拓扑相变与拓扑场论 (i) (2023/春季)

往年主页: 2021 , 2018

主讲老师: 龚明, 15156057001, 物质楼C812，mail: gongm@ustc.edu.cn

助教: 张诚江 mail: zcj12138@mail.ustc.edu.cn

上课时间: 1~15周, 1(1,2), 5(8,9)

上课地点: 2106

为方便同学们沟通, 助教创建了一个qq群, 群号:599140540.
qq群二维码

助教会在每周天晚前完成对于当周课程内容和参考资料的上传.

评分标准

平时作业占比 1/2
1. 作业要求纸质版, 一个月交一次(暂定每个月的第一个星期一的课, 可于当周补交. 之后按迟交处理).(题量较少)
2. 如有特殊情况可通过邮箱将作业以pdf的格式发给助教, 同时说明原因
3. 助教会在收作业的当周六晚(23:59)前, 在qq群的在线文档中登记当周的作业接受情况, 有问题的同学请及时联系, 若当周未及时联系, 后续关于作业接受情况的争议较大概率按迟交处理
4. 评分细则
Project 占比 1/2 阅读文献，可1/2/3人，约20页/人，ddl约7月底。
选题以公布在qq群在线文档，请同学们选题后填写在线文档并发送电子邮件给助教确定。 project登记表

下一次作业ddl

6/23 homework_0616
之前的作业: homework_220403 (参考答案 )， homework_220508

参考资料

教材
1. 《Gauge fields, Knots and Gravity》 John Baez
2. 《Geometry, topology, and physics》 Mikio Nakahara
3. 《Topological Insulators》沈顺清
4. 《a short course on topological insulators》 János K. Asbóth
5. 《Topological Insulators and Topological Supercondutors》 B. Andrel Bernevig
牛谦, 物理学中的几何相位,2(8,9,10), 2103, 课程主页
online course on topology in condense matter

课程内容^{随讲课进度更新}

03.06 笔记
1. wedge外微分 "\( \wedge \)"
  性质:
  1. dx \( \wedge \) sx = 0;
  2. dx \( \wedge \) dy = - dy \( \wedge \) dx;
  3. \(d(dx) = d^2x = 0 \).
2. form (形式)
  定义
  1. 1-form: \( w = \sum_i f_i dx_{i} \);
  2. 2-form: \( w = \sum_{ij} f_{ij} dx_{i}\wedge dx_{j} \);
  3. n-form: \( w = \sum_{i_1 i_2 ... i_n} f_{i_1 i_2 ... i_n} dx_{i_1}\wedge dx_{i_2} ... \wedge dx_{i_n} \);
3. Stokes定理
  1. 微积分中的定义(适用3维空间中的线积分)
    \(I = \int_{\gamma}\left( P dx + Q dy + R dz\right) =\iint_{\Sigma} (\frac{\partial R}{\partial y} -\frac{\partial Q}{\partial z} )dydz +(\frac{\partial P}{\partial z} -\frac{\partial R}{\partial x} )dzdx +(\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y} )dydz \) with \( \gamma =\partial \Sigma \).
  2. 外微分中的定义(适用n维空间中的积分)
    \( \int_{\partial \Sigma} \omega = \int_{\Sigma} d\omega \) 或者 \((\partial \Sigma,\omega) = (\Sigma,d\omega)\)
4. 作业1:
  证明: 设\( dx_{i}=a_{ij}dx'_{J}, 证明dx_{1}\wedge dx_{2} ... \wedge dx_{n}= J dx'_{1}\wedge dx'_{2} ... \wedge dx'_{n}, 其中J=det(a) \).
3.10 笔记
1. 形变下不变的特殊函数 \( F_{\lambda} (x_{1}, x_{2}, ..., x_{d}) \)
  例: winding number (绕数) \( I_{l} = \frac{1}{2 \pi i} \int \frac{dz}{z} \)
  形变不变的积分, 类比积分于路径无关仅于积分上下限有关. 其实就是全微分.
  \( \int_{a}^{b} f(x)dx = \int_{a}^{b} \frac{\partial F}{\partial x}dx = \int_{a}^{b} dF = F(b)-F(a) \). 如\( I_{l} = \frac{1}{2 \pi i} \int \frac{dz}{z} = \frac{1}{2 \pi i} \int d(ln z) \)
2. 映射(Map): \(f: X\rightarrow Y \) 其中 \( X,Y \) 是集合.
  分类:
  1. 单射(Injective Map): \(\forall x,x' \in X\), 若有 x' ≠ x ，则\( f(x) ≠ f(x') \);
  2. 满射(Surjective Map): \(\forall y \in Y \) , 至少存在一个\(x\in X\)有\(y = f(x)\);
  3. 双射(bijection)：映射既是单射也是满射(存在逆映射\(f^{-1}\)).
3. 拓扑(topology)基础概念
  1. 拓扑研究的是形变(连续变换)下, 不变的性质. <\li>
  2. 拓扑放弃了距离的概念, 通过元素(点)的相对关系定义开集(领域), 进而定义了拓扑空间.
  3. 拓扑空间的定义：\(X\)是集合，\(T\)是集合(X)中的一些子集构成的族\(T =\{u_i|u_i \subset X \}\). 则\((X,T)\)是一个拓扑空间，如果以下的性质成立
    1. \(\varnothing \)和\(X \)是\(T\)中的元素;
    2. \(T \)中任意元素的并集仍然是\(T\)中的元素;
    3. \(T \)中有限元素的交集仍然是\(T \)中的元素.
  4. 拓扑空间同胚 (topological isomorphism)
    1. 连续映射(continous Map): X, Y为两个拓扑空间, 映射 f(X)=Y. 若Y中任何一个开集的原象也是开集, 则称该映射为连续映射.
    2. 同胚映射(homeomorphic Map): 双射, 且映射本身, 逆映射都是连续的. 又称同胚.
    3. 拓扑空间同胚: 两个拓扑空间间存在同胚映射, 则称这两个拓扑空间同胚.
      拓扑性质: 在同胚映射(形变)下不变的性质, 如开集,连通,流形的维数等.
      eg. 单位开圆盘\( D^2 \) 和平面 \( R^2 \) 同胚. 映射(极坐标系) f: \( D^2 \rightarrow R^2 \), \( \theta'=\theta \), \( r' = -ln(1-r) \).
4. 流形: 局部同胚于n维欧几里得空间, 整体上并不一样.
3.13 笔记
1. 流形上的积分, \( I_{s} =\int dx_{1} dx_{2} ... dx_{d} F(\vec{X}) \)
  1. 注意积分空间, 通常为闭空间(闭曲面);
  2. x在物理中的对应:实空间坐标, 如拓扑缺陷; 动量, 如拓扑能带.
2. F的特殊性质以保证连续形变下不变
  1. \( I_{s} =\int_{\partial{V}} dx_{1} dx_{2} ... dx_{d} F(\vec{X}) = \int_{\partial{V}} \omega = \int_{V} d \omega \);
  2. \( \omega \rightarrow \omega + d \eta \), \( I_s \) 不变. \( \omega \sim \omega + d \eta \) 等价;
  3. 特殊解 \( \int d \eta = 0 \); 真实情况中, \(I_{s} \ne 0 \), 因为Stokes定理要求函数光滑, 在实际问题中不一定满足.
3. 高维 \(I_{s} \) 的构造
  1. 矩阵 \( \frac{1}{2\pi i} \int \frac{dz}{z} \rightarrow \frac{1}{2\pi i} \int Tr(A^{-1} dA) \);
  2. 高维矩阵 \( \int Tr(g^{-1} dg \wedge g^{-1} dg \wedge .. \wedge g^{-1} dg) =\int Tr((g^{-1} dg)^{d}) \);
  3. 角度 \( \frac{1}{2\pi} \int d\theta \);
  4. 立体角 \( \frac{1}{4\pi} \int d\Omega \), 其中\( d\Omega= \frac{1}{r^{3}} \vec{r} \cdot (\frac{\partial \vec{r}}{\partial x} \times \frac{\partial \vec{r}}{\partial y})dxdy =\vec{n} \cdot (\vec{n_x} \times \vec{n_y}) dxdy \), \( \vec{a} \cdot (\vec{b} \times \vec{c}) = \epsilon _{ijk} a_{i} b_{j} c_{k} \);
  5. 高维角 \( \epsilon_{i_{1} i_{2} ... i_{d}} n_{i_{1}} (\partial_{x_{1}} n_{i_{2}}) ... (\partial_{x_{d-1}} n_{i_{d}}) dx_{1} dx_{2} ... dx_{d-1} \).
4. 常见空间
  1. 实数空间 \( \mathbb{R}^{n} \);
  2. 球面 \( S^{d}= ((x_{1}, x_{2} ... x_{d+1}) | \sum_{i=1}^{d+1} x_{i}^2 = 1) \);
  3. 矩阵空间 A或g:
    1. U空间: \( UU^{\dagger} =1 \);
    2. O空间: \(O^{T}O = 1 \), 正交矩阵;
  4. d. 环面 \(T^{d} =S^{1} \times S^{1} ... \times S^{1} (d 个 S^{1}) \);
  5. 实射影空间 \( \mathbb{R}P^{n} = S^{n} / \mathbb{Z}_{2} \).
5. 作业
  1. 通过使用 \( \frac{1}{x+iy} =\frac{x-iy}{x^{2} +y^{2} + \epsilon^{2}} \), 其中 \(\epsilon \rightarrow 0 \), 并使用Stokes定理计算\( \frac{1}{2\pi i} \int \frac{dz}{z} \)
  2. 考虑非对角的矩阵 \( A_{n \times n} \), 计算 \( \frac{1}{2\pi i} \int Tr(A^{-1} dA) =\sum_{j} \frac{1}{2\pi i} \int \frac{d \lambda_{j}}{\lambda_{j}} \)

