Genuine pair density wave order on the kagome lattice

Using state-of-the-art functional renormalization group studies, this paper reports the discovery of a genuine primary pair density wave (PDW) phase in a two-orbital kagome Hubbard model, driven by sublattice and orbital polarized Bloch states that suppress zero-momentum pairing in favor of finite-momentum Cooper pairs, with potential realization in materials like CsCr$_3$Sb$_5$ and cold atom systems.

The Gist

The Big Idea: A New Kind of Superconductor

Imagine a superconductor as a massive dance floor where electrons (the dancers) pair up and move in perfect unison. Usually, these pairs move together in a straight line, like a marching band. This is standard superconductivity.

But what if the dancers didn't just march forward? What if they formed a pattern that rippled across the floor, getting stronger and weaker in a wave-like motion? This is called a Pair Density Wave (PDW).

For a long time, scientists thought PDWs were rare or only happened when you forced them with strong magnets. This paper says: "No! We found a way to make a 'pure' PDW happen naturally, without any magnets, just by arranging the dance floor and the dancers in a very specific way."

The Setting: The Kagome Lattice (The Dance Floor)

The researchers used a specific crystal structure called a kagome lattice.

• The Analogy: Imagine a floor tiled with triangles, where every triangle shares corners with others, looking like a woven basket or a starry sky.
• The Twist: On this floor, there are two types of "dance moves" (orbitals) available to the electrons. The researchers found that if you tune the energy just right, the electrons get confused about where to stand. They end up preferring specific corners of the triangles (sublattices) and specific dance moves.

The Problem: Why is this so hard?

Usually, electrons want to pair up and stand still (zero momentum). It's like two people holding hands and standing in the center of the room. It's the easiest, most stable thing to do.

To get a PDW, you have to force them to pair up while moving in a specific direction (non-zero momentum).

• The Challenge: It's like trying to get two people to hold hands and run in a circle instead of standing still. Nature usually resists this. Usually, if you try to make them run, they just break up or form a different kind of order (like a static pattern of charge).

The Solution: The "Traffic Jam" Trick

The authors discovered a clever trick using the unique geometry of the kagome lattice. Here is how they forced the electrons to dance in a wave:

1. The "Sublattice" Rule: The electrons are picky. They only want to hold hands with partners standing on specific corners of the triangles.
2. The "Zero-Momentum" Blockade: Because of this pickiness, if two electrons try to stand still and hold hands (the usual superconducting move), they are blocked by a strong electrical repulsion (like a bouncer kicking them out).
3. The "Wave" Escape: However, if the electrons pair up while moving in a specific direction (creating a wave), they can find partners on different parts of the floor that do match their picky requirements.
4. The Result: The "standing still" option is blocked, so the electrons are forced to choose the "moving wave" option. This creates a Genuine Pair Density Wave.

The Discovery: A Chiral Twist

The paper found that this wave doesn't just happen in one direction. It happens in three directions simultaneously (like a three-way intersection).

• The Analogy: Imagine three waves crashing into each other.
• The Magic: When these three waves combine, they create a Chiral PDW. "Chiral" means they have a "handedness"—they swirl like a corkscrew or a spiral galaxy. This swirling state breaks time-reversal symmetry, meaning if you played the movie backward, the physics would look different. It's a topological state, which is a fancy way of saying it's very robust and has special properties useful for future quantum computers.

Why Does This Matter?

1. It's Real: Before this, PDWs were mostly theoretical ideas or "secondary" effects (ripples caused by something else). This paper shows a "primary" PDW, where the wave is the main event, not just a side effect.
2. Real Materials: The researchers suggest this isn't just math. It could happen in real materials like CsCr₃Sb₅ (a chromium-based metal) or in cold atom experiments in labs.
3. New Physics: It proves that by mixing different types of electron orbitals and using specific lattice shapes, we can force nature into exotic states that were previously thought impossible.

Summary in One Sentence

By arranging electrons on a triangular (kagome) grid and using their natural "picky" preferences to block the usual way of superconducting, the researchers forced the electrons to form a swirling, wave-like superconducting state that exists naturally without magnets.

Technical Summary

1. Problem Statement and Motivation

The Pair Density Wave (PDW) is a superconducting state where Cooper pairs condense with a non-zero center-of-mass momentum (CMM), leading to a spatially oscillating superconducting order parameter. While the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state describes PDW induced by external magnetic fields, a genuine (primary) PDW is a spontaneous state emerging in the absence of external fields.

• The Challenge: Realizing a primary PDW as the ground state in microscopic models is extremely difficult.
  1. Competition with Uniform SC: Standard zero-CMM Cooper pairing (time-reversed states) typically has a divergent susceptibility, making it energetically favored over finite-CMM pairing.
  2. Secondary vs. Primary: Many observed PDW-like modulations in materials (cuprates, kagome metals) are "secondary," meaning they are induced by a pre-existing Charge Density Wave (CDW) or Spin Density Wave (SDW). A genuine primary PDW must emerge independently, without being a mere modulation of a pre-existing density wave.
• Goal: To identify a realistic microscopic model and mechanism where the primary PDW is the ground state, overcoming the dominance of uniform superconductivity and density wave orders.

2. Methodology

The authors employ a two-orbital Hubbard model on a kagome lattice and analyze it using the Singular-Mode Functional Renormalization Group (SM-FRG).

