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.
Electron-phonon (
e-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∼1/8). 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
e-ph coupling is required to stabilize the BOW phase in the small
U 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) optical SSH model.
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
U 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.
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.
Over the past several years, a new generation of quantum simulations has greatly expanded our understanding of charge density wave phase transitions in Hamiltonians with coupling between local phonon modes and the on-site charge density. A quite different, and interesting, case is one in which the phonons live on the bonds, and hence modulate the electron hopping. This situation, described by the Su-Schrieffer-Heeger (SSH) Hamiltonian, has so far only been studied with quantum Monte Carlo in one dimension. Here we present results for the 2D SSH model, show that a bond ordered wave (BOW) insulator is present in the ground state at half filling, and argue that a critical value of the electron-phonon coupling is required for its onset, in contradistinction with the 1D case where BOW exists for any nonzero coupling. We determine the precise nature of the bond ordering pattern, which has hitherto been controversial, and the critical transition temperature, which is associated with a spontaneous breaking of Z_{4} symmetry.
Electron-phonon coupling is typically not associated with magnetic order but with either charge density waves (patterns of high and low electron density) or valence-bond solids (patterns of strong and weak electronic bonds). Using quantum Monte Carlo simulations, the authors demonstrate that, within the paradigmatic two-dimensional Su-Schrieffer-Heeger model, the electron-phonon interaction favors antiferromagnetic order in large regions of the phase diagram. A valence-bond solid is observed in the adiabatic regime, while the two phases are connected by an apparently direct phase transition.
Most nonperturbative numerical studies of electron-phonon interactions focus on model Hamiltonians where the electrons interact with a phonon branch via a single type of microscopic mechanism. Two commonly explored couplings in this context are the Holstein and Su-Schrieffer-Heeger (SSH) interactions, which describe phonons modulating the on-site energy and intersite electron hopping, respectively. Many materials, however, have multiple phonon branches that can each interact with electronic degrees of freedom in different ways. We present here a determinant quantum Monte Carlo study of the half-filled two-dimensional (bond) SSH-Holstein Hamiltonian, where electrons couple to different phonon branches via either the Holstein or SSH mechanism. We map the model's phase diagram and determine the nature of the transitions between charge-density wave, bond-order wave, and antiferromagnetic order.