Superconductivity in the repulsive Hubbard model on different geometries induced by density-assisted hopping

This paper demonstrates that density-assisted hopping in repulsive Hubbard models on dimerized geometries, such as bilayers and ladders, induces effective attractive interactions in the bonding band that overcome onsite repulsion to drive a Berezinskii-Kosterlitz-Thouless transition into a complex, filling-dependent superconducting phase.

The Gist

The Big Picture: Turning Repulsion into a Dance

Imagine you are at a crowded party where everyone is extremely shy and hates being touched. In physics, we call this repulsion. If you try to get two people close together, they push each other away. In the world of electrons (the tiny particles that carry electricity), this is the standard rule: electrons usually hate being on top of each other.

For decades, scientists have been trying to figure out how to make these "shy" electrons dance together in a synchronized way to create superconductivity—a state where electricity flows with zero resistance, like a perfectly smooth ice rink. Usually, you need a "glue" (like a vibration in the material) to make them stick. But this paper suggests a clever new trick: density-assisted hopping.

The Analogy: The Two-Lane Highway

Think of the material in this study as a two-lane highway (a "ladder" structure in physics terms).

• Lane 1 (The Bonding Band): The inner lane, closer to the center.
• Lane 2 (The Anti-bonding Band): The outer lane.

Normally, electrons can drive in either lane. But in this specific setup, the "traffic rules" are changed by a special term called density-assisted hopping.

Here is the magic trick:
Imagine that the speed limit for a car (an electron) depends on how many other cars are in the other lane right next to it.

• If the other lane is crowded, the car in your lane is allowed to move faster or change lanes more easily.
• If the other lane is empty, the car moves slower.

The researchers found that this rule changes the nature of the traffic entirely.

1. In the Inner Lane (Bonding): The rule makes the cars feel like they want to be together. Even though they naturally hate each other (repulsion), this new rule creates an effective attraction. It's like a shy person suddenly finding a reason to hold hands with their neighbor.
2. In the Outer Lane (Anti-bonding): The rule makes them push away even harder.

The Result: A Superconducting Phase

Because the electrons in the inner lane suddenly feel an attraction, they start pairing up. In physics, when electrons pair up, they can flow without friction. This is superconductivity.

The paper shows that you don't need to change the material itself to get this effect; you just need to tune this "density-assisted hopping" parameter. It's like turning a dial on a radio.

• Turn the dial a little: The electrons are still shy and repulsive (a normal metal).
• Turn the dial past a critical point: Suddenly, the electrons in the inner lane lock hands and start flowing perfectly. The system undergoes a phase transition (like water turning to ice, but for electricity).

The "Complex Dance" (Pairing Structure)

Usually, when electrons pair up, they do a simple "s-wave" dance (holding hands directly). But because this material has two lanes, the pairing is more complex.

• Imagine a couple dancing. In a simple dance, they hold hands face-to-face.
• In this new dance, they are holding hands, but they are also spinning around a partner in the other lane. It's a superposition of different dance moves.
• This means the superconductivity here is "unconventional." It's not the standard kind; it's a richer, more complex form of superconductivity that emerges from the interaction between the two lanes.

How They Proved It

The scientists didn't just guess this; they used two methods:

1. Math (Analytical): They wrote down equations to predict what should happen. They realized that mathematically, this hopping term turns the "shy" electrons into "social" ones in the inner lane.
2. Supercomputers (Numerical): They used a powerful simulation technique called Matrix Product States (MPS) to simulate a giant ladder of electrons. They watched the system evolve as they turned the "density-assisted" dial.
   • They measured the "central charge" (a way to count how many independent ways the system can wiggle). When the superconductivity started, this number dropped, confirming a phase change.
   • They measured how far the "spin" of the electrons could travel before getting stuck. In the superconducting state, the spin gets "gapped" (stuck), which is a hallmark of this new phase.

Why This Matters

This is exciting for a few reasons:

• It's Robust: The effect happens even when the electrons are very strongly repulsive (which usually kills superconductivity).
• It's Realistic: The strength of this "density-assisted" effect needed to create superconductivity in their model is very similar to what is estimated to exist in cuprates (the materials used in high-temperature superconductors). This suggests that this mechanism might be the secret sauce behind why some real-world materials conduct electricity so well.
• New Pathways: It suggests that we might be able to engineer new superconductors by designing materials where electrons can "see" each other's density and adjust their movement accordingly, rather than just relying on vibrations.

Summary

The paper discovers that by adding a rule where electrons move differently depending on how crowded their neighbors are, you can trick repulsive electrons into pairing up. This turns a "shy" system into a "super-dancer" that conducts electricity perfectly, offering a new clue on how to build better superconductors for the future.

Technical Summary

1. Problem Statement

The theoretical description of high-temperature superconductivity, particularly in cuprates, remains a central challenge in condensed matter physics. The standard two-dimensional Hubbard model, featuring only on-site repulsion ($U$) and nearest-neighbor hopping, is known to host a rich phase diagram, but it is still debated whether it supports superconductivity in the purely repulsive regime without additional mechanisms.
While extensions like diagonal hopping ($t'$) have been shown to favor superconductivity, density-assisted hopping (where the hopping amplitude depends on the local density of opposite spins) is a crucial term arising naturally from the down-folding of the three-band Hubbard model to a single-band model. Although recent studies suggest this term is significant in cuprates (approx. 60% of normal hopping), its specific role in inducing superconductivity in repulsive systems across different geometries (specifically dimerized lattices like ladders and bilayers) requires further rigorous investigation, especially in the strong coupling regime.

