Charge stripe manipulation of superconducting pairing symmetry transition

The study addresses how charge stripes—ubiquitous in cuprates, iron-based, nickelate, and kagome superconductors—intertwine with superconductivity and influence pairing symmetry. Prior observations show variable stripe periods (P) and amplitudes (V0) across materials and tunability by external parameters such as pressure and defects. The Hubbard model, a paradigmatic framework for strongly correlated systems, captures d-wave pairing, stripe order, and antiferromagnetism, as well as their interplay. While induced stripes with P = 4 can enhance d-wave pairing via strengthened antiferromagnetic (AFM) correlations and π phase shifts, a comprehensive understanding of how varying P and V0 affect pairing symmetry across systems has been lacking. The purpose here is to determine whether and how P sets the dominant pairing symmetry and whether tuning doping (δ) and V0 can drive transitions between d- and s-wave states, and to uncover the associated magnetic-correlation changes. This has implications for engineering superconductivity by manipulating charge inhomogeneity.

Charge and spin stripes have been extensively reported in unconventional superconductors: cuprates (e.g., La2CuO4, RBa2Cu3O6, Bi2Sr2CaCu2O8), iron-based (FeSe), nickelates (infinite-layer and Ruddlesden–Popper phases), and kagome metals (CsV3Sb5, CsCr3Sb5). The stripe period and intensity vary across materials and can be modified by external conditions (pressure, defects). The Hubbard model has reproduced d-wave pairing, stripe order, and AFM order, and studies suggest stripe formation can be sensitive to parameters and boundary conditions; alternatively, charge modulations can be externally imposed to probe superconductivity. Prior theoretical work with P = 4 modulations reported enhanced d-wave pairing due to strengthened AFM correlations and π-phase shifts. However, systematic understanding over general P and V0, and the conditions for competing s-wave or PDW states, remained open.

Models and setup: The two-dimensional Hubbard model on a square lattice is studied with nearest-neighbor hopping t (set to 1) and on-site repulsion U. A periodic charge-stripe potential V0 is applied to rows with iy modulo P = 0, introducing externally imposed charge stripes of period P and tunable amplitude V0. The chemical potential μ controls doping δ. Both step-like and cosine-like stripe modulations are considered (the latter further lowers the critical V0 for the transition). The focus is on a minimal single-dxy-band model; multi-band checks confirm robustness. Lattice sizes include L×L with periodic boundary conditions for DQMC (e.g., L = 9, 12, 15) and cylinders (6×3, 6×6) for DMRG/CPQMC.

Observables: Effective pairing interactions Pα for extended s-wave and d-wave channels are computed by subtracting uncorrelated contributions from pair susceptibilities. Pair-density-wave (PDW) tendencies are probed via momentum-resolved PDW structure factors; a peak at finite momentum indicates PDW. Spin susceptibility χ(q) at zero frequency (z-component) is evaluated to examine magnetic correlations. DMRG also computes effective zero-momentum pair-pair structure factors Ss and Sd and analyzes two-particle density matrices to extract dominant Cooper pair modes and condensate wave-function patterns.

DQMC: Finite-temperature determinant quantum Monte Carlo evaluates pairing interactions and χ(q). Typical parameters include U/t = 3–5, temperatures down to T = t/12 (with main results at T = t/5), and doping δ ∈ [0.1, 0.3]. Simulations use 8000 warm-up sweeps and 10,000–1,200,000 measurement sweeps, binned for jackknife error analysis; two local updates occur between measurements. The inverse temperature β is discretized with Δτ = 0.1, keeping Trotter errors below statistical uncertainties. Sign-problem diagnostics indicate average sign > 0.55 for most runs and > 0.4 for lower-T explorations, supporting reliability of conclusions.

DMRG: Zero-temperature density-matrix renormalization group on cylinders (e.g., 6×3, 6×6) with up to 8192 SU(2) states (~25,000 U(1) states) and truncation error < 1e-5 evaluates effective static pair-pair structure factors at q = 0 for s and d channels and identifies PDW via finite-q peaks. It also diagonalizes the two-particle density matrix to obtain dominant Cooper pair modes and real-space bond-sign patterns near on-stripe and inter-stripe regions.

CPQMC: Constrained path quantum Monte Carlo checks long-range superconducting correlations in the s-wave channel through distance-resolved pair-pair correlations with vertex corrections, corroborating potential long-range order within investigated parameter regimes.

Phase-space exploration: Principal focus is P = 3; P = 2 and P = 4 are also examined for comparison. Doping δ ranges 0.1–0.3; V0 from 0 to 8; U/t varied (e.g., 3–8). Lattice-size effects are assessed and found weak for key trends.

