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In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
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Keywords:
- strongly-correlated system /
- topological order /
- symmetry-protected topological state /
- topological quantum field theory
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图 1 蒙特卡洛验证投影后得到的SPT波函数的拓扑纠缠熵为零, 摘自文献[86]
Figure 1. Monte Carlo verification of vanishing topological entanglement entropy of the SPT wave function obtained from the projective construction
图 2 一种将电子分成三个部分子的投影构造(即
$ n=1 $ ), 摘自文献[85]Figure 2. Parton decomposition of electron operators
图 3 “Twist缺陷和拓扑激发”的融合规则示意图 (a) 二维iTO的任意子和点缺陷的融合. (b) 三维iTO的点激发与线缺陷的融合. (c) 三维iTO的圈激发与线缺陷的融合. 摘自[85]
Figure 3. Diagrammatic illustration of fusion rules among twist defects and topological excitations. (a) Fusions between an anyon (quasiparticle) and a point-defect in a two-dimensional iTO. (b) Fusions between a particle excitation and a line defect in a three-dimensional iTO; (c) Fusions between a loop excitation and a line defect in a three-dimensional iTO.[85]
图 5 三维iTO中的点激发和圈激发示意图[138]
Figure 5. Illustration of point-like excitations and loop excitations in three-dimensional iTO
图 6 (a) 点粒子-圈之间的编织: 点粒子激发
$ e_i $ (携带单位$ \mathbb{Z}_{N_i} $ 规范荷) 绕着圈激发$ m_i $ (携带单位$ \mathbb{Z}_{N_i} $ 规范磁通)转一圈.$ e_i $ 的轨迹$ \gamma_{e_i} $ 与静止的圈$ m_i $ 形成一个Hopf环链. (b) 博罗梅安编织(点粒子-圈-圈编织): 点粒子$ e_k $ 绕着两个互相未链接的圈激发$ m_i, m_j $ 转一圈.$ e_k $ 的轨迹$ \gamma_{e_k} $ 与$ m_i $ ,$ m_j $ 一起形成博罗梅安环(Borromean Rings, 或更一般的Brunnian link)Figure 6. (a) Particle-loop braiding: a particle
$ e_i $ travels around a loop$ m_i $ such that the braiding trajectory$ \gamma_{e_i} $ and$ m_i $ form a Hopf link. (b) Borromean-Rings braiding: a particle$ e_k $ moves around two unlinked loops$ m_i, m_j $ such that$ m_i $ ,$ m_j $ and the trajectory$ \gamma_{e_k} $ form the Borromean rings (or generally the Brunnian link)
图 7 SEG的构造图. 摘自文献[48]
Figure 7. Illustration of SEG
图 8 (a) 公式(42)代表的拓扑响应现象的示意图.
$ \mathbb{Z}_{N_1} $ 的对称性畴壁$ D_1 $ 和$ \mathbb{Z}_{N_2} $ 的对称性畴壁$ D_2 $ 的交点携带分数角动量$ \mathcal{J} $ .$ A^1 $ 和$ A^2 $ 分别是垂直于畴壁$ D_1 $ 和$ D_2 $ 的规范联络. (b) 公式(44)代表的拓扑响应现象的示意图. 旋错线与$ \mathbb{Z}_{N_2} $ 对称性畴壁$ D_2 $ 的交点携带$ A^1 $ 规范场的分数规范荷$ \mathcal{Q}_1 $ .$ \omega $ 和$ A^2 $ 分别是垂直于旋错线和畴壁的规范联络[160]Figure 8. (a). Topological response for Eq. (42). The intersection of
$ \mathbb{Z}_{N_1} $ and$ \mathbb{Z}_{N_2} $ symmetry domain walls$ D_1 $ and$ D_2 $ carries the angular momentum$ \mathcal{J} $ .$ A^1 $ and$ A^2 $ are the gauge connections normal to the domain walls. (b). Topological response of Eq. (44). The intersection of disclination line and$ \mathbb{Z}_{N_2} $ symmetry domain walls$ D_2 $ carries the$ A^1 $ charge$ \mathcal{Q}_1 $ .$ \omega $ and$ A^2 $ are the gauge connections normal to the disclination line and domain wall, respectively
图 9 两个三维SPT拓扑响应现象示意图. 摘自文献[160]
Figure 9. Illustration of two examples of SPT topological response phenomena in three dimensions
表 1 二维投影构造中的部分子的拟设.