3.17 笔记

复射影空间 \( \mathbb{C}P^{n} =S^{2n+1} /U(1) \)

群的定义 < G,* >
集合G, 和二元运算 * , 满足:

含单位元, \( e \in G \);
有逆, \( \forall g_{i} \in G, g_{i}^{-1} \in G \);
封闭, \( \forall g_{i}, g_{j} \in G, g_{i} * g_{j} \in G \);
结合律, \( (g_{i} *g_{j}) *g_{k} =g_{i} *(g_{j} *g_{k}) \). 其中, 群元 \( g_{i} \) 可以是数, 操作\矩阵或映射.
乘法表

\( C_{3} \)群乘法表
	a =1	\( b =\exp{i2\pi /3} \)	\( c =\exp{i4\pi /3} \)
a	a	b	c
b	b	c	a
c	c	a	b

重排定理: \( \forall g \in G, g*G =G*g =G \).
子群: \( \forall H \subset G \), < H,* > 也构成群.
子群不唯一
陪集
1. 定义: 左陪集 \( 对于子集H, 群元g, gH=\left\{ g*h| h \in H \right\} \), 同理右陪集 Hg;
2. 陪集定理: \( \exists h_{1} \in g_{1}H, h_{2} \in g_{2}H, h_{1} =h_{2}, 则 g_{1}H =g_{2}H \)

