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Physical review. B./Physical review. B

Comprehensive study of bond bipolaron superconductivity on the triangular lattice

Chao Zhang

Published Nov 25, 2025• 2 citations
Publisher

Abstract

We employ the diagrammatic Monte Carlo method with a lattice path-integral formulation for both electron and phonon degrees of freedom to investigate the formation and properties of bond polarons and bipolarons on a two-dimensional triangular lattice. In the deep adiabatic regime ($\ensuremath{\omega}/t<1.0$), single polarons remain light with a small effective mass, while bipolarons remain compact and lightweight, resulting in a high superfluid transition temperature ${T}_{c}$. We systematically study the dependence of ${T}_{c}$ on electron-phonon coupling strength and on-site interaction in the bond bipolaron model. Our results show that a moderate on-site repulsion enhances ${T}_{c}$ by stabilizing compact yet lightweight bipolarons, leading to high-${T}_{c}$ superconductivity in the dilute limit. Notably, the triangular lattice sustains relatively high ${T}_{c}$ across a wide range of phonon frequencies, outperforming square lattice geometry. This enhancement arises from the higher coordination number and the bond-centered nature of the electron-phonon coupling. These findings suggest that triangular lattice geometry offers a promising platform for realizing high-${T}_{c}$ bipolaronic superconductivity.

DOI
10.1103/ckbn-jp9t

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We employ the diagrammatic Monte Carlo method with a lattice path-integral formulation for both electron and phonon degrees of freedom to investigate the formation and properties of bond polarons and bipolarons on a two-dimensional triangular lattice. In the deep adiabatic regime (\ensuremathω/t<1.0\ensuremath{\omega}/t<1.0\ensuremathω/t<1.0), single polarons remain light with a small effective mass, while bipolarons remain compact and lightweight, resulting in a high superfluid transition temperature Tc{T}_{c}Tc​. We systematically study the dependence of Tc{T}_{c}Tc​ on electron-phonon coupling strength and on-site interaction in the bond bipolaron model. Our results show that a moderate on-site repulsion enhances Tc{T}_{c}Tc​ by stabilizing compact yet lightweight bipolarons, leading to high-Tc{T}_{c}Tc​ superconductivity in the dilute limit. Notably, the triangular lattice sustains relatively high Tc{T}_{c}Tc​ across a wide range of phonon frequencies, outperforming square lattice geometry. This enhancement arises from the higher coordination number and the bond-centered nature of the electron-phonon coupling. These findings suggest that triangular lattice geometry offers a promising platform for realizing high-Tc{T}_{c}Tc​ bipolaronic superconductivity.
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 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 [C. Zhang, J. Sous, D. R. Reichman, M. Berciu, A. J. Millis, N. V. Prokof’ev, and B. V. Svistunov, Bipolaronic high-temperature superconductivity, ]. 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.
Polarons originating from phonon displacement modulated hopping have relatively light masses and thus are of significant current interest as candidates for the bipolaron mechanism of high-temperature superconductivity [Sous et al., Phys. Rev. Lett. 121, 247001 (2018)]. We observe that the bond model, when the dominant coupling comes from atomic vibrations on lattice bonds, can be solved by efficient sign-free Monte Carlo methods based on the path-integral formulation of the particle sector in combination with either the (real-space) diagrammatic or Fock-path-integral representation of the phonon sector. We introduce the corresponding algorithms and provide illustrative results for singlet bipolarons in two dimensions. The data suggest that the route towards high-temperature superconductivity (if any) in the multiparametric space of the model lies between the Scylla of large size of moderately light bipolarons and the Charybdis of large mass of compact bipolarons. As a result, on-site repulsion is helping sss-wave superconductivity in sharp contrast with existing expectations.
We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion UUU and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large UUU at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and UUU. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and∕or electron-electron interaction.
We study a lattice bipolaron on a staggered triangular ladder and triangular and hexagonal lattices with both long-range electron-phonon interaction and strong Coulomb repulsion using a novel continuous-time quantum Monte Carlo algorithm to solve the two-particle Coulomb-Fröhlich model. The algorithm is preceded by an exact integration over phonon degrees of freedom, and as such is extremely efficient. The bipolaron effective mass and radius are computed. Bipolarons on lattices constructed from triangular plaquettes have a novel crablike motion, and are small but very light over a wide range of parameters. We discuss the conditions under which such particles may form a Bose-Einstein condensate with high transition temperature, proposing a route to room temperature superconductivity.
We compute the properties of singlet and triplet bipolarons on two-dimensional lattices using the continuous-time quantum Monte Carlo algorithm. Properties of the bipolaron including the total energy, inverse mass, bipolaron radius, and number of phonons associated with the bipolaron demonstrate the qualitative difference between models of electron-phonon interaction with long-range interaction (screened Fr\"ohlich) and those with purely local (Holstein) interaction. A major result of our survey of the parameter space is the existence of extra-light hybrid singlet bipolarons consisting of an on-site and an off-site component on both square and triangular lattices. We also compute triplet properties of the bipolarons and the pair dispersion. For pair momenta on the edge of the Brillouin zone of the triangular lattice, we find that triplet states are more stable than singlets.
We review a series of exact, analytical and numerical results obtained on the adiabatic Holstein–Hubbard model, many of which are new and non-trivial. We study next the role of the quantum lattice fluctuations that were initially neglected. The possibility of having high- T c bipolaronic superconductors is analysed on the basis of these results. We suggest that models that involve only electron–phonon coupling are very unlikely to produce bipolaronic superconductivity, which is prevented by the spatial ordering of the bipolarons associated with a lattice instability. This is due to a very large effective mass of the bipolarons that can be related to the large Peierls–Nabarro energy barrier required to move these bipolarons through the lattice in the adiabatic limit. We conjecture that high- T c superconductivity originates specifically from the exceptionally well-balanced competition between electron–phonon coupling and electron–electron repulsion. In the restricted region in which, within the mean field approach, the energy of a bipolaron is close to those of two polarons, a new type of electron pairing occurs by formation of pairs of polarons in the spin singlet state. Such a polaron pair, called a spin resonant bipolaron (SRB), is not the standard bipolaron (both could exist in the adiabatic case). Its Peierls–Nabarro energy barrier can be sharply depressed, almost to zero. As a result of quantum lattice fluctuations, its tunnelling energy is sharply enhanced. Then, superconductivity could persist for a relatively large electron–phonon coupling, with an unusually large critical temperature as a Bose condensate of SRBs before becoming, at larger coupling and in any case, an insulating spin–Peierls polaronic phase.