$ A1, A2,\cdots, A4 $ 代表四种拟设. 每条完全填充的能带由箭头和正负号标记. 箭头表示自旋方向, 正负号代表陈数为1或–1. A1一共有8条填满的陈-能带. A2和A3都有4条填满的陈-能带. A4只用到了f1, 一共有两条陈-能带被填满. 括号里成对的数字表示单个元胞里的费米子f1或f2的填充数: (自旋向上的费米子数目, 自旋向下的费米子数目).Table 1. Parton ansatzes in the two-dimensional projective construction.
$ A1,A2, \cdots, A4 $ stand for four different ansatzes respectively. Each fully occupied band is labeled by a pair of arrow and plus/minus sign. The arrow represents the spin eigenvalue of$ S^z $ , and$ \pm $ represents Chern number$ \pm1 $ . In A1, there are 8 fully occupied Chern bands; There are 4 fully occupied Chern bands in each of A2 and A3. In A4, flavor index is not involved, so only one flavor, say,$ f_1 $ is taken into account. And there are two filled Chern bands. A pair of integers denote the filling number of either$ f_1 $ and$ f_2 $ in each unit cell: (fermion number with up spin, fermion number with down spin).拟设 完全被$f_1$填充的陈-能带 完全被$f_2$填充的陈-能带 自旋矢量$q^T_s$ 电荷矢量$q^T_c$ $A1$ $\uparrow+, \downarrow+, \uparrow-, \downarrow-$ $(2, 2)$ $\uparrow+, \downarrow+, \uparrow-, \downarrow-$ $(2, 2)$ $\left(\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}\right)$ $(1~~1~~1~~1~~1~~1~~1~~1)$ $A2$ $\uparrow+, \downarrow-$ $(1, 1)$ $\uparrow+, \downarrow-$ $(1, 1)$ $\left( {1}/{2}~~- {1}/{2}~~ {1}/{2}~~ -{1}/{2}\right)$ $(1~~1~~1~~1)$ $A3$ $\uparrow+, \downarrow-$ $(1, 1)$ $\downarrow+, \uparrow-$ $(1, 1)$ $\left(1/{2}~~- {1}/{2}~~- {1}/{2}~~ {1}/{2}\right)$ $(1~~1~~1~~1)$ $A4$ $\uparrow+, \downarrow-$ $(1, 1)$ 无 $\left( {1}/{2}~~- {1}/{2}\right)$ $(1~~1)$ 
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表 2 在大U极限下, 实空间每个格点上的不消耗U能量的占据状态形成了物理希尔伯特空间. 我们需要对费米子的总的填充数做限制. 限制之后, 所有格点都能够同时处于物理希尔伯特空间.
Table 2. At large U limit, the physical Hilbert space is formed by those occupancy bases without energy cost. We should restrict the total particle number of each flavor properly such that Hilbert space of every site is always in the physical Hilbert space
U 任意一个格点上的物理希尔伯特空间基矢$ [f_1]n_{i, 1 \uparrow}, n_{i, 1 \downarrow}, n_{i, 2 \uparrow}, n_{i, 2 \downarrow}[f_2] $ 费米子填充总数要求 $U_1$ $(0, 0, 0, 0)$, $(0, 1, 0, 1)$, $(0, 1, 1, 0)$, $(1, 0, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 1, 1)\, $ $N^{f1}=N^{f2} $ $U_2$ $(0, 0, 0, 0)$, $(0, 1, 1, 0)$, $(1, 0, 0, 1)$, $(1, 1, 1, 1)\, $ $N^{f1}_{\uparrow} = N^{f2}_{\downarrow}, $ $N^{f1}_{\downarrow}=N^{f2}_{\uparrow}$ $U_3$ $(0, 0, 0, 0)$, $(0, 1, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 1, 1)\, $ $N^{f1}_{ \uparrow}=N^{f2}_{ \uparrow}, $ $N^{f1}_{ \downarrow}=N^{f2}_{\downarrow}$ $U_4$ $(0, 0, 1, 1)$, $(0, 1, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 0, 0)$ $N^{f1}_{\uparrow}+N^{f2}_{\downarrow}=N_{\rm latt}$, $N^{f2}_{\uparrow}+ N^{f1}_{\downarrow}=N_{\rm att}$ $U_5$ $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$, $(0, 0, 0, 1)$ $ N^{f1} + N^{f2}=N_{\rm latt}$ $U_6$ $(1, 0)$, $(0, 1)$ $ N^{f1} =N_{\rm latt}$ $U_7$ $(0, 0)$, $(1, 1)$ $N^{f1}_{\uparrow} = N^{f1}_{\downarrow }$
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表 3 受到幺正阿贝尔群保护的“不可约”的三维SPT态的低能有效理论及其分类.