• The Model:

  • Lattice: Kagome lattice with three sublattices.
  • Orbitals: Two orbitals per site ($x$ and $y$), transforming under planar rotations. The $x$-orbital points toward the triangle center, and the $y$-orbital is orthogonal.
  • Hamiltonian: Includes tight-binding hopping (Slater-Koster rules) and electron-electron interactions.
  • Interactions: Standard multi-orbital Hubbard interactions including onsite Coulomb repulsion ($U$), inter-orbital repulsion ($U'$), Hund's coupling ($J_H$), and pair hopping ($J_P$).
  • Parameters: Tuned to mimic realistic materials like CsCr$_3$Sb$_5$, featuring specific Fermi surface topologies (hole-like pockets at $\Gamma$ and electron-like pockets at the $M$ points).

• The Method (SM-FRG):

  • An unbiased approach that treats the Superconducting (SC), Spin Density Wave (SDW), and Charge Density Wave (CDW) channels on equal footing.
  • It tracks the flow of effective interactions ($V_{eff}$) as the infrared energy cutoff $\Lambda$ decreases.
  • The first channel to diverge indicates the leading instability. The eigenmode of the divergent interaction reveals the structure of the emerging order parameter.
  • Advantage: SM-FRG exactly preserves orbital and sublattice polarization of Bloch states, which is crucial for capturing the specific interference effects in this model.

3. Key Contributions and Mechanisms

The paper identifies specific physical ingredients that allow the primary PDW to win over competing orders:

1. Sublattice and Orbital Selectivity:

   • The Fermi surfaces exhibit strong sublattice and orbital polarization.
   • Zero-Momentum Suppression: Time-reversed pairs ($|k\uparrow\rangle, |-k\downarrow\rangle$) on the same Fermi pocket occupy predominantly the same sublattice. Due to strong onsite Coulomb repulsion ($U$), onsite pairing is suppressed.
   • Finite-Momentum Promotion: Pairing between electrons on different sublattices (inter-pocket pairing) is favored. This naturally leads to Cooper pairs with non-zero total momentum ($Q \approx M_{1,2,3}$), as the momentum transfer connects different pockets on the Brillouin zone boundary.

2. Suppression of Competing Density Waves:

   • Particle-Hole (PH) Channel: While there is some quasi-nesting between the hole-like $\Gamma$ pocket and electron-like $M$ pockets, the sublattice selectivity significantly weakens the scattering amplitude in the PH channel.
   • Result: The susceptibility in the PH channel (driving CDW/SDW) is weaker than the Particle-Particle (PP) channel (driving SC) at the relevant momenta, overcoming the second challenge of realizing a primary PDW.

3. Intertwined Orders:

   • The PDW order is not isolated; it is strongly intertwined with subleading spin and charge correlations.
   • The model predicts a chiral PDW state formed by the linear combination of three degenerate PDW components at momenta $M_1, M_2, M_3$. This state breaks time-reversal symmetry and supports loop currents.

4. Key Results

• Ground State Identification:

  • For a wide range of interaction parameters (moderate $U$ and $J_H$), the SC channel diverges first, indicating a primary PDW ground state.
  • The PDW order parameter has three degenerate components at the $M$ points of the Brillouin zone boundary ($Q_{PDW} = M_{1,2,3}$).

• Order Parameter Structure:

  • The PDW consists of spin-singlet pairing on bonds connecting different sublattices.
  • The order parameter oscillates spatially with a $2 \times 2$ periodicity relative to the unit cell.
  • Chiral State: The three components can combine with a relative phase of $120^\circ$ (non-collinear), breaking time-reversal symmetry and creating a topologically non-trivial chiral PDW.

• Spectral Properties:

  • Gap Structure: The quasiparticle gap is highly anisotropic. The hole-like $\Gamma$ pocket is only weakly gapped (due to lack of quasi-nesting with $M$ pockets), while the $M$ pockets are strongly gapped.
  • Residual Fermi Surface: The weakly gapped regions result in "residual" Fermi arcs, a signature consistent with recent experimental observations in kagome superconductors (e.g., CsV$_3$Sb$_5$).
  • Density of States (DOS): The low-energy DOS is only partially suppressed, featuring spikes due to the anisotropic gap.

• Phase Diagram:

  • The PDW is the ground state in a large regime of moderate interactions.
  • At very strong interactions, the system transitions to a primary SDW state (ordering at $K$).
  • At weak interactions, the system remains a stable paramagnet.
  • Inside the PDW phase, there is a subleading instability toward loop currents (complex bond CDW), induced by non-collinear spin correlations.

5. Significance and Implications

• Theoretical Breakthrough: This is one of the first convincing demonstrations of a genuine primary PDW as the ground state in a realistic, unbiased microscopic model. It resolves the long-standing theoretical challenge of why PDW usually loses to uniform SC or secondary modulations.
• Material Realization: The authors propose that CsCr$_3$Sb$_5$ (a chromium-based kagome metal) and similar multi-orbital kagome materials are prime candidates for realizing this state. The model's Fermi surface topology closely matches these materials.
• New Quantum Matter: The discovery of a chiral PDW with intertwined loop currents and topological properties opens new avenues for understanding exotic superconductivity, fractional flux quantization, and vestigial orders in correlated electron systems.
• Experimental Signatures: The paper provides specific predictions for experimental verification, including:
  • Anisotropic gap structures observable via ARPES (Angle-Resolved Photoemission Spectroscopy).
  • Residual Fermi arcs.
  • Spontaneous time-reversal symmetry breaking (detectable via muon spin rotation or Kerr effect).
  • Spatial modulations of the superconducting gap observable via STM (Scanning Tunneling Microscopy).

In summary, the paper establishes that the unique combination of sublattice/orbital polarization and multi-orbital Fermi surface topology in kagome lattices creates a "sweet spot" where finite-momentum pairing is energetically favored over uniform pairing, leading to a robust, genuine Pair Density Wave ground state.