Superfluid transition of bond bipolarons with long-range Coulomb repulsion in two dimensions

Using numerically exact diagrammatic Monte Carlo simulations in the two-electron (single-bipolaron) sector, we explore the impact of long-range Coulomb repulsion on the dilute-limit Berezinskii–Kosterlitz–Thouless (BKT) transition temperature $T_c$ of bipolarons on a two-dimensional square lattice. We study the bond Su–Schrieffer–Heeger model, in which bond phonons modulate the electron hopping. In the absence of long-range repulsion, this model was shown to support small, light bipolarons with a comparatively high transition temperature \cite{PhysRevX.13.011010}. Here we find that long-range Coulomb repulsion suppresses the optimal $T_c$ but leaves it appreciable over a broad parameter window, including the adiabatic regime $ω/t=0.5$ at a representative Coulomb strength $V=U/10$ (with $U$ the on-site repulsion). Our results provide controlled single-bipolaron inputs for dilute-limit $T_c$ estimates in the presence of long-range repulsion.

💡 Research Summary

This paper investigates how long‑range Coulomb repulsion influences the Berezinskii–Kosterlitz–Thouless (BKT) transition temperature of bipolarons in a two‑dimensional square‑lattice bond Su–Schrieffer–Heeger (SSH) model. Using numerically exact diagrammatic Monte‑Carlo (DiagMC) simulations confined to the two‑electron sector, the authors extract the bipolaron dispersion, binding energy, effective mass (m*BP), and mean‑square radius (R²BP) for various electron‑phonon coupling strengths (λ), phonon frequencies (ω/t), and Coulomb strengths (V). The SSH coupling is parameterized by the dimensionless λ=g²/(2Dtω) (D=2), while the long‑range repulsion follows Vij=V/|ri−rj|. The on‑site Hubbard repulsion is fixed at U/t=8, a value known to maximize the transition temperature in the absence of long‑range interactions.

From the single‑bipolaron data the authors construct a dilute‑limit estimate of the BKT transition temperature. In a low‑density 2D Bose gas the BKT temperature is approximately Tc≈1.84 ρBP/m*BP, where ρBP is the bipolaron density. To account for Coulomb‑induced suppression they introduce a multiplicative factor C=0.85 (C=1 for V=0). The optimal density is taken as ρBP≈1/(πR²BP) when the bipolaron size exceeds the lattice spacing, leading to the compact formula (Eq. 10):

• If R²BP ≥ 1: Tc≈C·0.5·m*BP / R²BP
• If R²BP < 1: Tc≈C·0.5·m*BP

The authors first present results for the adiabatic ratio ω/t=1.0. For V=0 the Tc/ω curve shows a dome peaking at λ≈0.8 with Tc/ω≈0.12. Introducing V=U/20 and V=U/10 shifts the dome to larger λ and reduces the peak by roughly 30–40 %, yet the maximum remains above the conventional McMillan reference (Tc/ω≈0.05). In this regime the Coulomb repulsion weakens binding (ΔBP decreases) but paradoxically reduces the effective mass in parts of the parameter space, while expanding the bipolaron size. The product m*BP·R²BP therefore changes only modestly, explaining the relatively mild Tc suppression.

The study then turns to the more adiabatic case ω/t=0.5. Here the impact of Coulomb repulsion is stronger. For V≤U/10 the Tc/ω dome persists with peak values around 0.07–0.09, indicating that even deep in the adiabatic regime the SSH mechanism can sustain sizable superfluid stiffness. However, for V=U/4 the dome collapses: binding only appears at λ ≈ 1.3, the bipolaron becomes extremely compact (R²BP≈1) while its mass skyrockets (m*BP an order of magnitude larger than for weaker V). Consequently Tc/ω falls below 0.04, well under the McMillan benchmark.

Key insights emerge: (i) long‑range Coulomb repulsion pushes the onset of bipolaron formation to stronger electron‑phonon coupling, (ii) it expands the spatial extent of the pair but can simultaneously lighten its effective mass, and (iii) the BKT transition temperature in the dilute limit is governed by the ratio m*BP/R²BP, so competing trends can partially cancel, preserving a non‑negligible Tc even with appreciable repulsion.

Overall, the work demonstrates that bond‑SSH coupling remains a viable route to high‑temperature bipolaronic superfluidity in two dimensions, despite the unavoidable presence of long‑range Coulomb forces. The authors suggest future extensions to finite‑density simulations, inclusion of phonon dispersion, and material‑specific parameter mapping to bridge the gap between model calculations and experimental realizations.