$ a^I $ 和$ b^I $ 分别是1-形式 和2-形式$ {U(1)} $ 规范场. 系数p、$ p_1 $ 、$ p_2 $ 的取值满足一定的量子化条件和周期性. 系数的周期给出分类的结果.“($ {\mathbb{Z}}_{N_{12}} ) \cdots $ ”表示相应的分类. 其中, 符号$ N_{IJ\cdots} $ 表示$ N_I, N_J, \cdots $ 等整数的最大公约数. 受到$ {\mathbb{Z}}_N $ 或$ {U(1)}^k $ 或$ {\mathbb{Z}}_N\times {U(1)}^k $ 保护的SPT态都是平凡的, 因而没有列入表中.“不可约”是指对称群的所有子群都起着保护SPT的作用. 其他SPT都可以通过表格里的结果构造出来. 具体摘自[129].Table 3. A brief summary of irreducible 3D SPT phases with unitary Abelian symmetry.
$ a^I $ and$ b^I $ are 1-form and 2-form$ {U(1)} $ gauge fields, respectively. “($ {\mathbb{Z}}_{N_{12}} $ )$ \cdots $ ”denote the corresponding classifications, where$ N_{IJ\cdots} $ are greatest common divisors of$ N_I, N_J, \cdots $ . SPT phases with either$ {\mathbb{Z}}_N $ or${U(1)}^k $ or$ {\mathbb{Z}}_N\times {U(1)}^k $ are trivial and not included below. By “irreducible”, we means that all subgroups of symmetry group play nontrivial roles in protecting the nontrivial SPT phases. All other SPT's with unitary Abelian group symmetries can be obtained directly by using this table[129].对称群G 拓扑规范场论与分类 $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}$ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^2_Ib^I\wedge \, {\rm d}a^I+ p_1\displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^2~~(\mathbb{Z}_{N_{12}} )$ ; $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^2_Ib^I\wedge \, {\rm d}a^I+ p_2\displaystyle\int a^2\wedge a^1\wedge \, {\rm d}a^1 ~(\mathbb{Z}_{N_{12}}) $ $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3} $ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+ p_1 \displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^3~~(\mathbb{Z}_{N_{123}})$ ; $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+ p_2 \displaystyle\int a^2\wedge a^3\wedge \, {\rm d}a^1~~(\mathbb{Z}_{N_{123}}) $ $\prod^4_I\mathbb{Z}_{N_I} $ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^4_Ib^I\wedge \, {\rm d}a^I+p \displaystyle\int a^1\wedge a^2\wedge a^3\wedge a^4~~ ( \mathbb{Z}_{N_{1234}} )$ $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times {U}(1)$ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+p\displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^3~~ (\mathbb{Z}_{N_{12}})$
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表 4 部分三维SET的分类, 摘自[50].
Table 4. Classification of SET examples.
