Acta Physica Sinica, Volume. 69, Issue 7, 077102-1(2020)
Figures & Tables(16)
Fig. 1. Monte Carlo verification of vanishing topological entanglement entropy of the SPT wave function obtained from the projective construction
Fig. 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]
Fig. 4. Illustration of gauge-invariant Wilson operators in Eq. (
Fig. 5. Illustration of point-like excitations and loop excitations in three-dimensional iTO
Fig. 6. (a) Particle-loop braiding: a particle
travels around a loop
such that the braiding trajectory
and
form a Hopf link. (b) Borromean-Rings braiding: a particle
moves around two unlinked loops
such that
,
and the trajectory
form the Borromean rings (or generally the Brunnian link)
Fig. 8. (a). Topological response for Eq. (
and
symmetry domain walls
and
carries the angular momentum
.
and
are the gauge connections normal to the domain walls. (b). Topological response of Eq. (
symmetry domain walls
carries the
charge
.
and
are the gauge connections normal to the disclination line and domain wall, respectively
Fig. 9. Illustration of two examples of SPT topological response phenomena in three dimensions
Table 1. Parton ansatzes in the two-dimensional projective construction.
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
, and
represents Chern number
. 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,
is taken into account. And there are two filled Chern bands. A pair of integers denote the filling number of either
and
in each unit cell: (fermion number with up spin, fermion number with down spin).View tableView in ArticleTable 1. Parton ansatzes in the two-dimensional projective construction.
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
, and
represents Chern number
. 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,
is taken into account. And there are two filled Chern bands. A pair of integers denote the filling number of either
and
in each unit cell: (fermion number with up spin, fermion number with down spin).拟设 完全被 $f_1$ 完全被 $f_2$ 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 $\left( {1}/{2}~~- {1}/{2}\right)$ $(1~~1)$
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
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.
View tableView in ArticleTable 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
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.
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}$ $U_7$ $(0, 0)$ $(1, 1)$ $N^{f1}_{\uparrow} = N^{f1}_{\downarrow }$
Table 3. A brief summary of irreducible 3D SPT phases with unitary Abelian symmetry.
and
are 1-form and 2-form
gauge fields, respectively. “(
)
”denote the corresponding classifications, where
are greatest common divisors of
. SPT phases with either
or
or
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].View tableView in ArticleTable 3. A brief summary of irreducible 3D SPT phases with unitary Abelian symmetry.
and
are 1-form and 2-form
gauge fields, respectively. “(
)
”denote the corresponding classifications, where
are greatest common divisors of
. SPT phases with either
or
or
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} $ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^4_Ib^I\wedge \, {\rm d}a^I+p \displaystyle\int a^1\wedge a^2\wedge a^3\wedge a^4~~ ( \mathbb{Z}_{N_{1234}} )$ $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times {U}(1)$ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+p\displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^3~~ (\mathbb{Z}_{N_{12}})$
Table 4.
Classification of SET examples.
部分三维SET的分类, 摘自[50].
View tableView in ArticleTable 4.
Classification of SET examples.
部分三维SET的分类, 摘自[50].
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) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}$
Table 5.
Bulk and boundary theories of SET with anti-unitary symmetry (e.g., time-reversal symmetry).
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].
View tableView in ArticleTable 5.
Bulk and boundary theories of SET with anti-unitary symmetry (e.g., time-reversal symmetry).
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].
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( $\partial\varSigma^3$ 玻色理论 二维平面( $\varSigma^2$ $K_G$ ${\mathbb{Z}}_N \rtimes{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$ $\dfrac{\theta_c}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^c_\nu \partial_\lambda A^c_\rho$ ${\mathbb{Z}}^T_2$ $\partial\varSigma^3$ $\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$ ${\mathbb{Z}}^T_2$ $\varSigma^2$ $\left(\begin{array}{*{20}{c}} {2 p}&N\\ N&0 \end{array}\right)$ ${\mathbb{Z}}_N\!\times\!{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$ $\dfrac{\theta_s}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^s_\rho$ ${\mathbb{Z}}^T_2$ $\partial\varSigma^3$ $\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$ ${\mathbb{Z}}^T_2$ $\varSigma^2$ $ \left({\begin{array}{*{20}{c}} 2 p &N \\ N & 0 \end{array}} \right)$ ${\mathbb{Z}}_N \!\times\! [U(1)_{S^z}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_N\!\times\! U(1)_{S^z}$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c +$ $\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$ ${\mathbb{Z}}_2$ $\partial\varSigma^3$ $\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$ ${\mathbb{Z}}_2$ $\varSigma^2$ $ \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)$ $U(1)_C \!\times\! [{\mathbb{Z}}_N \rtimes{\mathbb{Z}}_2]$ $U(1)_C\!\times\!{\mathbb{Z}}_N$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$ $\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$ ${\mathbb{Z}}_2$ $\partial\varSigma^3$ $\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$ ${\mathbb{Z}}_2$ $\varSigma^2$ $ \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)$ ${\mathbb{Z}}_{N_1} \!\times\! [{\mathbb{Z}}_{N_2}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_{N_1}\!\times\! {\mathbb{Z}}_{N_2}$ ${\mathbb{Z}}_2$ $\dfrac{N_1}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$ $\dfrac{N_2}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$ $\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$ ${\mathbb{Z}}_2$ $\partial\varSigma^3$ $\dfrac{N_1}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j+$ $\dfrac{N_2}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$ ${\mathbb{Z}}_2$ $\varSigma^2$ $\begin{aligned} & {}\\ & \left({\begin{array}{*{20}{c}} 2 p_1 &N_1 & p_{12}& 0\\ N_1 & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & N_2\\ 0 & 0 &N_2 & 0 \end{array}} \right)\end{aligned}$
Table 6.
