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

By Xie Chen, Zhengxin Liu, X. Wen. 2011

288 citations recorded by the source corpus.

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 is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z{sub 2} group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z{sub 2} symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are usedmore » to construct a 2D fermionic model with onsite Z{sub 2} symmetry-protected topological order.« less

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6207963ff4f6e9bd49d3e1a4df40a9f06f4992df

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Citations in the corpus

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions (2017, methodology): “Soon after their classification in 1+1D, bosonic SPT states in higher dimensions were also classified based on group cohomology Hd+1(G,U(1)) and Hd+1(G× SO(∞), U(1)) [14–17], 1 or based on cobordism theory [18–20].”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “Third, MPOs play a fundamental role in describing symmetries of PEPS (Bultinck et al., 2017; Chen et al., 2011c; Şahinoğlu et al., 2014).”
Anyons and matrix product operator algebras (2015, background): “This connection between MPO group representations and three cocycles was first established in [54].”
Topological Phases Protected by Point Group Symmetry (2016, methodology): “[56], the edge is a 1D chain of Ising spins located at sites labeled by the integer j, with Pauli spin operators τj, τ x j .”
Gapless Symmetry-Protected Topological Order (2017, background): “[25] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen, “Twodimensional symmetry-protected topological orders and their protected gapless edge excitations,” Phys.”
Pre-thermal phases of matter protected by time-translation symmetry (2016, background): “[9] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen, “Twodimensional symmetry-protected topological orders and their protected gapless edge excitations,” Phys.”