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arXiv:2112.14744 · Physical review. B./Physical review. B

Robustness of antiferromagnetism in the Su-Schrieffer-Heeger Hubbard model

Xun Cai, Zixiang Li, Hong Yao

Published Aug 18, 2022• 31 citations
arXivHTMLPDF

Abstract

We recently unveiled that antiferromagnetism (AFM) order can be dominantly induced by the bond Su-Schrieffer-Heeger (SSH) electron-phonon coupling (EPC), which is beyond the conventional wisdom that AFM order is usually driven by strong Coulomb interactions. Nevertheless, many aspects of the interplay between EPC and strong electronic interactions on AFM ordering remain unexplored. Here, we investigate the Su-Schrieffer-Heeger-Hubbard (SSHH) model with bond SSH phonons and on-site Hubbard interactions by large-scale quantum Monte Carlo (QMC) simulations and obtain its ground-state phase diagram for various Hubbard interactions and EPC strength at half filling. Our results show that Hubbard interactions further enhance EPC-induced AFM, especially for small phonon frequency or in adiabatic limit, the regime most relevant to various quantum materials. This could shed further light to understanding the cooperative effect between EPC and electronic correlations in quantum materials.

DOI
10.1103/physrevb.106.l081115

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•Sep 22, 2025•0 citations
We recently unveiled that antiferromagnetism (AFM) order can be dominantly induced by the bond Su-Schrieffer-Heeger (SSH) electron-phonon coupling (EPC), which is beyond the conventional wisdom that AFM order is usually driven by strong Coulomb interactions. Nevertheless, many aspects of the interplay between EPC and strong electronic interactions on AFM ordering remain unexplored. Here, we investigate the Su-Schrieffer-Heeger-Hubbard (SSHH) model with bond SSH phonons and on-site Hubbard interactions by large-scale quantum Monte Carlo (QMC) simulations and obtain its ground-state phase diagram for various Hubbard interactions and EPC strength at half filling. Our results show that Hubbard interactions further enhance EPC-induced AFM, especially for small phonon frequency or in adiabatic limit, the regime most relevant to various quantum materials. This could shed further light to understanding the cooperative effect between EPC and electronic correlations in quantum materials.
Antiferromagnetism (AFM) such as Néel ordering is often closely related to Coulomb interactions such as Hubbard repulsion in two-dimensional (2D) systems. Whether Néel AFM ordering in two dimensions can be dominantly induced by electron-phonon couplings (EPC) has not been completely understood. Here, by employing numerically exact sign-problem-free quantum Monte Carlo (QMC) simulations, we show that bond Su-Schrieffer-Heeger (SSH) phonons with frequency ω and EPC constant λ can induce AFM ordering for a wide range of phonon frequency ω>ω_{c}. For ω<ω_{c}, a valence-bond-solid (VBS) order appears and there is a direct quantum phase transition between VBS and AFM phases at ω_{c}. The phonon mechanism of the AFM ordering is related to the fact that SSH phonons directly couple to electron hopping whose second-order process can induce an effective AFM spin exchange. Our results shall shed new light on understanding AFM ordering in correlated quantum materials.
Antiferromagnetism (AF) such as Neel ordering is often closely related to Coulomb interactions such as Hubbard repulsion in two-dimensional (2D) systems. Whether Neel AF ordering in 2D can be dominantly induced by electron-phonon couplings (EPC) has not been completely understood. Here, by employing numerically-exact sign-problem-free quantum Monte Carlo (QMC) simulations, we show that optical Su-Schrieffer-Heeger (SSH) phonons with frequency ωωω and EPC constant λλλ can induce AF ordering for a wide range of phonon frequency ω>ωcω>ω_cω>ωc​. For $ω
The Hubbard and Su-Schrieffer-Heeger (SSH) Hamiltonians are iconic models for understanding the qualitative effects of electron-electron and electron-phonon interactions, respectively. In the two-dimensional square lattice Hubbard model at half filling, the on-site Coulomb repulsion UUU between up and down electrons induces antiferromagnetic (AFM) order and a Mott insulating phase. On the other hand, for the SSH model, there is an AFM phase when the electron-phonon coupling $\ensuremath{\lambda}$ is less than a critical value ${\ensuremath{\lambda}}_{c}$ and a bond order wave when $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$. In this Letter, we perform numerical studies on the square lattice optical Su-Schrieffer-Heeger-Hubbard Hamiltonian, which combines both interactions. We use the determinant quantum Monte Carlo method which does not suffer from the fermionic sign problem at half filling. We map out the phase diagram and find that it exhibits a direct first-order transition between an antiferromagnetic phase and a bond-ordered wave as $\ensuremath{\lambda}$ increases. The AFM phase is characterized by two different regions. At smaller $\ensuremath{\lambda}$ the behavior is similar to that of the pure Hubbard model; the other region, while maintaining long-range AFM order, exhibits larger kinetic energies and double occupancy, i.e., larger quantum fluctuations, similar to the AFM phase found in the pure SSH model.
