High-temperature $η$-pairing superconductivity in the photodoped Hubbard model

Using steady-state dynamical mean-field theory on the real-frequency axis, this study demonstrates that photodoping a Mott insulating Hubbard model induces a distinct form of high-temperature $\eta$-pairing superconductivity, characterized by a high effective critical temperature and identifiable spectroscopic signatures in the spectral function and optical conductivity.

The Gist

The Big Idea: Turning a "Traffic Jam" into a "Superhighway" with a Flash of Light

Imagine a crowded city street where cars (electrons) are stuck in a massive traffic jam. They can't move because they are too afraid to bump into each other. In physics, this is called a Mott Insulator. It's a material that should conduct electricity (like a metal), but because the electrons repel each other so strongly, they freeze in place. It's an electrical dead end.

For decades, scientists have been trying to find a way to make these materials conduct electricity without resistance (superconductivity) at room temperature. Usually, this requires extreme cold (like liquid helium), which is expensive and impractical.

This paper proposes a radical new idea: Don't cool the traffic jam; blast it with a laser.

The Experiment: The "Flash" that Creates Order

The researchers used a computer simulation to study what happens when you shine a strong laser pulse on this "frozen" material.

1. The Photodoping Effect: When the laser hits the material, it gives some electrons a massive energy boost. These electrons jump to a higher energy level, leaving behind empty spots (holes) in the lower level.
   • Analogy: Imagine the traffic jam suddenly gets a burst of energy. Some cars zoom up to a "sky lane" (doublons), leaving empty spaces in the "ground lane" (holons).
2. The Problem: Usually, these excited cars would quickly crash back down, and the traffic jam would return. But in this specific material, the gap between the lanes is so huge that the cars stay up there for a long time. The system gets stuck in a "steady state" where it's neither a normal metal nor a frozen insulator.
3. The Surprise: In this chaotic, excited state, the electrons suddenly decide to dance in perfect sync. They form pairs and move together without any friction. This is Superconductivity.

The Secret Dance: "Eta-Pairing"

Most superconductors work like a waltz where partners hold hands and move in a circle (s-wave or d-wave pairing). This paper discovered a different, stranger dance called $\eta$-pairing (Eta-pairing).

• The Analogy: Imagine a dance floor where everyone is usually fighting. Suddenly, a laser flash makes the dancers pair up, but instead of holding hands, they do a synchronized "staggered" step. One person steps forward, their partner steps back, and they move across the room in a wave pattern that repeats every other step.
• Why it matters: This specific dance allows the material to become superconducting at extremely high temperatures. The paper predicts this state could survive at temperatures well above room temperature (over 1,000 Kelvin, or roughly 1,300°F), which is a game-changer for technology.

The "High-Tech" Math: Why They Needed a Better Calculator

To prove this works, the researchers had to solve incredibly complex math equations (the Hubbard model).

• The Old Way: Previous computer programs used "low-resolution" math (like looking at a blurry photo). They could guess the temperature where superconductivity starts, but they couldn't see the details of how the electrons were moving. They were like trying to read a book through a foggy window.
• The New Way: The authors developed a "high-resolution" solver (called TOA). It's like switching from a foggy window to a 4K microscope.
  • The Result: With this new tool, they didn't just see that superconductivity happened; they saw the spectral gap.
  • Analogy: If the superconducting state is a "quiet room" where no sound (energy) can pass through a certain range, the "gap" is the silence. The new math proved that this silence exists and is very wide. They also saw how the "optical conductivity" (how the material reacts to light) changes, giving scientists a way to actually see this state in a real lab experiment.

Why This Changes Everything

1. Room Temperature Superconductivity: This suggests we might not need to cool materials down to near absolute zero. We might be able to create superconductors that work in a hot room, just by shining a light on them.
2. A New Mechanism: This isn't the same kind of superconductivity we see in copper-oxide ceramics (cuprates) or in magnets. It's a completely new type of quantum state born from chaos and light.
3. The Path Forward: The paper gives experimentalists a "recipe." It tells them exactly what to look for:
   • Shine a laser on a specific type of insulator.
   • Look for a specific "gap" in the energy spectrum.
   • Check if the material conducts electricity without resistance at high temperatures.

Summary

Think of this paper as discovering a way to turn a frozen, chaotic traffic jam into a perfectly synchronized, frictionless superhighway just by flashing a strobe light. The researchers built a super-advanced "traffic camera" (the new math solver) to prove that this highway exists, that it works at scorching temperatures, and that we can identify it by the specific "silence" (energy gap) it creates. This opens the door to a future where we can control electricity with light, potentially revolutionizing power grids, computers, and transportation.

Technical Summary

1. Problem Statement

The paper addresses the challenge of achieving high-temperature superconductivity (HTS) in strongly correlated electron systems. While equilibrium mechanisms (e.g., spin fluctuations in cuprates) typically yield critical temperatures ($T_c$) far below room temperature, nonequilibrium routes offer a promising alternative. Specifically, the authors investigate $\eta$-pairing superconductivity, a state predicted by C.N. Yang in the Hubbard model where Cooper pairs possess a finite center-of-mass momentum ($\mathbf{Q} = (\pi, \pi)$).

The central problem is the lack of quantitative understanding regarding:

• The stability and critical temperature of $\eta$-pairing states in realistic, higher-dimensional (2D/3D) photodoped Mott insulators.
• The existence of a superconducting gap with clear spectroscopic signatures in these nonequilibrium steady states.
• The numerical accuracy required to describe these states, as previous studies were limited to 1D or infinite-dimensional Bethe lattices using lower-order approximations that may fail to capture essential spectral features.