3.20 笔记
1. 陪集
  1. 定义: 左陪集 \( 对于子集H, 群元g, gH=(g*h| h \in H) \), 同理右陪集 Hg;
  2. 陪集定理: \( \exists h_{1} \in g_{1}H, h_{2} \in g_{2}H, h_{1} =h_{2}, 则 g_{1}H =g_{2}H \)
  3. Lagrange 定理: \( |G| =|G/H| |H| \)
  4. 正规子群(Normal subgroup): \( \forall g \in G, gN =Ng \).
2. 群表示论: 利用矩阵表示抽象的群元(同态/同构)
3. 同态/同构基本定理
  1. 同态(homomorphic): \( f:G \rightarrow H, \forall a,b \in G, f(a) \times f(b) =f(ab) \);
  2. 同构(isomorphism): 同态映射f为双射;
  3. 同态/同构基本定理: \( f:G \rightarrow H 为群同态, 则 \)
    1. \( f(G) \subset H \);
    2. \( kerf 为G 的正规子群 \);
    3. \( G/kerf \simeq Im(f) \).
    其中, \( kerf =\left\{ g| g \in G, f(g) =1_{H} \right\} \).
3.24 笔记
1. 使用的课程资料:
  1. nakahara_chapter3 , nakahara_chapter6
  2. homomorphisms
2. 阿贝尔群(Abelian groups)
  交换不变, < G,+ >, e=0;
  1. 生成子 \( x_{1} ... x_{r} \)
    群中元素由生成子生成 \( g = n_{1}x_{1} + ... + n_{r}x_{r} \)
    1. 有限生成群: 生成子个数有限;
    2. 线性独立: \( g = n_{1}x_{1} + ... + n_{r}x_{r} =0\) 当且仅当 \( n_{1} = n_{2} =...= n_{r} = 0 \);
  2. 自由阿贝尔群: 由r个线性独立的生成子生成的群
  3. 循环群: 由1个生成子生成的群
  4. 有限生成群都可表示为: \( G \simeq \mathbb{Z}^{r} \oplus \mathbb{Z}_{k_1} \oplus ... \oplus \mathbb{Z}_{k_m} \)
3. 单(纯)形(simplex): 由r+1个几何独立的顶点构成的多面体内的所有点(含边界) 记为 \( \sigma_{r} = (p_{1}, p_{2}, ..., p_{r}) \)
  从中选出q个点构成的单形 \( \sigma_{q} \), 称为 \( \sigma_r \)的面, 记为 \( \sigma_{q} \leq \sigma_{r} \)
4. 单(纯)复形: 由有限个单形构成的集合, 记为 K, 且满足下列规则:
  1. \( \forall \sigma \in, \sigma' \leq \sigma, \sigma' \in K \);
  2. \( \forall \sigma, \sigma' \in K, \sigma \cap \sigma' = \emptyset 或 \leq \sigma 或 \leq \sigma' \)
  多面体 |k| 为所有 K 中单形的点的集合.
5. 定向单形(记为: (...))
  \( (p_{i_{0}} p_{i_{1}} ... p_{i_{r}}) = sgn(P) (p_{0} p_{1} ... p_{r}) \)
  \[ P = \begin{pmatrix} 0 & 1 & ... & r \\ i_{0} & i_{1} & ... & i_{r} \end{pmatrix} \]
6. 同调群(Homology group)
  1. chain group: 由K中 r-simplexes(共 \( I_{r} \) 个) 为生成子的自由阿贝尔群, 记为 \( C_{r}(K) \)
    \( c = \Sigma^{I_{r}}_{i=1} c_{i} \sigma_{r,i} \)
    \( C_{r}(K) \simeq \mathbb{Z}^{I_{r}} \)
  2. 边界算子 \( \partial_{r} \)
    \( \partial_{r} \sigma_r = \Sigma^{r}_{i=0} (-1)^{i}(p_{0} ... \hat{p_{i}} ... p_{r}) \), 其中 \( \hat{p_{i}} \) 意为忽略该点.
  3. r-cycle group \( Z_{r}(K) = \left\{ c | \partial_{r}c = 0, c \in C_{r}(K) \right\} \)
  4. r_boundary group \( B_{r}(K) = \left\{ c | \partial_{r+1}d, d \in C_{r+1}(K) \right\} \)
  5. homology group \( H_{r}(K) = Z_{r}(K) / B_{r}(K) \)
3.27 笔记
今日讲课内容为对上次课的补充, 计算和一些应用的初步举例, 所以不在主页再次重复.
3.31 笔记
1. 课上提及的两个资料
  1. online course on topology in condense matter
  2. 《a short course on topological insulators》 János K. Asbóth
2. 路径(path)(对应群的元素):\( \alpha :t \in [0, 1] \rightarrow \alpha (t) \), 为X中的一条曲线
  1. 环路(loop): \( \alpha (0) =\alpha (1) =x_{0}, x_{0} \) 称为基点(basepoint)
  2. 常值路径(constant path, 记为 \( C_{x} \) ): \( \alpha (t) \equiv x_0 \)
3. 粘接(对应群的乘法): \( (\alpha \times \beta) = \left\{ \begin{aligned} \alpha (2s) & \quad & 0\leq s \leq 1/2 \\ \beta (2s-1) & \quad & 1/2 \leq s \leq 1 \end{aligned} \right. \)
4. 群的逆元: \( \alpha^{-1}(s) =\alpha(1-s) \)
5. 同伦类(homotopy class)
  1. 同伦关系(homotopy equivalence):可通过连续形变相互转换 \( \exists F(x,t), F(x,0)=f, F(x,1)=g \)
  2. 同伦类: 同一个同伦关系的道路的集合 \( [\alpha] =\left\{ \beta |\alpha \sim \beta \right\} \)
6. 同伦群(homotopy group)
  1. \([C_{x}] =e \)
  2. \( [\alpha] \times [\beta] =[\alpha \times \beta] \)
  3. \( ([\alpha] \times [\beta]) \times [\gamma] =[\alpha] \times ([\beta] \times [\gamma]) \)
7. 我们关注\( \Pi_{d}(X) =\left\{ f| f: S^{d} \rightarrow X \right\} \)
4.03 笔记
1. \(\Pi_{d}(X) \) 的计算
  1. \(\Pi_{d}(S_{n}) =0 (n > d) \)
  2. \(\Pi_{d}(S_{n}) =\mathbb{Z} (n =d) \)
  3. \(\Pi_{d}(S_{n}) \neq 0 (n < d) \)
  4. Hopf map \(\Pi_{3}(S^{2}) =\mathbb(Z) \)
2. 立体角:d维 \( d\Omega =\epsilon^{i_{0}i_{1}...i_{d}} n_{i_{0}} \partial_{x_{1}} n_{i_{1}} ... \partial_{x_{d}} n_{i_{d}} dx_{i_{1}} ... dx_{i_{d}} \)
3. Stokes定理: \( \int_{V} dx_{x_{0}} dx_{i_{1}} ... dx_{i_{d}} \sim \int_{\partial V} \epsilon^{i_{0}i_{1}...i_{d}} x_{i_0} dx_{i_{1}} ... dx_{i_{d}} \)
4. \(\Pi_{d}(X) \)的应用: XY model 相变
5. 作业
  1. 计算 Hppf map: \(\Pi_{3}(S^{2}) =\mathbb{Z} \)(参考mathematica)
  2. 阅读文章 Dirac's monopole and the Hopf map, 可推导文章内容或阅读其它内容写一些感想. 网址 , pdf
4.07 笔记
1. 课堂资料
2. 同伦群的应用
  1. \(\Pi_{1}(S^{1}) =\mathbb{Z} \): 描述XY model 中的缺陷
  2. \(\Pi_{1}(\mathbb{R}P^{d}) \):
    1. 描述液晶中的缺陷
      由于\(\vec{n} \sim -\vec{n} \), 所以n空间为\(S^{d} /\mathbb{Z}_{2} =\mathbb{R}P^{d} \)
    2. 描述费米子/玻色子
      由于\(\Pi_{1}(\mathbb{R}P^{1}) =\mathbb{Z} \), 所以二维时存在任意子;
