Bogoliubov flat bands in twisted layered materials

This paper proposes a new paradigm of superconducting twistronics by demonstrating that twisting layered $d$-wave superconductors with an odd-symmetry order parameter can engineer Bogoliubov flat bands near the rotation axis, where the twist angle serves as a key tuning parameter for creating gapless flat-band superconductors.

The Gist

Imagine you have two sheets of a very special, super-conductive fabric. In the world of physics, these are called "superconductors," materials that let electricity flow without any resistance. Now, imagine you take one sheet, twist it slightly relative to the other, and stack them on top of each other.

This simple act of twisting creates a complex, repeating pattern between the two layers, kind of like the moiré pattern you see when you overlap two window screens or two combs. Scientists call this a "moiré superlattice."

For a long time, physicists have been obsessed with these twisted stacks (especially with graphene) because the twist can create "flat bands." Think of a flat band like a perfectly flat, endless parking lot for electrons. Usually, electrons zoom around like race cars on a bumpy track (high energy, high speed). But in a flat band, the track is so flat that the electrons lose all their speed and just sit there, packed tightly together. When electrons are forced to sit close together, they start talking to each other intensely, leading to wild, new quantum behaviors like superconductivity or magnetism.

The Big Discovery in This Paper

Until now, scientists mostly looked for these flat parking lots in the "normal" state of materials (before they become superconductors). This paper, however, asks a new question: What happens if we look for these flat parking lots inside the superconductor itself?

The authors, a team from Japan, studied twisted layers of a specific type of superconductor (called a "d-wave" superconductor, which is common in high-temperature superconductors). They discovered something amazing: By twisting the layers just right, they can create "Bogoliubov flat bands."

Here is the breakdown of what they found, using simple analogies:

1. The "Magic Twist"

In the world of twisted graphene, there is a "magic angle" (about 1.1 degrees) where the flat bands appear. In this new study, the researchers found that for superconductors, the "magic" isn't just about the angle; it's about the symmetry of the twist.

They found that if the superconducting properties change sign (flip from positive to negative) when you rotate the material 180 degrees, the twist acts like a tuning knob. By adjusting the twist angle, you can engineer a situation where the energy of the particles becomes perfectly flat.

2. The "Traffic Jam" of Particles

In a normal superconductor, the particles (called Bogoliubov quasiparticles) move at different speeds depending on their direction. It's like a highway with cars speeding up and slowing down.

In this twisted system, the authors found a specific spot (along a specific line in the material's momentum space) where the "traffic" comes to a complete, dead stop. The speed of these particles drops to zero.

• The Analogy: Imagine a river flowing fast. If you build a dam just right, the water in front of it becomes perfectly still and flat. That "still water" is the flat band. The particles aren't moving; they are frozen in a state of high potential energy, ready to interact with each other.

3. The "Berry Connection" Compass

How did they know exactly where to look for this flat band? They used a mathematical tool called the Berry Connection.

• The Analogy: Imagine the electrons are hikers walking on a mountain. The "Berry Connection" is like a compass that tells the hikers which way the wind is blowing. The authors discovered that if the "wind" (the mathematical phase of the wave) blows in a specific direction relative to the twist axis, the hikers get stuck in a valley where the ground is perfectly flat. They used this compass to predict exactly where the flat band would form without needing to simulate the whole mountain.

4. Why Does This Matter?

This is a new paradigm called "Superconducting Twistronics."

• Before: We twisted materials to find flat bands in normal metals.
• Now: We can twist superconductors to create flat bands inside the superconducting state.

This is a big deal because flat bands are the "fertile soil" for discovering new quantum phases. If you can create a superconductor where the particles are stuck in a flat band, you might be able to:

• Create Majorana particles (particles that are their own antiparticles), which are the holy grail for building stable quantum computers.
• Understand how high-temperature superconductivity works better.
• Design materials where you can turn "gapless" superconductivity (superconductivity that doesn't have an energy gap) on and off just by twisting the layers.

The Bottom Line

The authors have shown that twisting two layers of a specific superconductor is like turning a dial on a radio. By finding the right "frequency" (twist angle), you can tune the material so that its internal particles stop moving and form a flat, calm sea. This calm sea is the perfect place for new, strange, and useful quantum phenomena to emerge. It opens the door to a new era of engineering quantum materials simply by twisting them.

Technical Summary

1. Problem Statement

Flat electronic bands have become a central platform for exploring strongly correlated and topological quantum phases, most notably in twisted bilayer graphene (TBG) and transition metal dichalcogenides (TMDs). However, existing research has primarily focused on flat bands within the normal-state electronic structures.

The authors address a significant gap in the literature: the possibility of realizing flat bands directly within the superconducting Bogoliubov quasiparticle spectrum. Specifically, they investigate whether the "twistronics" paradigm—where a small twist angle reconstructs the band structure—can be applied to nodal $d$-wave superconductors (analogous to cuprates) to engineer "Bogoliubov flat bands." They aim to elucidate the energy dispersion, the mechanism of formation, and the conditions required for these gapless flat bands to emerge.

