Pseudogaps in strongly correlated metals: A generalized dynamical mean-field theory approach

We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale $\xi$ via a momentum dependent external self-energy ${\Sigma }_{\mathbf{k}}$. This external self-energy describes nonlocal dynamical correlations induced by the short-ranged collective spin density wave–like antiferromagnetic spin (or the charge density wave–like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with a finite correlation length. This generalized $\mathrm{DMFT}+{\Sigma }_{\mathbf{k}}$ approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with the nearest and the next nearest neighbor hopping. The effective single impurity problem in this generalized $\mathrm{DMFT}+{\Sigma }_{\mathbf{k}}$ is solved by the numerical renormalization group. Both types of the strongly correlated metals, namely: (i) The doped Mott insulator and (ii) the case of the bandwidth $W\lesssim U$ ($U$—value of the local Coulomb interaction) are considered. The densities of states, the spectral functions, and the angle resolved photoemission spectra calculated within the $\mathrm{DMFT}+{\Sigma }_{\mathbf{k}}$ show a pseudogap formation near the Fermi level of the quasiparticle band.

Figure 1

(Color online) Comparison of DOS obtained from $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations for different combinatorical factors (SF–spin-fermion model, commensurate), inverse correlation lengths $\left({\xi }^{-1}\right)$ in units of the lattice constant, temperatures $\left(T\right)$, and the value of pseudogap potential $\Delta =2t$. The left column corresponds to $t^{\prime }∕t=-0.4$, the right column to $t^{\prime }=0$. In all graphs the Coulomb interaction is $U=4t$ and $n=1$. The Fermi level corresponds to zero.

Figure 2

(Color online) Comparison of DOS obtained from $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations for a filling $n=0.8$, other parameters as in Fig. 1.

Figure 3

(Color online) Comparison of DOS obtained from $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations for $t^{\prime }∕t=-0.4$, $T=0.088t$, $U=40t$, $\Delta =2t$, and filling $n=0.8$.

Figure 4

One-eighth of the bare Fermi surfaces for the occupancy $n=0.8$ and different combinations $(t,t^{\prime })$ used for the calculation of spectral functions $A(\mathbf{k},\omega )$. The diagonal line corresponds to the Umklapp surface. The full circle marks the so-called hot spot.

Figure 5

(Color online) Real (dashed line) and imaginary (full line) parts of the self-energy $\Sigma (\mathbf{k},\omega )$ for $t∕t^{\prime }=-0.4$, $U=4t$ (left column), and $U=40t$ (right column) for characteristic $\mathbf{k}$ points: $\Gamma$, hot-spot (see Fig. 4) and cold-spot (point $B$ in Fig. 4). For all graphs the filling is $n=0.8$, temperature $T=0.088t$, inverse correlation length ${\xi }^{-1}=0.1$, value of pseudogap potential $\Delta =2t$, and SF combinatorics.

Figure 6

Spectral functions $A(\mathbf{k},\omega )$ obtained from the $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations along the directions shown in Fig. 4. Model parameters were chosen as $U=4t$, $n=0.8$, $\Delta =2t$, ${\xi }^{-1}=0.1$, and temperature $T=0.088t$. The hot-spot $\mathbf{k}$ point is marked as a fat dashed line. The Fermi level corresponds to zero.

Figure 7

Spectral functions $A(\mathbf{k},\omega )$ obtained from the $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations for $U=40t$; other parameters as in Fig. 6.

Figure 8

Spectral functions $A(\mathbf{k},\omega )$ obtained from the $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations along high-symmetry directions of the first Brillouin zone $\Gamma (0,0)-X(\pi ,0)-M(\pi ,\pi )-\Gamma (0,0)$, $\mathit{SF}$ combinatorics (left row) and commensurate combinatorics (right column). Other parameters are $U=4t$, $n=0.8$, $\Delta =2t$, ${\xi }^{-1}=0.1$, and temperature $T=0.088t$. The Fermi level corresponds to zero.

Figure 9

ARPES spectra simulated by the multiplication of the spectral functions obtained from $\mathrm{DMFT}\left(\mathrm{NRG}\right)+{\Sigma }_{\mathbf{k}}$ calculations for $U=4t$ and $n=0.8$ in Fig. 6 with the Fermi function at $T=0.088t$ plotted along the lines in the first BZ as depicted by Fig. 4. All other parameters are the same as in Fig. 6.

Figure 10

Local skeleton diagrams for the DMFT self-energy $\Sigma$. Wavy lines represent the local (Hubbard) Coulomb interaction $U$; full lines denote the local Green function $G_{ii}$.

Figure 11

Typical skeleton diagrams for the self-energy in the $\mathrm{DMFT}+{\Sigma }_{\mathbf{k}}$ approach. The first two terms are DMFT self-energy diagrams; the middle two diagrams show contributions to the nonlocal part of the self-energy from spin fluctuations (see Sec. 3) represented as dashed lines; the last diagram (b) is an example of the neglected diagram leading to the interference between the local and nonlocal parts.

Figure 12

(Color online) Filling dependence of the pseudogap potential $\Delta$ calculated with the DMFT(QMC) for the varying Coulomb interaction $\left(U\right)$ and the temperature $\left(T\right)$ on a two-dimensional square lattice with two sets of $(t,t^{\prime })$.