2020-02-14 Kitaev model on a hex

作者: 低维量子系统 | 来源:发表于2020-02-14 23:00 被阅读0次

2020-02-14 Kitaev model on a hex
2020-02-25 Kitaev model on a hex
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JSONModel源码解析
js rgb转hex
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将透明度的百分比转化为十六进制的字符串
Sass、Scss 中颜色转换 rgbaToHex、rgbToH

Kitaev model on the hexagonal lattice

这两天摈弃了所有杂念，踏踏实实学习Kitaev model。找到的一篇能够follow的文献是Chen在2008年发表的论文。

标题：Exact results of the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations
作者：Han-Dong Chen and Zohar Nussinov
全文链接：论文

笔记的目的
- 日后关于Jordan-Wigner变换，Wick定理，傅立叶变换等，可以在这里温习。
- 如果日后恰好有人学到这篇文献，希望这篇笔记能够帮到他（她）。

Hamiltonian

$H=-J_x\sum_{x-links}\sigma_j^x\sigma_k^x-J_y\sum_{y-links}\sigma_j^y\sigma_k^y-J_z\sum_{z-links}\sigma_j^z\sigma_k^z$

pauli operators

$\sigma^x=[0,1;1,0]$

$\sigma^y=[0,-i;i,0]$

$\sigma^z=[1,0;0,-1]$
$\sigma_x^2=\sigma_y^2=\sigma_z^2=I$
$\sigma_x\sigma_y\sigma_z=I$

$\sigma_y\sigma_z\sigma_x=I$

$\sigma_z\sigma_x\sigma_y=I$
$\sigma_z\sigma_y\sigma_x=-i$

$\sigma_y\sigma_x\sigma_z=-i$

$\sigma_x\sigma_z\sigma_y=-i$

fermion operators

$\{a^{\dagger}_j,a_j\}=1$

$\{a^{\dagger}_j,a_k\}=0$

$\{a_j,a_k\}=0$

$\{a^{\dagger}_j,a^{\dagger}_k\}=0$

$a_j^2=0$

${a^{\dagger2}_j}=0$

Jordan-Wigner transformation from spin operators to fermion operators

$\sigma^x_{\ell}=[\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot(c^{\dagger}_{\ell}+c_{\ell})$

$\sigma^y_{\ell}=-i\cdot[\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot(c^{\dagger}_{\ell}-c_{\ell})$

$\sigma^z_{\ell}=2c^{\dagger}_{\ell}c_{\ell}-1$
communication relations

$[\sigma^z_{\ell},c^{\dagger}_j]$

$=[2c^{\dagger}_{\ell}c_{\ell}-1,c^{\dagger}_j]$

$=(2c^{\dagger}_{\ell}c_{\ell}-1)\cdot c^{\dagger}_j - c^{\dagger}_j \cdot (2c^{\dagger}_{\ell}c_{\ell}-1)$

$=(2c^{\dagger}_{\ell}c_{\ell}\cdot c^{\dagger}_j- c^{\dagger}_j) - (2 c^{\dagger}_j \cdot c^{\dagger}_{\ell}c_{\ell}-c^{\dagger}_j )$

$=2c^{\dagger}_{\ell}c_{\ell}\cdot c^{\dagger}_j - 2 c^{\dagger}_j \cdot c^{\dagger}_{\ell}c_{\ell}$

$=0$

$[\sigma^z_{\ell},c^{\dagger}_{\ell}]$

$=[2c^{\dagger}_{\ell}c_{\ell}-1,c^{\dagger}_{\ell}]$

$=(2c^{\dagger}_{\ell}c_{\ell}-1)\cdot c^{\dagger}_{\ell} - c^{\dagger}_{\ell} \cdot (2c^{\dagger}_{\ell}c_{\ell}-1)$

$=(2c^{\dagger}_{\ell}c_{\ell}\cdot c^{\dagger}_{\ell}- c^{\dagger}_{\ell}) - (2 c^{\dagger}_{\ell} \cdot c^{\dagger}_{\ell}c_{\ell}-c^{\dagger}_{\ell} )$

