Quantum Monte Carlo Simulation of Bipolaron Superconductivity in Extended Hubbard--Holstein models on Face-Centered-Cubic and Body-Centered-Cubic Lattices

We investigate superlight pairing of bipolarons driven by electron-phonon interactions (EPIs) in face-center-cubic (FCC) and body-center-cubic (BCC) lattices using a continuous-time path-integral quantum Monte Carlo (QMC) algorithm. The EPIs are of the Holstein and extended Holstein types, and a Hubbard interaction is also included. Effects of adiabaticity are calculated. The number of phonons associated with the bipolaron, inverse mass, and radius are calculated and used to construct a phase diagram for bipolaron pairing (identifying the regions of pairing into intersite bipolarons and onsite bipolarons). From the inverse mass we determine that for the extended interaction, there is a region of light pairing associated with intersite bipolarons formed in both BCC and FCC lattices. Intersite bipolarons in the extended model at intermediate phonon frequency and large Coulomb repulsion become superlight due to first order hopping effects. We estimate the transition temperature, determining that intersite bipolarons are associated with regions of high transition temperatures.

💡 Research Summary

This paper presents a comprehensive quantum Monte‑Carlo (QMC) investigation of bipolaron formation and superconductivity in face‑centered‑cubic (FCC) and body‑centered‑cubic (BCC) lattices within extended Hubbard–Holstein models. The authors employ a continuous‑time path‑integral QMC algorithm to treat two electrons on a 20³ lattice at inverse temperature βt = 20, exploring both the conventional Holstein‑Hubbard model (HHM) with on‑site electron‑phonon coupling and the extended Holstein‑Hubbard model (EHHM) where the electron‑phonon interaction extends to nearest‑neighbour sites. In addition to the hopping amplitude t, the Hamiltonian includes a phonon frequency ω, a dimensionless electron‑phonon coupling g₍ᵢⱼ₎, and an on‑site Hubbard repulsion U.

The theoretical framework begins with a Lang‑Firsov transformation in the anti‑adiabatic limit (ℏω ≫ t) to obtain effective on‑site (U′ = U − 2Wλ) and nearest‑neighbour (V′ = −2Wλ Φ_NN/Φ₀₀) interactions, where λ = Φ₀₀ W is the dimensionless electron‑phonon strength, W = z t the half‑bandwidth, and Φ encodes the phonon‑mediated coupling. For the HHM, the electron‑phonon deformation is strictly local, leading to an exponential increase of the single‑polaron mass m* ∝ exp(γWλ/ℏω) with γ = 1. Consequently, bipolaron motion is a second‑order process: the on‑site singlet (S₀) bipolaron moves via virtual intermediate states, giving a bipolaron mass m** ∝ (m*)² Δ, where Δ is the binding energy. This makes HHM bipolarons extremely heavy and unsuitable for high‑temperature superconductivity.

In contrast, the EHHM spreads the lattice deformation over neighbouring sites, reducing γ < 1 and producing “light” polarons. When the Hubbard repulsion U is sufficiently large, the two electrons are forced onto adjacent sites, forming an intersite singlet (S₁). On the FCC lattice the geometry permits the S₁ bipolaron to hop in first order (single‑electron hopping t) because a closed three‑hop loop exists that returns the pair to its original configuration. This yields a “super‑light” bipolaron with m** ∝ m*, dramatically lighter than the HHM case. On the BCC lattice a similar but less pronounced effect occurs; the S₁ bipolaron still moves primarily via second‑order processes, so the mass reduction is modest.

The QMC implementation uses binary kink updates (insertion/removal of electron hops) and, crucially for the FCC lattice, a three‑kink update that respects the l₁ + l₂ + l₃ = 0 constraint. This ensures ergodicity because certain configurations with an odd number of kinks cannot be reached by binary updates alone. Twisted boundary conditions are applied to the path ends to extract the effective bipolaron mass from the winding number statistics.

Simulation results are presented across a grid of electron‑phonon coupling λ and Hubbard repulsion U for both lattice types and for both Holstein and extended Holstein interactions. Key observables include (i) the average number of phonons bound to the bipolaron, (ii) the bipolaron radius (derived from the spatial correlation of the two electrons), and (iii) the inverse effective mass (or mobility).

Findings:

1. In the EHHM, increasing λ raises the phonon cloud size but simultaneously reduces the bipolaron radius for the S₁ configuration, indicating a compact intersite pair.
2. The inverse mass maps reveal a distinct region of “light” or “super‑light” bipolarons at intermediate phonon frequencies (ℏω ≈ t) and large U. This region is broader for the FCC lattice because the three‑hop loop enables first‑order hopping.
3. Using the ideal‑gas Bose‑Einstein condensation formula T_BEC ≈ 3.31 ℏ² n^{2/3}/(k_B m**) and a close‑packing estimate for the bipolaron volume, the authors estimate transition temperatures T* that are an order of magnitude larger for the super‑light FCC S₁ bipolarons (T* ≈ 0.1–0.2 t, corresponding to tens of kelvin for realistic hopping scales) than for HHM bipolarons, whose T* is essentially zero.

The authors argue that these results provide a microscopic basis for the relatively high T_c observed in alkali‑doped fullerides (e.g., Cs₃C₆₀), which can adopt both BCC (A15) and FCC structures under pressure. The extended electron‑phonon interaction, combined with strong on‑site repulsion, yields intersite bipolarons that are both light and tightly bound, a prerequisite for high‑temperature superconductivity in the real‑space bipolaron (BEC) regime.

Limitations are acknowledged: the study treats only the two‑electron problem, assuming dilute bipolaron density so that bipolaron‑bipolaron interactions can be neglected (estimated to introduce < 40 % error in T_c). Moreover, the simulations are performed at a fixed inverse temperature βt = 20 and on finite lattices, so extrapolation to the true zero‑temperature limit and to higher carrier concentrations requires further work.

In summary, the paper demonstrates that extending the electron‑phonon coupling beyond a single site dramatically lightens bipolarons, especially on FCC lattices where geometry enables first‑order hopping of intersite pairs. This “super‑light” bipolaron regime leads to substantially higher Bose‑Einstein condensation temperatures, suggesting a viable pathway toward designing or discovering new high‑T_c superconductors based on strong, non‑local electron‑phonon interactions.