• Stripe-period control of pairing symmetry: The dominant superconducting pairing symmetry depends decisively on stripe period P. For P ≥ 4, d-wave prevails; for P ≤ 3, both s and d waves can occur, with a transition between them. P = 3 is the critical point for the d–s transition.
• d–s transition at P = 3: In DQMC at U/t = 4 and T = t/5, a clear transition from d-wave to s-wave appears for V0 exceeding a critical V0c ≈ 3.25 over a broad doping range δ ≈ 0.10–0.23. At V0 = 4, Ps surpasses Pd across 0 < δ ≤ 0.22; d-wave is fully suppressed up to δ ≈ 0.20 and converts into a d-wave PDW competing with s-wave. At V0 = 3, s-wave robustness increases relative to d-wave. At V0 = 8, d-wave remains more stable than s-wave for moderate δ (0.05 < δ ≤ 0.15). For sufficiently large δ, d-wave dominates over s-wave regardless of V0.
• Period dependence: For P = 2, a similar d–s transition occurs at even smaller V0c with a sharper transition slope. For P = 4, only d-wave appears in the same δ range; no d–s transition is seen. The P-dependent behavior is robust in single- and multi-band models.
• Ground-state confirmation and V0c–U scaling: DMRG on 6×3 and 6×6 cylinders shows a robust d–s transition at zero temperature. At δ = 0.111 and U/t = 8 on 6×6, Ss dominates for V0 ≲ 6.2, while Sd overtakes above ~6.2. Extracted V0c scales nearly linearly with U for both 6×3 and 6×6 systems; for U/t = 4, DMRG V0c values align with DQMC within finite-size differences.
• Temperature, U, and size effects: At δ = 0.3 (d-wave region), Pd increases upon cooling and is enhanced by larger V0 and larger U; lattice-size effects (L = 9, 12, 15) are weak. At δ = 0.18 (s-wave region), Ps is positive and strengthened by larger V0 (5–7), while Pd is negative at V0 = 5–6 and only becomes positive at V0 ≥ 7; s-wave remains more stable than PDW. CPQMC and DMRG indicate possible long-range s-wave order in the explored regime.
• Local selection rule from condensate wave function: DMRG condensate analysis shows stripe-induced domain walls generate s-wave pairing locally near on-stripe regions (horizontal and vertical bond signs equal) at moderate V0 and δ, while inter-stripe regions favor d-wave (opposite bond signs). Smaller P increases the fraction of s-wave-favoring regions, enabling global s-wave for P ≤ 3; for P ≥ 4, inter-stripe d-wave dominance yields global d-wave.
• Magnetism–pairing interplay: In the d-wave regime (e.g., δ = 0.3), χ(q) exhibits strong AFM correlations at (π,π) that are enhanced as V0 increases, with additional incommensurate subpeaks at q1 = π ± π/P (2π/3 and 4π/3 for P = 3) that weaken with larger V0. In the s-wave regime (e.g., δ = 0.18), AFM correlations at (π,π) are weakened; χ(q) shows a dumbbell distribution that becomes more pronounced with increasing V0. Temperature dependences of (π,π) magnetism and dominant pairing track each other, evidencing intertwined pairing-symmetry and magnetic-correlation transitions.
• Modulation style: Cosine-like stripe modulation further reduces V0c for the d–s transition compared to step-like modulation.
• Robustness: Findings hold across temperatures down to T = t/12, various lattice sizes, and in multi-band checks, indicating generality for P-dependent superconducting systems.

The results demonstrate that externally imposed charge stripes can be used to engineer superconducting pairing symmetry via control of stripe period P, amplitude V0, and carrier doping δ. A stripe-induced local selection rule—s-wave favored near domain walls (on-stripe), d-wave in inter-stripe areas—explains why P = 3 is a critical threshold and why s-wave emerges for P ≤ 3 when charge modulation is sufficiently strong. The d–s transition is accompanied by a magnetic-correlation transition: AFM fluctuations at (π,π) are enhanced in the d-wave regime but suppressed in the s-wave regime, with distinct momentum-space signatures. The near-linear dependence of V0c on U provides a quantitative guideline for materials or experiments where U and charge inhomogeneity can be estimated or tuned. The conclusions are robust to model variations (single vs multi-band, modulation profiles) and suggest that PDW tendencies can arise where d-wave is suppressed; however, s-wave remains more stable in the explored regimes, offering an explanation for the difficulty of detecting PDW in some nickelates. Overall, the work highlights charge-stripe engineering as a route to manipulate and understand pairing mechanisms across different unconventional superconductors.

This study establishes that the charge-stripe period governs the dominant superconducting pairing symmetry in an inhomogeneous Hubbard model: d-wave for P ≥ 4 and a tunable competition between d- and extended s-wave for P ≤ 3, with P = 3 as a critical point. By adjusting doping and stripe amplitude, a robust d–s transition is achieved; the transition threshold V0c scales nearly linearly with U. The underlying mechanism is a stripe-induced local selection rule for pairing—s-wave near on-stripe domain walls and d-wave in inter-stripe regions—leading to global symmetry control via P. The d–s transition is intertwined with a magnetic-correlation change, providing a coherent picture linking charge, spin, and superconductivity. Future directions include experimental realization of stripe-engineered pairing symmetry through pressure/defect tuning, exploration of broader modulation profiles and multi-band effects, investigation of long-range order and PDW stability across larger systems and lower temperatures, and studying the coupling to spin stripes and other competing orders.

• The model assumes pre-existing charge stripes and does not address the microscopic origin of stripe formation.
• Finite-size effects: DMRG uses cylinder geometries (e.g., 6×3, 6×6) and DQMC finite lattices (L up to 15). While trends are robust and size effects appear weak, thermodynamic-limit extrapolation is indirect.
• Sign problem in DQMC limits accessible temperatures and parameter ranges; most results are at T = t/5 with checks at T = t/12.
• PDW is inferred from finite-q peaks in structure factors and remains subdominant to s-wave in explored regimes; definitive experimental correspondence requires further validation.
• Spin stripes and other competing orders are not explicitly included, and only the z-component of spin susceptibility at zero frequency is analyzed.
• The minimal single-band focus, though cross-checked with multi-band models, may not capture all material-specific complexities.