A Numerical Renormalization Group Study of the Superconducting Instability in Holstein--Hubbard Model

Ka-Ming Tam, Shan-Wen Tsai, David K Campbell

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.

Hiroki Goto, Takashi Hotta, Yasutami Takada +3 more

Masaki Tezuka, Ryotaro Arita, Hideo Aoki

Soh Koike

C.-H. Pao, H.-B. Schüttler

The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed numerical renormalization-group (NRG) methods for Hubbard-type purely electronic models are applicable, while brute force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. Here, we describe a NRG technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.

Walter Metzner

We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the effective low-energy action is generated from the bare action of the microscopic model by a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions, the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities are computed. This and similar RG calculations by others conclusively establish the existence of d-wave superconductivity in the Hubbard model.