Evidence for interior-gap pair-density-wave state in Kondo-Heisenberg chains

Using infinite and finite density-matrix-renormalization-group calculations, this study provides evidence that one-dimensional Kondo-Heisenberg chains realize an exotic interior-gap pair-density-wave state where strong correlations dynamically generate a reconstructed momentum distribution and dominant superconducting correlations, distinct from conventional mismatch-based scenarios.

The Gist

Imagine a long, one-dimensional train track made of atoms. On this track, you have two types of passengers:

1. The Commuters: Electrons that can zip back and forth freely (conduction electrons).
2. The Locals: Atoms stuck in place that have their own tiny magnetic spins (localized spins).

Usually, when these two groups interact, they either ignore each other or get stuck in a rigid, non-moving pattern. But in this specific "Kondo-Heisenberg" setup, something magical and strange happens when the interaction between the Commuters and Locals gets strong. They form a special kind of superconducting state, but it's not the usual kind you see in textbooks.

Here is what the paper discovered, explained simply:

1. The "Interior-Gap" Mystery

In a normal superconductor, electrons pair up and move without resistance, creating a smooth, empty "gap" in their energy levels. It's like a highway where all the cars are moving in perfect sync, and there are no obstacles.

In this study, the researchers found a state called an "Interior-Gap Pair-Density-Wave" (PDW).

• The Analogy: Imagine a highway where most cars are moving in perfect pairs (superconducting), but right in the middle of the highway, there are still some lonely, single cars driving around freely.
• Usually, physicists thought this "gap with holes" (interior gap) could only happen if you forced two different groups of cars with different speeds to mix. But here, the researchers found this state happening naturally in a single group of cars, created entirely by the strong "social pressure" (correlations) between the Commuters and the Locals.

2. The "Two-Face" Electron

The most surprising finding is about the "shape" of the electrons' movement.

• The Old View: Think of the electrons as having a single "home base" or a single favorite speed (a single Fermi surface).
• The New Discovery: The paper shows that the strong interaction creates a second home base out of thin air.
  • For a specific type of chain (where the local spins are "3/2"), the electrons' behavior changes so drastically that their distribution looks like a dip (a valley) in the middle of the road.
  • This "dip" proves that the electrons have reorganized themselves into two distinct groups moving at different speeds, even though they started as just one group. It's as if a single crowd of people suddenly split into two distinct dance circles without anyone telling them to.

3. The "Hump" vs. The "Dip"

The researchers tested two versions of this train track: one with "light" local spins (Spin 1/2) and one with "heavier" local spins (Spin 3/2).

• Spin 1/2: The electrons showed a small, blurry "hump" in their movement. It was hard to tell exactly what was happening.
• Spin 3/2: The "hump" sharpened into a clear, deep "dip."
• Why it matters: This clear dip is the "smoking gun" evidence. It confirms that the electrons have truly reconstructed their internal structure into this exotic "interior-gap" state. The heavier spins made the effect so strong it became impossible to miss.

4. The "Boundary" Problem (The Mirror Effect)

One of the biggest challenges in studying these tiny atomic chains is that the ends of the chain mess up the data.

• The Analogy: Imagine trying to hear a quiet song in a room with echoey walls. The sound bouncing off the walls (boundary effects) makes it hard to hear the actual song (the bulk physics).
• In previous studies, scientists used finite chains (short tracks with ends). The "echoes" from the ends made it look like different types of order were competing, and it was hard to tell which one was the winner.
• The Solution: This paper used a special mathematical trick (infinite DMRG) to simulate a track with no ends at all.
  • When they removed the "echoes," the answer became clear: The "Pair-Density-Wave" (the electrons pairing up in a wavy pattern) is the undisputed champion.
  • They also showed that the "echoes" in the shorter chains were actually hiding the true nature of the electrons, making the "dip" look like a "hump" or vice versa.

5. The "Ghost" Momentum

There is a famous rule in physics (the YOA constraint) that says if you have these magnetic spins, the system must have a specific amount of "momentum" (a kind of push).