规范群$G_g$ twisted拓扑项 对称群$G_s$ SET分类 ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 n + 1}$ ${{\mathbb{Z}}_1}$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 n}$ $ {({\mathbb{Z}}_2)^2}\oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_3$ – ${\mathbb{Z}}_{3 n}$ $({\mathbb{Z}}_3)^2\oplus {\mathbb{Z}}_1 \oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_3$ – ${\mathbb{Z}}_{3 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n+2}$ $ ({\mathbb{Z}}_2)^2\oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n}$ $ ({\mathbb{Z}}_4)^2\oplus ({\mathbb{Z}}_2)^2\oplus {\mathbb{Z}}_1 \oplus {\mathbb{Z}}_1 $ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6\oplus ({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^2$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 2) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (0, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 4) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (0, 0) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2\oplus 2({\mathbb{Z}}_4)^2\oplus4({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^6$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 0) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 4) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m + 1} \times {\mathbb{Z}}_{2 n + 1}$ $({\mathbb{Z}}_{2\gcd(2 m + 1, 2 n + 1)})^2$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m + 1} \times {\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_{\gcd(2 m + 1, 2 n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(2 m + 1, 2 n)})^2 $ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m} \times {\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6\times({\mathbb{Z}}_{2\gcd(m, n)})^2\oplus ({\mathbb{Z}}_{2\gcd(2 m, n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(m, 2 n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(m, n)})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2 n + 1} \times {\mathbb{Z}}_{2 n + 1}$ $16({\mathbb{Z}}_{2 n + 1})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2(2 n + 1)} \times {\mathbb{Z}}_{2(2 n + 1)}$ $4({\mathbb{Z}}_2)^6 \times ({\mathbb{Z}}_{2(2 n + 1)})^2\oplus 12({\mathbb{Z}}_{2(2 n + 1)})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n} \times {\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_4)^6 \times ({\mathbb{Z}}_{4 n})^2\oplus 12({\mathbb{Z}}_{4 n})^2 \oplus 3[ ({\mathbb{Z}}_{4 n})^2\times ({\mathbb{Z}}_2)^6]$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}\oplus 6({\mathbb{Z}}_2)^8 \oplus 3({\mathbb{Z}}_2)^6 \oplus 6({\mathbb{Z}}_2)^4$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 2) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}$
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表 5 部分含有反幺正对称群(时间反演)的SET的体内理论与边界理论, 摘自[51].
Table 5. Bulk and boundary theories of SET with anti-unitary symmetry (e.g., time-reversal symmetry).
投影对称群(PSG) 规范群$G_g$ 对称群$G_s$ 三维体内($\varSigma^3$)
的规范理论表面($\partial\varSigma^3$)的反常
玻色理论二维平面($\varSigma^2$)的正常Chern-Simons理论的$K_G$-矩阵 ${\mathbb{Z}}_N \rtimes{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$
$\dfrac{\theta_c}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^c_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}^T_2$破缺的 $\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$${\mathbb{Z}}^T_2$破缺的$\varSigma^2$:
$\left(\begin{array}{*{20}{c}} {2 p}&N\\ N&0 \end{array}\right)$${\mathbb{Z}}_N\!\times\!{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_s}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^s_\rho$${\mathbb{Z}}^T_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}^T_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p &N \\ N & 0 \end{array}} \right)$${\mathbb{Z}}_N \!\times\! [U(1)_{S^z}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_N\!\times\! U(1)_{S^z}$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c +$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p_1 &N & p_{12}& 0\\ N & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & 0\\ 0 & 0 &0 & 0 \end{array}} \right)$$U(1)_C \!\times\! [{\mathbb{Z}}_N \rtimes{\mathbb{Z}}_2]$ $U(1)_C\!\times\!{\mathbb{Z}}_N$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p_1 &0 & p_{12}& 0\\ 0 & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & N\\ 0 & 0 &N & 0 \end{array}} \right)$${\mathbb{Z}}_{N_1} \!\times\! [{\mathbb{Z}}_{N_2}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_{N_1}\!\times\! {\mathbb{Z}}_{N_2}$ ${\mathbb{Z}}_2$ $\dfrac{N_1}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$
$\dfrac{N_2}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N_1}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j+$
$\dfrac{N_2}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$\begin{aligned} & {}\\ & \left({\begin{array}{*{20}{c}} 2 p_1 &N_1 & p_{12}& 0\\ N_1 & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & N_2\\ 0 & 0 &N_2 & 0 \end{array}} \right)\end{aligned}$
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表 6 带整数自旋和电荷的玻色SPT的电荷和自旋响应理论[51].
