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

TL;DR

This paper introduces a modified $k ext{ extperiodcentered}p$ method incorporating $ ext{ extbardbl}Z_2 ext{ extbardbl}$ gauge fields, revealing new topological phases in artificial systems like photonic and acoustic crystals.

Contribution

It develops a gauge-field extended $k ext{ extperiodcentered}p$ approach accounting for $ ext{ extbardbl}Z_2 ext{ extbardbl}$ gauge fields, leading to predictions of novel topological phases in artificial systems.

Findings

Higher-dimensional irreducible representations lead to degenerate Fermi points.

Breaking primitive translations transforms Fermi points into topological phases.

Models demonstrate graphene-like semimetals and second-order nodal-line semimetals.

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.