Pair density wave, infinite-length stripes, and holon Wigner crystal in single-band Hubbard model on diagonal square lattice

Using large-scale GPU-accelerated DMRG simulations on a diagonally oriented square lattice, this study reveals a doping-dependent phase diagram in the hole-doped Hubbard model featuring distinct diagonal stripe, holon Wigner crystal, and infinite-length stripe phases, providing the first controlled numerical evidence of a dominant pair density wave emerging from the interplay between charge order and short-range superconductivity.

The Gist

Imagine you are trying to understand how a crowd of people behaves in a giant, crowded room. In the world of physics, these "people" are electrons, and the "room" is a crystal lattice (a grid of atoms). Usually, electrons are shy and avoid each other, but in certain materials (like the superconductors found in high-temperature experiments), they get very crowded and start doing something strange: they dance together in pairs to conduct electricity with zero resistance. This is superconductivity.

For decades, scientists have been trying to figure out the exact steps of this dance. A major mystery has been a specific, complex dance move called a Pair Density Wave (PDW).

Here is a simple breakdown of what this paper discovered, using everyday analogies.

The Problem: The "Short Hallway" Limit

To study these electron dances, scientists use powerful computer simulations. Imagine trying to watch a long, winding parade.

• The Old Way: Previous simulations were like watching the parade through a narrow, short hallway (a "cylinder"). The parade (the electron stripes) would hit the end of the hallway and stop. Because the hallway was too short, the scientists couldn't see the full pattern of the dance. They could only see the beginning, and it looked like the dancers were just standing still or moving in short, jerky bursts.
• The New Trick: The authors of this paper decided to rotate the room. Instead of a standard square grid, they tilted the grid diagonally (like a diamond shape).
  • Why this matters: In this tilted room, the "parade" can stretch infinitely long without hitting the walls. It's like turning a short hallway into a long, straight highway. This allowed them to see the full, continuous dance for the first time.

The Discovery: Three Different Dance Styles

By using this new "tilted room" and super-powerful computers (running on GPUs, the same chips that power advanced video games), they watched how the electrons behaved as they added more "holes" (empty spots) to the crowd. They found three distinct phases, or dance styles, depending on how crowded the room was:

1. The "Diagonal Line" Phase (Low Crowd)

• What happens: The electrons form short, diagonal lines.
• The Dance: They pair up, but the connection is weak and short-lived. It's like people holding hands in a small circle, but the circle breaks apart quickly.

2. The "Crystal Grid" Phase (Medium Crowd)

• What happens: The electrons arrange themselves into a checkerboard pattern of empty spots (called a "Holon Wigner Crystal").
• The Dance: The electrons are still paired, but the rhythm is weird. The strength of their connection goes up and down in a wave as you move across the room. It's like a dance where the music gets louder and softer in a repeating pattern, but the dancers can't quite keep a steady beat across the whole room.

3. The "Infinite Highway" Phase (High Crowd) — The Big Breakthrough

• What happens: This is the most exciting part. When the crowd gets dense enough, the electrons form infinite-length stripes that stretch all the way across the simulation.
• The Dance (The PDW): Here, the electrons finally perform the elusive Pair Density Wave.
  • Imagine a long line of dancers. Instead of everyone moving in perfect sync (which is normal superconductivity), they move in a wave.
  • The "hand-holding" (superconductivity) is strong in some spots, weak in others, and even reverses direction (like a wave going left, then right) as you move down the line.
  • Crucially, this wave pattern is stable and stretches across the entire system. This is the first time a computer simulation has definitively shown this specific "wavy" superconductivity in a simple, single-band model.

Why Does This Matter?

You might ask, "So what? It's just a computer simulation."