$=2c^{\dagger}_{\ell}c_{\ell}\cdot c^{\dagger}_{\ell}$

$=2c^{\dagger}_{\ell}\cdot (1-c^{\dagger}_{\ell} c_{\ell})$

$=2c^{\dagger}_{\ell}$
Jordan-Wigner transformation for $\sigma^x_{\ell}\sigma^x_{\ell+1}$

$\sigma^x_{\ell}\sigma^x_{\ell+1}$

$=[\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot(c^{\dagger}_{\ell}+c_{\ell}) \cdot [\Pi_{m=1...\ell}\sigma^z_m ] \cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})$

$=(c^{\dagger}_{\ell}+c_{\ell}) \cdot[\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot [\Pi_{m=1...\ell}\sigma^z_m ]\cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})$

$=(c^{\dagger}_{\ell}+c_{\ell}) \cdot \sigma^z_{\ell}\cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})$ ------(1)

Since $\sigma^z_{\ell}=2c^{\dagger}_{\ell}c_{\ell}-1$ , one will have

$(c^{\dagger}_{\ell}+c_{\ell})\cdot \sigma^z_{\ell}$

$= (c^{\dagger}_{\ell}+c_{\ell})\cdot(2c^{\dagger}_{\ell}c_{\ell}-1)$

$= c^{\dagger}_{\ell} \cdot(2c^{\dagger}_{\ell}c_{\ell}-1) +c_{\ell}\cdot(2c^{\dagger}_{\ell}c_{\ell}-1)$

$=- c^{\dagger}_{\ell} +c_{\ell}\cdot(2c^{\dagger}_{\ell}c_{\ell}-1)$

$=- c^{\dagger}_{\ell} +2c_{\ell}\cdot c^{\dagger}_{\ell}c_{\ell}-c_{\ell}$

$=- c^{\dagger}_{\ell} +2c_{\ell}\cdot (1- c_{\ell}c^{\dagger}_{\ell})-c_{\ell}$

$= - c^{\dagger}_{\ell} + c_{\ell}$ ------(2)

According to (1) and (2), we finally have

$\sigma^x_{\ell}\sigma^x_{\ell+1}=- (c^{\dagger}_{\ell}- c_{\ell})\cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})$
Jordan-Wigner transformation for $\sigma^y_{\ell}\sigma^y_{\ell+1}$

$\sigma^y_{\ell}\sigma^y_{\ell+1}$

$=-[\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot(c^{\dagger}_{\ell}-c_{\ell})\cdot [\Pi_{m=1...\ell}\sigma^z_m ] \cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-(c^{\dagger}_{\ell}-c_{\ell})\cdot [\Pi_{m=1...\ell-1}\sigma^z_m ] \cdot [\Pi_{m=1...\ell}\sigma^z_m ] \cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-(c^{\dagger}_{\ell}-c_{\ell})\cdot \sigma^z_{\ell} \cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-(c^{\dagger}_{\ell}-c_{\ell})\cdot (2c^{\dagger}_{\ell}c_{\ell}-1)\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[c^{\dagger}_{\ell}\cdot (2c^{\dagger}_{\ell}c_{\ell}-1)-c_{\ell}\cdot (2c^{\dagger}_{\ell}c_{\ell}-1)]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ (c^{\dagger}_{\ell}\cdot2c^{\dagger}_{\ell}c_{\ell}-c^{\dagger}_{\ell})-(2c_{\ell}\cdot c^{\dagger}_{\ell}c_{\ell}-c_{\ell})]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ (0-c^{\dagger}_{\ell})-(2c_{\ell}\cdot c^{\dagger}_{\ell}c_{\ell}-c_{\ell})]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ -c^{\dagger}_{\ell}-2c_{\ell}\cdot c^{\dagger}_{\ell}c_{\ell}+c_{\ell}]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ -c^{\dagger}_{\ell}-2c_{\ell}\cdot (1-c_{\ell}c^{\dagger}_{\ell})+c_{\ell}]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ -c^{\dagger}_{\ell}-2c_{\ell}+c_{\ell}]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=-[ -c^{\dagger}_{\ell}-c_{\ell}]\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$=(c^{\dagger}_{\ell}+c_{\ell})\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$
Jordan-Wigner transformation for $\sigma^z_{\ell}\sigma^z_{\ell+1}$