      由于\(\forall d>2, \Pi_{1}(\mathbb{R}P^{d-1}) =\mathbb{Z}_{2} \), 所以高维时仅存在玻色子和费米子.
  3. \(\Pi_{2}(S^{2}) =\mathbb{Z} \): 描述Qi-Wu-Zhang(QWZ) model
    \(H(k_{x}, k_{y}) =\lambda sin(k_{x}) \sigma_{x} +\lambda sin(k_{y}) \sigma_{y} + (m -cos(k_{x}) -cos(k_{y})) \sigma_{z} \)
    k空间同构于\(T^{2} \), 但仅其中较小区域对积分有贡献, 所以可以近似为 \(S^{2} \)
3. 作业
  1. QWZ model 的能量可视为\(\vec{n} =\frac{1}{A} (\lambda sin(k_{x}), \lambda sin(k_{y}), m -cos(k_{x}) -cos(k_{y})), \left\| \vec{n} \right\| =1 \), 计算关于 n 的立体角 \(\vec{n} \cdot (\frac{\partial \vec{n}}{\partial k_{x}} \times \frac{\partial \vec{n}}{\partial k_{y}}) \), 画图(density plot 或等高线图) (要求标注姓名+学号防止抄袭,并注明画图参数).(推荐使用mathematica)
  2. 画相图, \(\int \vec{n} \cdot (\frac{\partial \vec{n}}{\partial k_{x}} \times \frac{\partial \vec{n}}{\partial k_{y}}) dk_{x} dk_{y}\) ,关于\(\lambda,m \) 的图(要求同上).
4.10 笔记
1. 自发对称性破缺(spontaneous symmetry breaking)
  1. \(U =\alpha (T-T_{c}) |\Delta|^{2} +\beta |\Delta|^{4} \)
  2. Quench 过程产生topo defects
    1. \(\Delta \) 为实数, 产生Domain(筹, 例:磁畴)
    2. KZ机制
2. topological excitation
  1. isotropy group (迷向群)
    1. Group \(G =\left\{g |F(\psi) =F(g\psi) \right\} \)
    2. Isotropy Group \(H =\left\{h |\psi) =h\psi \right\} \)
    3. order parameter manifold \(M =G/H \)
  2. 例: spin-1 BECs, \(\psi =(1,0,0)^{T}, M =\frac{U(1) \times SO(3)}{U(1)} =SO(3) \),
3. 课程资料:
  1. cosmological experiment in superfluid helium
  2. Ueda Ch12 topological excitation
4.17 笔记
1. 纤维丛 (Fibre Budle)
  1. 纤维(Fiber):\(映射f:X \rightarrow Y, Y 中元素 y 的原象称为纤维，即 F^{-1}(y) = \left\{x \in X | f(x) = y \right\} \)
  2. 纤维丛: 映射\(f:E \rightarrow B \), 其中\(B\)称为底空间. (课堂中, \(\vec{X} 为底空间, 映射为 \Psi^{-1}: \mathbb{C}^{n} \rightarrow \vec{X} \))
2. \(\Pi_{d}(X) \) 的几何图像: \(\vec{X} \in S^{d}, \Psi_{\vec{X}} \in X, \Psi_{\vec{X}} 随空间 \vec{X} 变化而变化, 具有几何的图像 \)
3. 正合序列 (exact sequence)
  1. 正合: \(f:A \rightarrow B,\ g:B \rightarrow C\) 满足 \(\mathrm{Im}f = \ker g\)
  2. 短正合序列:
    - \(0\rightarrow A\rightarrow B\rightarrow 0 \Rightarrow A \simeq B \)
    - \(0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0 \Rightarrow B \simeq A \times C\ \text{or}\ B/A \simeq C \)
  3. 长正合序列: 可通过分拆为短正合序列计算
  4. 计算 \(\Pi_{d}(G/H) \) \[ \begin{matrix} \vdots & \vdots & \vdots \\ \rightarrow \Pi_2(H) &\rightarrow \Pi_2(G)& \rightarrow \Pi_2(G/H) \rightarrow \\ \rightarrow \Pi_1(H) &\rightarrow \Pi_1(G)& \rightarrow \Pi_1(G/H) \rightarrow \\ \rightarrow \Pi_0(H) &\rightarrow \Pi_0(G)& \rightarrow \Pi_0(G/H) \end{matrix} \]
    一般前两列可通过查表得到, 利用短正合序列求最后一列.
4. 几何相, berry phase
  1. 绝热过程:
    1. 经典力学: 绝热不变量\(I = \frac{1}{2\pi} \oint pdq \) (量子力学中也具有相同的性质?)
    2. 量子力学: 一般意味着不会发生跃迁, 但能级可能发生变化\(\lambda_{n}(t) \)
  2. 几何相(推导详见笔记): \(\gamma_{n}(T) = i\int^{T}_{0} <\psi_{n}| \frac{\partial}{\partial t} |\psi_{n}> dt = \int^{R(T)}_{R(0)} <\psi_{n} |i\nabla_{R} |\psi_{n}> dR = \oint <\psi_{n} |i\nabla_{R} |\psi_{n}> dR (整圈) \)
  3. 推荐阅读: Holonomy, the quantum adiabatictheorem, and Berry's phase 和 Quantal phase factors accompanying adiabatic changes
5. 作业: \(设 H = B_{1} \sigma_{z} + B_{2} cos\theta \sigma_{x} + B_{2} sin\theta \sigma_{y}, 计算\)
  1. 本征态 \(|\psi (\theta)> \)
  2. \(\frac{\partial}{\partial \theta} |\psi(\theta)> \)
  3. \(\int^{2\pi}_{0} <\psi| \frac{\partial}{\partial \theta} |\psi(\theta)> d\theta \)
4.21 笔记
1. berry 论文 Quantal phase factors accompanying adiabatic changes
  1. introduction
  2. Geometry Phase \(\gamma_{n}(c) = i \oint_{c} \langle \psi_{n} |\nabla_{R}| \psi_{n} \rangle d\vec{R} \)
  3. spin model \(H = \vec{B} \cdot \vec{\sigma} = B(cos \theta \sigma_{z} + sin \theta cos \phi \sigma_{x} + sin \theta sin \phi \sigma_{y}) \)
    \(\gamma_{c} = -a^{2} \oint d\phi = -2\pi a^{2} \) 立体角
2. Zak Berry's Phase for Energy Band in Solid
  1. 1d Bloch band \(\gamma_{n} = \int^{\pi/a}_{-\pi/a} \chi_{nn}(k) dk \), 其中 \(\chi_{nn}(k) = \frac{2\pi}{a} \int^{a}_{0} U^{*}_{nk} (i\frac{\partial}{\partial k}) U_{nk} dx \sim \int x |a_{n}(x)|^{2} dx \)，即波包中心
  2. \(\gamma = 2\pi \frac{q}{a} \)
3. Geometry phase 意义
  1. \(G = ie^{2} /h \)
  2. \(H = \vec{B} \cdot \vec{\sigma} \)
  3. Zak phase \(\gamma = 2\pi \frac{q}{a} \)
4. 作业：阅读文献 Appearance of Gauge Structure in Simple Dynamical Systems, Frank Wilczek and A. Zee, 1984
4.24 笔记
1. 课程资料
  1. Topological insulators and superconductors: tenfold way and dimensional hierarchy
  2. Unpaired Majorana fermions in quantum wires
2. 拓扑分类表
  