Table 6. Charge and spin response of spin-1 and charge-1 boson systems.
轴子角 对称群 三维体内($\varSigma^3$)的响应 二维表面($\partial\varSigma^3$)的反常响应 二维平面($\varSigma^2$)的响应 $ \theta_c=2{\text{π}}+4{\text{π}} k$
(带电玻色系统)$U(1)_C\rtimes{\mathbb{Z}}^{\rm T}_2$ 电荷-威腾效应:
$N^c=n^c+N^c_m$量子电荷霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\partial\varSigma^3$):
$\widetilde{\sigma}^{c}=(1+2 k)\dfrac{1}{2{\text{π}}}$量子电荷霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\varSigma^2$)
$\sigma^c=2 k\dfrac{1}{2{\text{π}}}$$ \theta_s=2{\text{π}}+4{\text{π}} k$
(整数自旋的
玻色系统)$U(1)_{S^z} \times {\mathbb{Z}}^{\rm T}_2$ 自旋-威腾效应:
$N^s=\displaystyle \sum_i q_in_i^s+N^s_m\sum_{i}q_i^2$量子自旋霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\partial\varSigma^3$):
$\widetilde{\sigma}^{s}=(1+2 k)\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$量子自旋霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\varSigma^2$)
$\sigma^s=2 k\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$$ \theta_0={\text{π}}+2{\text{π}} k$
(带电和整数自旋
的玻色系统)$U(1)_C \!\times\! [U(1)_{S^z} \!\rtimes\! {\mathbb{Z}}_2]$ 交互-威腾效应: $N^c=n^c+\dfrac{1}{2}N^s_m$;
$N^s=n^s_{+}-n^s_{-}+\dfrac{1}{2}N^c_m$量子电荷-自旋/
自旋-电荷效应
(${\mathbb{Z}}_2$破缺的 $\partial\varSigma^3$):
$\widetilde{\sigma}^{cs}=\widetilde{\sigma}^{sc}=\left(\dfrac 1 2+k\right)\dfrac{1}{2{\text{π}}}$量子电荷-自旋/
效应 自旋-电荷
(${\mathbb{Z}}_2$破缺的$\varSigma^2$):
$\sigma^{cs}=\sigma^{sc}=k\dfrac{1}{2{\text{π}}}$
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表 7 推广的Wen-Zee拓扑项, 摘自[160].
Table 7. Generalized Wen-Zee terms.
时空维度 空间对称群$G_s$ 内部对称群$G_i$ 不可约的Wen-Zee拓扑项$S$ 角动量/自旋${\cal{J}}$ $(2 + 1)$维 $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$, $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$, $k \in \mathbb{Z}_{N_{01}}$, $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} $ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A^1 \wedge A^2$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^2} A^1 \wedge A^2$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A \wedge {\rm d}A$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int _{M^3} A \wedge {\rm d}A$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int A \wedge \omega \wedge {\rm d} \omega$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{2{\text{π}}^2 N_{01}} \displaystyle\int_{M^3} A \wedge {\rm d}\omega$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times U(1)$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$维 $SO(2)$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int A^1 \wedge A^2 \wedge {\rm d} \omega$, $k \in \mathbb{Z}_{N_{12}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int_{M^3} {\rm d} (A^1 \wedge A^2)$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int \omega \wedge A^2 \wedge {\rm d}A^1$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int_{M^3} A^2 \wedge {\rm d}A^1$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} \times \mathbb{Z}_{N_3}$ $k \dfrac{N_0 N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int \omega \wedge A^1 \wedge A^2 \wedge A^3$,
$k \in \mathbb{Z}_{N_{0123}}$$k \dfrac{ N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int_{M^3} A^1 \wedge A^2 \wedge A^3$ $(3 + 1)$维($*$) $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$, $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$维($*$) $C_{N_0}$ ${\mathbb{Z}}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$, $k \in \mathbb{Z}_{N_{01}}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$维($*$) $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A \wedge B$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^3} A \wedge B$
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