The Su-Schrieffer-Heeger (SSH) model, with bond phonons modulating electron tunneling, is a paradigmatic electron-phonon model that hosts an antiferromagnetic order to bond order transition at half-filling. In the presence of a repulsive Hubbard interaction, the antiferromagnetic phase is enhanced, but the phase transition remains first order. Here, we explore the physics of the SSH model with an attractive Hubbard interaction, which hosts an interesting interplay among charge order, sss-wave pairing, and bond order. Using the numerically exact determinant quantum Monte Carlo method, we show that both charge order, present at weak electron-phonon coupling, and bond order, at large coupling, give way to sss-wave pairing when the system is doped. Furthermore, we demonstrate that the SSH electron-phonon interaction competes with the attractive Hubbard interaction and reduces the sss-wave pairing correlation.
Increasing numerical studies showed that the simplest Hubbard model on the square lattice with strong repulsion may not exhibit high-temperature superconductivity (SC). It is desired to look for other possible microscopic mechanism of realizing high-temperature SC . Here, we explore the interplay between the Su–Schrieffer–Heeger (SSH) electron–phonon coupling (EPC) and the Hubbard repulsion by density-matrix-renormalization-group (DMRG) simulations. Our state-of-the-art DMRG study showed convincingly that the interplay between strong Hubbard U and moderate Su–Schrieffer–Heeger EPC λ can induce robust d -wave SC. The SSH-type EPC can generate effective antiferromagnetic spin-exchange interactions between neighboring sites, which plays a crucial role in the interplay of inducing robust d -wave SC. Specifically, for U = 8 t , we find that d -wave SC emerges when λ > λ c with a moderate critical value λ c = 0.1–0.2. Our results might shed new light to understanding high-temperature SC in cuprates as well as pave a possible new route in looking for high-temperature SC in other quantum materials with both strong U and moderate λ .
This thesis investigates the interplay between electron-electron and electron-phonon interactions in low-dimensional lattice models, specifically within the Hubbard model incorporating Su-Schrieffer-Heeger (SSH) coupling with phonons. SSH interactions, which modulate the electron hopping amplitude through lattice displacements, have recently attracted renewed interest due to their potential relevance in strongly correlated materials. The main objective of this work is to determine whether the inclusion of SSH couplings in the Hubbard model can stabilize superconducting or spin-gapped metallic states, as well as other exotic phases of matter. While the half-filled SSH-Hubbard model has been extensively studied in one dimension and only partially examined in two dimensions, its phase diagram in the doped regime remains largely unexplored. This thesis aims to fill this gap, focusing on the emergence of correlated phases away from half-filling in both one and two spatial dimensions. We use the variational Monte Carlo method with a trial wave function defined in a hybrid electron-phonon Hilbert space. In addition to Jastrow factors that account for correlation effects, the wave function includes an antisymmetric component that captures backflow-inspired electron-phonon correlations. This approach enables us to accurately model both electron-electron and electron-phonon interactions. Moreover, since the variational method avoids the sign problem, we can simulate the system in the doped regime, an area typically inaccessible to exact quantum Monte Carlo methods (QMC), which are limited to half-filling. In one dimension, we observe the stabilization of a Luther-Emery liquid, a metallic phase with gapless charge excitations and a finite spin gap, upon doping a Peierls insulating phase. This finding resolves a long-standing debate regarding its realization in SSH chains. Benchmarking with density-matrix renormalization group results confirms the reliability of our approach. In two dimensions, we investigate the SSH-Hubbard model on the square lattice, obtaining results that are consistent with previous exact QMC studies at half-filling. Upon doping, we observe that SSH phonons can induce s-wave superconductivity in the weakly correlated regime. However, we suggest that the enhancement of d-wave superconductivity by SSH coupling in the strongly correlated regime may significantly depend on the specific implementation of the SSH interaction. In one particular case, we find that d-wave superconductivity is slightly suppressed by SSH coupling. These results are further confirmed by a Hartree-Fock analysis performed in the anti-adiabatic limit. Overall, this thesis highlights the crucial role of lattice effects in correlated electron systems, shedding light on the distinct contributions of SSH-type couplings in one and two dimensions. Our findings provide new insights into the microscopic mechanisms behind exotic metallic phases and set the stage for further studies of phonon-mediated phenomena in correlated systems.