| Dynamical mean field theory | |
|---|---|
| Name | Dynamical mean field theory |
| Field | Condensed matter physics |
| Introduced | 1989 |
| Derived from | Mean field theory |
| Keywords | Strongly correlated electrons, Hubbard model, Mott insulator |
Dynamical mean field theory
Dynamical mean field theory (DMFT) is a non-perturbative theoretical framework for treating strong local electronic correlations in lattice models of condensed matter physics. It maps a quantum lattice problem onto a self-consistent quantum impurity model embedded in a dynamical bath, capturing local temporal fluctuations while approximating spatial correlations. DMFT has been central to understanding phenomena such as the Mott transition, heavy fermion behavior and correlation-driven metal–insulator transitions in realistic materials.
Dynamical mean field theory was developed to extend classical mean field theory methods into the quantum and strongly interacting regime, preserving frequency-dependent self-energy effects omitted by static approximations. The method became prominent following seminal work by Georges, Kotliar, Krauth and Rozenberg in the early 1990s, and it forms a cornerstone of modern approaches for correlated electron systems alongside techniques like density functional theory (DFT) and quantum Monte Carlo (QMC). DMFT is used both as a conceptual tool for model Hamiltonians and combined with electronic structure methods in the so-called DFT+DMFT framework to study real materials such as transition metal oxides, rare earth compounds, and actinides.
DMFT is founded on the limit of infinite coordination number (or infinite dimensions) where the lattice self-energy becomes purely local. This observation allows mapping lattice models to an effective Anderson impurity model (AIM) subject to a self-consistency condition. The central objects are the local Green's function and the dynamical self-energy, which encode frequency-dependent quasiparticle renormalization and incoherent spectral weight (e.g., Hubbard bands). DMFT preserves temporal quantum fluctuations, enforces causality, and respects sum rules; it connects to diagrammatic methods such as Feynman diagram expansions and to non-perturbative renormalization group perspectives like the numerical renormalization group (NRG).
The most common lattice Hamiltonian studied with DMFT is the single-band Hubbard model with on-site interaction U and hopping t, used to investigate the Mott insulator and correlation-driven metal-insulator transitions. Multi-orbital extensions incorporate Hund's coupling J and crystal-field splitting to study materials with partially filled d- and f-shells (e.g., iron pnictides, nickelates, cerium compounds). The periodic Anderson model and Kondo lattice model are treated to explore heavy fermion physics and Kondo screening. Extensions include coupling to phonons (Holstein–Hubbard), long-range interactions via extended DMFT (EDMFT), and spin-orbit coupled models relevant to topological correlated materials and actinides like uranium compounds.
Solving the quantum impurity problem at the heart of DMFT requires accurate impurity solvers. Common numerical solvers include continuous-time quantum Monte Carlo (CT-QMC) algorithms (hybridization and interaction expansions), the numerical renormalization group (NRG) for low-energy spectral properties, exact diagonalization (ED) for discrete bath representations, and matrix product state (MPS)/density matrix renormalization group (DMRG)-based solvers for real-frequency spectra. Each solver balances tradeoffs in temperature range, real-frequency resolution, multi-orbital complexity, and computational cost. Diagrammatic impurity solvers and iterative perturbation theory (IPT) provide approximate but efficient options. Solver development often leverages high-performance computing facilities at institutions like Argonne National Laboratory, Max Planck Institute for Solid State Research, and collaborative software projects such as TRIQS and ALPS.
To include non-local spatial correlations absent in single-site DMFT, cluster extensions were developed: cellular DMFT (CDMFT), dynamical cluster approximation (DCA), and cluster perturbation theory hybrids. These cluster methods map to multi-impurity problems capturing short-range correlations important for pseudogap formation, unconventional superconductivity, and magnetism in cuprates and other low-dimensional systems. Diagrammatic extensions like dual fermion, dual boson, and GW+DMFT combine DMFT with diagrammatic corrections (e.g., GW approximation) to incorporate long-range screening and momentum-dependent self-energies. These approaches bridge DMFT with methods used in quantum chemistry and materials modeling.
DMFT has been applied to a wide range of phenomena: the finite-temperature Mott transition in V2O3, spectral weight redistribution in photoemission spectroscopy (PES) and angle-resolved photoemission spectroscopy (ARPES), optical conductivity in correlated metals, mass enhancement in heavy fermion compounds, and orbital-selective Mott transitions in strongly correlated multiorbital systems like iron-based superconductors. Combined DFT+DMFT studies address real-material properties including magnetic ordering, structural stability, and thermoelectric response in complex oxides, actinides, and rare earth systems. DMFT predictions are routinely compared with experiments at facilities such as European Synchrotron Radiation Facility and Advanced Photon Source.
Limitations of DMFT include the neglect of long-range nonlocal correlations in single-site formulations, sign problems in QMC impurity solvers at low temperatures or away from particle-hole symmetry, and challenges in treating very large multi-orbital systems with full rotationally invariant interactions. Ongoing developments address these issues via improved impurity solvers (tensor networks, machine learning acceleration), diagrammatic extensions, real-time and nonequilibrium DMFT for pump-probe spectroscopy, and tighter integration with ab initio methods (GW+DMFT, self-consistent DFT+DMFT). Active research groups at universities such as Princeton University, University of Cambridge, and institutions like Oak Ridge National Laboratory continue to push methodological frontiers and applications toward quantum materials discovery and functional correlated systems.