Second order topology in a band engineered Chern insulator

Abstract

Haldane model is a celebrated tight binding toy model of a Chern insulator in a 2D honeycomb lattice that exhibits quantized Hall conductance in the absence of an external magnetic field. In our work, we deform the bands of the Haldane model smoothly by varying one of its three nearest neighbour hopping amplitudes (t1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_1$$\\end{document}), while keeping the other two (t) fixed. This breaks the C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} symmetry of the Hamiltonian, while the Mx∗T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_x*T$$\\end{document} symmetry is preserved. The symmetry breaking causes the Dirac cones to shift from the K and the K′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$'$$\\end{document} points in the Brillouin zone (BZ) to an intermediate M point. This is evident from the Berry curvature plots which show a similar shift in the corresponding values as a function of the deformation parameter, namely t1t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{t_1}{t}$$\\end{document}. We observe two different topological phases of which, one is a topological insulator (TI) and the other is a second order topological insulator (SOTI). The Chern number (C) remains perfectly quantized at a value of C=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=1$$\\end{document} for the TI phase and it goes to zero in the SOTI phase. Furthermore, the evolution of the Wannier charge center (WCC) as the band is smoothly deformed shows a jump in the TI phase indicating the presence of conducting edge modes. We also study the SOTI phase and diagonalize the real space Hamiltonian on a rhombic supercell which shows the presence of in-gap zero energy corner modes. The polarization of the system, namely px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document}, are evaluated, along the x and the y directions, respectively. We see that both px\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_x$$\\end{document} and py\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_y$$\\end{document} are quantized in the SOTI phase owing to the presence of the inversion symmetry of the system. Finally we establish the SOTI phase as an example of a topological phase with zero Berry curvature and provide an analogy with the two dimensional Su–Schrieffer–Heeger model.