Superconductivity, antiferromagnetism, and phase separation in the two-dimensional Hubbard model: A dual-fermion approach

The dual-fermion approach offers a way to perform diagrammatic expansion around the dynamical mean field theory. Using this formalism, the influence of antiferromagnetic fluctuations on the self-energy is taken into account through ladder-type diagrams in the particle-hole channel. The resulting phase diagram for the (quasi-)two-dimensional Hubbard model exhibits antiferromagnetism and $d$-wave superconductivity. Furthermore, a uniform charge instability, i.e., phase separation, is obtained in the low-doping regime around the Mott insulator. We also examine spin/charge density wave fluctuations including $d$-wave symmetry. The model exhibits a tendency towards an unconventional charge density wave, but no divergence of the susceptibility is found.

Figure 1

A phase diagram at half filling, $\delta =0$.

Figure 2

Phase diagrams under doping $\delta =1-n$ for $U=8$.

Figure 3

Diagrammatic representations of (a) bare interaction $\tilde{V}$ for dual fermions, (b) the dual self-energy $\tilde{\Sigma }$ in the ladder approximation, and (c) the equation for the renormalized vertex $\Gamma$.

Figure 4

Self-energy diagrams: (a) Example of a leading order diagram in terms of $1/d$ but not included in the present approximation for physical reasons (see text), (b) an example of the next-leading contributions in the $1/d$ expansion, and (c) zero contribution by the self-consistency condition.

Figure 5

Temperature dependence of (a) the static AFM susceptibility ${\chi }_{\nu =0,\mathit{Q}}^{\mathrm{sp}}$ and (b) the largest eigenvalue ${\lambda }^{\mathrm{sp}}$ of the matrix $\widehat{A}$ at half filling $n=1$. The closed symbols and open symbols show results in the present approximation and within DMFT, respectively.

Figure 6

A scaling plot: $1-{\lambda }^{\mathrm{sp}}$ as a function of $1/T$. Results for different system sizes, $N=32\times 32,64\times 64$, and $128\times 128$, are shown for comparison. The solid lines indicate the scaling $1-{\lambda }^{\mathrm{sp}}\propto exp(-\Delta /T)$.

Figure 7

The pairing interaction (the irreducible vertex for the pairing susceptibility) ${\Gamma }^{\mathrm{pp}}$ in the ladder approximation. The box with stripes stands for the renormalized vertex $\Gamma$ in Fig. 3.

Figure 8

Momentum dependence of the eigenfunctions ${\varphi }_{{\omega }_0,\mathit{k}}$ of Eq. (30) for $U=8,\delta =0.14$, and $T=0.1$, where ${\omega }_0=\pi T$. Either the even-frequency part ${\varphi }_{{\omega }_0,\mathit{k}}^{\mathrm{even}}$ or the odd-frequency part ${\varphi }_{{\omega }_0,\mathit{k}}^{\mathrm{odd}}$ is plotted depending on which is allowed by the Pauli principle.

Figure 9

Temperature dependence of the eigenvalues ${\lambda }^{\mathrm{SC}}$ of Eq. (30) for $U=8$ and $\delta =0.14$.

Figure 10

Temperature dependence of the chemical potential $\mu$. The doping $\delta$ is varied from 0 to 0.2 in 0.02 steps. The inset shows $\mu$ as a function of $\delta$ for fixed $T$.

Figure 11

The local-current configuration in the d-DW state.

Figure 12

(a) An exemplary susceptibility diagram which contributes to the unconventional CDW. (b) A diagram for the dual-fermion irreducible vertex ${\Gamma }^{\prime }$ in Eqs. (36) and (37). The box with stripes stands for the renormalized vertex $\Gamma$ in Fig. 3.

Figure 13

Temperature dependence of the eigenvalues ${\lambda }^{\mathrm{DW}}$ of Eq. (35) with $\mathit{q}=\mathit{Q}$. The parameters are $U=8$ and $\delta =0.08$.