By using the density-matrix renormalization-group numerical method, we investigate the bond order wave (BOW) and energy gap (EG) in a one-dimensional half-filling bond-charge attraction model, which can be mapped from the electron-phonon interacting Su-Schrieffer-Heeger (SSH) Hamiltonian at the antiadiabatic limit. The obtained results are compared with that of the SSH model at the adiabatic limit as well as that within the mean-field approximation. It is shown that a simple linear relation between the BOW and the EG is kept even at the antiadiabatic limit while the reduction of the BOW is much smaller than that of the EG.
By using the density-matrix renormalization-group numerical method, we investigate the bond order wave (BOW) and energy gap (EG) in a one-dimensional half-filling bond-charge attraction model, which can be mapped from the electron-phonon interacting Su-Schrieffer-Heeger (SSH) Hamiltonian at the antiadiabatic limit. The obtained results are compared with that of the SSH model at the adiabatic limit as well as that within the mean-field approximation. It is shown that a simple linear relation between the BOW and the EG is kept even at the antiadiabatic limit while the reduction of the BOW is much smaller than that of the EG.
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.
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, s-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 s-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 s-wave pairing correlation.
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.
This study explores the ground-state phase diagram and topological properties of the spinless 1D Su-Schrieffer-Heeger (SSH) model with nearest-neighbor (NN) interactions at quarter filling. We analyze key physical quantities such as the local electron density distribution, correlation functions for bond-order-wave (BOW) and charge-density-wave (CDW) -- by integrating twisted boundary conditions with the Lanczos technique and employing high-precision numerical diagonalization methods, complemented by a mean-field approximation (MFA) based on bond-order and charge-density modulation analysis. This approach enables precise identification of phase transition critical points. Our results indicate that the system exhibits a topologically trivial band insulating (BI) phase for strong attractive interactions, with its upper boundary forming a downward-opening curve peaking at V/t≃−2.3 and extending to V/t≃−2.6. Within −2.6≤V/t≤−0.5, a BOW phase emerges for ∣δt/t∣>0.45, with its boundaries converging as ∣δt/t∣ decreases, terminating at a single point at ∣δt/t∣≃0.45. In other parameter regions, a CDW phase is realized. Through this analysis, we elucidate the topological properties of the interacting spinless SSH model at quarter filling, highlighting the competition among CDW, BOW, and BI phases. By tuning V and δt, the system exhibits diverse correlated phenomena, offering new insights into one-dimensional quantum phase transitions and the interplay between topology and order.
Theoretical and computational studies of the quantum phase diagram of the one-dimensional half-filled extended Hubbard model (EHM) indicate a narrow bond-order wave (BOW) phase with finite magnetic gap Em for on-site repulsion $U<{U}^{\ensuremath{\ast}}$, the critical point, and nearest-neighbor interaction ${V}_{c}\ensuremath{\approx}U/2$ near the boundary of the charge-density wave (CDW) phase. Potentials with more extended interactions that retain the EHM symmetry are shown to have a less cooperative CDW transition with higher ${U}^{\ensuremath{\ast}}$ and wider BOW phase. Density-matrix renormalization group is used to obtain Em directly as the singlet-triplet gap, with finite Em marking the BOW boundary Vs(U). The BOW/CDW boundary Vc(U) is obtained from exact finite-size calculations that are consistent with previous EHM determinations. The kinetic energy or bond order provides a convenient new estimate of ${U}^{\ensuremath{\ast}}$ based on a metallic point at Vc(U) for $U<{U}^{\ensuremath{\ast}}$. Tuning the BOW phase of half-filled Hubbard models with different intersite potentials indicates a ground state with large charge fluctuations and magnetic frustration. The possibility of physical realizations of a BOW phase is raised for Coulomb interactions.
Density matrix renormalization group methods are used to investigate the quantum phase diagram of a one-dimensional half-filled ionic Peierls-Hubbard model at the antiadiabatic limit where quantum phonon fluctuations are taken into account partially. We found that two continuous phase transitions always exist from dimerized spin-gapped (bond-order-wave) state to band-insulator and undimerized spin-gapless (Mott-insulator) phase while undimerized spin-gapless phase vanishes at the adiabatic limit. Our results indicate that quantum phonon fluctuations, electron-electron interaction, and ionic potential combine in the formation of the bond-order wave phase.