d-Wave pair density wave superconductivity in a two-orbital model

This paper investigates two-orbital models on a square lattice and demonstrates that interband pairing drives the emergence of incommensurate $d$-wave pair density wave superconductivity, which competes with uniform pairing and density wave orders, particularly in correlated multi-orbital materials and specific lattice geometries.

The Gist

Imagine you are running a busy dance floor in a crowded club. Usually, in a superconductor (a material that conducts electricity with zero resistance), the "dancers" (electrons) pair up and waltz across the floor in perfect sync, moving in the same direction with zero momentum. This is the standard "uniform" superconductivity.

But what if the dancers decided to pair up, but instead of moving in a straight line, they started doing a synchronized, rhythmic shuffle that creates a pattern of waves across the floor? Some pairs move left, some move right, creating a "Pair Density Wave" (PDW). This is a more exotic, wavy form of superconductivity.

This paper by Samuel Vadnais and Arun Paramekanti explores how to get electrons to do this wavy dance, specifically in materials where electrons have two different "outfits" or "roles" (called orbitals) they can wear.

Here is the breakdown of their discovery using simple analogies:

1. The Two-Orbital Dance Floor

Most simple models of superconductors imagine electrons as having only one type of shoe. But in complex materials (like certain high-temperature superconductors or cold atom experiments), electrons can wear two different types of shoes: let's call them Red Shoes ($p_x$) and Blue Shoes ($p_y$).

The authors set up a "square lattice" dance floor where every dancer can wear either Red or Blue shoes. Crucially, the dance floor has a special rule: the Red and Blue shoes interact with each other. If a Red dancer is next to a Blue dancer, they feel a magnetic "tug" or a repulsive "push."

2. The "Traffic Jam" of Electrons

The researchers looked at how the electrons move when the dance floor is only partially full (low density) versus when it's crowded.

• The Low-Density Crowd (The "Highway" Effect): When there are few electrons, the Red and Blue dancers tend to stay in their own lanes. The Red dancers like to move East-West, and the Blue dancers like to move North-South. Because they are so different, they don't want to pair up and move in the same direction (zero momentum).
• The Solution: Instead, a Red dancer pairs with a Blue dancer, but they agree to move in opposite directions to balance each other out. This creates a wave. It's like a couple on a dance floor where one partner steps forward while the other steps back, creating a rhythmic oscillation across the room. This is the Pair Density Wave (PDW).

3. The "Tug-of-War" (Competition)

The paper studies a competition between three different "dance styles":

1. The Wavy PDW: Electrons pair up but move in a wave pattern (the main discovery).
2. The Uniform Waltz: Electrons pair up and move together in a straight line (standard superconductivity).
3. The Magnetic Stare: Electrons stop dancing and just stare at each other, creating a magnetic pattern (magnetism).

The Finding:

• At low density: The "Wavy PDW" wins! The unique way the Red and Blue lanes interact forces the electrons into this exotic wave pattern.
• At medium density: The "Uniform Waltz" takes over. The lanes mix enough that the electrons can just dance together normally.
• At high density: The "Magnetic Stare" wins. The electrons get too crowded and stop dancing entirely, becoming magnetic.

4. The "Strong Grip" (Strong Coupling)

The authors also looked at what happens when the "tug-of-war" between the Red and Blue dancers is extremely strong (Strong Coupling).

Imagine the dancers are holding hands so tightly that they can't move individually. They become a single unit (a "Cooper pair").

• When they are forced to move as a unit, the math shows they naturally prefer to hop in a specific pattern: Left-Right, Left-Right.
• This creates a perfect, repeating pattern every two steps (a "checkerboard" or period-2 wave).
• The authors used a clever mathematical shortcut (Gutzwiller ansatz) to prove that in this "strong grip" scenario, the PDW is the most stable state for a wide range of conditions.

Why Does This Matter?

You might ask, "Who cares about wavy electron dances?"

1. New Materials: This helps explain weird behaviors in real-world materials like iron-based superconductors, nickelates, and twisted graphene (Moiré materials), where scientists have seen hints of these waves but didn't know why.
2. Cold Atoms: Scientists can now build "artificial crystals" using lasers and cold atoms. This paper gives them a blueprint: "If you set up your atoms with two types of orbitals and tune the interactions just right, you can force them to create this exotic PDW state."
3. The Future of Tech: Understanding how to stabilize these exotic states might help us design better superconductors that work at higher temperatures, potentially revolutionizing power grids and electronics.

The Bottom Line

The paper shows that by giving electrons two different roles (orbitals) and letting them interact, you can naturally force them into a wavy, rhythmic superconducting state (PDW) instead of the boring, straight-line state. It's like realizing that if you put two different types of dancers on a floor, they don't just walk in a line; they create a beautiful, complex wave pattern that is actually more stable than walking in a straight line.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the emergence of Pair Density Wave (PDW) superconductivity in multi-orbital systems. While PDW states (where the superconducting order parameter has finite momentum, $Q \neq 0$) have been proposed in various contexts, their stabilization in correlated multi-orbital materials remains a key theoretical challenge.

The authors focus on a two-orbital model on a square lattice, representing either $(p_x, p_y)$ or $(d_{xz}, d_{yz})$ orbitals. The central questions addressed are:

• Can inter-orbital interactions in a multi-band system drive a transition from uniform superconductivity to a finite-momentum PDW state?
• How does the Fermi surface topology and orbital character influence the momentum of the Cooper pairs?
• What is the nature of the competition between PDW, uniform $d$-wave superconductivity (SC), and magnetic/charge density wave (CDW) orders?