2. Methodology

The authors employ a combination of tight-binding modeling, numerical diagonalization, and low-energy effective Hamiltonian analysis:

• Model System: They consider a twisted bilayer system composed of two-dimensional square lattices representing nodal $d$-wave superconductors. The layers are rotated by angles $\pm \theta/2$ relative to the $z$-axis.
• Hamiltonian Formulation:
  • A tight-binding model is constructed with nearest-neighbor in-plane hopping ($t$) and interlayer hopping ($t_z$) between vertically aligned atoms.
  • The system is analyzed using the Bogoliubov-de Gennes (BdG) formalism. The BdG Hamiltonian includes the normal-state Hamiltonian ($\hat{H}_n$) and the pair potential ($\hat{\Delta}$), assuming in-plane $d$-wave pairing.
  • The authors utilize a Partially Diagonalized (PD) basis to simplify the Hamiltonian, transforming the interlayer coupling into a uniform term ($-\tilde{t}_z$).
• Effective Hamiltonian: To understand the mechanism, they derive a $4 \times 4$ effective low-energy Hamiltonian. This model focuses on the subspace of bands relevant to low-energy quasiparticle excitations near the Fermi level, specifically analyzing the degeneracy along the $C_2$ rotation axis ($k_x = \pm k_y$).
• Analysis: They numerically solve the BdG equation to calculate quasiparticle excitation energies, Fermi surfaces, and the density of states (DOS). They also analytically derive the group velocity near nodal points to identify conditions for band flattening.

3. Key Contributions

• Discovery of Bogoliubov Flat Bands: The paper demonstrates that twisting nodal $d$-wave superconductors can induce the flattening of quasiparticle bands (Bogoliubov flat bands) near the rotation axis, distinct from the flat bands found in normal-state graphene.
• Role of Interlayer Hopping ($t_z$): The authors identify that interlayer hopping is the critical tuning parameter. While $t_z=0$ yields standard $d$-wave nodes, introducing finite $t_z$ creates new nodal points on the $C_2$ rotation axis and suppresses the energy gap between them.
• Berry Connection Criterion: A major theoretical contribution is the derivation of a clear criterion for flat band formation based on the Berry connection of the single-layer system. They show that the group velocity vanishes when the Berry connection vector at the nodal point is parallel to the $C_2$ rotation axis.
• Twist Angle as a Tuning Knob: The work establishes that the twist angle acts as a powerful control parameter to align the Fermi surface with the specific momentum points where the Berry connection satisfies the flat-band condition.

4. Key Results

• Emergence of Nodes: Numerical results show that as interlayer hopping ($t_z$) increases, new nodal points appear at $\phi = \pi/4$ (along the $C_2$ axis). When $|t_z| \gtrsim 0.6|\eta|$ (where $\eta$ is the pairing amplitude), these nodes are created.
• Group Velocity Suppression: The group velocity of the quasiparticles along the Fermi surface tangential direction becomes zero at a critical $t_z$ (approx. $0.6t$). This results in a "flat" dispersion relation, analogous to the magic-angle flat bands in TBG.
• Density of States (DOS): The presence of these flat bands significantly alters the low-energy DOS. While the single-layer system exhibits a V-shaped DOS (characteristic of $d$-wave nodes), the twisted system with optimal $t_z$ shows a lifted V-shape with a maximum at zero energy ($E=0$), indicating a high density of states due to the flat band.
• Analytical Mechanism: Using the $4 \times 4$ effective Hamiltonian, the authors prove that the group velocity $v_{kn}$ is proportional to the gradient of the phase $\phi(k)$ of the complex quantity $\varepsilon(k) + i\Delta(k)$.
  • The condition for zero velocity ($v_{kn}=0$) is met when the gradient of this phase (which relates to the Berry connection $\mathbf{A}(k)$) is parallel to the rotation axis.
  • Since the Berry connection rotates by $-2\pi$ around the nodal point, there exists a specific line in momentum space where the connection is parallel to the axis. Tuning $t_z$ moves the Fermi surface to intersect this line, creating the flat band.

5. Significance and Future Outlook

• New Paradigm of Superconducting Twistronics: This work establishes "superconducting twistronics" as a viable field, demonstrating that twist angles can be used to design gapless flat-band superconductors. This opens pathways for engineering novel quantum phases in twisted superconducting heterostructures.
• Topological and Correlated Physics: The existence of Bogoliubov flat bands suggests enhanced interaction effects. The authors propose future research directions, including:
  • Investigating the relationship between these flat bands and the quantum metric (which dominates superfluid weight in flat band systems).
  • Studying the emergence of odd-frequency pairing, which is theoretically linked to gapless superconducting states.
  • Exploring the impact of these flat bands on zero-energy edge states (Andreev bound states) in $d$-wave superconductors.

In conclusion, the paper provides a rigorous theoretical framework showing that twisting layered $d$-wave superconductors can engineer Bogoliubov flat bands, offering a new route to explore strongly correlated superconductivity and topological phenomena in twisted quantum materials.