Multiband dual fermion approach to quantum criticality in the Hubbard honeycomb lattice

We study the Hubbard model on the honeycomb lattice in the vicinity of the quantum critical point by means of a multiband formulation of the dual fermion approach. Beyond the strong local correlations of the dynamical mean field, critical fluctuations on all length scales are included by means of a ladder diagram summation. Analysis of the susceptibility yields an estimate of the critical interaction strength of the quantum phase transition from a paramagnetic semimetal to an antiferromagnetic insulator, in good agreement to other numerical methods. We further estimate the crossover temperature to the renormalized classical regime. Our data imply that, at large interaction strengths, the Hubbard model on the honeycomb lattice behaves like a quantum nonlinear $\sigma$ model, while displaying signs of non-Fermi-liquid behavior.

Figure 1

(a) In a cluster approach lattice symmetry is broken since self-energies obtained from the cluster (thick lines) or perturbatively (thin lines) treat a priori equivalent bonds unequally. (b) In the multiband approach all nonlocal self-energies are calculated on equal footing.

Figure 2

Sketch of the bipartite honeycomb lattice consisting of sublattices $A$ and $B$.

Figure 3

Diagrammatic representation of the Schwinger-Dyson equation (first line), the three lowest order contributions to the dual self-energy within the ladder dual fermion approach (second line), and the Bethe-Salpeter equation. The second line follows from the first line if the vertex is expanded according to the Bethe-Salpeter equation.

Figure 4

Left column: Log scale plot of $(1-\lambda )\beta$ against inverse temperature $\beta$ for different numbers of bosonic frequencies entering the BSE (a) $N_W=1$ (first row), (b) $N_W=4$ (second row), (c) $N_W=16$ (third row), and various values of $U$ in steps of $0.1t$. For $U=U_c,(1-\lambda )\beta$ is nearly constant and decays exponentially for $U>U_c$. Black solid lines are exponential fits to the data. Right column: Estimate for the crossover temperature $T_x$ to the renormalized classical regime against interaction $U$. Data points are obtained from intersections of fits to $(1-\lambda )\beta$ with constants 0.1 (red) and 0.05 (blue) (horizontal lines in left column). Solid lines are fits $T_x\propto {(U/U_c-1)}^{\nu }$, with $U_c$ and $\nu$ as fit parameters. Fit results for $U_c$ agree well with $U_c$ from left column, while results for $\nu$ are in good agreement with the literature as they are close to $\nu =0.7$. The values for $U_c$ and $\nu$ shown in the plot correspond to the blue line fit.

Figure 5

Size dependence of the inverse AFM susceptibility (left column) and $\tilde{\lambda }=(1-\lambda )\beta$ (right column) for (a) $N_W=1$, (b) $N_W=4$, and (c) $N_W=16$ bosonic frequencies. Different point types indicate sizes $24\times 24$ (triangles), $48\times 48$ (squares), $96\times 96$ (bullets), and $144\times 144$ (diamonds). Different values of $U$ are indicated by different colors according to the legend in the mid panel. Points are connected with lines as a guide to the eye. For $U>U_c$ these quantities are linear for sufficient lattice sizes in the log scale plot and have a clear size dependence. For $U<U_c$ curves for different sizes are almost on top of one another and $\tilde{\lambda }$ approaches a constant. According to this plot we estimate the critical couplings to be (a) $U_c=3.6\pm 0.1$, (b) $U_c=3.7\pm 0.1$, and (c) $U_c=3.8\pm 0.1$, reproducing the results obtained from the fit to $T_x$ of Fig. 4 rather well.

Figure 6

Quantity $Z_{\mathbf{k}}\left(T\right)$ defined in (13) at the Dirac point $\mathbf{k}=\mathbf{K}$ for different values of Hubbard interaction $U$ against temperature according to LDFA for (a) $N_W=1$, (b) $N_W=4$, and (c) $N_W=16$ bosonic frequencies. Polynomial extrapolation is used to determine the value of the Hubbard interaction $U_{\mathrm{SMIT}}$ for which the quasiparticle weight vanishes at $T=0$ and the system becomes insulating. The resulting values (a) $U_{\mathrm{SMIT}}=3.9$, (b) $U_{\mathrm{SMIT}}=4.0$, and (c) $U_{\mathrm{SMIT}}=4.1$ trail the critical $U_c$ by $0.3t$.

Figure 7

Analysis of the finite temperature quasiparticle weight (top panels) and self-energy at the Dirac-point (bottom panels). Top left panel: (a) Quasiparticle weight Z from Eq. (14) versus the Hubbard interaction for different temperatures ($\beta =10–30$) using $N_W=16$. $Z$ exhibits an unphysical increase with $U$ for $U>3.8$. Top mid panel: (b) This behavior is seen as well for $N_W=1$ (green triangles $U>3.5$) and $N_W=4$ (red circles) ($\beta =10$). Top right panel: (c) Example of the imaginary part of the self-energy $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ versus real frequencies, as obtained from Padé approximants. It shows a finite imaginary part at $\omega =0$ indicating non-Fermi-liquid behavior, where the real part shown in the inset has an inflection point. Bottom left panel: (d) The finite imaginary part at zero frequency is confirmed by polynomial extrapolation with ninth order polynomials. Bottom mid panel: (e) The double-log plot in the inset shows $\mathrm{Im}\mathrm{\Sigma }$ for $N_W=16,\beta =30$ and interaction strengths $U=2.5–3.9$ and fortifies the assumption that $\mathrm{Im}\mathrm{\Sigma }\left(i{\nu }_n\right)\propto {\left(i{\nu }_n\right)}^{\alpha }$. The data for $U=3.9$ deviates from this behavior and does not extrapolate to zero at zero frequency. Bottom right panel: (f) Power-law exponent $\alpha$ extracted from fits to $\mathrm{Im}\mathrm{\Sigma }\left(i{\nu }_n\right)$ on the Matsubara axis show that the system leaves the Fermi-liquid regime as $U$ increases with roughly a ${\nu }^{1/2}$ behavior along the critical line.