Peng Ye. Gauge theory of strongly-correlated symmetric topological Phases [J]. Acta Physica Sinica, 2020, 69(7): 077102-1
- Acta Physica Sinica
- Vol. 69, Issue 7, 077102-1 (2020)
Fig. 1. Monte Carlo verification of vanishing topological entanglement entropy of the SPT wave function obtained from the projective construction
Fig. 2. Parton decomposition of electron operators
![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]](/Images/icon/loading.gif){kind=icon}
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. (26 )
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)
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. 7. Illustration of SEG
Fig. 8. (a). Topological response for Eq. (42 ). The intersection of
and
symmetry domain walls
and
carries the angular momentum
.
and
are the gauge connections normal to the domain walls. (b). Topological response of Eq. (44 ). The intersection of disclination line and
symmetry domain walls
carries the
charge
.
and
are the gauge connections normal to the disclination line and domain wall, respectively
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
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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).
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).
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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.
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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].
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].
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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].
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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].
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