The interplay of bond and charge correlations is studied in a one-dimensional model with both Holstein and Su-Schrieffer-Heeger (SSH) couplings to quantum phonons. The problem is solved exactly by quantum Monte Carlo simulations. If one of the couplings dominates, the ground state is a Peierls insulator with long-range bond or charge order. At weak coupling, the results suggest a spin-gapped and repulsive metallic phase arising from the competing order parameters and lattice fluctuations. Such a phase is absent from the pure SSH model even for quantum phonons. At strong coupling, evidence for a continuous transition between the two Peierls states is presented.
The interplay of bond and charge correlations is studied in a one-dimensional model with both Holstein and Su-Schrieffer-Heeger (SSH) couplings to quantum phonons. The problem is solved exactly by quantum Monte Carlo simulations. If one of the couplings dominates, the ground state is a Peierls insulator with long-range bond or charge order. At weak coupling, the results suggest a spin-gapped and repulsive metallic phase arising from the competing order parameters and lattice fluctuations. Such a phase is absent from the pure SSH model even for quantum phonons. At strong coupling, evidence for a continuous transition between the two Peierls states is presented.
We study one-dimensional Su-Schrieffer-Heeger (SSH) models with quantum phonons using a continuous-time quantum Monte Carlo method. Within statistical errors, we obtain identical results for the SSH model with acoustic phonons, and a related model with a coupling to an optical bond phonon mode. Based on this agreement, we first study the Peierls metal-insulator transition of the spinless SSH model, and relate it to the Kosterlitz-Thouless transition of a spinless Luttinger liquid. In the Peierls phase, the spectral functions reveal the single-particle and charge gap, and a central peak related to long-range order. For the spinful SSH model, which has a dimerized ground state for any nonzero coupling, we reveal a symmetry-related degeneracy of spin and charge excitations, and the expected spin and charge gaps as well as a central peak. Finally, we study the SSH-UV model with electron-phonon and electron-electron interaction. We observe a Mott phase with critical spin and bond correlations at weak electron-phonon coupling, and a Peierls phase with gapped spin excitations at strong coupling. We relate our findings to the extended Hubbard model, and discuss the physical origin of the agreement between optical and acoustic phonons.
The interplay between electron-electron and electron-phonon interactions is studied in a one-dimensional lattice model by means of a variational Monte Carlo method based on generalized Jastrow-Slater wave functions. Here, the fermionic part is constructed by a pair-product state, which explicitly depends on the phonon configuration, thus including the electron-phonon coupling in a backflow-inspired way. We report the results for the Hubbard model in the presence of the Su-Schrieffer-Heeger coupling to optical phonons, both at half filling and upon hole doping. At half filling, the ground state is either a translationally invariant Mott insulator, with gapless spin excitations, or a Peierls insulator, which breaks translations and has fully gapped excitations. Away from half filling, the charge gap closes in both Mott and Peierls insulators, turning the former into a conventional Luttinger liquid (gapless in all excitation channels). In the latter case, instead, a finite spin gap remains at small doping. Even though consistent with the general theory of interacting electrons in one dimension, the existence of such a phase (with gapless charge but gapped spin excitations) has never been demonstrated in a model with repulsive interaction and with only two Fermi points. Since the spin-gapped metal represents the one-dimensional counterpart of a superconductor, our results furnish evidence that a true off-diagonal long-range order may exist in the two-dimensional case.
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 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.
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.