We examine the low energy effective theory of phase oscillations in a two-dimensional granular superconducting sheet where the grains are arranged in honeycomb lattice structure. Using the example of graphene we present the evidence for the engineered Dirac nodes in the bosonic excitations: the spectra of the collective bosonic modes cross at the K and K points in the Brillouin zone and form Dirac nodes. We show how two different types of collective phase oscillations are obtained and that they are analogous to the Leggett and the Bogoliubov-Anderson-Gorkov modes in a two-band superconductor. We show that the Dirac node is preserved in the presence of an inter-grain interaction, despite induced changes of the qualitative features of the two collective modes. Finally, breaking the sublattice symmetry by choosing different on-site potentials for the two sublattices leads to a gap opening near the Dirac node, in analogy with Fermionic Dirac materials. Dirac node dispersion of bosonic excitations is thus expanding the discussion of the conventional Dirac cone excitations to the case of bosons. We call this case as a representative of Bosonic Dirac Materials (BDM), similar to the case of Fermionic Dirac materials extensively discussed in the literature.
Valley-resolved high harmonics probe symmetry breaking beyond the Dirac cone
Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学物理学院粒子与核物理研究所）；Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile（智利南方大学物理与数学科学研究所）；Istituto Nazionale di Fisica Nucleare, Bologna branch（意大利国家核物理研究所博洛尼部分所）；Department of Physics, Guangdong Technion-Israel Institute of Technology（广东以色列理工学院物理系）；Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology（广东以色列理工学院广东省能源转换材料与技术重点实验室）
