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

We employ large-scale density-matrix renormalization group (DMRG) simulations to investigate the quantum phase diagram of the hole-doped Hubbard model on square lattices. By implementing a diagonally oriented square lattice and GPU-accelerated DMRG with up to $48000$ states, we identify three distinct quantum phases across $δ= 5%$ to $15%$ doping: (i) A diagonal stripe phase with short-range uniform superconductivity (SC) at lower doping $δ\lesssim 9%$; (ii) An intermediate holon Wigner crystal (WC*) phase exhibiting bidirectional charge-density order and short-range SC with spatial oscillating correlations; (iii) An unprecedented infinite-length stripe (i-stripe) phase at $δ\gtrsim 12%$ hosting long stripes spanning the whole lattice. Remarkably, as doping increases, the short-range SC in WC* phase evolves into a 2D-like pair density wave (PDW) with divergent susceptibility in the i-stripe phase, constituting probably the first controlled numerical evidence of dominant PDW in the single-band square-lattice Hubbard model. The established 2D-like PDW and its interplay with charge orders provide new perspectives on dynamical layer decoupling phenomena in cuprates and multifaceted relationships between charge, spin and SC orders in quantum materials.

💡 Research Summary

In this work the authors perform large‑scale density‑matrix renormalization group (DMRG) calculations, accelerated by GPUs, on a hole‑doped single‑band Hubbard model defined on a square lattice that has been rotated by 45° (a “diagonal” lattice). By exploiting the unique symmetries of this geometry—its built‑in d‑wave nodal line, the presence of multiple quantized momenta that intersect the nodal line, and the ability to host stripes that extend across the entire cylinder—the study overcomes several limitations that have hampered previous investigations of two‑dimensional superconductivity in Hubbard‑type models.

The simulations are carried out on cylinders with open boundaries along the diagonal direction (ê₁) and periodic boundaries along the orthogonal direction (ê₂), with widths L₂ = 6 and 8 and lengths up to L₁ = 20. The on‑site repulsion is set to U = 12 (with additional data for U = 8), and next‑nearest‑neighbor hopping t′ is taken negative (t′ = −0.1 … −0.3) to mimic cuprate band structures. Up to 48 000 states are retained in each DMRG block, yielding truncation errors below 5 × 10⁻⁵.

Across the hole‑doping range δ = 5 %–15 % three distinct ground‑state phases emerge:

1. Diagonal‑stripe phase (δ ≲ 9 %) – Charge and spin densities form diagonal stripes with ordering vector Q_c = Q_s = (4πδ, 4πδ). The stripes are short (length √2 L₂) and coexist with short‑range d‑wave superconducting (SC) correlations that decay rapidly (exponent K_sc ≈ 2.5).

2. Holon Wigner‑crystal (WC) phase (δ ≈ 10 %)* – Holes crystallize into a bidirectional charge‑density wave (CDW) with wavevectors Q_ch,1 = (0, 2π/3) and Q_ch,2 ≈ (2π/3 − δQ, δQ). Spin remains unidirectional, evidencing spin‑charge separation. SC correlations are still short‑ranged but develop a sign‑alternating oscillation at longer distances, a precursor to pair‑density‑wave (PDW) order.

3. Infinite‑length‑stripe (i‑stripe) phase (δ ≳ 12 %) – Charge density organizes into period‑3 stripes that run the full length of the cylinder, breaking translation symmetry along ê₁ but restoring it along ê₂. Spin exhibits antiphase domain walls across each stripe. In this regime the SC pair‑pair correlator along the stripe direction follows
   Φ_xx(x) ≈ A x^{−K_sc} cos(Q_p x + θ)
   with K_sc ≈ 1.6 and Q_p ≈ 0.55 π (wavelength ≈ 3.5 lattice spacings). The power‑law envelope indicates quasi‑long‑range order, and the exponent predicts a divergent SC susceptibility χ_sc ∝ T^{−(2−K_sc)} as T → 0. Correlations perpendicular to the stripes display the same exponent, confirming that the superconductivity is genuinely two‑dimensional rather than a collection of decoupled one‑dimensional chains.

The PDW wavevector Q_p matches the x‑component of the CDW vector Q_ch,2, suggesting a tight coupling between charge‑density fluctuations and the emergent PDW. Detailed analysis of various bond‑pair correlators (Φ_xx, Φ_xy, Φ_yx) demonstrates that the PDW possesses local d‑wave symmetry (Φ_xy ≈ −α Φ_xx with α < 1) and that the oscillatory components on adjacent stripes are in phase, consistent with the order‑parameter form Δ_SC(x,y) ∼ cos(Q_p x + θ)