  对称性：
  1. T: time-reversal symmetry 时间反演对称性
  2. C: charge-conjugate / particle-hole 电子空穴对称性
  3. S: chiral symmetry 手征对称性
3. 高量基础 \(c_{l} = \frac{1}{\sqrt(N)} \sum_{k} e^{ik(la)} c_{k} \)(自旋不影响\(e^{ika} \)项 )
  1. \( \sum_{l} c^{\dagger}_{l} c_{l} = \sum_{k} c^{\dagger}_{k} c_{k} \)
  2. \( \sum_{l} c^{\dagger}_{l} c_{l+1} = \sum_{k} e^{ika} c^{\dagger}_{k} c_{k} \)
  3. \( \sum_{l} c^{\dagger}_{l} c^{\dagger}_{l} = \sum_{k} c^{\dagger}_{k} c^{\dagger}_{-k} \)
  4. \( k \rightarrow 0 \) 时，做泰勒展开，\(2cosk_{x} \approx 2-k^{2}_{x}, sink_{x} \approx k_{x}, \sum_{\vec{k}} k_{x}k_{y}c^{\dagger}_{\vec{k}} c_{\vec{k}} \approx \sum_{\vec{k}} sink_{x} sink_{y} c^{\dagger}_{\vec{k}} c_{\vec{k}} \sim \sum_{i,j} (c^{\dagger}_{i,j} c_{i+1,j+1} + h.c.) \)
4. Kitaev model \(H = -w \sum_{j} (a^{\dagger}_{j} a_{j+1} + h.c.) - \mu \sum_{j} (a^{\dagger}_{j} a_{j} - \frac{1}{2} ) + \sum_{j} (\Delta a_{j} a_{j+1} \Delta^{*} a^{\dagger}_{j+1} a^{\dagger}_{j} ) \)
  1. majorana 表示：\(c_{2j-1} = a_{j} + a^{\dagger}_{j}, c_{2j} = \frac{a_{j} - a^{\dagger}_{j} }{i} \)
    性质：
    1. \(c_{j} = c^{\dagger}_{j} \)
    2. \(c^{2}_{j} = 1 \)
    3. \(c_{i} c_{j} = - c_{j} c_{i} (i \neq j) \)
  2. \(H = \sum_{j} \frac{i}{2} (-\mu c_{2j-1} c_{2j} +(w+\Delta) c_{2j}c_{2j+1} +(\Delta-w) c_{2j-1}c_{2j+2} ) \)
5. 作业：编程计算一个 \(200 \times 200 \)格点的QWZ model，分析其相变，边界态。
4.28 笔记
1. Kitaev model \(H = -w \sum_{j} (a^{\dagger}_{j} a_{j+1} + h.c.) - \mu \sum_{j} (a^{\dagger}_{j} a_{j} - \frac{1}{2} ) + \sum_{j} (\Delta a_{j} a_{j+1} \Delta^{*} a^{\dagger}_{j+1} a^{\dagger}_{j} ) \)
  1. majorana 表示：\(c_{2j-1} = a_{j} + a^{\dagger}_{j}, c_{2j} = \frac{a_{j} - a^{\dagger}_{j} }{i} \)
    性质：
    1. \(c_{j} = c^{\dagger}_{j} \)
    2. \(c^{2}_{j} = 1 \)
    3. \(c_{i} c_{j} = - c_{j} c_{i} (i \neq j) \)
  2. \(H = \sum_{j} \frac{i}{2} (-\mu c_{2j-1} c_{2j} +(w+\Delta) c_{2j}c_{2j+1} +(\Delta-w) c_{2j-1}c_{2j+2} ) \)
  3. 图像、极限分析：特殊解 \(\mu = 0, \Delta = -w , H = \sum_{j} \frac{i}{2} 2\Delta c_{2j-1}c_{2j+2} \)
    1. 考虑单独一项： \( c_{2j-1} = a_{j} + a^{\dagger}_{j}, c_{2j} = \frac{a_{j} - a^{\dagger}_{j} }{i}, c_{2} c_{3} = \frac{1}{i} (f^{\dagger} f - f f^{\dagger}) = \frac{1}{i} (2f^{\dagger} f - 1) \)
    2. 边界态：\( c_{1},c_{10} \) 可视为 \( \lim_{\epsilon \rightarrow 0 } \epsilon c_{1} c_{10} \) 在 \(0 \) 处两重简并，且和其余态具有能隙，可以保持稳定。
    3. 局域态和拓展态的耦合\(\sim \frac{e^{-\frac{L}{\xi}}}{\Delta} \)
    4. Transverse Ising model 详见笔记
  4. Pfaffian: \(Pf(A) = \frac{1}{2^{N} N!} \sum_{\tau \in S_{2N}} sgn(\tau) A_{\tau (1) ,\tau (2)} ... A_{\tau (2N-1) ,\tau (2N)} \)
    1. \((Pf (B))^{2} = det (B) \)
    2. \(B = W^{T} \Lambda W, Pf(B) = Pf(\Lambda) det(W) = Pf(\Lambda) \)
    3. \(\mathcal{M}(B) = sgn[Pf B(0)] sgn[Pf B(\pi)] \)
  5. 守恒量 \(P = \prod_{j} (-ic_{2j-1}c_{2j}) \)
    1. \(P = P^{\dagger} \)
    2. \(P^{2} = 1 \)
    3. \([H,P] = 0 \)
  6. 4\(\pi\) 周期
2. BdG 方程 Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures
  \(H = h_{ij} c^{\dagger}_{i} c_{j} + \Delta_{ij}c^{\dagger}_{i} c^{\dagger}_{j} + h.c. \)
  矩阵形式 \( (c^{\dagger}_{1} ... c^{\dagger}_{N} c_{1} ... c_{N} ) \left( \begin{array}{cc} h \quad & \Delta \\ \Delta^{T} \quad & -h^{*} \end{array} \right) \left( \begin{array}{c} c_{1} \\ ... \\ c_{N} \\ c^{\dagger}_{1} \\ ... \\ c^{\dagger}_{N} \end{array} \right) \)
  其中\(\Delta^{T} = - \Delta \)
3. TR symmetry sakurai
  1. \(T^{2} = -1, T = \sigma_{y} K , THT^{-1} = H\)
  2. \(T^{2} = 1, T = K / \sigma_{x}K / \sigma_{z} K \)，为实表示
4. 作业：利用 BdG 方程计算 \( H = \sum_{i} (t c^{\dagger}_{i} c_{i+1} + \Delta c^{\dagger}_{i} c^{\dagger}_{i+1} + h.c. - \mu c^{\dagger}_{i} c_{i}) \)
  1. 计算相图与边界态；
  2. 改变化学势\(\mu c^{\dagger}_{i} c_{i} \rightarrow Vcos(Qi) c^{\dagger}_{i} c_{i} \)，其中\(Q = \frac{\sqrt{5} -1}{2} \);
  3. 尝试任意的 \(V_{i} c^{\dagger}_{i} c_{i} \)
5.05 笔记
1. particle-hole symmetry \(CH_{k}C^{-1} = -H_{-k} \)
  1. e.g. \(C = \sigma_{x} K, \sigma_{y} K. \)
  2. 性质：若\(H_{k} |\psi\rangle = E_{k} |\psi\rangle, H_{-k}C |\psi\rangle = -E_{k}C |\psi\rangle. \)
    对于时间反演对称性, \(H_{-k}T |\psi\rangle = -E_{k}T |\psi\rangle. \)
2. 手征对称性 \(SH_{k}S^{-1} = -H_{k} \)
  同时具有\(T, C \) 时具有手征对称性，或都不具备时，可能具有手征对称性。（十重分类）