2. Model and Methodology

The Model

The authors define a Hamiltonian $H = H_t + H_{int}$ on a square lattice with two orbitals ($X, Y$) per site:

• Kinetic Term ($H_t$): Includes intra-orbital hopping ($t_\parallel, t_\perp$) and inter-orbital hopping ($t_L$). The dispersion relations $\epsilon^X_k$ and $\epsilon^Y_k$ exhibit quasi-one-dimensional character, with mixing occurring primarily along the $\Gamma-M$ direction.
• Interaction Term ($H_{int}$): Includes on-site inter-orbital spin exchange ($J \vec{S}^X_i \cdot \vec{S}^Y_i$) and density-density repulsion ($V n^X_i n^Y_i$).
• Symmetry: The model possesses $C_4$ rotation symmetry (interchanging $X$ and $Y$ with a sign change) and time-reversal symmetry.

Methodological Approaches

The study employs a multi-scale theoretical approach:

1. Random Phase Approximation (RPA): Used in the weak-coupling regime to calculate orbital-resolved magnetic, charge, and pairing susceptibilities. This identifies the leading instability wavevectors ($Q$) and the nature of the order parameter.
2. Mean-Field Theory (MFT):
   • Momentum Space: Assumes a Fulde-Ferrell (FF) type PDW where pairs carry momentum $Q$. The authors solve the self-consistent Bogoliubov-de Gennes (BdG) equations to find the ground state energy for various $Q$.
   • Real Space: Uses a Bogoliubov-de Gennes approach to check for Larkin-Ovchinnikov (LO) type modulations (density waves) and to probe commensurate unit cells (e.g., $2\times2$, $4\times4$).
3. Strong Coupling Expansion: For large $J/t$, the system is mapped onto an effective Hamiltonian of hard-core bosons (Cooper pairs).
4. Gutzwiller Ansatz: Applied to the effective bosonic Hamiltonian to study the ground state phases, including Mott insulators and CDW states, beyond the non-interacting pair limit.

3. Key Contributions and Results

A. Weak Coupling Regime (RPA and MFT)

• Low Filling ($n < n_{vH} \approx 0.105$): The Fermi surfaces consist of nearly pure $p_x$ and $p_y$ character with minimal mixing. The dominant instability is an incommensurate $d_{xy}$ Pair Density Wave (PDW).
  • The pairing is driven by inter-band pairing between $X$ and $Y$ orbitals.
  • The optimal momentum $Q$ shifts continuously from $(\pi, \pi)$ at very low density to $(0,0)$ as filling approaches the van Hove singularity.
  • The order parameter transforms as $d_{xy}$ (changes sign under $C_4$ rotation), making it a $d$-wave PDW.
• Intermediate Filling ($n_{vH} < n \lesssim 0.45$): The Fermi surface reconstructs with strong $p_x/p_y$ hybridization. The system favors uniform $d_{xy}$ superconductivity ($Q=0$).
  • The uniform state has point nodes (gapless excitations) at weak coupling but opens a full gap at strong coupling.
• High Filling ($n \gtrsim 0.45$): The system becomes unstable towards incommensurate magnetic order (spin density wave) with $Q \approx (\pi, \pi)$.
• Role of Interactions: While $J$ drives magnetic order, the inter-orbital density interaction $V$ suppresses magnetism but enhances pairing in the orbital-triplet spin-singlet (OTSS) channel.

B. Strong Coupling Regime ($J/t \gg 1$)

• Effective Hamiltonian: The system is described by hard-core bosons (inter-orbital singlet pairs) with effective hopping terms derived from second-order perturbation theory.
• $Q=(\pi, \pi)$ PDW: The effective Cooper pair hopping amplitude has a "wrong sign" (negative Josephson coupling) due to the opposite signs of $X$ and $Y$ hopping on nearest neighbors. This energetically favors a commensurate period-2 PDW with $Q=(\pi, \pi)$ over a wide range of fillings.
• Phase Diagram:
  • Quarter-filling ($n=0.25$): The system forms an insulating checkerboard Charge Density Wave (CDW).
  • Half-filling ($n=0.5$): The system forms a trivial Mott insulator with a gap at every site.
  • General Filling: The ground state is predominantly the commensurate $(\pi, \pi)$ PDW.

C. Excitation Spectra

• Uniform SC: Exhibits $d$-wave point nodes where the gap function $\sin(k_x)\sin(k_y)$ vanishes.
• $(\pi, \pi)$ PDW: The quasiparticle spectrum shows closed contours of gapless excitations, resembling a Bogoliubov Fermi surface with multiple pockets, distinct from the point nodes of the uniform state.

4. Significance and Implications

1. Mechanism for PDW: The paper establishes that orbital content and multi-band Fermi surface topology are sufficient to stabilize PDW states via purely local interactions, without needing long-range Coulomb forces or phonon mediation.
2. Material Relevance: The results are directly applicable to:
   • Correlated multi-orbital materials with quasi-1D bands.
   • Square-octagon lattice Hubbard models.
   • Cold atom systems where $p$-orbital physics can be engineered.
   • Moiré materials and iron-based/nickel-based superconductors where multi-orbital physics is prominent.
3. Theoretical Insight: It highlights the competition between uniform $d$-wave SC and PDW, showing that the transition is controlled by filling and interaction strength. The strong-coupling derivation provides a robust analytical explanation for the emergence of the $(\pi, \pi)$ PDW, a state often sought in high-$T_c$ cuprates and twisted bilayer graphene.
4. Experimental Prediction: The study predicts specific signatures, such as the shift from incommensurate to commensurate PDW with increasing interaction strength, and the existence of Bogoliubov Fermi surfaces in the PDW phase, which can be probed via ARPES or STM.

In conclusion, Vadnais and Paramekanti demonstrate that multi-orbital models naturally host robust $d$-wave PDW states, driven by inter-band pairing and stabilized by strong correlations, offering a new perspective on the physics of unconventional superconductivity.