2. Methodology

The authors employ Steady-State Dynamical Mean-Field Theory (DMFT) implemented directly on the real-frequency axis. This approach avoids the need for analytic continuation from imaginary time, which is crucial for resolving fine spectral features in nonequilibrium systems.

• Model: The two- and three-dimensional Hubbard model with strong on-site repulsion ($U = 16t$), representing a Mott insulator.
• Photodoping: Instead of chemical doping, the system is driven into a nonequilibrium steady state by photoexcitation. This creates doublon-holon pairs (doublons in the upper Hubbard band, holons in the lower). The distribution is modeled by shifting effective Fermi levels to $\pm \omega_F$ while maintaining a common effective temperature $T_{\text{eff}}$.
• Impurity Solvers: A critical methodological advancement is the use of high-order strong-coupling impurity solvers based on Quantics Tensor Cross Interpolation (QTCI). The authors compare:
  • NCA (Non-Crossing Approximation): First-order.
  • OCA (One-Crossing Approximation): Second-order.
  • TOA (Third-Order Approximation): Third-order.
  • Rationale: Lower-order solvers (NCA/OCA) were found to violate causality constraints in the superconducting phase, leading to numerical instabilities. TOA is required for a causal and accurate description of the Green's functions.
• Symmetry Handling: The DMFT equations are formulated in a Nambu basis adapted for $\eta$-pairing, where the Nambu spinor couples momenta $\mathbf{k}$ and $\mathbf{Q}-\mathbf{k}$ (with $\mathbf{Q}=(\pi, \pi)$).

3. Key Contributions

1. High-$T_c$ Prediction in 2D/3D: The study provides the first quantitative phase diagram for $\eta$-pairing in 2D and 3D photodoped Mott insulators, demonstrating that effective critical temperatures ($T_c^{\text{eff}}$) can exceed room temperature even at moderate photodoping levels ($\delta \approx 0.1$).
2. Spectroscopic Signatures: The authors identify clear experimental signatures of the $\eta$-pairing state, including a superconducting gap in the momentum-resolved spectral function and a characteristic $1/\omega$ divergence in the imaginary part of the optical conductivity.
3. Methodological Necessity of TOA: The paper establishes that third-order strong-coupling corrections are not merely quantitative improvements but qualitatively necessary for $\eta$-pairing states. Lower-order solvers violate the causality constraint $|\text{Im}\Sigma_N| \ge |\text{Im}\Sigma_A|$ near the transition, causing numerical collapse and failing to resolve the gap.
4. Mechanism Clarification: The work elucidates that while the attractive Hubbard model relies on explicit on-site attraction, the photodoped Mott insulator develops an effective $\eta$-pair attraction via a Schrieffer-Wolff transformation in the strong-coupling limit, driven by the ferromagnetic coupling in the $\eta$-spin sector.

4. Key Results

• Phase Diagram: The effective critical temperature $T_c^{\text{eff}}$ exhibits a dome-shaped dependence on photodoping concentration $\delta$.
  • At $\delta \approx 0.1$, $T_c^{\text{eff}}$ remains above 300 K (room temperature).
  • At optimal doping, $T_c^{\text{eff}}$ reaches ~1400–1600 K, comparable to or exceeding the attractive Hubbard model's equilibrium $T_c$.
  • NCA significantly underestimates $T_c$, while OCA and TOA agree well on the phase boundary, though TOA is required for spectral accuracy.
• Spectral Functions:
  • Gap Opening: The momentum-resolved spectral function $A(\mathbf{k}, \omega)$ shows a clear depletion of spectral weight near the effective Fermi levels ($\pm \omega_F$), indicating a superconducting gap.
  • Band Sharpening: Quasiparticle bands in the superconducting state become significantly sharper than in the normal state due to a reduction in the scattering rate ($\text{Im}[\Sigma_N - \Sigma_A]$).
• Optical Conductivity:
  • Real Part ($\text{Re}\sigma$): Shows suppression at low frequencies (indicating reduced dissipation) and a negative value at $\omega \to 0$ (a signature of the nonequilibrium steady state).
  • Imaginary Part ($\text{Im}\sigma$): Exhibits a $1/\omega$ divergence at low frequencies, signaling the emergence of finite superfluid stiffness and a superconducting response.
• Dimensionality: The high-$T_c$ $\eta$-pairing state and its spectral gap are robust features in both 2D square and 3D cubic lattices, suggesting they are generic to higher-dimensional photodoped Mott insulators.

5. Significance

• Controllable HTS: The work proposes a viable route to controllable, high-temperature superconductivity using light. Unlike equilibrium HTS, which is limited by material complexity, this state is driven by electronic correlations in a Mott insulator and can be tuned via laser intensity (photodoping).
• Experimental Guidance: The identification of specific spectroscopic fingerprints (gap in ARPES, $1/\omega$ divergence in optical conductivity) provides concrete targets for experimentalists working with laser-driven correlated materials (e.g., cuprates, fullerides).
• Theoretical Advancement: The paper highlights the limitations of standard perturbative impurity solvers (NCA/OCA) in nonequilibrium superconducting phases and establishes TOA/QTCI as a necessary tool for studying causality-preserving, high-order correlation effects in driven quantum matter.
• Fundamental Distinction: It demonstrates that nonequilibrium superconductivity can be fundamentally distinct from equilibrium $s$-wave or $d$-wave pairing, offering a new paradigm for understanding superconducting mechanisms in strongly correlated systems.