• The Expectation: Usually, this momentum shows up as a giant, single "Fermi surface" (a big, obvious circle of electrons).
• The Reality: In this system, the momentum is there, but it's hidden. It doesn't show up as a big circle of single electrons. Instead, it shows up as a "ghost" wave in the density of the electrons and as a "composite" pair.
• The Takeaway: The system satisfies the rule, but it does so in a sneaky, complex way that defies the simple "big circle" expectation. The momentum is carried by a mix of the electron and a neutral "ghost" wave, rather than by a single electron.

Summary

The paper proves that in a specific one-dimensional chain of magnets and electrons, strong interactions create a weird, exotic superconducting state.

1. It creates a "gap with holes" (interior-gap) where some electrons move freely even while others are paired up.
2. It forces the electrons to split into two distinct groups (creating a "dip" in their movement pattern), even though they started as one group.
3. This state is the dominant behavior of the system, but you can only see it clearly if you look at the system without the "noise" of the ends (using infinite simulations).

It's a discovery that shows how strong "social pressure" between particles can completely rewrite the rules of how they move, creating a state of matter that is more complex and intertwined than anyone expected.

Technical Summary

1. Problem Statement

The paper addresses the long-standing theoretical question of whether interior-gap superconductivity (also known as the breached-pair state) can be realized in canonical strongly correlated electron models.

• Context: Interior-gap superconductivity typically arises in systems with mismatched Fermi surfaces, where pairing coexists with gapless Fermi-surface-like structures, leading to a single-particle momentum distribution distinct from conventional BCS superconductors.
• Challenge: While theoretically proposed, such states are often unstable in simple pairing models (prone to phase separation or finite-momentum pairing). It remains unclear if they exist in realistic, strongly correlated 1D models.
• Specific System: The authors investigate the one-dimensional Kondo-Heisenberg model, which consists of conduction electrons coupled to localized spins (Kondo coupling) with an additional antiferromagnetic Heisenberg interaction between the localized spins. Previous studies suggested a spin-gapped phase with Pair-Density-Wave (PDW) correlations, but the nature of the single-particle spectrum and the dominance of the PDW order in the thermodynamic limit were ambiguous due to boundary effects in finite-size simulations.

2. Methodology

The authors employ a combination of advanced numerical techniques to distinguish intrinsic bulk properties from finite-size artifacts:

• Infinite Density Matrix Renormalization Group (iDMRG): Used to directly access the thermodynamic limit ($L \to \infty$) without physical boundaries. This allows for the precise calculation of bulk correlation functions and momentum distribution functions, free from Friedel oscillations.
• Finite DMRG: Performed on open chains to analyze boundary-induced effects and real-space correlation modulations.
• Model Parameters:
  • Spin values: $S = 1/2$ and $S = 3/2$ (chosen to preserve the non-trivial Yamanaka-Oshikawa-Affleck (YOA) momentum structure).
  • Filling: $n = 7/8$.
  • Couplings: Fixed at $J_K = J_H = 2t$ (a regime known to support PDW states).
• Observables Calculated:
  • Correlation functions for various orders: Charge Density Wave (CDW), Charge-2e/4e pairing, Composite pairing, and Bond-pairing (PDW).
  • Single-particle correlation functions and momentum distribution functions $n(k)$.
  • Entanglement entropy to extract the central charge $c$.

3. Key Contributions and Results

A. Dominance of PDW Order in the Bulk

Using iDMRG, the authors identified the dominant quasi-long-range order in the spin-gapped phase:

• Bond-pairing (PDW) is dominant: The bond-pairing correlation function $\langle O_B^\dagger(i)O_B(j) \rangle$ exhibits the slowest power-law decay (smallest exponent $\alpha$) among all competing channels.
  • For $S=1/2$: $\alpha_B \approx 1.11$.
  • For $S=3/2$: $\alpha_B \approx 1.56$.
• Oscillatory Nature: The PDW correlations oscillate with wavevectors $Q=\pi$ and $Q' = 2k_F = 7\pi/8$, confirming the state is a PDW rather than a uniform superconductor.
• Contrast with Finite DMRG: In finite systems with open boundaries, the hierarchy of correlations is obscured by boundary-induced modulations, making it difficult to definitively identify the dominant order without iDMRG.

