Impact of nonlocal correlations over different energy scales: A dynamical vertex approximation study

In this paper, we investigate how nonlocal correlations affect, selectively, the physics of correlated electrons over different energy scales, from the Fermi level to the band edges. This goal is achieved by applying a diagrammatic extension of dynamical mean field theory (DMFT), the dynamical vertex approximation $\left(\mathrm{D}\mathrm{\Gamma }\mathrm{A}\right)$, to study several spectral and thermodynamic properties of the unfrustrated Hubbard model in two and three dimensions. Specifically, we focus first on the low-energy regime by computing the electronic scattering rate and the quasiparticle mass renormalization for decreasing temperatures at a fixed interaction strength. This way, we obtain a precise characterization of the several steps through which the Fermi-liquid physics is progressively destroyed by nonlocal correlations. Our study is then extended to a broader energy range, by analyzing the temperature behavior of the kinetic and potential energy, as well as of the corresponding energy distribution functions. Our findings allow us to identify a smooth but definite evolution of the nature of nonlocal correlations by increasing interaction: They either increase or decrease the kinetic energy w.r.t. DMFT depending on the interaction strength being weak or strong, respectively. This reflects the corresponding evolution of the ground state from a nesting-driven (Slater) to a superexchange-driven (Heisenberg) antiferromagnet (AF), whose fingerprints are, thus, recognizable in the spatial correlations of the paramagnetic phase. Finally, a critical analysis of our numerical results of the potential energy at the largest interaction allows us to identify possible procedures to improve the ladder-based algorithms adopted in the dynamical vertex approximation.

Figure 1

Phase diagrams of the half-filled Hubbard model on a simple square and cubic lattice in $3d$ (left panel) and $2d$ (right panel), respectively. Data points of the main panels are reproduced from Refs. [38, 39] $\left(3d\right)$ and [45] $\left(2d\right)$, respectively, except for the new $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ estimate of $T_N$ at $U=4.0$ in $3d$. Right panel: The metal-to-insulator (MIT) transition marked in blue is the one observed in paramagnetic DMFT calculations. Because of the strong nonlocal correlations captured by $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ calculations, in $2d$ the MIT is transformed into a sharp crossover (red shaded area) which extends down to $U=0$ at low $T$ [45, 69]. Inset: $log\left[\xi \right]$, computed in $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ for $U=0.75$, as a function of the inverse temperature $\beta =1/T$ for different sizes of the momentum grid $(2N_{\mathbf{q}}$ is the number of $\mathbf{q}$ points in one direction) used for the determination of $\lambda$. The fit (orange line) has been performed in the low-$T$ regime according to the exponential function in Eq. (14a).

Figure 2

Self-energies and spectral functions of the $2d$ Hubbard model at the nodal point $[{\mathbf{k}}_N=(\pi /2,\pi /2)]$ for three (decreasing) temperatures $\beta =35$ (first row), $\beta =45$ (second row), and $\beta =50$ (third row). In the first column the Matsubara self-energy from DMFT (blue squares) and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ (purple pentagons) and their corresponding Padé fits (continuous lines) are shown as a function of the (fermionic) Matsubara index $2n+1=\nu \beta /\pi$. In the second column the real and imaginary parts of the (retarded) self-energy are reported. The corresponding spectral functions of DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, respectively, are plotted in the third column of the figure.

Figure 3

Temperature dependence of low-energy coefficients of the self-energy for $U=0.75$ in $2d$. Upper panel: Scattering factors ${\gamma }_{\mathbf{k}}\left(T\right)$ [for the definition see Eq. (12b)] for $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ at the nodal $[{\mathbf{k}}_N=(\pi /2,\pi /2)]$ and the antinodal $[{\mathbf{k}}_A=(\pi ,0)]$ point compared to the corresponding DMFT data; lower panel: (Negative) slope of the real part of the self-energy at $\omega =0$, i.e., ${\alpha }_{\mathbf{k}}\left(T\right)$ [Eq. (12a)] for $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ at ${\mathbf{k}}_N=(\pi /2,\pi /2)$ and ${\mathbf{k}}_A=(\pi ,0)$ compared to the corresponding DMFT data. The continuous lines represent the fits of the data points with the analytical function given in the plot (for the analytical derivations see the main text).