1. Solving a Mystery: For years, scientists have seen hints of this "wavy" superconductivity in real materials (like cuprates, which are used in MRI machines and maglev trains), but they couldn't prove it was the main driver. This paper provides the "smoking gun" evidence that this complex dance is a natural result of electron interactions, not just a fluke.
2. The "Layer Decoupling" Puzzle: In real superconductors, the material is made of many thin layers stacked on top of each other. Sometimes, these layers stop talking to each other, and the superconductivity breaks.
   • The Analogy: Imagine two rows of dancers. If they are doing a normal dance, they hold hands across the gap between rows. But if they are doing this "wavy" PDW dance, the waves in Row A might be perfectly out of sync with the waves in Row B. When they try to hold hands, they cancel each other out!
   • This paper explains why the layers sometimes disconnect: the wave pattern naturally causes them to interfere destructively.

The Takeaway

The authors didn't just find a new dance; they built a better stage (the diagonal lattice) to see the dance clearly. They proved that when electrons are crowded enough, they naturally organize into long, wavy stripes where superconductivity flows in a rhythmic, oscillating pattern.

This gives us a new map for understanding how high-temperature superconductors work, potentially helping us design materials that can carry electricity without loss at room temperature—a "holy grail" for energy technology.

Technical Summary

1. Problem Statement

The paper addresses the long-standing challenge of identifying and characterizing the Pair Density Wave (PDW) state in the single-band Hubbard model, a paradigmatic model for high-temperature superconductors (cuprates).

• The Challenge: While PDW (a superconducting state with finite-momentum Cooper pairs) has been observed experimentally in various materials, conclusive numerical evidence of a dominant PDW in the simple single-band Hubbard model has remained elusive.
• Limitations of Previous Approaches: Standard Density Matrix Renormalization Group (DMRG) simulations on regular square lattices are typically restricted to cylinder geometries. On these cylinders, stripe orders are constrained by the cylinder width, preventing the formation of "infinite-length" stripes. This geometric constraint limits the ability to probe long-distance correlations along the stripe direction, often resulting in only short-range superconducting (SC) correlations in previous studies.
• Goal: To overcome finite-size limitations and uncover the interplay between charge density waves (CDW), spin density waves (SDW), and unconventional superconductivity, specifically searching for a dominant PDW phase.

2. Methodology

The authors employ large-scale, GPU-accelerated Density Matrix Renormalization Group (DMRG) simulations to investigate the hole-doped Hubbard model on a diagonally oriented square lattice.

• Model: The Hamiltonian includes nearest-neighbor ($t$) and next-nearest-neighbor ($t'$) hopping, with on-site Coulomb repulsion $U$. The study focuses on the hole-doped regime with $t' < 0$ (relevant to cuprates).
• Lattice Geometry: The lattice is rotated by $\pi/4$ relative to the conventional orientation.
  • Advantages:
    1. Symmetry: Allows unambiguous discrimination between $d$-wave and $s$-wave SC and between vertical stripes and bidirectional CDW.
    2. Momentum Space: Quantized momenta naturally include a $d$-wave nodal line and multiple Fermi points, facilitating low-energy physics exploration without fine-tuning.
    3. Infinite-Length Stripes (i-stripes): Crucially, this geometry permits stripes to span the entire length of the cylinder ($L_1$) without being cut off by the finite width ($L_2$), circumventing the geometric constraints of regular lattices.
• Simulation Parameters:
  • System Size: Cylinders with widths $L_2 = 6$ and $8$, and lengths up to $L_1 = 20$.
  • Doping ($\delta$): Ranges from 5% to 15%.
  • Interaction: $U = 8$ and $12$;$t' = -0.1$ to $-0.3$.
  • Computational Power: Simulations utilize up to 48,000 states (bond dimension $m$) with over 100 sweeps, leveraging GPU acceleration. Finite-truncation-error extrapolation is performed to ensure reliability.

3. Key Contributions

1. Novel Lattice Design: The implementation of the $\pi/4$-rotated diagonal square lattice to enable the formation of infinite-length stripes (i-stripes), a state geometrically prohibited on regular lattices.
2. Discovery of a 2D-like PDW: Providing what is likely the first controlled numerical evidence of a dominant, quasi-long-range PDW in the single-band Hubbard model.
3. Comprehensive Phase Diagram: Mapping a rich quantum phase diagram across doping levels, identifying three distinct phases: a diagonal stripe phase, a Holon Wigner Crystal phase, and the i-stripe phase.
4. Mechanism of Interlayer Decoupling: Offering a microscopic explanation for dynamical layer decoupling in cuprates based on the specific spatial oscillation of the PDW order parameter.

