantum Material）: 量子限域效应显著的低维纳米结构

    为了理论的便利，通常假设周期边界以及体系有一定的旋转和反演对称

## Grounding

* 多尺度框架 (Length Scale)

    0. _ab initio_ (10-9)
    0. **_Effective_** (9-6)
    0. **Classical** (6-3)

* 简化 (Reduction)

    1. 以有效模型 **_复现_** 从头算的结果
    1. **_投影_** 至低维表示
    1. 策略  :fa-search-plus:  降解、拆分、细化、提取

* 目标 (Objective)  :fa-flag: 

    1. 带结构 E(k) & 波函数 ψ(k)  :fa-arrow-right:  介电函数  ε(q,ω)

## Advances

理论的自底向上（研究／推导）和自顶向下（综述／诠释）

### Bottom-up

    微扰论是近代物质科学演变的核心，数值模拟是探索机理的有力工具

> 等离激元计算的简便途径是“单电子近似能带+随机相位近似（Random Phase Approximation）”
> 1. 响应方法的关键是电子态相关联的矩阵元（Selection Rules），其余则由能带和观测窗口确定的联合态密度（joint Density of States）以及其上布居数和平衡分布假设共同决定的泡利阻塞（Pauli Blocking）；
> 1. 石墨烯由栅极电压而偏置，其载流子的集体振荡模式（Plasmon）由两个对顶的圆锥面型能带而确定，不同于半导体量子阱有质量电子气的抛物面型能带；(Hwang2007)
> 1. 伴随着超导实验，石墨烯亚稳叠层，如转角双层石墨烯（Twisted Bilayer Graphene）和菱方多层石墨烯引起了凝聚态物理学界的关注，众多理论工作指向了石墨烯体系结构和性质的手性（Chirality）；
> 1. 叠层结构的平坦能带（Flat bands）让多体交互的试验有诸多施展的余地：例如实验观测到的晶格弛豫使体系对称性降低，在紧束缚能带上体现为远端带隙（Remote band gap）以及粒子空穴不对称（Particle-hole asymmetry）(Kuang2025)
> 1. 实验制备、样品表征；（“噪声”：无序成分，尚未分类，未寻见恰当表象，可以一战！）

(Hwang2007) Dielectric function, screening, and plasmons in two-dimensional graphene
E. H. Hwang and S. Das Sarma
Phys. Rev. B 75, 205418 (2007)

(Kuang2025) Review of the tight-binding method applicable to the properties of moiré superlattices.
Xueheng Kuang, …, Zhen Zhan. 
Phys. Chem. Chem. Phys., 2025, 27, 25232


### Top-down

实验是物理研究的根本。为了重现(implement)或预计(expect)观测结果，
在理论上做出相应的接口(interface)是必要的。在较大自由度的完备框架中，
体系本身的约束和适当的假设将研究限定在少数几个维度的谱(spectrum)上。
所关注的物理量性质能给到一些线索来检查(inspect)计算本身的逻辑。
研究应当有重点(Prominence)，其余自由度的变化可附在补充(Supplement)。
在我看(perspective)来，如此论文有希望(prospect)。

    熊就要有个熊样，例如

- **非线性**  :fa-angle-double-up:  Quantum frequency doubling in the topological insulator Bi2Se3. [Nat Commun 12, 698 (2021)](https://doi.org/10.1038/s41467-021-20983-1)

- **ab initio DeePMD**  :fa-terminal:  Transfer learning relaxation, electronic structure and continuum model for twisted bilayer MoTe2. [Commun Phys 7, 262 (2024)](https://doi.org/10.1038/s42005-024-01754-y). 

- **variational Monte Carlo**  :fa-terminal:  Is attention all you need to solve the correlated electron problem? arXiv:2502.05383
    1. (_benchmark_) band-projected exact diagonalization
    1. (VMC) minimal self-attention NN **_vs._** SlaterNet


## 化繁为简

1. **微分**  :fa-scissors:  **物化**  :fa-share:  The linear free energy relationship predicts (in some cases) that structural changes in the starting molecule, such as the introduction of substituent X, have _**proportional**_ effects on the thermodynamics (equilibrium) and kinetics (rate) of chemical reactions.

    - Hammett parameters are dimensionless constants quantifying substituent effects on organic compound properties. They correlate substituent electronic effects with ionization equilibria in benzoic acids and establish a linear relationship ($\log(k/k_0) = \sigma\rho$) between rate constants $k,k_0$ and substituent characteristics $\sigma$ in reaction kinetics. This dual application demonstrates consistent electronic effect propagation across thermodynamic equilibria and kinetic processes, forming a _**quantitative**_ framework in physical organic chemistry. 
    - Although chemists usually prefer to explain linear free energy relationships in terms of chemical pictures such as **entropy enthalpy complementarity**, there are more essential mathematical implications behind this linear relationship.  Assuming that the **reactivity** $f_r$ is a function uniquely determined by the **substituent property** $\sigma$, we can always _**Taylor expand**_ $f_r$ with respect to $\sigma$ between substituents $X$ and $H$ and retain only one term as long as the functional relationship is not hopelessly poor.  If $f_r = \lg k_r$ and $\sigma(H) = 0$, we obtain the Hammett equation. 
    - 吉布斯自由能的变化 $\Delta G$ 定义为焓变 $\Delta H$ 减去温度与熵变的乘积 $T\Delta S$. 即使一个过程的焓变不利 (吸热，$\Delta H > 0$)，如果其熵变有利且温度足够高，整个过程仍可能自发进行 (即 $\Delta G < 0$). 反之亦然. 因此，"熵焓互补性" 描述的是熵和焓在判断过程自发性时的相互作用和补充关系. 
    - 若将微扰论比作“解方程的策略”，泰勒展开则是其中“构造近似解的工具”之一. 

