[Paper Review] Density-matrix-renormalization-group-based downfolding of the three-band Hubbard model: the importance of density-assisted hopping

Shengtao Jiang, D. J. Scalapino+1 more|arXiv (Cornell University)|Mar 1, 2023

Physics of Superconductivity and Magnetism54 references9 citations

TL;DR

The paper uses DMRG-computed natural orbitals to downfold the three-band Hubbard (Emery) model to a single-band model, revealing a substantial density-assisted hopping term (t_n) that significantly affects hole-doped pairing and mobility beyond mean-field expectations.

ABSTRACT

Typical Wannier-function downfolding starts with a mean-field or density functional set of bands to construct the Wannier functions. Here we carry out a controlled approach, using DMRG-computed natural orbital bands, to downfold the three-band Hubbard model to an effective single band model. A sharp drop-off in the natural orbital occupancy at the edge of the first band provides a clear justification for a single-band model. Constructing Wannier functions from the first band, we compute all possible two-particle terms and retain those with significant magnitude. The resulting single-band model includes two-site density-assisted hopping terms with $t_n \sim 0.6 t$. These terms lead to a reduction of the ratio $U/t_{ m eff}$, and are important in capturing the doping-asymmetric carrier mobility, as well as in enhancing the pairing in a single-band model for the hole-doped cuprates.

Motivation & Objective

• Justify downfolding from the three-band Hubbard model to a single-band model using DMRG-derived natural orbitals.
• Identify and retain significant two-particle terms in the Wannier transformation.
• Show that density-assisted hopping terms renormalize effective hopping and influence pairing.
• Demonstrate that these terms are essential for capturing doping asymmetry and superconducting tendencies.
• Assess limitations of mean-field treatments for the density-assisted hopping terms.

Proposed method

• Compute natural orbitals from DMRG for the three-band Hubbard model to identify occupied bands.
• Construct Cu-centered Wannier functions from the first natural band and derive a Wannier Hamiltonian.
• Transform the three-band Hamiltonian into the Wannier basis and truncate to significant single-particle and two-particle terms.
• Retain density-assisted hopping terms t_n and its neighbors (t_n', t_n'') based on magnitude criteria.
• Compare the downfolded single-band model (with t_n) to a mean-field effective Hubbard model (t_eff) to assess impact on mobility and pairing.
• Evaluate superconducting phase stiffness under edge-pair fields to gauge pairing tendencies.

Figure 1: (a): The three-band Hubbard model and our phase convention for the orbital basis. (b): Charge and spin structure on a $12\times 5$ cylinder at a hole doping $\sim 0.15$ . The length of the arrows and the diameter of the circles represent $\langle S^{z}\rangle$ and local doping, respectivel

Experimental results

Research questions

• RQ1 Can a single-band model accurately reproduce key features of the three-band Hubbard model for cuprates?
• RQ2 What is the role and magnitude of density-assisted hopping terms in the downfolded model?
• RQ3 How do t_n and related terms affect hole-doped mobility and pairing compared to a mean-field t_eff?
• RQ4 Do density-assisted hoppings qualitatively alter superconducting tendencies in hole-doped systems?
• RQ5 Is the downfolding robust across different three-band parameter regimes and system sizes?

Key findings

• A sharp drop-off in natural orbital occupancies after the first band justifies a single-band downfolding.
• The resulting Wannier single-band model includes sizable two-site density-assisted hopping terms t_n ~ 0.6 t, and related t_n' and t_n'' terms.
• Density-assisted hopping reduces the effective U/t ratio (via t_eff = t + t_n ⟨n⟩) and enhances hole mobility, promoting pairing.
• Mean-field treatment of t_n underestimates hole-pair mobility and cannot capture the enhanced pairingseen in the full t-t_n-U model.
• Edge-field tests show the t-t_n-U model exhibits substantially larger superconducting phase stiffness in hole-doped cases than the t_eff-U model.
• The downfolded model maintains similar WF overlaps across hole and electron doping, indicating the downfolding is governed by local physics.

Figure 2: At a hole doping of 0.15 (a): occupancies of the natural orbitals obtained by diagonalizing the single-particle correlation matrix $M_{\alpha\beta}=\sum_{\sigma}\langle C^{\dagger}_{\alpha\sigma}C_{\beta\sigma}\rangle$ , with $C^{\dagger}=\{d^{\dagger},p_{x}^{\dagger},p_{y}^{\dagger}\}$ .