2, Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois, United States
3, School of Physics, Georgia Institute of Technology, Atlanta, Georgia, United States
The Berry phase is a geometrical phase that is known to provide the essential physics behind several intriguing materials properties such as the electric polarization, anomalous Hall effect, orbital magnetization, etc. It is expected that this geometrical phase also reflects the intrinsic topological properties of one-dimensional (1D) insulators, because the 1D Brillouin zone (BZ) integral forms a natural loop in the k-space. In this work, we study the system of gapped graphene nanoribbons (GNRs) with spatial symmetries (e.g., inversion) and show that a symmetry-protected Z2 topological phase exists. Although the Berry phase turns out to be -quantized in the presence of the chiral symmetry, it does not provide the correct Z2 correspondence as expected. It is found that only the origin-independent part of the Berry phase gives the correct bulk-boundary correspondence by its -quantized values, with the relevant Z2 invariant dependent on the choice of the bulk unit cell (namely, ribbon truncation) and connected to the symmetry eigenvalues of the wave functions at the center and boundary of the BZ. Using the cove-edged GRNs as examples, we demonstrate the existence of localized states at the end of some GNR segments and at the junction between two GNRs based on topological analysis. The current results are expected to shed light on the design of electronic devices based on GNRs as well as the understanding of the topological features in 1D systems.