Intertwined fluctuations and isotope effects in the Hubbard-Holstein model on the square lattice from functional renormalization

Abstract

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half-filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the $d$-wave superconducting susceptibility towards larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the $s$-wave superconducting susceptibility change sign and eventually lead to a reduction of $s$-wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities.

Figures (24)

• Figure 1: Representation of Hubbard-Holstein model. Electrons hop with amplitude $t$ to their nearest neighbors and with amplitude $t'$ to their second-nearest neighbors. They interact instantaneously via the Hubbard repulsion $U$. Lattice sites vibrate with frequency $\omega_0$ and the effect of an electron visiting a vibrating site leaves an imprint felt by an electron passing by that site at a later time. This induces an effective retarded interaction of strength $V_H$ between the electrons.
• Figure 2: Leading $s$-wave susceptibilities at half filling for $t'=0$. The superconducting $s$-wave susceptibility $\chi^{\mathrm{SC}}$ is indicated by $\bigcirc$, the $s$-wave density susceptibility $\chi^\mathrm{D}$ by $\square$, and the $s$-wave magnetic susceptibility $\chi^{\mathrm{M}}$ by $\triangle$. Parameters are $\beta = 20/t$, $t' = 0$ and $\mu = 0$. Different phonon frequencies $\omega_0$ are indicated above the panels. Overlayed symbols correspond to equal values.
• Figure 3: Leading $s$-wave susceptibilities at finite doping. We consider a filling $n=0.39$ with model parameters $t' = -0.25t$ and $\mu = 4t'$, corresponding to van-Hove filling of the noninteracting Fermi surface at $\beta = 20/t$. Symbols are chosen as in Fig. \ref{['fig:phase_diagram_halffilling']}.
• Figure 4: Inverse ($s$-wave) susceptibilities, evaluated at the respective maxima, as a function of $\omega_0$, at $\beta=20/t$ and half filling. Left panel: $U_\textrm{eff} > 0$ for $U=2.2t$ and $V_H = 1.4t$. Right panel: $U_\textrm{eff} < 0$ for $U=1.5t$ and $V_H = 2t$.
• Figure 5: Inverse susceptibilities as in Fig. \ref{['fig:halffilling_inverse_susceptibility_omega0_influence']}, but at finite doping with filling $n =0.39$ , for $t'=-0.25t$ and $\mu=4t'$. Left panel: $U_\textrm{eff} > 0$ for $U=2.5t$ and $V_H = 1t$. Right panel: $U_\textrm{eff} < 0$ for $U=1.7t$ and $V_H = 2.2t$.