DMFT Impurity Mapping and Self-Consistency

Dynamical mean-field theory (DMFT) replaces a lattice problem by a quantum impurity embedded in a self-consistent bath. The mapping becomes controlled when coordination tends to infinity with hopping scaled as $t_{ij}\sim t^*/\sqrt z$, making the self-energy local while retaining its full frequency dependence.

Required background. Use the Hubbard model and its limits. Helpful background. Conserving self-energy constructions and quantum impurity models provide useful context.

Local self-energy and impurity action

In the large-coordination limit, nonlocal self-energy diagrams vanish after the hopping rescaling, so $\Sigma_{ij}(z)=\delta_{ij}\Sigma(z)$. The local lattice Green function is

$$G_{\mathrm{loc}}(z)=\int\mathrm d\epsilon\, \frac{\rho_0(\epsilon)}{z+\mu-\epsilon-\Sigma(z)}.$$

Define a Weiss field $\mathcal G_0$ through

$$\mathcal G_0^{-1}(z)=G_{\mathrm{loc}}^{-1}(z)+\Sigma(z).$$

The corresponding impurity action is

$$S_{\mathrm{imp}}=-\sum_\sigma\int_0^\beta\mathrm d\tau\,\mathrm d\tau' c_\sigma^\dagger(\tau)\mathcal G_0^{-1}(\tau-\tau')c_\sigma(\tau') +U\int_0^\beta\mathrm d\tau\,n_\uparrow n_\downarrow.$$

Solving the impurity produces $G_{\mathrm{imp}}$ and $\Sigma_{\mathrm{imp}}=\mathcal G_0^{-1}-G_{\mathrm{imp}}^{-1}$. Self-consistency requires $G_{\mathrm{imp}}=G_{\mathrm{loc}}$ and $\Sigma=\Sigma_{\mathrm{imp}}$. This cavity construction and its infinite-dimensional proof are developed in Georges et al. 1996, §§ II–III.

Bethe-lattice checkpoint

For a semicircular noninteracting density of states of half-bandwidth $2t^*$,

$$\Delta(z)=t^{*2}G_{\mathrm{loc}}(z), \qquad \mathcal G_0^{-1}(z)=z+\mu-\Delta(z).$$

At $U=0$, $G=1/(z-t^{*2}G)$, so

$$G(z)=\frac{z-\sqrt{z^2-4t^{*2}}}{2t^{*2}},$$

where the branch is fixed by $G(z)\sim1/z$. This exactly reproduces the semicircular density of states and checks normalization. At finite $U$, metallic and insulating fixed points can coexist over a temperature-dependent region, but locating them requires a converged impurity solution and a declared iteration protocol.

Reliability conditions

Track residuals in $G$, $\Sigma$, and observables, initialize both metallic and insulating branches, and verify causality and spectral moments. Bath discretization, Monte Carlo noise, impurity truncation, and analytic continuation are separate errors. DMFT does not become controlled in two dimensions merely because $U$ is large; short-range spatial correlations then motivate the cluster extensions.

Exercises

Select the physical square-root branch in the noninteracting Bethe equation.

Solution

For large $z$, $\sqrt{z^2-4t^{*2}}=z-2t^{*2}/z+\cdots$ on the analytic branch outside the band. The minus sign gives $G\sim1/z$; the plus sign grows as $z/t^{*2}$ and violates the spectral normalization.

References

• Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, “Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions,” Reviews of Modern Physics 68 (1996) 13–125, §§ II–III, doi:10.1103/RevModPhys.68.13.