Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials.
In such materials, the approximation of independent electrons, which is used in density functional theory and usual band structure calculations, breaks down.
Dynamical mean-field theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the nearly free electron gas limit and the atomic limit of condensed-matter physics.
[2] While the lattice problem is in general intractable, the impurity model is usually solvable through various schemes.
The only approximation made in ordinary DMFT schemes is to assume the lattice self-energy to be a momentum-independent (local) quantity.
[3] One of DMFT's main successes is to describe the phase transition between a metal and a Mott insulator when the strength of electronic correlations is increased.
[6] In the Ising model, the lattice problem is mapped onto an effective single site problem, whose magnetization is to reproduce the lattice magnetization through an effective "mean-field".
While the N-site Ising Hamiltonian is hard to solve analytically (to date, analytical solutions exist only for the 1D and 2D case), the single-site problem is easily solved.
This also points to the major difference between the Ising MFT and DMFT: Ising MFT maps the N-spin problem into a single-site, single-spin problem.
The Hubbard model [7] describes the onsite interaction between electrons of opposite spin by a single parameter,
denote the creation and annihilation operators of an electron on a localized orbital on site
The following assumptions have been made: The Hubbard model is in general intractable under usual perturbation expansion techniques.
This hybridization function describes the dynamics of electrons hopping in and out of the bath.
It is related to the non-interacting Green's function by the relation: Solving the Anderson impurity model consists in computing observables such as the interacting Green's function
There exists a number of ways to solve the AIM, such as The self-consistency condition requires the impurity Green's function
The only DMFT approximations (apart from the approximation that can be made in order to solve the Anderson model) consists in neglecting the spatial fluctuations of the lattice self-energy, by equating it to the impurity self-energy: This approximation becomes exact in the limit of lattices with infinite coordination, that is when the number of neighbors of each site is infinite.
Indeed, one can show that in the diagrammatic expansion of the lattice self-energy, only local diagrams survive when one goes into the infinite coordination limit.
Thus, as in classical mean-field theories, DMFT is supposed to get more accurate as the dimensionality (and thus the number of neighbors) increases.
Put differently, for low dimensions, spatial fluctuations will render the DMFT approximation less reliable.
Spatial fluctuations also become relevant in the vicinity of phase transitions.
The most widespread way of solving this problem is by using a forward recursion method, namely, for a given
DMFT has several extensions, extending the above formalism to multi-orbital, multi-site problems, long-range correlations and non-equilibrium.
DMFT can be extended to Hubbard models with multiple orbitals, namely with electron-electron interactions of the form
The combination with density functional theory (DFT+DMFT)[4][8] then allows for a realistic calculation of correlated materials.
In order to improve on the DMFT approximation, the Hubbard model can be mapped on a multi-site impurity (cluster) problem, which allows one to add some spatial dependence to the impurity self-energy.
The Typical Medium Dynamical Cluster Approximation (TMDCA) is a non-perturbative approach for obtaining the electronic ground state of strongly correlated many-body systems, built on the dynamical cluster approximation (DCA).
[10] Spatial dependencies of the self energy beyond DMFT, including long-range correlations in the vicinity of a phase transition, can be obtained also through diagrammatic extensions of DMFT[11] using a combination of analytical and numerical techniques.
DMFT has been employed to study non-equilibrium transport and optical excitations.
[13] Here, the reliable calculation of the AIM's Green function out of equilibrium remains a big challenge.
DMFT has also been applied to ecological models in order to describe the mean-field dynamics of a community with a thermodynamic number of species.