This paper predicts that twisting two layers of a d‑wave superconductor can produce nearly flat energy bands for the superconductor’s quasiparticles. Those quasiparticles are called Bogoliubov quasiparticles — the particle‑like excitations that carry energy in a superconductor. The authors show that if the superconducting order parameter changes sign under an in‑plane C2 rotation (a 180° turn), twisting plus interlayer coupling can create new band nodes and flatten the quasiparticle bands near the rotation axis.

To reach this result the researchers built and solved a tight‑binding model for a twisted bilayer on a square lattice. They set up the Bogoliubov–de Gennes (BdG) equations for superconductivity, diagonalized the resulting matrices, and also derived a reduced 4×4 effective Hamiltonian to capture the low‑energy physics. Numerical examples used a chemical potential µ = −2.5 t and a tiny pairing scale η± = 10−3 t. The key tuning parameter was the interlayer hopping tz and the relative twist angle between layers.

They found that turning on interlayer hopping creates new nodes in the superconducting gap on the C2 rotation axis (angles labelled φ = π/4 in their plots). As tz grows, the quasiparticle group velocity at those nodes drops and becomes nearly zero around tz ≈ 0.6 t. This is the flattening that the authors call Bogoliubov flat bands. The calculated density of states keeps a V‑shaped minimum (typical of nodal superconductors) but the minimum is lifted by tz, and the zero‑energy density of states peaks near tz ≈ 0.55 t. The paper also argues that a quantity called the Berry connection of the single layer — roughly the way quantum wave functions twist across momentum space — gives a clear criterion for when these Bogoliubov flat bands form.

This work matters because flat bands reduce the kinetic energy of quasiparticles. That can make interaction effects stronger and open the door to new, strongly correlated or topological states in a superconducting setting. The idea extends concepts from moiré and twist engineering in normal metals, such as twisted bilayer graphene, into what the authors call “superconducting twistronics.” In this view, the twist angle and interlayer coupling become design knobs to shape superconducting quasiparticle spectra.