4. Key Results

The study identifies three distinct quantum phases as doping ($\delta$) increases:

A. Diagonal Stripe Phase ($\delta \lesssim 9\%$)

• Structure: Short-range diagonal charge and spin stripes with a locked ordering vector $Q_c = 2Q_s = (4\pi\delta, 4\pi\delta)$.
• Superconductivity: Exhibits short-range uniform $d$-wave SC. The pair correlations decay exponentially.

B. Holon Wigner Crystal Phase ($\delta \sim 10\%$)

• Structure: A bidirectional charge density order (WC*) with wavevectors $Q_{h,1} = (0, 2\pi/3)$ and $Q_{h,2} \approx (2\pi/3, 0)$. Spin order remains unidirectional, indicating spin-charge separation (a hallmark of holons).
• Superconductivity: SC correlations remain short-ranged but develop sign-alternating oscillations at long distances, serving as a precursor to the PDW.

C. Infinite-Length Stripe (i-stripe) Phase ($\delta \gtrsim 12\%$)

• Structure: The system forms infinite-length stripes spanning the entire cylinder. The charge density profile shows a period-3 modulation (or period-4 in wider systems with cuprate parameters), breaking translation and mirror symmetries.
• Superconductivity (The Major Finding):
  • Divergent Susceptibility: The short-range SC in the WC* phase evolves into a 2D-like Pair Density Wave (PDW).
  • Quasi-Long-Range Order: Pair-pair correlations along the stripe ($\Phi_{XX,0}$) decay as a power law ($x^{-K_{sc}}$) with exponent $K_{sc} \approx 1.6$, implying divergent SC susceptibility as $T \to 0$.
  • Oscillatory Nature: The correlations exhibit spatial oscillations with a wavelength of $\sim 3.5$ lattice spacings (matching the charge stripe period).
  • 2D Character: Correlations perpendicular to the stripes also show quasi-long-range power-law decay with the same exponent, indicating the order is not merely 1D but possesses 2D-like characteristics.
  • Symmetry: The state exhibits local $d$-wave pair symmetry.

Evolution and Stability

• The transition from WC* to i-stripe involves the "melting" of the bidirectional charge order into unidirectional infinite stripes, which enhances SC coherence.
• The i-stripe + PDW phase is robust across a wide range of $U$ ($8-12$) and $t'$ ($-0.1$ to $-0.3$).
• In systems with cuprate-relevant parameters ($\delta=1/8, t' \approx -0.2$), the model reproduces period-4 "half-filled" charge stripes with period-8 spin stripes, aligning with experimental observations in Bi2212 and Bi2201.

5. Significance

• Resolution of the PDW Puzzle: This work provides strong numerical evidence that the single-band Hubbard model, when free from finite-size constraints on stripe length, naturally hosts a dominant PDW state.
• Understanding Layer Decoupling: The authors propose a mechanism for the vanishing interlayer Josephson coupling in stripe-ordered superconductors (like LBCO). The intrinsic spatial oscillation of the PDW along the i-stripes leads to destructive interference between adjacent layers (either orthogonal or parallel stacking), explaining the "dynamical layer decoupling" observed experimentally.
• Role of Charge Fluctuations: The results suggest that fluctuating charge density orders (partially melted CDW) are crucial for facilitating PDW, whereas fully developed static CDW (as in the WC* phase) suppresses SC.
• New Platform: The diagonal square lattice is established as a superior computational platform for studying intertwined orders in quantum materials, overcoming the geometric limitations of traditional cylinder geometries.

In summary, the paper demonstrates that by removing geometric constraints on stripe length, the single-band Hubbard model reveals a rich phase diagram culminating in a robust, 2D-like Pair Density Wave state, offering new microscopic insights into the complex interplay of charge, spin, and superconductivity in high-$T_c$ materials.