Strained Lieb-Kagome lattices: evolution of the electronic spectrum and topological phase transitions

Abstract

We systematically investigate the effects of simple shear and uniaxial strains, applied along various crystallographic directions, as well as biaxial and pure shear strains, on the electronic spectra of Lieb and Kagome lattices using a tight-binding model. This model employs a general Hamiltonian that characterizes both lattice types through a single control parameter, θ. Our findings indicate that such deformations do not open an energy gap in their electronic spectra but can lead to (i) convergence of energy cones, (ii) anisotropy in energy levels, and (iii) deformation of the flat band. Consequently, the triply degenerate Dirac point in the Lieb lattice transforms into two doubly degenerate Dirac points. Our analysis of hypothetical strain scenarios, in which the hopping parameters are unchanged, shows that effects such as the flat band deformation and the splitting of the triply degenerate Dirac point result solely from strain-induced changes in hopping parameters. Additionally, we identify cases where non-zero strain-induced pseudovector potentials arise in Lieb and Kagome lattices. Moreover, when considering intrinsic spin-orbit coupling, these lattices exhibit twodimensional topological insulator behavior with a Z2 topological classification. Our comprehensive study reveals that such deformations can induce topological phase transitions by altering the structural lattice angle, strain amplitude, and the magnitude of the intrinsic spin-orbit coupling. These transitions are evidenced by the evolution of Berry curvature and shifts in the Chern number when the gap closes. By analyzing hypothetical strain scenarios where the hopping and intrinsic spin-orbit coupling parameters remain intentionally unchanged, we demonstrate that the strain-induced phase transitions stem from simultaneous modifications in the hopping and intrinsic spin-orbit coupling parameters. Further analysis extends to finite-size effects on the topological properties of these lattices, evaluating the energy spectrum for nanoribbons with straight, bearded, and asymmetric edges. The results confirm straindriven topological phase transitions, supported by the bulk-edge correspondence. Additionally, the evolution of edge states under strain suggests the generation of opposite spin currents.