The gauge-field extended $k\cdot p$ method and novel topological phases

Abstract

Although topological artificial systems, like acoustic/photonic crystals and cold atoms in optical lattices were initially motivated by simulating topological phases of electronic systems, they have their own unique features such as the spinless time-reversal symmetry and tunable ${\mathbb{Z}}_2$ gauge fields. Hence, it is fundamentally important to explore new topological phases based on their unique features. Here, we point out that the ${\mathbb{Z}}_2$ gauge field leads to two fundamental modifications of the conventional $k\cdot p$ method: (i) The little co-group must include the translations with nontrivial algebraic relations; (ii) The algebraic relations of the little co-group are projectively represented. These give rise to higher-dimensional irreducible representations and therefore highly degenerate Fermi points. Breaking the primitive translations can transform the Fermi points to interesting topological phases. We demonstrate our theory by two models: a rectangular $\pi$-flux model exhibiting graphene-like semimetal phases, and a graphite model with interlayer $\pi$ flux that realizes the real second-order nodal-line semimetal phase with hinge helical modes. Their physical realizations with a general bright-dark mechanism are discussed. Our finding opens a new direction to explore novel topological phases unique to artificial systems and establishes the approach to analyze these phases.

arXiv comment18 pages, 10 figures, to be published in Physical Review Letters

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