2D symmetry protected topological orders and their protected gapless edge excitations (1106.4752v2)

Abstract: 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 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 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 $Z_2$ 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 $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.

• The paper introduces a novel 2D spin model with on-site Z2 symmetry that exhibits gapless edge excitations distinguishing nontrivial SPT orders.
• It extends traditional WZW theory to discrete groups using group cohomology and MPUOs, enhancing the classification of topological phases.
• The findings suggest that the identified 3-cocycle structure serves as a phase transition marker, paving the way for new quantum material explorations.

Analysis of 2D Symmetry Protected Topological Orders and Their Gapless Edge Excitations

The paper under consideration explores the field of two-dimensional symmetry-protected topological (SPT) orders as realized in strongly interacting fermionic and bosonic systems, focusing on models that exhibit nontrivial distinctions from trivial states when symmetry is conserved. It extends the understanding of SPT orders beyond the simpler framework of non-interacting, free fermion systems, aiming to characterize the complex interplay of symmetries and topologies in strongly interacting systems.

Key Findings and Contributions

The researchers provide a novel model within a two-dimensional spin system imbued with an on-site $Z_2$ symmetry characterized by non-triviality; this is delineated by the presence of gapless boundary excitations when such symmetry remains unbroken. The work generalizes the result from the well-known Wess-Zumino-Witten (WZW) theory, which applies to continuous symmetry groups, to discrete symmetry groups like $Z_2$. It represents a significant advancement by bringing discrete symmetries into the fold, which have often been challenging to integrate into the theoretical framework of topological orders.

The model is linked intricately to a non-trivial 3-cocycle of the symmetry group, effectively utilizing the formalism of group cohomology to infer potential classes of SPT orders in both boson and fermion systems across any dimensionality. This approach allows the development of a clearer classification of topological phases, aligning with higher-order cocycles of symmetry groups to infer topological distinctions and classifications.

Implications and Theoretical Insights

On a closed manifold, the presented model attains a unique gapped ground state with gapless edge states that emerge when the system is open-ended, conditioned on the preservation of the $Z_2$ symmetry. The researchers leverage matrix product unitary operators (MPUOs) to paper boundary symmetries, articulating a methodology that reveals a deeper 3-cocycle structure in the boundary transformations. This method provides insights into how boundary degrees of freedom in these models can signify a deeper topological ordering unseen in the bulk alone.

Interestingly, the paper suggests that this 3-cocycle relationship presents itself as a potential phase transition marker when considering interactions and boundary conditions, indicating surfaces where symmetry protection could offer robust theoretical and experimental phenomena.

Future Prospects in Research

The concepts introduced here offer promising avenues for future theoretical exploration, notably in the development of new models in higher dimensions where symmetry protection may unveil novel quantum states with exciting experimental possibilities. Moreover, extrapolating these ideas to other discrete symmetry groups and examining their corresponding cocycles could enrich the understanding of potential quantum materials and metastructures.

In essence, this work bridges a key gap in the current understanding of SPT orders in interacting systems, laying groundwork that could impact both prospective research into the nature and cataloging of SPT phases and the exploration of practical, real-world quantum topological materials. As experimental methods evolve, the insights gleaned here could be pivotal in the identification and harnessing of novel quantum states of matter informed by the interplay of topology and symmetry.