Charge and spin response of spin-1 and charge-1 boson systems.
带整数自旋和电荷的玻色SPT的电荷和自旋响应理论[51].
View tableView in ArticleTable 6.
Charge and spin response of spin-1 and charge-1 boson systems.
带整数自旋和电荷的玻色SPT的电荷和自旋响应理论[51].
轴子角 对称群 三维体内( $\varSigma^3$ 二维表面( $\partial\varSigma^3$ 二维平面( $\varSigma^2$ $ \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 ( ${\mathbb{Z}}^{\rm T}_2$ $\partial\varSigma^3$ $\widetilde{\sigma}^{c}=(1+2 k)\dfrac{1}{2{\text{π}}}$ 量子电荷霍尔效应 ( ${\mathbb{Z}}^{\rm T}_2$ $\varSigma^2$ $\sigma^c=2 k\dfrac{1}{2{\text{π}}}$ $ \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 ( ${\mathbb{Z}}^{\rm T}_2$ $\partial\varSigma^3$ $\widetilde{\sigma}^{s}=(1+2 k)\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$ 量子自旋霍尔效应 ( ${\mathbb{Z}}^{\rm T}_2$ $\varSigma^2$ $\sigma^s=2 k\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$ $ \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 ( ${\mathbb{Z}}_2$ $\partial\varSigma^3$ $\widetilde{\sigma}^{cs}=\widetilde{\sigma}^{sc}=\left(\dfrac 1 2+k\right)\dfrac{1}{2{\text{π}}}$ 量子电荷-自旋/ 效应 自旋-电荷 ( ${\mathbb{Z}}_2$ $\varSigma^2$ $\sigma^{cs}=\sigma^{sc}=k\dfrac{1}{2{\text{π}}}$
Table 7.
Generalized Wen-Zee terms.
推广的Wen-Zee拓扑项, 摘自[160].
View tableView in ArticleTable 7.
Generalized Wen-Zee terms.
推广的Wen-Zee拓扑项, 摘自[160].
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 ${\cal{J}}$ $(2 + 1)$ $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$ $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$ $k \in \mathbb{Z}_{N_{01}}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} $ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A^1 \wedge A^2$ $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^2} A^1 \wedge A^2$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A \wedge {\rm d}A$ $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int _{M^3} A \wedge {\rm d}A$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int A \wedge \omega \wedge {\rm d} \omega$ $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{2{\text{π}}^2 N_{01}} \displaystyle\int_{M^3} A \wedge {\rm d}\omega$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times U(1)$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$ $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$ $SO(2)$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int A^1 \wedge A^2 \wedge {\rm d} \omega$ $k \in \mathbb{Z}_{N_{12}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int_{M^3} {\rm d} (A^1 \wedge A^2)$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$ $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int \omega \wedge A^2 \wedge {\rm d}A^1$ $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int_{M^3} A^2 \wedge {\rm d}A^1$ $(3 + 1)$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} \times \mathbb{Z}_{N_3}$ $k \dfrac{N_0 N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int \omega \wedge A^1 \wedge A^2 \wedge A^3$ $k \in \mathbb{Z}_{N_{0123}}$ $k \dfrac{ N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int_{M^3} A^1 \wedge A^2 \wedge A^3$ $(3 + 1)$ $*$ $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$ $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$ $*$ $C_{N_0}$ ${\mathbb{Z}}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$ $k \in \mathbb{Z}_{N_{01}}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$ $*$ $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A \wedge B$ $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^3} A \wedge B$
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Peng Ye.
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Received: Feb. 10, 2020
Accepted: --
Published Online: Nov. 20, 2020
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