Dual fermion approach to susceptibility of correlated lattice fermions

In this paper, we show how the two-particle Green function can be obtained within the framework of the dual fermion approach. This facilitates the calculation of the susceptibility in strongly correlated systems where long-ranged nonlocal correlations cannot be neglected. We formulate the Bethe–Salpeter equations for the full vertex in the particle-particle and particle-hole channels and introduce an approximation for practical calculations. The scheme is applied to the two-dimensional Hubbard model at half-filling. The spin-spin susceptibility is found to strongly increase for the wave vector $\mathbf{q}=(\pi ,\pi )$, which indicates the antiferromagnetic instability. We find a suppression of the critical temperature compared to the mean-field result due to the incorporation of the nonlocal spin fluctuations.

Figure 1

The first two lowest-order diagrams for the dual self-energy ${\Sigma }^d$.

Figure 2

(Color online) Illustration of the dual fermion calculation procedure.

Figure 3

Local density of states for the 2D Hubbard model at half-filling obtained within DMFT (dashed lines) and dual fermion calculations (solid lines) for $U∕t=4$ (left) and $U∕t=8$ (right) at inverse temperature $\beta =4.5$.

Figure 4

(Color online) Contour plot of the real part of the lattice self-energy on the first Matsubara frequency $\mathfrak{R}\Sigma (\omega =\pi T,\mathbf{k})$ (centered at the $\Gamma$ point) for $U∕t=4$ (left) and $U∕t=8$ (right), calculated on a grid of $256\times 256$ $k$-points at inverse temperature $\beta =3.5$.

Figure 5

(Color online) Contour plot of the imaginary part of the lattice self-energy $\mathfrak{I}\Sigma (\omega =\pi T,\mathbf{k})$ (centered at the $\Gamma$ point) for $U∕t=4$ (left) and $U∕t=8$ (right) at inverse temperature $\beta =3.5$.

Figure 6

Spectral function for the 2D Hubbard model at half-filling obtained within DMFT (left) and DF (right) calculations for $U∕t=8$ at inverse temperature $\beta =4.5$. From bottom to top, the curves are plotted along the high-symmetry lines $\Gamma \to X\to M\to \Gamma$. The high-symmetry points $X$ and $M$ are marked by dashed lines.

Figure 7

(Color online) Comparison of the DMFT Weiss field $g_0\left(i\omega \right)={[i\omega +\mu -\Delta \left(i\omega \right)]}^{-1}$ and its modification after a fully self-consistent dual fermion (DF) calculation, for $U∕t=4$ (left) and $U∕t=8$ (right). The dashed curves correspond to the inverse temperature $\beta =3.5$ and the solid curves to $\beta =4.5$.

Figure 8

The Bethe–Salpeter equation in three different channels. In the calculations, we considered the equations for the particle-hole channel.

Figure 9

Diagrams for the susceptibilities in two different particle-hole channels.

Figure 10

(Color online) Nontrivial part of the dual susceptibility ${\tilde{\chi }}^{zz}=\frac{1}{2}({\tilde{\chi }}^{\uparrow \uparrow }-{\tilde{\chi }}^{\uparrow \downarrow })$ for different $k$ points as a function of temperature. The divergence at $\mathbf{q}=(\pi ,\pi )$ indicates the antiferromagnetic instability.

Figure 11

(Color online) The antiferromagnetic correlations visualized in real space for $\beta =4.25$ and $\beta =4.5$. At temperatures close to the transition, the correlations become essentially independent of distance.

Figure 12

(Color online) Momentum dependence of the static susceptibility (centered at the $M$ point) for $U=4$ and different temperatures calculated on a $256\times 256$ lattice. Top row from left to right: $\beta =0.5$, $\beta =1.0$, and $\beta =3.0$. Bottom row: $\beta =4.0$, $\beta =4.5$, and $\beta =4.65$.

Figure 13

(Color online) Inverse of the antiferromagnetic susceptibility ${\chi }_{\mathrm{AF}}=⟨S_z{S}_z⟩$ $\mathbf{(}\Omega =0,\mathbf{q}=(\pi ,\pi )\mathbf{)}$ as a function of temperature as obtained from DMFT and within the dual fermion approach for $U=4$ and different lattice sizes. Here, e.g., the label $64\times 64$ indicates the number of $k$ points used in the Brillouin zone integration. The inset compares the fully self-consistent DF result (dashed line) with the one obtained from the first iteration of the outer loop (solid line) shown in Fig. 2.

Figure 14

(Color online) The antiferromagnetic susceptibility for $U=8$. The inset shows the same on a larger scale. For high temperatures, the dual fermion result converges to the DMFT result.