Classification of Gapped Topological Phases in 1D Interacting System

Lin Zhang, Long Zhang, Sen Niu +1 more

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or onsite unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also considering the possibility of symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

Emmanuel V. C. Lopes

Chen Yang Wang, Jingping Xu, Ce Wang +1 more

Abstract Floquet engineering provides a unique platform for realizing novel non-equilibrium topological phases. However, the intrinsic multi-gap structure of Floquet systems poses significant challenges for their classification using conventional analytical methods. Here, we propose an unsupervised machine learning framework that employs a kernel defined directly in momentum-time ( k , t ) space, constructed from Floquet–Bloch states. This data-driven approach does not require prior knowledge of topological invariants. By using a compact computational procedure that works directly with the system’s eigenstates, the method can identify topologically distinct phases associated with both the 0 -gap and the π -gap across various symmetry classes (1D AIII, 1D D, and 2D A). Our results demonstrate that key topological characteristics of Floquet systems are directly accessible from their eigenstates in ( k , t ) space, providing a tool for the systematic classification and study of non-equilibrium topological phases without a priori invariant information.

Raditya Weda Bomantara

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.