Renormalized mean-field analysis of antiferromagnetism and $d$-wave superconductivity in the two-dimensional Hubbard model

We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\Lambda }_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the $d$-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\pi$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\pi$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.

Figure 1

Diagrammatic representation of the flow equation for $V_m^{\Lambda }$ in the Wick ordered version of the functional RG. The line with a slash corresponds to $\partial G_0^{<\Lambda }∕\partial \Lambda$, the others to $G_0^{<\Lambda }$; all possible pairings leaving $m$ incoming and $m$ outgoing external legs have to be summed.

Figure 2

(Color online) A typical Fermi surface (bold line) and the support of the propagator $G_0^{<\Lambda }(k_0,\mathbf{k})$ (dotted region) in the first quadrant of the Brillouin zone. The dashed lines mark the boundaries of the “patches” used for the discretization of the momentum dependence of the effective two-particle interaction.

Figure 3

Flow equation for the effective two-particle interaction ${\Gamma }^{\Lambda }$ in one-loop approximation with the particle-particle channel (PP) and the two particle-hole channels (PH and ${\mathrm{PH}}^{\prime }$).

Figure 4

(Color online) Dispersion of quasiparticle energies $E_{\mathbf{k}}^{\pm }$ on the magnetic Brillouin zone boundary (“umklapp surface”) in the antiferromagnetic state for a momentum independent $A_{\mathbf{k}}$ and $t^{\prime }<0$.

Figure 5

(Color online) Effective Fermi surfaces in the antiferromagnetic state for $t^{\prime }=-0.2$, $\mu =-0.6$, and various momentum independent choices of the antiferromagnetic gap function $A_{\mathbf{k}}$: 0 (solid line), 0.15 (dashed line), and 0.3 (dashed-dotted line).

Figure 6

(Color online) Topology of the effective Fermi surfaces in the plane spanned by the chemical potential $\mu$ and the antiferromagnetic gap $A$. No Fermi surface exists in the fully gapped regime.

Figure 7

(Color online) Mean-field solution for the antiferromagnetic gap as a function of $\mu$ for the Hubbard model with $t^{\prime }=-0.2$ and $U=2.25$. Regions with different Fermi surface topology are indicated as in Fig. 6.

Figure 8

(Color online) Antiferromagnetic gap as a function of the cutoff ${\Lambda }_{\mathrm{MF}}$ for the Hubbard model with pure nearest neighbor hopping $(t^{\prime }=0)$ and $U=2$ at half-filling. Due to the weak momentum dependence of $A_{\mathbf{k}}$ the curves corresponding to different patches are very close to each other. The intersection of the largest $A_{\mathbf{k}}$ (patch 1) with the straight line given by ${\Lambda }_{\mathrm{MF}}=2A_{\mathbf{k}}$ yields a suitable choice of ${\Lambda }_{\mathrm{MF}}$ leading to a reasonable estimate of the true gap size.

Figure 9

(Color online) Superconducting gap as a function of the cutoff ${\Lambda }_{\mathrm{MF}}$ for the Hubbard model in a regime where superconductivity is the only instability. Parameters: $U=2.5$, $t^{\prime }=-0.2$, and $\mu =-0.9265$.

Figure 10

(Color online) Amplitudes of the superconducting gap ${\Delta }_{\mathbf{k}}$ and the antiferromagnetic gap $A_{\mathbf{k}}$ as a function of $\mu$ for $U=2.5$ and $t^{\prime }=-0.15$. The dotted lines are the separatrices between different Fermi surface topologies as specified in Fig. 6.

Figure 11

(Color online) Amplitudes of ${\Delta }_{\mathbf{k}}$ and $A_{\mathbf{k}}$ as a function of density for $U=2.5$ and $t^{\prime }=-0.15$. Filled colored symbols represent the results obtained for the combined theory with two order parameters, while in the results represented by open symbols only one order parameter, either antiferromagnetic or superconducting, was allowed in the mean-field calculation. The two slightly different results for the antiferromagnetic gap at half-filling correspond to two different choices of $\mu$ within the gap; this $\mu$ dependence is an artifact of the approximations.

Figure 12

(Color online) Top: Angular dependence of the $d$-wave superconducting and antiferromagnetic order parameters at three different values of the chemical potential. Bottom: Fermi surfaces in the magnetic Brillouin zone (solid blue lines); in the antiferromagnetic state close to half-filling the Fermi surface forms a hole pocket around $(\pi ∕2,\pi ∕2)$. The corresponding bare Fermi surfaces, backfolded with respect to the umklapp surface, are shown as broken lines. The straight lines indicate the patching scheme. The parameters $U=2.5$ and $t^{\prime }=-0.15$ are the same as in Figs. 10, 11.

Figure 13

(Color online) Points with antiferromagnetism, superconductivity, and coexistence of both in the $\mu \text{−}U$ plane for a fixed $t^{\prime }=-0.1$; “coexistence” means that the amplitude of ${\Delta }_{\mathbf{k}}∕A_{\mathbf{k}}$ is at least ${10}^{-3}$.

Figure 14

(Color online) Superconducting gap ${\Delta }_{\mathbf{k}}$ as a function of the cutoff ${\Lambda }_{\mathrm{MF}}$ for the attractive Hubbard model ($U=-1.5$, $t^{\prime }=-0.1$) at quarter-filling. Due to the weak momentum dependence of ${\Delta }_{\mathbf{k}}$ curves corresponding to different patches lie almost on top of each other. The inset shows ${\Delta }_{\mathbf{k}}$ for ${\Lambda }_{\mathrm{MF}}⩽1$ with a higher resolution.