B. Interior-Gap Single-Particle Reconstruction

The most significant finding is the characterization of the single-particle momentum distribution function $n(k)$:

• Emergence of Two Fermi Points: The single-particle correlation function reveals two distinct gapless modes at momenta $k_{F1} \approx 0.44\pi$ and $k_{F2} \approx 0.56\pi$ (where $k_{F2} = \pi - k_{F1}$).
• Interior-Gap Signature:
  • For $S=1/2$, $n(k)$ shows a "hump-like" structure around $k_{F2}$.
  • For $S=3/2$, this structure evolves into a clear dip at $k \approx \pi/2$.
  • The difference $\Delta n = n(\pi/2) - n(\pi)$ changes sign from positive ($S=1/2$) to negative ($S=3/2$), providing strong evidence for an interior-gap-like reconstruction where the single-particle spectrum is gapped in certain regions but retains gapless pockets (Fermi points).
• Absence of Large Fermi Surface: Crucially, there is no singularity at the "large" Fermi surface momentum $k^*_F = k_F + \pi/2$ (which would be expected in a standard large-Fermi-surface Kondo lattice). This indicates the system does not follow the conventional large-Fermi-surface scenario.

C. Central Charge and Low-Energy Physics

• Central Charge $c=1$: For the $S=1/2$ system, the entanglement entropy scaling yields a central charge $c \approx 1.00$.
• Implication: Despite the presence of two Fermi points ($k_{F1}$ and $k_{F2}$), the system possesses only one gapless bosonic degree of freedom in the charge sector. This implies that the two fermionic excitations are not independent; they are intertwined components of a single correlated mode.
• YOA Momentum Constraint: The YOA argument requires a gapless excitation at momentum $P_{YOA} = 2k_F + \pi$. The authors show this momentum appears in the density correlations (as a scalar mode) but not as a conventional single-particle large Fermi surface.

D. Boundary Effects

Finite DMRG results highlight that open boundaries induce strong Friedel oscillations and modify real-space correlations:

• In $S=3/2$, boundary effects significantly suppress PDW correlations near the edges due to density mismatches, further complicating the identification of the bulk phase in finite systems.
• This underscores the necessity of iDMRG for identifying intrinsic bulk properties in gapless 1D systems.

4. Significance and Conclusion

• Realization of Interior-Gap PDW: The paper provides the first numerical evidence that a canonical strongly correlated 1D model (Kondo-Heisenberg) realizes an interior-gap PDW state. Unlike conventional scenarios where interior-gap states arise from pre-existing mismatched Fermi surfaces, here the additional low-energy structure ($k_{F2}$) emerges dynamically through the Kondo coupling.
• Intertwined Order: The results suggest that the PDW order, the reconstructed single-particle spectrum, and the charge-neutral mode at $Q=\pi$ are not separate cause-and-effect phenomena but intertwined manifestations of the same strong-coupling correlated phase.
• Theoretical Distinction: The work clarifies the distinction between the YOA momentum constraint (which guarantees a gapless mode at a specific momentum) and the existence of a conventional large Fermi surface. In this phase, the YOA momentum is realized via a composite scalar mode rather than a single-particle singularity.
• Methodological Impact: The study demonstrates that combining iDMRG (for bulk properties) and finite DMRG (for boundary analysis) is essential for correctly interpreting strongly correlated 1D systems where boundary effects can mask the true nature of the ground state.

In summary, the authors establish that the spin-gapped phase of the Kondo-Heisenberg chain is a strongly correlated interior-gap PDW state, characterized by a unique single-particle reconstruction and a unified low-energy description where multiple order parameters are intrinsically linked.