Figure 4

Same as in Fig. 3 but in $3d$ for $U=1.25$ with ${\mathbf{k}}_N=(\pi /2,\pi /2,\pi /2)$ and ${\mathbf{k}}_A=(\pi ,\pi /2,0)$. $T_N$ denotes the finite transition temperature to the antiferromagnetic phase obtained in $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ (see Ref. [38]).

Figure 5

Kinetic energies for the noninteracting case $\left(U=0\right)$, DMFT, $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, and the atomic limit (AL) as a function of temperature in two (main panels) and three (insets) dimensions at half filling for different values of the interaction parameter $U=0.75$ (left panel), $U=2.0$ (middle panel), and $U=4.0$ (right panel).

Figure 6

Energy distributions $n\left(\varepsilon \right)$ for the $2d$ Hubbard model at three different values of the interaction parameter $U$ for the lowest considered temperature $\left(\beta =80\right)$. In the insets the difference $\mathrm{\Delta }\left(\varepsilon \right)$ between the energy distributions of DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ as well as the contribution $\varepsilon \mathrm{\Delta }\left(\varepsilon \right)$ to the corresponding difference of the kinetic energies are shown.

Figure 7

Same as in the middle panel of Fig. 6, but for $\beta =31.4$.

Figure 8

Potential energy of the half-filled Hubbard model computed in DMFT, $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, and the atomic limit (AL) as a function of temperature in two (main panels) and three (insets) dimensions for different values of the interaction parameter $U=0.75$ (left panel), $U=2.0$ (middle panel), and $U=4.0$ (right panel).

Figure 9

Diagrams for $F_{\uparrow \downarrow }^{k{k}^{\prime }q}$ that do not depend on $\nu$ (and $\mathbf{k})$. Note that the four-vector notation $k\widehat{=}(i\nu ,\mathbf{k})$ and $q\widehat{=}(i\mathrm{\Omega },\mathbf{q})$ has been adopted here.

Figure 10

All Feynman diagrams for $F_{r,\mathbf{q}}^{\nu {\nu }^{\prime }\mathrm{\Omega }}$ of class a. Note that this contribution to the full vertex is independent of the fermionic frequency $\nu$.

Figure 11

Examples for diagrams of class b.

Figure 12

Bethe-Salpeter-like equation for constructing diagrams of type c.

Figure 13

Self-energies and spectral functions for the antinodal point of the $2d$ Hubbard model at half filling $[{\mathbf{k}}_A=(\pi ,0)]$. The conventions adopted are the same as those of Fig. 2 in Sec. 3a.

Figure 14

Energy distributions $n\left(\varepsilon \right)$ for the three-dimensional Hubbard model at two different values of the interaction parameter $U$ for temperatures slightly above the $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ phase transition to the antiferromagnetically ordered phase. In the inset the difference $\mathrm{\Delta }\left(\varepsilon \right)$ between the energy distributions of DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ as well as the contribution $\varepsilon \mathrm{\Delta }\left(\varepsilon \right)$ to the corresponding difference of the kinetic energies are shown.

Figure 15

Energy distributions $n\left(\varepsilon \right)$ for the two-dimensional Hubbard model at $U=2.375$ for $\beta =80$ corresponding to the coexistence region between an metallic and an insulating state in DMFT. In the inset the difference $\mathrm{\Delta }\left(\varepsilon \right)$ between the energy distributions of DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ as well as the contribution $\varepsilon \mathrm{\Delta }\left(\varepsilon \right)$ to the corresponding difference of the kinetic energies are shown.