1. Because increasing parameters necessarily sacrifices the _**chemical picture**_ of the equation, it reduces the interpretability of the model.

    - Incomplete induction: An attempt to extract an experience or rule from limited experimental facts in the hope that it will generalize to an unknown sample.
    - The validity of inductive reasoning has long been a controversial issue in the history of philosophy.

郑超：大道至简vs多者异也，物理有机化学通往何处？

---

> 超导材料, 二维材料, 拓扑材料, 超构材料

## 多者异也
**Statistical Physics of Multiscale, Nonlinear, Disordered Systems and Emergence Theory**

1. The current success of artificial intelligence largely relies on **an empirical black box** and _**brute-force computation**_. There remains a lack of deep understanding, and many statistical physics laws are yet to be discovered, just as there are numerous principles in neuroscience awaiting exploration from a statistical physics perspective.
2. Condensed matter physics often studies artificially engineered materials —— in fact, most are “human-made” materials. These materials (e.g., many _**superconducting**_ materials, _**2D**_ materials, _**topological**_ materials, _**metamaterials**_, etc.) exhibit rich physical phenomena. If similar systems are later found in nature, they are perceived as novel. Quantum computing further involves designing artificial systems to realize theoretical quantum algorithms.
    - Studying statistical physics in complex systems —— such as artificial neural networks, cells, biological populations, human crowds, traffic flow, and economics —— follows a logic similar to **quantum simulation**.
    - However, systems used for quantum simulation are mostly simple, composed directly of atoms or electrons. Some are more complex, such as superconducting qubits (where a superconducting loop implements a single qubit) or exploratory topological quantum computing.
    - Metamaterials for electromagnetic wave modulation are larger-scale artificial structures.
3. Since humans are part of nature, so-called “artificial” systems are ultimately “natural.”

Reductionism and emergence theory are two sides of the same coin. Reductionism, as the foundation of emergence, has brought brilliance to physics. Emergence is not exclusive to macroscopic scales but occurs across all energy scales. Spontaneous symmetry breaking and vacuum structures in high-energy physics, akin to their condensed matter counterparts, are emergent phenomena. The formation of cosmic structures across multiple scales, as well as the origin and evolution of life and intelligence, are also manifestations of emergence. From atomism to elementary particles, reductionism has driven physics’ glory. Yet even as progress in fundamental interactions and particles slows temporarily, emergence theory remains a vast frontier for exploration.


Wir müssen wissen,  
Wir werden wissen.


施郁：统计物理和层展论的胜利：为何 2024 年诺贝尔物理学奖授予神经网络研究

## 科学指导

### 范式 —— 思考问题的逻辑方式
1. _实验科学_ 范式：文明早期，人们只能依靠 **经验** 来描述自然现象并处理问题。
1. _理论科学_ 范式：为避免自然环境影响，人们通过实验设计与 **演算**，归纳总结而得出各种 **_模型_**。
1. _计算科学_ 范式：为重现更复杂的环境，人们借助 **_数值计算_** 来 **模拟** (仿真)。
1. _数据科学_ 范式：对 **海量数据** 进行挖掘分析。**数据** 依靠信息设备收集或模拟产生，依靠软件处理，用计算机进行存储，使用专用的数据管理和统计软件进行分析。

> 1. 牛奔，耿爽，王红著,数据科学导论,中国经济出版社,2022.02,第7页
> 1. 程显毅，任越美主编,大数据技术导论：第2版,机械工业出版社,2022.08,第7页

#### Sarfaraz K. Niazi (Pharmacist Against Data Science)
1. 实验 - 数据 - 模型
    * It is necessary to change the direction of research to experimental design because billions of dollars are invested in AI-based data transfer, which affects experimental design. 
1. 置信度 - 可迁移？
    *  The algorithm learns from known frozen structures, so it can only provide some ideas about frozen structures.  Based on Anfinsen's conclusion, it is not surprising that algorithms do not work well for predicting new structures, and confidence is low even for known structures. 
1. 科学 / 技术
    *  AI uses algorithms that are essentially "efficient data pattern recognition" rather than breakthrough discoveries based on physicochemical principles.  Its existing success owes more to computational power and data size than to a fundamental understanding of the mechanisms of protein folding. 
    *  AI-assisted protein structure prediction is closer to a technical optimization than a basic scientific breakthrough. 

### Steven Weinberg 四个建议

1. $\texttt{sink or swim}$
1. $\texttt{rough water only}$
1. $\texttt{becalmed}$
1. $\texttt{history}$

### Walther Bothe 五点忠告

1. $\texttt{Fruitful thoughts}$
1. $\texttt{simple and flexible}$
1. $\texttt{Schedule}$
1. $\texttt{Keep records properly}$
1. $\texttt{Write early}$