Dynamical vertex approximation: A step beyond dynamical mean-field theory

We develop a diagrammatic approach with local and nonlocal self-energy diagrams, constructed from the local irreducible vertex. This approach includes the local correlations of dynamical mean-field theory and long-range correlations beyond. It allows us, for example, to describe (para)magnons and weak localization effects—in strongly correlated systems. As a first application, we study the interplay between nonlocal antiferromagnetic correlations and the strong local correlations emerging in the vicinity of a Mott-Hubbard transition.

Figure 1

(a) From the reducible vertex, we directly obtain the self-energy. (b) and (c) The two particle-hole channels contributing to the self-energy (longitudinal and transversal) in the ladder approximation. Instead of the bare interaction, ladder diagrams are constructed from the irreducible local vertices (crosshatched) in $\mathrm{D}\Gamma \mathrm{A}$.

Figure 2

Graphical notation of the parquet Eqs. (3, 5, 6, 7). $C$, $Z$, and $\tilde{Z}$ denote the contributions of the particle-particle, particle-hole, and interaction channels to the nonlocal vertex ${\Gamma }_{{\mathbf{k}}_1,k_4,k_3-k_1}^{{\nu }_1,{\nu }_4,{\nu }_3-{\nu }_1,\uparrow \downarrow }$ (the momentum and frequency ${\mathbf{k}}_2,{\nu }_2$ are determined by conservation laws).

Figure 3

Diagrams for the self-energy in terms of the vertex to order $1∕d^0$ [(a), i.e., DMFT] and $1∕d^{1∕2}$ [(b)–(f)]. The ladder diagram (g) shows how diagrams (b)–(e) can be constructed from a fully irreducible local two-particle vertex. In contrast, for diagram (f) a local three-particle vertex is needed, see (h). Hence the contribution (f) is the only one not included in the $\mathrm{D}\Gamma \mathrm{A}$.

Figure 4

(Color online) Dependence of the local vertex ${\Gamma }_{s,\mathrm{ir}}^{\nu =\pi T,{\nu }^{\prime },\omega =0}$ on the incoming fermionic frequency ${\nu }^{\prime }$, for the three different values of $U$ and $T$ indicated as symbols in the left inset, which shows the DMFT phase diagram with paramagnetic metallic (PM), insulating (PI), and antiferromagnetic (AF) phase. Right inset: Same as the main panel but at fixed $U=1D$ and at different $\omega$’s (circles: $\omega =0$; triangles: $\omega =6\pi T$; squares: $\omega =12\pi T$).

Figure 5

(Color online) Self-energy (left: real, middle: imaginary part) and spectral function (right) at $\mathbf{k}=(\pi ∕2,\pi ∕2,\pi ∕2)$, $U=1D$ (top), $1.5D$ (central), and $2D$ (bottom) and the same $T$’s of Fig. 4. Compared to DMFT (light blue line), quasiparticles are damped through scattering at nonlocal spin fluctuations in $\mathrm{D}\Gamma \mathrm{A}$ (dark red line), and the system is more insulating. At $U=1.5D$ and $2D$, these effects are drastically enhanced because of strong local correlations reflected in ${\Gamma }_{\mathrm{s},\mathrm{ir}}$