3. non-Abelian geometry phase \(A^{ab} = \langle \psi_{a} |d| \psi_{b} \rangle \)
  1. \(\gamma = i\int_{\partial \Sigma} \mathrm{Tr}(A) = i\int_{\Sigma} \mathrm{Tr}(F), F = dA + A^{2} \)
  2. 设\(\psi_{a}\rangle = g_{ai} |\psi_{j}\rangle, A' = g^{-1}Ag + g^{-1} dg \)
  3. \(d=2n+2, ch_{n+1}(F) = \frac{1}{(n+1)!} tr(\frac{iF}{2\pi})^{n+1}, Ch_{n+1}(F) = \int_{BZ} ch_{n+1}(F) \)
  4. \(ch_{n+1} = dQ_{2n+1}(A,F), Q_{2n+1}(A,F) = \frac{1}{n!} (\frac{i}{2\pi})^{n+1} \int^{1}_{0} dt \mathrm{Tr} (AF^{n}t), Ft = tdA + t^{2} A^{2} = tF + (t^{2} - t) A^{2} \)
4. 如何使用Stokes定理
  1. 计算球面上的积分 \(\oint \vec{B} \cdot d\vec{S} \)，分为上下两个半球面 \(\int_{\epsilon^{+} } \vec{B} \cdot d\vec{S} + \int_{\epsilon^{-} } \vec{B} \cdot d\vec{S} = \oint_{l} (\vec{A}_{+} - \vec{A}_{-} ) \cdot d\vec{l} \)
5. 作业：阅读文献（2选1）
  1. A four-dimension generation of quantum hall effect
  2. Dirac's monopole wihtout strings: classical lagrange theory
5.08 笔记
1. 奇数维量子系统的拓扑数
  1. 手征对称性：设 \(H = \left( \begin{array}{cc} a \quad & b \\ b^{\dagger} \quad & c \end{array} \right) \)，由手征对称性 \(\{H, S\} = 0, H = \left( \begin{array}{cc} 0 \quad & b \\ b^{\dagger} \quad & 0 \end{array} \right) \)
  2. winding number: \(\frac{1}{2\pi i} \oint \mathrm{Tr} (q^{-1}dq) = \frac{1}{2\pi i} \oint \mathrm{Tr} (q^{\dagger}dq) \)
  3. Chein-Simons invarient: \(CS_{2n+1} = \int_{BZ^{2n+1}} Q_{2n+1} (A, F) \)
  4. 拓扑不变量\(W_{2n+1} = e^{i2\pi CS_{2n+1}} \)
2. 计算 \(\Pi_{d} (G/H) \)
  1. 资料： Concise Tables of James Number And some Homotopy ...
  2. stiefel manifold: \(\forall X = U(N+M)/U(N) or SP(N+M)/SP(N) or O(N+M)/O(N), \Pi_{d}(X) = 0 \)
  3. \[ \begin{matrix} d & U(M) & U(N+M)/U(N) & U(N+M)/U(N) \times U(M) \\ 3 & \mathbb{Z} & 0 & 0 \\ 2 & 0 & 0 & \mathbb{Z} \\ 1 & \mathbb{Z} & 0 & 0 \\ 0 & 0 & 0 & \\ \end{matrix} \]
3. 作业：
  1. \(if |\psi_{\alpha} \rangle = g_{\alpha j} |\psi_{j}\rangle, A' = g^{-1} \wedge g + g^{-1} dg, F' = g^{-1} F g \) 证明：\(Q_{2n+1}(A', F') - Q_{2n+1}(A, F) = Q_{2n+1}(g^{-1}dg, 0) +d\alpha_{2n-2}, Q_{2n+1}(g^{-1}dg, 0) = \omega_{2n+1}(g) eq24,25\) NJP,12,065010(2010)
  2. 利用 \( lundell Table 1, steifel manifold \), 推导 NJP,12,065010(2010) Table 3
5.12 笔记
1. Dimension reduction
  1. \(d=2n+3, chiral sym, \vec{k} = (k_{1}, k_{2}, ..., k_{2n+3}), H = \sum^{2n+3}_{i=1} k_{i} \Gamma^{i}_{2n+3}, E = \pm |\vec{k}| \)；
    \(d=2n+2 \)，设\( k_{2n+3} = m, H = \sum^{2n+2}_{i=1} k_{i} \Gamma^{i}_{2n+3} + m\Gamma^{2n+3}_{2n+3} \)，对称性改变；
    \(d=2n+1 \)，设\( k_{2n+2} = 0, H = \sum^{2n+1}_{i=1} k_{i} \Gamma^{i}_{2n+3} + m\Gamma^{2n+3}_{2n+3} \)，对称性改变。
  2. eg: \(d=3, H = \vec{k} \cdot \vec{\sigma} \)，具有时间反演对称性；
    \(d=2 \)，设\( k_{3} = m, H = k_{x} \sigma_{x} + k_{y} \sigma_{y} + m \sigma_{z},CH_{1}(F) = \frac{1}{2} sgn(m) \)，无对称性；
    \(d=1 \)，设\( k_{2} = 0, H = k_{x} \sigma_{x} + m \sigma_{z}, W = \frac{1}{2} sgn(m) \)，具有手征对称性。
2. \(why \enspace \mathbf{Z}_{2}?\)
  \(q_{1}(k) \sim q_{2}(k) \) 同伦等价，通过构造可以证明从偶数维度(AIII)升高一个维度后，整个系统可以分为两个等价类，即\(\mathbf{Z}_{2} \)。
3. 物理实现
  Jackin_Rebbi model, PRD,13,3398(1976)
  1. \(H = k\sigma_{x} + m\sigma_{z}, E = \pm \sqrt{k^{2} + m^{2}} \)
  2. \(H = P\sigma_{x} + m\sigma_{z}, H^{2} = P^{2} + m^{2} + 2\sigma_{y} m \delta(x) \rightarrow P^{2} + m^{2} + 2 m \delta(x) \)，存在束缚态。
5.15 笔记
1. SSH model soliton in polyacetylene , QMC 2d SSH model
  1. \(H = - \sum_{n} t_{n+1,n} (c^{\dagger}_{n+1} c_{n} + h.c. ) + \frac{1}{2} m \dot{u}^{2}_{n} + \frac{1}{2} k (u_{n+1} - u_{n})^{2} \)
  2. 采用半经典近似 \(H = - \sum_{n} (t_{0} + (-1)^{n} 2\alpha u) (c^{\dagger}_{n+1} c_{n} + h.c. ) + 2 nk u^{2} \)
  3. 转到k空间 \(H = - \sum_{k}(t_{0} + 2\alpha u) (A^{\dagger}_{k} B_{k} + h.c. ) - \sum_{k}(t_{0} - 2\alpha u) (B^{\dagger}_{k} A_{k} e^{ik} + h.c. ) \)
  4. 设\(\sum_{k} cos(k) = 0 \)，则\(E_{g} = const + 2u^{2} nk - 4\alpha un \)，能量极小点发生漂移，破坏对称性。
2. Haldane model Model for a Quantum Hall Effect without Landau Levels
  1. \(H = - \sum_{< ij >} t_{0} (A^{\dagger}_{i} B_{j} + h.c. ) - \sum_{<< ij >>} t_{1} (A^{\dagger}_{i} A_{j} e^{i\phi /3} + h.c. ) - \sum_{<< ij >>} t_{1} (B^{\dagger}_{i} B_{j} e^{i\phi /3} + h.c. ) \)
  2. 转到k空间，做好区分，分别转换，最后汇总所有相互作用。详细计算见笔记。
3. 作业：
  推导 SSH model 论文 eq.2.1 ~ eq.3.19，可参考 Su-Schrieffer-Heeger model applied to chains of finite length 。
5.19 笔记
