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arXiv:1502.01981 · Physical Review B

Nature of ground states in one-dimensional electron-phonon Hubbard models at half filling

H. Bakrim, C. Bourbonnais

Published Feb 19, 2015• 25 citations
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Abstract

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 $(U,{g}_{\mathrm{ph}})$ plane for the phonon-mediated interaction ${g}_{\mathrm{ph}}$ 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 $(U,{g}_{\mathrm{ph}})$ 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.

DOI
10.1103/physrevb.91.085114

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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)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. 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 \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical U∗\ensuremath≈gph∗{U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}U∗\ensuremath≈gph∗​, followed at larger U\ensuremath∼gphU\ensuremath{\sim}{g}_{\mathrm{ph}}U\ensuremath∼gph​ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(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 \ensuremathωD{\ensuremath{\omega}}_{D}\ensuremathωD​ decreases. A power-law dependence gph\ensuremath∼U2\ensuremathη{g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}gph​\ensuremath∼U2\ensuremathη 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 VVV is added to the Hubbard interaction.
We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove —in the course of a careful finite-size scaling analysis— recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequencies the metallic region is even more extended than previously expected, and subdivided into ordinary Luttinger-liquid and bipolaronic-liquid phases.
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 ggg to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction UUU drives the fermions into a Mott antiferromagnet. At low ggg and UUU, 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 UUU 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 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.
By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping ttt, the on-site electron-electron interaction UUU, the phonon energy ${\ensuremath{\omega}}_{0}$, and the electron-phonon coupling ggg. At half filling, the ground state is an antiferromagnetic insulator for $U\ensuremath{\gtrsim}2{g}^{2}/{\ensuremath{\omega}}_{0}$, while it is a charge-density-wave (or bipolaronic) insulator for $U\ensuremath{\lesssim}2{g}^{2}/{\ensuremath{\omega}}_{0}$. In addition to these phases, we find a superconducting phase that intrudes between them. For ${\ensuremath{\omega}}_{0}/t=1$, superconductivity emerges when both U/tU/tU/t and $2{g}^{2}/t{\ensuremath{\omega}}_{0}$ are small; then, by increasing the value of the phonon energy ${\ensuremath{\omega}}_{0}$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.
The one-dimensional Holstein model and its generalizations have been studied extensively to understand the effects of electron-phonon interaction. The half-filled case is of particular interest, as it describes a transition from a metallic phase with a spin gap due to attractive backscattering to a Peierls insulator with charge-density-wave order. Our quantum Monte Carlo results support the existence of a metallic phase with dominant power-law charge correlations, as described by the Luther-Emery fixed point. We demonstrate that for Holstein and also for purely fermionic models the spin gap significantly complicates finite-size numerical studies, and explains inconsistent previous results for Luttinger parameters and phase boundaries. On the other hand, no such complications arise in spinless models. The correct low-energy theory of the spinful Holstein model is argued to be that of singlet bipolarons with a repulsive, mutual interaction. This picture naturally explains the existence of a metallic phase, but also implies that gapless Luttinger liquid theory is not applicable.
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.