[Paper Review] Charge Order in the half-filled bond-Holstein Model

Charles Jordan, George Issa+4 more|arXiv (Cornell University)|Jan 19, 2026

Organic and Molecular Conductors Research0 citations

TL;DR

The authors study the half-filled bond-Holstein model on a square lattice using determinant quantum Monte Carlo and machine learning, finding a stronger CDW tendency and higher CDW transition temperature than the site-Holstein model, with Tc scaling insights from the atomic limit.

ABSTRACT

We use determinant quantum Monte Carlo to study the half-filled `bond-Holstein' model on a square lattice. We find that the model exhibits a charge-density-wave (CDW) phase transition with a critical temperature $T_\mathrm{cdw}$ considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor $S_{ m cdw}$, we obtain $T_\mathrm{cdw}$ to greater than one percent accuracy. At the same time, local observables also show clear signatures consistent with the transition temperatures inferred from our scaling analysis. We attribute the enhanced CDW tendencies to a phonon-mediated nearest-neighbor electron repulsion that is directly proportional to the dimensionless electron-phonon coupling $λ$ in the atomic ($t ightarrow 0$) limit. This behavior contrasts with the site-Holstein case, where the same limit yields only an on-site attraction. We supplement our analysis with results from several unsupervised machine learning methods, which not only confirm our estimates of $T_\mathrm{cdw}$ but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.

Motivation & Objective

• Motivate the study of electron-phonon interactions beyond the canonical site-Holstein model by exploring bond-based phonons.
• Quantify the CDW transition temperature and its dependence on the electron-phonon coupling in the bond-Holstein model.
• Compare bond-Holstein and site-Holstein CDW behavior, highlighting the role of intersite phonon-mediated interactions.
• Characterize high-temperature crossovers and bipolaron tendencies using both traditional observables and unsupervised ML methods.

Proposed method

• Define and contrast the site-Holstein and bond-Holstein Hamiltonians with explicit coupling forms.
• Use determinant quantum Monte Carlo (DQMC) with hybrid Monte Carlo and global swap updates to sample phonon fields.
• Compute the charge-density-wave structure factor S_cdw(q) and perform finite-size scaling to extract T_cdw.
• Analyze local observables (kinetic energy, electron-phonon energy, double occupancy) across temperatures.
• Supplement standard analysis with unsupervised machine learning methods (PCA, t-SNE, learning by confusion) to identify phase boundaries and crossovers.

Experimental results

Research questions

• RQ1 What is the CDW transition temperature T_cdw in the half-filled bond-Holstein model on a square lattice?
• RQ2 How does T_cdw in the bond-Holstein model compare to the site-Holstein model at equivalent couplings?
• RQ3 What is the nature of the high-temperature crossover between metallic and bipolaron regimes, and can ML methods detect it?
• RQ4 How does the t=0 atomic limit inform the CDW behavior and Tc in the bond-Holstein model?
• RQ5 Can unsupervised ML approaches reliably locate phase boundaries and crossovers in this system?

Key findings

• The bond-Holstein model exhibits a charge-density-wave phase transition at a higher temperature than the site-Holstein model for the same coupling, e.g., T_cdw ≈ 0.63 t for bond versus ≈ 0.248 t for site at the illustrated parameter set.
• In the atomic limit, the bond-Holstein model yields a nonzero T_cdw ≈ 1.76 |U_eff|, unlike the site-Holstein model where CDW order does not persist at t=0.
• The enhanced CDW tendency is attributed to a phonon-mediated nearest-neighbor electron repulsion that scales with the dimensionless coupling λ in the atomic limit.
• Local observables (kinetic energy, e-ph energy, double occupancy) show clear signatures aligned with the inferred transition temperatures and CDW ordering.
• Unsupervised ML methods (PCA, t-SNE, LBC, and a reference method) corroborate the Tcw estimates and reveal high-temperature crossovers between metallic and bipolaron liquid regimes.