We use the continuous-time interaction expansion (CT-INT) quantum Monte Carlo method to calculate the phonon spectral function of the one-dimensional Holstein-Hubbard model at half-filling. Our results are consistent with a soft-mode Peierls transition in the adiabatic regime, and the existence of a central peak related to long-range order in the Peierls phase. We explain a previously observed feature at small momenta in terms of a hybridization of charge and phonon excitations. Tuning the system from a Peierls to a metallic phase with a nonzero Hubbard interaction suppresses the central peak, but a significant renormalization of the phonon dispersion remains. In contrast, the dispersion is only weakly modified in the Mott phase. We discuss finite-size effects, the relation to the dynamic charge structure factor, as well as additional sum rules and their implications. Finally, we reveal the existence of a discrete symmetry in a continuum field theory of the Holstein model, which is spontaneously broken in the Peierls phase.
We use the continuous-time interaction expansion (CT-INT) quantum Monte Carlo method to calculate the phonon spectral function of the one-dimensional Holstein-Hubbard model at half-filling. Our results are consistent with a soft-mode Peierls transition in the adiabatic regime, and the existence of a central peak related to long-range order in the Peierls phase. We explain a previously observed feature at small momenta in terms of a hybridization of charge and phonon excitations. Tuning the system from a Peierls to a metallic phase with a nonzero Hubbard interaction suppresses the central peak, but a significant renormalization of the phonon dispersion remains. In contrast, the dispersion is only weakly modified in the Mott phase. We discuss finite-size effects, the relation to the dynamic charge structure factor, as well as additional sum rules and their implications. Finally, we reveal the existence of a discrete symmetry in a continuum field theory of the Holstein model, which is spontaneously broken in the Peierls phase.
Single- and two-particle excitation spectra of the one-dimensional, half-filled Holstein-Hubbard model are calculated using the continuous-time quantum Monte Carlo method. In the metallic phase, the results are consistent with a Luther-Emery liquid that has gapped spin and single-particle excitations but a gapless charge mode. However, given the initially exponential dependence of the spin gap on the backscattering matrix element, the numerical excitation spectra appear gapless in the weak-coupling regime, and therefore resemble those of a Luttinger liquid. The Mott phase has the expected charge gap and gapless spin excitations. The Peierls state shows a charge, spin, and single-particle gap, a soft phonon mode, backfolded shadow bands, and soliton excitations. Arguments and numerical evidence for the existence of a nonzero spin gap throughout the metallic phase are provided in terms of equal-time spin and charge correlation functions.
The Hubbard-Holstein model describes fermions on a discrete lattice, with on-site repulsion between fermions and a coupling to phonons that are localized on sites. Generally, at half-filling, increasing the coupling g to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction U drives the fermions into a Mott antiferromagnet. At low g and U, or when doped, the system is metallic. In one dimension, using quantum Monte Carlo simulations, we study the case where fermions have a long-range coupling to phonons, with characteristic range $\ensuremath{\xi}$, interpolating between the Holstein and Fr\"ohlich limits. Without electron-electron interaction, the fermions adopt a Peierls state when the coupling to the phonons is strong enough. This state is destabilized by a small coupling range $\ensuremath{\xi}$ and leads to a collapse of the fermions, and, consequently, phase separation. Increasing interaction U will drive any of these three phases (metallic, Peierls, phase separation) into a Mott insulator phase. The phase separation region is once again present in the $U\ensuremath{\ne}0$ case, even for small values of the coupling range.
The quantum phase transition between a repulsive Luttinger liquid and an insulating Peierls state is studied in the framework of the one-dimensional spinless Holstein model. We focus on the adiabatic regime but include the full quantum dynamics of the phonons. Using continuous-time quantum Monte Carlo simulations, we track, in particular, the dynamic charge structure factor and the single-particle spectrum across the transition. With increasing electron-phonon coupling, the dynamic charge structure factor reveals the emergence of a charge gap and a clear signature of phonon softening at the zone boundary. The single-particle spectral function evolves continuously across the transition. Hybridization of the charge and phonon modes of the Luttinger liquid description leads to two modes, one of which corresponds to the coherent polaron band. This band acquires a gap upon entering the Peierls phase, whereas the other mode constitutes the incoherent, high-energy spectrum with backfolded shadow bands. Coherent polaronic motion is a direct consequence of quantum lattice fluctuations. In the strong-coupling regime, the spectrum is described by the static, mean-field limit. Importantly, whereas finite electron density in general leads to screening of polaron effects, the latter reappear at half-filling due to charge ordering and lattice dimerization.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph) plane for the phonon-mediated interaction gph and the Coulomb interaction U. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on U. For the Peierls-Hubbard model, the region of the (U,gph) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term V is added to the Hubbard interaction.
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.
We calculate the spectral function of the one-dimensional Hubbard-Holstein model using the time-dependent density matrix renormalization group, focusing on the regime of large local Coulomb repulsion, and away from electronic half-filling. We argue that, from weak to intermediate electron-phonon coupling, phonons interact only with the electronic charge, and not with the spin degrees of freedom. For strong electron-phonon interaction, spinon and holon bands are not discernible anymore and the system is well described by a spinless polaronic liquid. In this regime, we observe multiple peaks in the spectrum with an energy separation corresponding to the energy of the lattice vibrations (i.e., phonons). We support the numerical results by introducing a well controlled analytical approach based on Ogata-Shiba's factorized wave function, showing that the spectrum can be understood as a convolution of three contributions, originating from charge, spin, and lattice sectors. We recognize and interpret these signatures in the spectral properties and discuss the experimental implications.