1. SSH model: 通过不同的方式打开gap (结构相变、不同的原子)
2. Haldane: 添加磁通，添加对角项，产生gap。\(\sqrt{|q|^{2} + |\Delta|^{2} } \)
3. Spin Hall Effect: Z2 Topological Order and the Quantum Spin Hall Effect, Kane and Mele
4. Qi-Wu-Zhang model 参考文献
  1. \( H = sink_{x} \sigma_{x} - sink_{y} \sigma_{y} + c(2 - ocsk_{x} - cosk_{y} - e_{s}) \sigma_{z} \)
  2. \( \sigma_{xy} = \left\{ \begin{array}{cc} 1/2\pi, \quad & 0 < e_{s} < 2, \\ -1/2\pi, \quad & 2 < e_{s} < 4, \\ 0, \quad & otherwise. \end{array} \right. \)
  3. \( \sigma^{\Gamma}_{xy} = \frac{n}{\pi} (n = 0, \pm 1). \)
5. 作业：推导文章 Finite Size Effects on Helical Edge States in a Quantum Spin-Hall System, Zhou Bin, Shunqin Shen.
5.22 笔记
1. 课程资料
2. 几何相的计算：\( \gamma = i \oint \langle \psi | \nabla | \psi \rangle \cdot d\Omega = \sum_{R} i [\langle \psi(R) | \psi(R+\delta R) \rangle - 1] = i\sum_{R} [ln(\langle \psi(R) | \psi(R+\delta R) \rangle)] = i ln \Pi_{n} \langle \psi_{n} | \psi_{n+1} \rangle \)
3. Spin HE 物理实现 Quantum Spin Hall Insulator Statein HgTe Quantum Wells
  1. Strong SOC, 小的 Energy gap
4. 3d system and weyl semimetal \( H = \left( \begin{array}{cc} \epsilon_{k} & Ak_{-} \\ Ak_{+} & -\epsilon_{k} \end{array} \right), \epsilon_{k} = \frac{k^{2}_{x} + k^{2}_{y} +k^{2}_{z} }{2 m} - \mu \)
  1. 无能隙零点： \(k_{x} = k_{y} = 0, k_{z} + \pm \sqrt(2m\mu) \)
  2. 扰动，零点移动
  3. 拓扑因子：取\(k_{z} = \lambda \) 为参数，\(H(k_{x}, k_{y} ) \)，计算chen number
5.29 笔记
1. 背景
  1. Ehrenfest theorem \( \begin{equation} \left\{ \begin{split} \frac{ d \langle x \rangle }{ dt} = \frac{ \langle p \rangle }{m} \\ \frac{ d \langle p \rangle }{ dt} = - \langle \frac{\partial V }{\partial x} \rangle \end{split} \right. \end{equation}\)
  2. Drude model \( \begin{equation} \left\{ \begin{split} \frac{ d x }{ dt} = \frac{ p }{m} \\ \frac{ d p }{ dt} = F - \gamma P \end{split} \right. \end{equation}\)
  3. 固体物理 Kittle \( \begin{equation} \left\{ \begin{split} \frac{ d r }{ dt} = \frac{\partial \epsilon }{\partial k} \\ \frac{ d k }{ dt} = -e (E + v \times B) \end{split} \right. \end{equation}\)
    Bloch oscillation
  4. Bloch电子在磁场中的行为：peierls 替换 \(P \rightarrow P - eA \)
    Hofstadter butterfly
2. effective motion of bloch wavepacket
  1. 参考文献：
  2. \(L = P_{c} \dot{r}_{c} - H = \langle \psi | i\frac{d}{dt} |\psi \rangle - \langle \psi | H |\psi \rangle\)
  3. \( \langle \psi | x |\psi \rangle = [i \frac{\partial}{\partial q} \delta_{nn ' } + \langle u_{nq} |i\frac{\partial u_{n '}}{\partial q '} \rangle ] \delta_{qq '} \)
3. 作业：推导文章公式和结论 Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides_Di Xiao
6.02 笔记
1. effective motion of bloch wavepacket
  1. \( \langle \psi | x |\psi \rangle = [i \frac{\partial}{\partial q} \delta_{nn ' } + \langle u_{nq} |i\frac{\partial u_{n '}}{\partial q '} \rangle ] \delta_{qq '} = - \left( \frac{\partial r}{\partial k} \right) |_{k=k_{c}} + |w(k)|^{2} A(k) \)
  2. \(L = P_{c} \dot{r}_{c} - H = \langle \psi | i\frac{d}{dt} |\psi \rangle - \langle \psi | H |\psi \rangle = eA \cdot r + \hbar k \cdot \dot{r} - \hbar A \cdot \dot{k} - E(k), \quad E(k) = \epsilon(k) + \frac{e}{2m} \delta B \cdot L \)
2. 课程资料
  1. Time reversal polarization and a Z2 adiabatic spin pump, Liang Fu and C. L. Kane
  2. Topological Insulator Materials, Yoichi Ando, JPSP
3. 反线性算子 \(A(\lambda_{1} \phi_{1} + \lambda_{2} \phi_{2}) = \lambda^{\ast}_{1} A \phi_{1} + \lambda^{\ast}_{2} A \phi_{2} \)
  1. \( T=UK, \langle \phi |T| \phi \rangle = - \langle \phi |T| \phi \rangle =0, \langle Tx|Ty \rangle = \langle y|x \rangle \)
6.05 笔记
1. Time reversal sysmetry
  1. W矩阵 \(W_{\alpha \beta}(k) = \langle u_{\alpha, -k} |\theta| u_{\beta, k} \rangle \)
    1. \(W^{\dagger} W = 1 \)
    2. \(W_{\alpha \beta}(-k) = - W_{\beta \alpha}(k) \)
  2. \(a_{\alpha \beta}(k) = -i \langle u_{\alpha, k} |\nabla_{k}| u_{\beta, k}, \quad a(-k) = W(k) a(k) W^{\dagger}(k) + i W(k) \nabla_{k} W^{\dagger}(k) \)
  3. \(P_{\theta} = \int^{\pi}_{-\pi} dk [a_{11}(k) - a_{22}(k) ], \quad (-1)^{P_{\theta}} = \Pi_{j} \frac{Pf(W(\Lambda_{j})) }{\sqrt{Pf^{2}(W(\Lambda_{j}))} } \)
6.09 笔记
1. 课程资料：
  1. Douglas R. Hofstadter, PRB 14,2239(1976)
  2. Xiaogang Wen, Nuclear Physics B316 (1989) 641-662
2. 出发点：磁场中的平移对称性 \(\vec{P} = \vec{P} - e \vec{A} \)
