dmft

Dynamical mean-field theory (DMFT), sometimes referred to by the acronym dmft, is a nonperturbative approach to strongly correlated electron systems. It maps a lattice Hamiltonian with local interactions onto a single-impurity Anderson model embedded in a self-consistently determined electronic bath. The method became widely used after a 1996 review by Georges, Kotliar, Krauth, and Rozenberg, which laid out its formalism and applications.

In DMFT the self-energy is assumed to be local, Σ(k,ω) ≈ Σ(ω). This locality becomes exact in the limit of infinite spatial dimensions or coordination number. The lattice Green’s function is G(k,ω) = 1/[ω+μ − ε_k − Σ(ω)], and the local Green’s function is G_loc(ω) = ∑_k G(k,ω). The impurity problem is defined by an effective bath, described by the Weiss function 𝒢0(ω). The impurity Green’s function is G_imp(ω) = 1/[𝒢0(ω)^{-1} − Σ(ω)]. The self-consistency condition requires that G_loc(ω) = G_imp(ω). This yields 𝒢0(ω) from G_loc and Σ via 𝒢0(ω)^{-1} = G_loc(ω)^{-1} + Σ(ω). In practice, an iterative loop is used: guess Σ(ω), compute G_loc, determine a new 𝒢0, solve the impurity problem to obtain a new Σ(ω), and repeat until convergence.

Solving the impurity problem employs methods such as continuous-time quantum Monte Carlo, exact diagonalization, the numerical renormalization group, or perturbative solvers. DMFT can be extended to clusters (cluster DMFT, DCA) to include short-range nonlocal correlations and to multi-orbital settings. Realistic materials studies often combine DMFT with density functional theory (LDA+DMFT).

Applications include describing the Mott metal-insulator transition, finite-temperature spectral properties, and correlation effects in transition metal oxides and f-electron systems. Limitations include neglect of long-range spatial correlations in standard DMFT and dependence on the chosen impurity solver and computational cost for complex systems.