Antiferromagnetic and $d$-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group

Using the dynamical mean-field theory (DMFT) as a “booster rocket”, the functional renormalization group (fRG) can be upgraded from a weak-coupling method to a powerful computation tool for strongly interacting fermion systems. The strong local correlations are treated nonperturbatively by the DMFT, while the fRG flow can be formulated such that it is driven exclusively by nonlocal correlations, which are more amenable to approximations. We show that the full frequency dependence of the two-particle vertex needs to be taken into account in this approach, and demonstrate that this is actually possible—in spite of the singular frequency dependence of the vertex at strong coupling. We are thus able to present results obtained from the DMFT-boosted fRG for the two-dimensional Hubbard model in the strongly interacting regime. We find strong antiferromagnetic correlations from half filling to 18% hole doping and at the lowest temperature we can access, a sizable $d$-wave pairing interaction driven by magnetic correlations at the edge of the antiferromagnetic regime.

Figure 1

Notation of the two-particle vertex.

Figure 2

$g^{\mathrm{\Lambda }}\left(\nu \right)$ as defined in Eq. (22) as a function of $\mathrm{\Lambda }$ for the first Matsubara frequency ${\nu }_0=\pi T$. Parameters are $n=0.82,U=8t,T=0.08t$, and $t^{\prime }=-0.2t$.

Figure 3

Contribution from three-particle vertex to the flow of the two-particle vertex.

Figure 4

Left panel: DMFT Néel temperature as a function of $U$ (black line) for $n=1$ and $t^{\prime }=0$. The shadowed area depicts the range of transition temperatures obtained from a simplified parametrization of the vertex with a single bosonic frequency variable in each channel. Right panel: Spin susceptibility ${\chi }^s$ along a path in the BZ zone as computed from RPA with DMFT vertex and self-energy (black solid line), and by the single-channel $\mathrm{DMF}{}^2\mathrm{RG}$ (blue symbols). Here $U=12t$ and $T=0.038t$, corresponding to the black dot in the left panel.

Figure 5

Left panel: Flow of the maximum of the magnetic fluctuation term as function of the flow parameter $\mathrm{\Lambda }$. Center panel: Flow of the magnetic susceptibility at $\mathbf{Q}=(\pi ,\pi )$ and $\mathrm{\Omega }=0$. Right panel: Frequency dependence of the magnetic fluctuation term for momentum $\mathbf{Q}=(\pi ,\pi )$ and vanishing bosonic frequency $\mathrm{\Omega }=0$. Parameters: $U=16t,T=0.29t,t^{\prime }=0$, and $n=1$.

Figure 6

Inverse of the static magnetic susceptibility for $\mathbf{Q}=(\pi ,\pi )$ as a function of the temperature for $U=4t$ in $\mathrm{DMF}{}^2\mathrm{RG}$ and in RPA with DMFT vertices for $n=1$ and $t^{\prime }=0$.

Figure 7

Real part of the self-energy as a function of momentum at the lowest Matsubara frequency ${\omega }_0=\pi T$. Parameters: $U=8t,T=0.5t,t^{\prime }=0$, and $n=1$.

Figure 8

Left axis: Critical flow parameter ${\mathrm{\Lambda }}_{\mathrm{c}}$ for the antiferromagnetic instability as a function of doping $\delta =1-n$ in full $\mathrm{DMF}{}^2\mathrm{RG}$ (blue circles) and in single-channel $\mathrm{DMF}{}^2\mathrm{RG}$ (orange circles). Right axis: Maximum of the $d$-wave pairing interaction $\mathcal{D}$ from the full $\mathrm{DMF}{}^2\mathrm{RG}$ (blue stars) and in a decoupling approximation (red stars). The lines connecting the symbols are guides to the eye. Parameters are $U=8t,T=0.08t$, and $t^{\prime }=-0.2t$.

Figure 9

Static magnetic susceptibility in DMFT-RPA (black line) and in full $\mathrm{DMF}{}^2\mathrm{RG}$ (blue points) along a specific path in the BZ. Parameters: $U=8t,T=0.08t,t^{\prime }=-0.2t$, and $\delta =0.18$.

Figure 10

Frequency dependence of the magnetic fluctuation channel at weak (left) and strong (right) coupling close to half filling for $T=0.08t$ and $t^{\prime }=-0.2t$.

Figure 11

Imaginary part of the self-energy as a function of Matsubara frequency for different points in the BZ along the noninteracting Fermi surface (see inset). The local DMFT self-energy is shown in black. Parameters: $U=8t,T=0.08t,t^{\prime }=-0.2t$, and $\delta =0.18$.

Figure 12

Inverse $d$-wave channel as a function of flow parameter $\mathrm{\Lambda }$ for various fillings. Parameters: $U=8t,T=0.08t$, and $t^{\prime }=-0.2$.

Figure 13

Flow of the $d$-wave pairing channel at higher and lower temperature for $U=8t$ and $t^{\prime }=-0.2$.