# Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

@article{Chen2011TwodimensionalST, title={Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations}, author={Xie Chen and Zhengxin Liu and Xiao-Gang Wen}, journal={Physical Review B}, year={2011}, volume={84}, pages={235141} }

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z{sub 2} symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry…

## 202 Citations

### Symmetry-Protected Topological Orders in Interacting Bosonic Systems

- PhysicsScience
- 2012

Just as group theory allows us to construct 230 crystal structures in three-dimensional space, group cohomology theory is used to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.

### Higher-order symmetry-protected topological states for interacting bosons and fermions

- PhysicsPhysical Review B
- 2018

Higher-order topological insulators have a modified bulk-boundary correspondence compared to other topological phases: instead of gapless edge or surface states, they have gapped edges and surfaces,…

### Lattice models that realize Zn -1 symmetry-protected topological states for even n

- Mathematics, PhysicsPhysical Review B
- 2020

Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high-energy scales while other topological excitations have…

### Symmetry protected topological orders and the group cohomology of their symmetry group

- Physics
- 2013

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the…

### Symmetry-Protected Topological Phases

- PhysicsQuantum Information Meets Quantum Matter
- 2019

Short-range entangled states can all be connected to each other through local unitary transformations and hence belong to the same phase. However, if certain symmetry is required, they break into…

### Quantum phase transitions between a class of symmetry protected topological states

- Physics, Mathematics
- 2015

### Topological superconductivity by doping symmetry-protected topological states

- Physics
- 2020

We propose an exotic scenario that topological superconductivity can emerge by doping strongly interacting fermionic systems whose spin degrees of freedom form bosonic symmetry protected topological…

### Symmetry-protected topological phases in spinful bosons with a flat band

- Physics
- 2020

We theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on lattices which have a flat band at the bottom…

### Gapless Symmetry-Protected Topological Order

- Physics
- 2017

We introduce exactly solvable gapless quantum systems in $d$ dimensions that support symmetry protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points…

### Subsystem symmetry enriched topological order in three dimensions

- Mathematics
- 2020

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as…

## References

SHOWING 1-3 OF 3 REFERENCES

### this is a reasonable restriction because we want to study systems at fixed point where the correlation length in the bulk is zero

### Note2, the (d + 1)-cohomology group of Z2 H d+1 (G, U (1)) is trivial for odd d and is a Z2 group for even d

### But this is not a problem as we can redefinẽ T (CZX) = iT (CZX) and the extra minus sign

- Note3, the mapping actually reduces T (CZX, CZX) to −T (I)