The search for realistic one-dimensional (1D) models that exhibit dominant superconducting (SC) fluctuations effects has a long history. In these 1D systems, the effects of commensurate band fillings---strongest at half-filling---and electronic repulsions typically lead to a finite charge gap and the favoring of insulating density wave ordering over superconductivity. Accordingly, recent proposals suggesting a gapless metallic state in the Holstein-Hubbard (HH) model, possibly superconducting, have generated considerable interest and controversy, with the most recent work demonstrating that the putative dominant superconducting state likely does not exist. In this paper we study a model with nonlocal electron-phonon interactions, in addition to electron-electron interactions. This model unambiguously possesses dominant superconducting fluctuations at half filling in a large region of parameter space. Using both the numerical multi-scale functional renormalization group (MFRG) for the full model and an analytic conventional renormalization group for a bosonized version of the model, we demonstrate the existence of these dominant SC fluctuations and show that they arise because the spin-charge coupling at high energies is weakened by the nonlocal electron-phonon interaction and the charge gap is destroyed by the resultant suppression of the Umklapp process. The existence of the dominant SC pairing instability in this half-filled 1D system suggests that nonlocal boson-mediated interactions may be important in the superconductivity observed in the organic superconductors.
The search for realistic one-dimensional (1D) models that exhibit dominant superconducting (SC) fluctuations effects has a long history. In these 1D systems, the effects of commensurate band fillings---strongest at half-filling---and electronic repulsions typically lead to a finite charge gap and the favoring of insulating density wave ordering over superconductivity. Accordingly, recent proposals suggesting a gapless metallic state in the Holstein-Hubbard (HH) model, possibly superconducting, have generated considerable interest and controversy, with the most recent work demonstrating that the putative dominant superconducting state likely does not exist. In this paper we study a model with nonlocal electron-phonon interactions, in addition to electron-electron interactions. This model unambiguously possesses dominant superconducting fluctuations at half filling in a large region of parameter space. Using both the numerical multi-scale functional renormalization group (MFRG) for the full model and an analytic conventional renormalization group for a bosonized version of the model, we demonstrate the existence of these dominant SC fluctuations and show that they arise because the spin-charge coupling at high energies is weakened by the nonlocal electron-phonon interaction and the charge gap is destroyed by the resultant suppression of the Umklapp process. The existence of the dominant SC pairing instability in this half-filled 1D system suggests that nonlocal boson-mediated interactions may be important in the superconductivity observed in the organic superconductors.
We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent K(rho) >1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency.
Asymptotic properties of nearly half-filled one-dimensional conductors coupled with phonons are studied through a renormalization-group method. Due to spin-charge coupling via an electron-phonon interaction, the spin correlation varies with filling as well as the charge correlation. Depending on the relation between cutoff energy scales of the umklapp process and of the electron-phonon interaction, various phases appear. We find a metallic phase with a spin gap and a dominant charge-density-wave correlation near half filling between a gapless density-wave phase (like in the doped repulsive Hubbard model) and a superconductor phase with a spin gap. The spin gap is produced by phonon-assisted backward scatterings that interfere with the umklapp process constructively or destructively depending on the character of electron-phonon coupling. \textcopyright{} 1996 The American Physical Society.
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.
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 leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g (dxy) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN, the latter induces pairing in other symmetry channels, including, e.g., A2g (g-wave), nodal A1g (extended s-wave), or nodal Eu (p-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu (p-wave) spin-triplet or B1g (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
Abstract Using the singular-mode functional renormalization group method, we studied the doping-dependent phase diagram of the two-dimensional extended Hubbard model on square lattice. At half-filling, a variety of states strongly compete, including spin-density wave, charge-density wave, phase separation, as well as d -wave and s -wave superconductivity (SC). Upon electron doping, the system exhibits a strong preference for superconducting states. At slight dopings, the spin-density wave phase is strongly suppressed, and d -wave SC region expands into positive U , V region. At moderate dopings, a region of degenerated p -wave superconductivities emerge for positive U and negative V . Our results demonstrate how doping tunes the dominant fluctuations, suppressing density waves and stabilizing unconventional superconducting states in two dimensions.