[Paper Review] Robust d-wave superconductivity from the Su-Schrieffer-Heeger-Hubbard model: possible route to high-temperature superconductivity

Haoxin Wang, Yi‐Fan Jiang+1 more|arXiv (Cornell University)|Nov 16, 2022

Physics of Superconductivity and Magnetism7 citations

TL;DR

This paper proposes that the interplay between strong Hubbard repulsion (U) and moderate Su-Schrieffer-Heeger (SSH) electron-phonon coupling (λ) in the SSH-Hubbard model on a square lattice induces robust d-wave superconductivity (SC), which emerges when λ exceeds a critical value λc ≈ 0.1–0.2 at U = 8t. The SSH coupling generates effective antiferromagnetic spin exchange, enabling d-wave pairing despite the absence of SC in the pure Hubbard model at this U.

ABSTRACT

Increasing numerical studies showed that the simplest Hubbard model on the square lattice with strong repulsion may not exhibit high-temperature superconductivity (SC). It is desired to look for other possible microscopic mechanism of realizing high-temperature SC. Here, we explore the interplay between the Su-Schrieffer-Heeger (SSH) electron-phonon coupling (EPC) and the Hubbard repulsion by density-matrix-renormalization-group (DMRG) simulations. Our state-of-the-art DMRG study showed convincingly that the interplay between strong Hubbard $U$ and moderate Su-Schrieffer-Heeger EPC $λ$ can induce robust $d$-wave SC. The SSH-type EPC can generate effective antiferromagnetic spin-exchange interactions between neighboring sites, which plays a crucial role in the interplay of inducing robust $d$-wave SC. Specifically, for $U=8t$, we find that $d$-wave SC emerges when $λ>λ_c$ with a moderate critical value $λ_c=0.1\sim 0.2$. Our results might shed new light to understanding high-temperature SC in cuprates as well as pave a possible new route in looking for high-temperature SC in other quantum materials with both strong $U$ and moderate $λ$.

Motivation & Objective

• To investigate whether electron-phonon coupling (EPC) can stabilize d-wave superconductivity in the Hubbard model, where pure Hubbard models fail to produce d-wave SC at strong U.
• To determine the role of SSH-type EPC in generating effective antiferromagnetic spin exchange and enabling d-wave pairing.
• To establish the quantum phase diagram of the SSH-Hubbard model under finite hole doping (δ = 1/8) and identify conditions for robust d-wave SC.
• To clarify whether the interplay between strong electron-electron repulsion (U) and moderate EPC (λ) can overcome stripe order and phase separation, common in pure Hubbard models.
• To provide a new microscopic mechanism for high-temperature superconductivity in cuprates and other correlated materials by combining strong correlations and phonon-mediated pairing.

Proposed method

• Density-matrix-renormalization-group (DMRG) simulations on four-leg cylinders to study the ground state of the SSH-Hubbard Hamiltonian at finite doping (δ = 1/8).
• The model includes nearest-neighbor hopping (t), on-site Coulomb repulsion (U), SSH-type electron-phonon coupling (λ), and phonon degrees of freedom with frequency ωD = 5.
• The SSH EPC is modeled via a Holstein-like coupling that modulates hopping integrals based on local lattice distortions, generating effective spin-exchange interactions.
• Finite-size scaling and correlation function analysis (pair-pair, spin-spin, charge-density) are used to identify superconducting, CDW, and magnetic orders.
• Luttinger parameter Ksc ≈ 0.94 is extracted from power-law decay of SC correlations, indicating a divergent SC susceptibility and Luther-Emery-like behavior.
• Spin-spin and charge-density correlation functions are computed to characterize magnetic and charge order, with correlation lengths ξs = 3.54 and ξsc = 3.3 in the respective phases.

Experimental results

Research questions

• RQ1 Can the interplay between strong Hubbard repulsion (U) and moderate SSH electron-phonon coupling (λ) induce robust d-wave superconductivity in the square-lattice Hubbard model?
• RQ2 What is the critical value λc of the electron-phonon coupling at which d-wave SC transitions from a stripe-ordered phase in the SSH-Hubbard model at U = 8t?
• RQ3 How does the SSH-type EPC generate effective antiferromagnetic spin exchange that promotes d-wave pairing?
• RQ4 Does the d-wave SC phase exhibit characteristics of a Luther-Emery liquid, such as divergent SC susceptibility and power-law decay of correlations?
• RQ5 Can this mechanism explain the emergence of high-temperature d-wave superconductivity in cuprates, given their known strong U and moderate EPC?

Key findings

• For U = 8t, d-wave superconductivity emerges when the electron-phonon coupling λ exceeds a critical threshold λc ≈ 0.1–0.2, while stripe charge-density-wave (CDW) order dominates at λ < λc.
• The d-wave SC phase exhibits a Luttinger parameter Ksc ≈ 0.94, indicating a divergent superconducting susceptibility χsc ∼ T^{Ksc−2} as T → 0, consistent with a Luther-Emery liquid.
• The spin-spin correlation function decays exponentially with a finite correlation length ξs = 3.54, confirming a finite spin gap in the d-wave SC phase.
• The charge-density wave (CDW) order has a wavelength of 8 at δ = 1/8 and ordering momentum QCDW = 2π/8, with long-range order in the stripe phase at small λ.
• Finite-size scaling confirms dominant d-wave SC correlations and subdominant CDW correlations, with SC pair-pair correlation decaying as ∼ r^{−Ksc} with Ksc ≈ 0.94.
• The SSH-type EPC generates effective antiferromagnetic spin exchange interactions between neighboring sites, which are crucial for stabilizing d-wave pairing in the presence of strong U.