Comparing pertinent effects of antiferromagnetic fluctuations in the two- and three-dimensional Hubbard model

We use the dynamical vertex approximation $\left(\text{D}\Gamma \text{A}\right)$ with a Moriyaesque $\lambda$ correction for studying the impact of antiferromagnetic fluctuations on the spectral function of the Hubbard model in two and three dimensions. Our results show the suppression of the quasiparticle weight in three dimensions and dramatically stronger impact of spin fluctuations in two dimensions where the pseudogap is formed at low enough temperatures. Even in the presence of the Hubbard subbands, the origin of the pseudogap at weak-to-intermediate coupling is in the splitting of the quasiparticle peak. At stronger coupling (closer to the insulating phase) the splitting of Hubbard subbands is expected instead. The $\mathbf{k}$ dependence of the self-energy appears to be also much more pronounced in two dimensions as can be observed in the $\mathbf{k}$-resolved $\text{D}\Gamma \text{A}$ spectra, experimentally accessible by angular resolved photoemission spectroscopy in layered correlated systems.

Figure 1

(Color online) Graphical representation of the contribution of (a) bare Coulomb interaction and (b) spin (charge) fluctuations to the self-energy in the $\text{D}\Gamma \text{A}$ approach, Eq. (8). Solid lines correspond to the electronic Green’s function $G_{\mathbf{k},\nu }$, dashed line to the bare Hubbard interaction $U$, wiggly lines—to the spin (charge) susceptibility ${\chi }_{\mathbf{q},\omega }^{\text{s}\left(\text{c}\right)}$; the triangle corresponds to the interaction vertex ${\gamma }_{\text{s}\left(\text{c}\right),\mathbf{q}}^{\nu ,\omega }$.

Figure 2

(Color online) Frequency dependence of the spin and charge three-point vertex Eq. (6) at $U=1$, $\beta =1/T=15$ (left), and $U=2$, $\beta =10$ (right), $\omega =0$, at the antiferromagnetic wave vector $\mathbf{Q}=\left(\pi ,\pi \right)$. All energies are in units of half the effective bandwidth $D\equiv 4t$.

Figure 3

The spectral functions in $d=2$ as obtained from the approximate self-energies including local [dashed lines, Eq. (14)] and nonlocal [solid lines, Eq. (13)] fluctuations for (a) $\kappa =0$, (b) 0.1, (c) 0.3, (d) 0.5, (e) 0.9, and (f) 1.0.

Figure 4

(Color online) $\text{D}\Gamma \text{A}$ self-energy on the Matsubara axis calculated with and without Moriya $\lambda$ correction for two different points of the Fermi surface in the two-dimensional Hubbard model (at $U=D=4t$ and $\beta =1/T=17$); also shown is the DMFT self-energy for comparison. Notice that, without introducing the Moriya $\lambda$ correction, one always observes a deviation of the high-frequency $\Sigma \left(\mathbf{k},i{\nu }_n\right)$ from the correct asymptotic behavior $\sim U^{2}n\left(1-\frac{n}{2}\right)/\left(2i{\nu }_n\right)=U^2/\left(4i{\nu }_n\right)$ which is consistent with the self-energy sum rule (see text).

Figure 5

(Color online) Temperature dependence of the $\lambda$-corrected inverse antiferromagnetic susceptibility in two dimensions (triangles) for $U=D=4t$. The data display an exponential temperature dependence, consistent with the expected behavior of $\xi$ (see text). The DMFT Néel temperature corresponding to this set of parameters is marked with an arrow.

Figure 6

(Color online) DMFT self-energies and spectral functions (gray dashed line) at ${\mathbf{k}}_F=\left(\pi /2,\pi /2,\pi /2\right)$ for the Hubbard model in $d=3$ at $U=1.5D$ $\left(D=2\sqrt{6}t\right)$ and $\beta =11.2$ (i.e., slightly above $T_N^{\text{DMFT}}$) are compared with the corresponding $\text{D}\Gamma \text{A}$ results with (lower row; solid blue line) and without (upper row; black dotted line) Moriya $\lambda$ correction. Note that (i) the nonlocal fluctuations modify only quantitatively the shape of the QP, but no pseudogap appears, and (ii) nonlocal correlation effects are further reduced by the inclusion of the Moriya $\lambda$ correction.

Figure 7

(Color online) DMFT self-energies and spectral functions at ${\mathbf{k}}_F=\left(\pi /2,\pi /2,\pi /2\right)$ for the Hubbard model in $d=3$ at $U=1.5D\left(D=2\sqrt{6}t\right)$ and $\beta =15$ compared with the corresponding $\text{D}\Gamma \text{A}$ ones with $\lambda$ correction. Lowering $T=1/\beta$ toward $T_N^{\text{D}\Gamma \text{A}}$, qualitatively similar results as without $\lambda$ correction at higher $T$ (upper row of Fig. 6, $\beta =11.2$) are obtained.

Figure 8

(Color online) $\text{D}\Gamma \text{A}$ results for the half-filled two-dimensional Hubbard model at $U=D=4t$, $\beta =17$ (just slightly above $T_N^{\text{DMFT}}$) computed without (first and third row; black dotted line) and with (second and fourth row; blue solid line) the Moriya $\lambda$ correction, and compared with DMFT (gray dashed line). The $\text{D}\Gamma \text{A}$ calculations in the non-self-consistent scheme show a clear pseudogap opening for the $\mathbf{k}$ points of the noninteracting FS, but more pronounced in the antinodal direction. Within the $\lambda$-corrected scheme, one can still notice a pseudogap opening but only in the antinodal direction, while in the nodal direction a strongly damped QP appears.

Figure 9

(Color online) Temperature evolution of the $\text{D}\Gamma \text{A}$ results for the half-filled two-dimensional Hubbard model at $U=D=4t$ in the nodal direction, computed with the Moriya $\lambda$ correction, and compared with DMFT. A clear pseudogap emerges at the lowest temperature $\left(\beta =60\right)$, similarly to the results of the non-self-consistent scheme in the proximity of $T_N^{\text{DMFT}}$ (see Fig. 8).

Figure 10

(Color online) Same as in Fig. 9 in the antinodal direction. As expected, a very pronounced pseudogap characterizes the lowest temperature results. The behavior of the spectral functions is not completely monotonous, as the pseudogap seems to disappear at $\beta =25$. At all temperatures, however, the pseudogap features are always more marked in the antinodal than in the nodal direction.

Figure 11

(Color online) $\text{D}\Gamma \text{A}$ results with $\lambda$ corrections for the half-filled two-dimensional Hubbard model at $U=2D=8t$, $\beta =40$ compared with the corresponding DMFT ones.

Figure 12

(Color online) $\mathbf{k}$-resolved $\text{D}\Gamma \text{A}$ spectra along the nodal (left) and antinodal (right) direction for the half-filled two-dimensional Hubbard model at $U=D=4t$ and $\beta =17$, calculated with the Moriya $\lambda$ correction.