2. Methodology

The authors employ a combination of analytical and numerical techniques to study the extended Hubbard model on dimerized lattices (specifically two-leg ladders and bilayers).

• Model Hamiltonian: The system is defined by the Fermi-Hubbard model extended with a density-assisted hopping term ($H_{DA}$) within the dimers (rungs). The total Hamiltonian is $H = H_{\parallel} + H_{\perp} + H_U + H_{DA}$.
  • $H_{DA} = -t_{n\perp} \sum_{\sigma, j} (\hat{n}_{1,\sigma,j} + \hat{n}_{2,\sigma,j}) (\hat{c}^{\dagger}_{1,\sigma,j}\hat{c}_{2,\sigma,j} + \text{H.c.})$
• Analytical Approach:
  • Basis Transformation: The authors transform the site basis into bonding ($k_\perp=0$) and antibonding ($k_\perp=\pi$) band bases.
  • Effective Hamiltonian: By performing a mean-field approximation and second-order perturbation theory (assuming the antibonding band is largely empty), they derive an effective Hamiltonian for the bonding band. They show that the density-assisted hopping induces an effective attractive interaction ($\tilde{U}$) in the bonding band, even when the original on-site interaction $U$ is repulsive.
  • Bosonization: Used to predict the low-energy physics, identifying a transition from a gapless Luttinger liquid ($C_1S_1$) to a spin-gapped Luther-Emery phase ($C_1S_0$) as the density-assisted hopping ratio ($c_n$) increases.
• Numerical Approach:
  • Matrix Product States (MPS): The authors use the Density Matrix Renormalization Group (DMRG) algorithm (via the ITensor library) to simulate the full Hamiltonian on ladders with up to $L=96$ rungs.
  • Observables: They calculate the central charge ($c$) via entanglement entropy scaling, spin correlation length ($\xi_{sz}$), momentum distribution discontinuity ($Z_{k_F}$), and power-law exponents for density, pair, and spin correlation functions.

3. Key Contributions

1. Mechanism of Induced Attraction: The paper analytically demonstrates that density-assisted hopping generates a sign change in the effective interaction within the bonding band. Specifically, it creates an attractive interaction ($\tilde{U} < 0$) in the lower band while remaining repulsive in the upper band, effectively mapping the repulsive system to an attractive Hubbard model in the relevant low-energy sector.
2. Complex Pairing Structure: The induced superconductivity is not simple $s$-wave. In the original site basis, the pairing corresponds to a superposition of on-site $s$-wave pairs and inter-site dimer singlets (often associated with $d$- or $p$-wave symmetry in ladder geometries). This results in a non-local pairing structure.
3. Robustness in Strong Coupling: The study confirms that the transition to superconductivity, predicted by weak-coupling bosonization, persists and is robust even in the strong interaction regime ($U = 4t_\parallel$ to $20t_\parallel$), far beyond the strict validity of the perturbative effective model.
4. Phase Transition Characterization: The transition is identified as a Berezinskii-Kosterlitz-Thouless (BKT) transition, characterized by the exponential opening of a spin gap and a change in the central charge from $c=2$ to $c=1$.

4. Key Results

• Phase Diagram: The authors map out the phase diagram of the two-leg ladder. For a fixed effective perpendicular hopping ($\tilde{t}_\perp = 3t_\parallel$) and filling ($n=0.9375$), increasing the density-assisted hopping ratio ($c_n$) drives the system from a Luttinger Liquid ($C_1S_1$) to a Spin-Gapped Superconductor ($C_1S_0$).
• Critical Threshold: The transition occurs at a critical ratio $c_n^* \approx 0.24 - 0.26$. This value is consistent across different observables (central charge, spin gap opening, and momentum discontinuity) and aligns closely with the analytical prediction derived from the effective Hamiltonian.
• Correlation Functions:
  • In the superconducting phase ($c_n > c_n^*$), the pair correlation function decays as a power law with an exponent $\eta_d$ that is significantly smaller (slower decay) than the density correlation exponent $\eta_n$, indicating dominant superconducting correlations.
  • The spin correlation function decays exponentially, confirming the opening of a spin gap.
• Momentum Distribution: At the critical point, the momentum distribution $n(k)$ exhibits a sharp step function with a discontinuity $Z_{k_F} \approx 1$, signaling the effective non-interacting nature of the bonding band at the transition point before interactions drive the pairing.
• Generalizability: The results hold for various interaction strengths ($U$) and fillings ($n$). The phase diagram shows that the superconducting region is robust and extends over a wide parameter space.

5. Significance

• New Pairing Mechanism: The work identifies a distinct pairing mechanism driven by correlated hopping (density-assisted) rather than spin fluctuations alone. This provides a new route to understanding high-$T_c$ superconductivity in repulsive systems.
• Relevance to Real Materials: The magnitude of the density-assisted hopping required to induce superconductivity in the model ($c_n \approx 0.24$) is comparable to values estimated for cuprates, suggesting this mechanism could be physically relevant for real materials like nickelates ($La_3Ni_2O_7$) and cuprates.
• Experimental Feasibility: The paper highlights that these effects can be realized in ultracold atom experiments, where density-assisted hopping can be engineered via Feshbach resonances or lattice modulation, offering a pathway to experimentally verify these theoretical predictions.
• Geometric Universality: While demonstrated on ladders, the analytical derivation suggests this mechanism applies generally to dimerized structures in 2D and 3D (e.g., bilayers), potentially explaining superconductivity in multilayered materials.

In conclusion, this paper provides compelling evidence that density-assisted hopping can overcome on-site repulsion to induce robust, unconventional superconductivity in dimerized lattices, bridging the gap between weak-coupling analytical theories and strong-coupling numerical realities.