2021/02/22 by Jintae Kim, Hyun-Yong Lee, Hyun‐Yong Lee +1
Mathematics · Physics and Astronomy · #Bioinformatics #Biology #Combinatorics #Degeneracy (biology) #Discrete mathematics #Geometry #Graph #Hamiltonian (control theory) #Homogeneous space #Mathematical optimization #Mathematical physics #Mathematics #Operator (biology) #Order (exchange) #Physics #Quantum and electron transport phenomena #Quantum many-body systems #Symmetry (geometry) #Theoretical physics #Topological Materials and Phenomena #Topology (electrical circuits) #cond-mat.str-el #quant-ph
paper · pdf · doi:10.1103/physrevb.103.195124
published as Phys. Rev. B 103, 195124 (2021) · 8 pages, 5 figures
arxiv created 2021/02/22 · openalex created_date 2021/03/01 · openalex publication_date 2021/05/12 · arxiv updated 2021/05/19 · openalex updated_date 2026/08/05
We write down and analyze a model demonstrating the co-existence of conventional symmetry-breaking order and symmetry-protected topological (SPT) order in the one-dimensional chain. When appropriately generalized to a model on a graph, the SPT and symmetry-breaking orders exist for each individual loop, or cycle, of the graph. It arises as a consequence of the kind of ``global'' symmetry operator responsible for SPT and the local-order parameter defining the Ginzburg-Landau order, both of which exist in our model and commute with the Hamiltonian. The anti-commuting character of these two-order parameters is responsible for the ground state degeneracy (GSD). As such operators and their anti-commuting relations can be defined for each independent loop, the GSD grows exponentially with the first Betti number for a graph.