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arXiv:2208.07036 · Physical review. B./Physical review. B

Functional renormalization group study of the two-dimensional Su-Schrieffer-Heeger-Hubbard model

Qing-Geng Yang, Da Wang, Qiang-Hua Wang

Published Dec 21, 2022• 17 citations
arXivHTMLPDF

Abstract

We study the Hubbard model on the square lattice coupled in addition to the optical Su-Schrieffer-Heeger (SSH) phonons, using the singular-mode functional renormalization group method. At half filling and in the absence of the Hubbard interaction $U$, we find the degenerate spin-density-wave--charge-density-wave--$s$-wave-superconductivity state at smaller electron-phonon coupling strength $\ensuremath{\lambda}$ and higher phonon frequency $\ensuremath{\omega}$, and the valence bond solid (VBS) state at larger $\ensuremath{\lambda}$ and lower $\ensuremath{\omega}$. After switching on a positive $U$, the VBS state is suppressed, while the spin-density wave (SDW) state is enhanced. At finite doping, the SSH phonon is found to favor $s$-wave superconductivity (sSC) at $U=0$. With increasing positive $U$, we find $d$-wave superconductivity (dSC) and incommensurate SDW states. In a narrow window of moderate $U$ and $\ensuremath{\lambda}$, we also find the incommensurate VBS state. The sSC and dSC here can be naturally related to the charge-density wave (CDW) and SDW fluctuations, both of which can be triggered by the SSH phonons. In contrast, the repulsive interaction $U$ enhances SDW but suppresses CDW fluctuations. Our results at half filling are consistent with quantum Monte Carlo (QMC) and provide insights at finite doping where QMC may suffer from the minus sign problem.

DOI
10.1103/physrevb.106.245136

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We study the Hubbard model on the square lattice coupled in addition to the optical Su-Schrieffer-Heeger (SSH) phonons, using the singular-mode functional renormalization group method. At half filling and in the absence of the Hubbard interaction UUU, we find the degenerate spin-density-wave--charge-density-wave--sss-wave-superconductivity state at smaller electron-phonon coupling strength \ensuremathλ\ensuremath{\lambda}\ensuremathλ and higher phonon frequency \ensuremathω\ensuremath{\omega}\ensuremathω, and the valence bond solid (VBS) state at larger \ensuremathλ\ensuremath{\lambda}\ensuremathλ and lower \ensuremathω\ensuremath{\omega}\ensuremathω. After switching on a positive UUU, the VBS state is suppressed, while the spin-density wave (SDW) state is enhanced. At finite doping, the SSH phonon is found to favor sss-wave superconductivity (sSC) at U=0U=0U=0. With increasing positive UUU, we find ddd-wave superconductivity (dSC) and incommensurate SDW states. In a narrow window of moderate UUU and \ensuremathλ\ensuremath{\lambda}\ensuremathλ, we also find the incommensurate VBS state. The sSC and dSC here can be naturally related to the charge-density wave (CDW) and SDW fluctuations, both of which can be triggered by the SSH phonons. In contrast, the repulsive interaction UUU enhances SDW but suppresses CDW fluctuations. Our results at half filling are consistent with quantum Monte Carlo (QMC) and provide insights at finite doping where QMC may suffer from the minus sign problem.
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.
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.
Abstract Using the singular-mode functional renormalization group method, we studied the doping-dependent phase diagram of the two-dimensional extended Hubbard model on square lattice. At half-filling, a variety of states strongly compete, including spin-density wave, charge-density wave, phase separation, as well as d -wave and s -wave superconductivity (SC). Upon electron doping, the system exhibits a strong preference for superconducting states. At slight dopings, the spin-density wave phase is strongly suppressed, and d -wave SC region expands into positive U , V region. At moderate dopings, a region of degenerated p -wave superconductivities emerge for positive U and negative V . Our results demonstrate how doping tunes the dominant fluctuations, suppressing density waves and stabilizing unconventional superconducting states in two dimensions.
Electron-phonon (eee-ph) interactions arise in many strongly correlated quantum materials from the modulation of the nearest-neighbor hopping integrals, as in the celebrated Su-Schrieffer-Heeger (SSH) model. Nevertheless, relatively few non-perturbative studies of correlated SSH models have been conducted in dimensions greater than one, and those that have been done have primarily focused on bond models, where generalized displacements independently modulate each hopping integral. We conducted a sign-problem free determinant quantum Monte Carlo study of the optical SSH-Hubbard model on a two-dimensional square lattice, where site-centered phonon modes simultaneously modulate pairs of nearest-neighbor hopping integrals. We report the model's low-temperature phase diagram in the challenging adiabatic regime (Ω/EF∼1/8Ω/E_\mathrm{F} \sim 1/8Ω/EF​∼1/8). It exhibits insulating antiferromagnetic Mott and bond-order-wave (BOW) phases with a narrow region of coexistence between them. We also find that a critical eee-ph coupling is required to stabilize the BOW phase in the small UUU limit. Lastly, in stark contrast to recent findings for the model's bond variant, we find no evidence for a long-range antiferromagnetism in the pure (U/t=0)(U/t=0)(U/t=0) optical SSH model.
We investigate the two-dimensional optical Su-Schrieffer-Heeger (SSH) model, in which the electron hopping amplitude is modulated by the difference between neighboring phonon coordinates. Using sign-problem-free auxiliary-field quantum Monte Carlo simulations, complemented by mean-field analysis, we determine the long-range ordered phases as a function of the electron-phonon coupling and phonon frequency. By examining both adiabatic and antiadiabatic regimes, we reveal the emergence of staggered and armchair valence bond solid (VBS) phases, as well as the O(4) antiferromagnetic phase. In addition, finite-temperature simulations show that the VBS transition occurs at critical temperatures significantly higher than in models with local electron-phonon coupling, consistent with the presence of lighter polarons in the metallic regime. These findings establish the ground-state and finite-temperature phase diagrams of the optical SSH model, which emphasize its similarities and contrasts with other electron-phonon systems.