  \(T_{x} = e^{-ia(p_{x} - eA_{x})}, T_{y} = e^{-ia(p_{y} - eA_{y})}, T_{x} T_{y} = e^{-ia^{2}B} T_{y}T_{x}. \)
  若 \(a^{2}B = 2\pi p/q, [T_{x}, T^{q}_{y}] = 0 \)，具有平移对称性(\(T_{x}, T_{y} \)的次数不唯一)。
3. Harper eq
  1. \(H = -t \sum_{< ij >} e^{-i\int^{j}_{i} \vec{A} \cdot d\vec{l}} a^{\dagger}_{i} a_{j} \)
  2. 设 \(\vec{A} = (0, Bx, 0), Eu_{i} = -t(u_{i-1} + u_{i+1} ) - 2t cos(iBa^{2} -k_{y} a ) u_{i}, H = -t(a^{\dagger}_{i} a_{i+1} + h.c. ) - 2v cos(i \phi - \theta) a^{\dagger}_{i} a_{i}. \)
4. Wen Xiaogang \(H = v(k_{x} \sigma_{x} + k_{y} \sigma_{y}), \Gamma = \sigma_{z}, \nu = \frac{1}{4\pi i} \oint \mathrm{Tr} (\Gamma H^{-1} dH ) \)
5. 作业：推导计算 Ganeshan et al. PRL,110,180403(2013) 。
6.12 笔记
1. Xiaogang Wen \(H = t\sum_{k} c^{\dagger}_{k+w} c_{k} e^{ik_{x}a} + c^{\dagger}_{k-w} c_{k} e^{-ik_{x}a} + 2cos(k_{y}a) c^{\dagger}_{k} c_{k}, \quad w = (0, \theta) = (0, 2\pi q/p) \)
  1. \(\nu = \frac{1}{2\pi i} \oint \frac{dz}{z} \)
  2. 算子形式，令\(a = 1 \)，\(H = e^{-ik_{y}} A + e^{ik_{x}} B + h.c. \)
    1. \(A^{p} = B^{p} = 1, AB = BA e^{i2\pi q/p} \)，A、B的具体形式见笔记;
    2. \(A H(k_{x}, k_{y}) A^{-1} = H(k_{x} + 2\pi q/p, k_{y}, B H(k_{x}, k_{y}) B^{-1} = H(k_{x}, k_{y} + 2\pi q/p ) \)
    3. p、q互质，存在\(nq - mp = 1 \)，设\(\tilde{A} = A^{n}, \tilde{B} = B^{n} \)， \(\tilde{A} H(k_{x}, k_{y}) \tilde{A}^{-1} = H(k_{x} + 2\pi /p, k_{y}, \tilde{B} H(k_{x}, k_{y}) \tilde{B}^{-1} = H(k_{x}, k_{y} + 2\pi /p ) \)
    4. 若\(H(k_{x}, k_{y}) \psi = \epsilon \psi, \tilde{A} H \tilde{A}^{-1} \tilde{A} \psi = \epsilon \tilde{A} \psi \)，存在p重简并。
2. Kitaev
  1. 课程资料
    1. Fault-tolerant quantum computation by anyons, A. Yu. Kitaev
    2. Anyons in an exactly solved model and beyond, Alexei Kitaev
  2. Toric-code model \(H = -\sum_{s} A_{s} - \sum_{p} B_{p} \)
    1. \(s-star A_{s} = \Pi_{j \in star(s)} \sigma^{x}_{j}, \quad p-plaquette B_{p} = \Pi_{j \in boundary(p)} \sigma^{z}_{j} \)
    2. 所有\(A_{s}, B_{p} \)都对易；
    3. groud-state\(- (N_{s} + N_{p}) \)，简并度/能否实现？与欧拉定理关联。
6.19 笔记
1. Toric-code model \(H = -\sum_{s} A_{s} - \sum_{p} B_{p} \)
  1. 基本性质
    1. \(A_{s}, B_{p} \) 都对易，且\([A_{s}, H] = [B_{p}, H] = 0 \)
    2. \(A_{s} = A^{\dagger}_{s}, B_{p} = B^{\dagger}_{p}, A^{2}_{s} = B^{2}_{p} = 1, \Pi_{s} A_{s} = \Pi_{p} B_{p} = 1 \)
  2. \(n=2k^{2} \) 个节点，希尔伯特空间 \(2^{2k^{2}} \) 维，考虑约束 \(\Pi_{s} A_{s} = \Pi_{p} B_{p} = 1 \)，应为 \(2^{2k^{2}-2} \) 维，4重简并\(\mathbb{Z}_{2} \times \mathbb{Z}_{2} \)
  3. Groud State \(|G\rangle = \Pi_{p} \frac{1 + B_{p}}{2} \Pi_{s} \frac{1 + A_{s}}{2} |\phi \rangle \)
  4. Loop(contractible or not)：详细图像见笔记
  5. \(\mathbb{Z}_{2} \) gauge theory, Ising model
    \(Z = Tr[e^{-\beta J \sum_{< ij > } \sigma_{i} \sigma_{j}}] = Tr[1 - \beta J \sum_{< ij >} \sigma_{i} \sigma_{j} + (\beta J \sum_{< ij >} \sigma_{i} \sigma_{j})^{2}/2] \)
    考虑是否可以形成环，1、3阶为0，2、4阶非0。图像见笔记。
2. KiteaV honeycomb lattice \(H = -J_{x} \sum_{ij} \sigma^{x}_{i} \sigma^{x}_{j} - J_{y} \sum_{ij} \sigma^{y}_{i} \sigma^{y}_{j} - J_{z} \sum_{ij} \sigma^{z}_{i} \sigma^{z}_{j} \)
  1. \(W_{p} = \sigma^{x}_{1} \sigma^{y}_{2} \sigma^{z}_{3} \sigma^{x}_{4} \sigma^{y}_{5} \sigma^{z}_{6} \)
  2. \(\sigma^{\alpha} = i b^{\alpha} c \), \( b^{\alpha} \)、c 为 majorana fermion， \( \sigma^{x} \sigma^{y} \sigma^{z} = i b^{x} b^{y} b^{z} c = i D \)
  3. \(u_{ij} = i b^{\alpha}_{i} b^{\alpha}_{j} \)，\(\alpha \) 由ij决定，\(W_{p} = \Pi u_{ij} \)
  4. \(H = \frac{i}{4} \sum A_{ij} c_{j} c_{k}, A_{ij} = 2i J_{\alpha} b^{\alpha}_{i} b^{\alpha}_{j} \)，对\(groud\ state, i b^{\alpha}_{i} b^{\alpha}_{j} = 1 \)
  5. 最终 \(H = \left( \begin{array}{cc} 0 & f(q) \\ f^{*}(q) & 0 \end{array} \right), f(q) = J_{z} + J_{x} e^{i \vec{q} \cdot \vec{R}_{1}} + J_{y} e^{i \vec{q} \cdot \vec{R}_{2}} \)

拓扑相变与拓扑场论 (i) (2023/春季)

往年主页: 2021 , 2018

主讲老师: 龚明, 15156057001, 物质楼C812，mail: gongm@ustc.edu.cn

助教: 张诚江 mail: zcj12138@mail.ustc.edu.cn

上课时间: 1~15周, 1(1,2), 5(8,9)

上课地点: 2106

评分标准

下一次作业ddl

参考资料

课程内容随讲课进度更新

03.06 笔记

3.10 笔记

3.13 笔记

3.17 笔记

3.20 笔记

3.24 笔记

3.27 笔记

3.31 笔记

4.03 笔记

4.07 笔记

4.10 笔记

4.17 笔记

4.21 笔记

4.24 笔记

4.28 笔记

5.05 笔记

5.08 笔记

5.12 笔记

5.15 笔记

5.19 笔记

5.22 笔记

5.29 笔记

6.02 笔记

6.05 笔记

6.09 笔记

6.12 笔记

6.19 笔记

课程内容^{随讲课进度更新}