- The paper presents a method using group cohomology to construct SPT phases in interacting bosonic systems across various dimensions.
- It demonstrates explicit models with exactly solvable Hamiltonians, ground state wavefunctions, and symmetry-protected gapless edge modes in 2D and 3D.
- The findings extend classification beyond free fermions, offering a unified framework that informs quantum material design and further theoretical research.
Symmetry Protected Topological Orders in Interacting Bosonic Systems
The paper by Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen, titled "Symmetry protected topological orders in interacting bosonic systems," tackles a fundamental challenge in understanding topologically ordered phases, particularly in interacting bosonic systems. While symmetry-protected topological (SPT) phases are well understood in free fermion systems, such as topological insulators, the behavior of these phases in interacting systems, especially bosonic ones, has remained unclear until this investigation.
Core Contributions
The central contribution of the paper is the systematic construction of SPT phases in interacting bosonic systems. This construction employs group cohomology theory to extend the classification of SPT phases beyond non-interacting fermionic systems. The authors identify new SPT phases in both two-dimensional (2D) and three-dimensional (3D) systems, which are protected by symmetries such as particle number conservation and time-reversal symmetry. They demonstrate that group cohomology enables the construction of various interacting bosonic SPT phases in any dimension for any symmetry group.
In particular, the paper outlines a method to construct topological terms in the path integral description of a system from nontrivial group cohomology classes. This leads to exactly solvable Hamiltonians, explicit ground state wavefunctions, and symmetry-protected gapless edge excitations. A critical finding is that nontrivial cohomology classes directly correspond to the existence of distinct SPT phases, thereby offering a unified framework for classifying these phases across different systems.
Numerical Results and Implications
The paper reveals that in one dimension, the construction mirrors the known classification results. Importantly, in higher dimensions, they uncover novel SPT phases not previously identified. For example, they find one type of bosonic topological insulator in 2D and three types in 3D, all under the symmetries of boson number conservation and time reversal. This discovery not only expands the landscape of known SPT orders but also provides a robust methodology for exploring unknown phases in similar settings. The paper provides Table I that summarizes the flavors of SPT phases based on certain symmetries, showing structured existence patterns depending on spatial dimensions.
Theoretical and Practical Impact
Theoretically, the paper pushes the boundaries of understanding topological phases by firmly establishing a bridge between group theory and quantum phases. Practically, it offers a toolkit that could be applied to design new quantum materials or quantum computation systems, potentially impacting fields like condensed matter physics and materials science.
Future Directions
While the authors focus on bosonic systems and explore the stability of these phases under symmetric interactions in one and two dimensions, the general applicability of their approach opens avenues for further research in higher-dimensional systems and more complex symmetry groups. The insight provided by group cohomology might lead to discovering even more exotic phases of matter or provide new ways to harness quantum states for technological applications.
In conclusion, this paper contributes significantly to the theoretical understanding of symmetry-protected phases in interacting systems, particularly in the bosonic domain, laying new groundwork for research and potential technological advances in topologically ordered systems.