$\sigma^z_{\ell}\sigma^z_{\ell+1}=(2c^{\dagger}_{\ell}c_{\ell}-1)(2c^{\dagger}_{\ell+1}c_{\ell+1}-1)$
Finally, according to the sequence in Fig. 2 of Chen2008, we have

$H=-J_x\sum_{x-links}[- (c^{\dagger}_{\ell}- c_{\ell})\cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})]$

$-J_y\sum_{y-links}(c^{\dagger}_{\ell}+c_{\ell})\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$

$-J_z\sum_{z-links}(2c^{\dagger}_{\ell}c_{\ell}-1)(2c^{\dagger}_{\ell+1}c_{\ell+1}-1)$

$H=J_x\sum_{x-links}(c^{\dagger}_{\ell}- c_{\ell})\cdot(c^{\dagger}_{\ell+1}+c_{\ell+1})$
$-J_y\sum_{y-links}(c^{\dagger}_{\ell}+c_{\ell})\cdot(c^{\dagger}_{\ell+1}-c_{\ell+1})$
$-J_z\sum_{z-links}(2c^{\dagger}_{\ell}c_{\ell}-1)(2c^{\dagger}_{\ell+1}c_{\ell+1}-1)$

$H=J_x\sum_{x-links}(c^{\dagger}- c)_w\cdot(c^{\dagger}+c)_b$
$-J_y\sum_{y-links}(c^{\dagger}+c)_b\cdot(c^{\dagger}-c)_w$
$-J_z\sum_{z-links}(2c^{\dagger}c-1)_b(2c^{\dagger} c -1)_w$ ------(4)

Majorana fermions

For white sites,

$A_w = \frac{1} {i}(c-c^{\dagger})_w$

$B_w = (c+c^{\dagger})_w$

For black sites

$B_b = \frac{1} {i}(c-c^{\dagger})_b$

$A_b = (c+c^{\dagger})_b$

$B_b A_b= \frac{1} {i}(c-c^{\dagger})_b(c+c^{\dagger})_b$

$=-i(c^2+cc^{\dagger}-c^{\dagger}c-c^{\dagger2})$

$=-i(cc^{\dagger}-c^{\dagger}c)$

$=-i(1-2c^{\dagger}c)$

$=i(2c^{\dagger}c-1)$

$J_x\sum_{x-links}(c^{\dagger}- c)_w\cdot(c^{\dagger}+c)_b$

$=J_x\sum_x (-iA_w) \cdot A_b$

$=-iJ_x\sum_x A_w \cdot A_b$

$-J_y\sum_{y-links}(c^{\dagger}+c)_b\cdot(c^{\dagger}-c)_w$

$=-J_y\sum_y A_b \cdot (-iA_w)$

$=iJ_y\sum_y A_b \cdot A_w$

$-J_z\sum_{z-links}(2c^{\dagger}c-1)_b(2c^{\dagger} c -1)_w$

$=-J_z\sum_z (B_b A_b)(B_w A_w)$

$=J_z\sum_z (B_b B_w) A_bA_w$

$=-iJ_z\sum_z (iB_b B_w) A_bA_w$

$=-iJ_z\sum_z \alpha_r A_bA_w$

$H\{\alpha\}=-iJ_x\sum_x A_w \cdot A_b+iJ_y\sum_y A_b \cdot A_w -iJ_z\sum_z \alpha_r A_bA_w$

5.1 Diagonalization

$d=(A_w+iA_b)/2$

$d^{\dagger}=(A_w-iA_b)/2$

** 公式（18）的证明

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2020-02-14 Kitaev model on a hex

Kitaev model on the hexagonal lattice

Hamiltonian

pauli operators

fermion operators

Jordan-Wigner transformation from spin operators to fermion operators

Majorana fermions

5.1 Diagonalization

相关文章

2020-02-14 Kitaev model on a hex

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2020-02-16 Kitaev model note 2 :

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爬坑记录

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OC_UIColor 类扩展支持十六进制

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Sass、Scss 中颜色转换 rgbaToHex、rgbToH

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