Dynamical Mean-Field Theory and the IPT Solver

DMFT maps the Hubbard model onto a self-consistent Anderson impurity problem — exact in infinite dimensions — retaining the full frequency dependence of the self-energy that the perturbative limits miss. With Landau Fermi-liquid theory for interpretation and the Iterated Perturbation Theory solver for the impurity, it tracks the metal across to the Mott insulator in one framework.

Methods Dynamical mean-field theoryIterated perturbation theory

Why a more sophisticated method, if we already have weak- and strong-coupling solutions? The weak-coupling mean-field gives an insulator only once antiferromagnetic order sets in — yet cuprates remain insulating above the Néel temperature (a paramagnetic Mott insulator), which that picture cannot explain. The strong-coupling expansion yields a Heisenberg model of localized spins but no itinerant/metallic behaviour. Both miss the intermediate-coupling regime, and crucially neither retains the frequency dependence of the self-energy — which governs quasiparticle lifetimes, spectral-weight transfer, and the Hubbard bands.

Dynamical Mean-Field Theory (DMFT) is a non-perturbative method that treats local quantum dynamics exactly while approximating spatial fluctuations — analogous to mean-field theory for the Ising model, or DFT. It tracks weak to strong coupling in one framework, putting metal and insulator on equal footing.

The DMFT mapping

The idea is to represent the lattice by a single local site coupled to an effective bath.

In the limit of infinite dimensions (large coordination ), the Hubbard model can be reformulated exactly as a self-consistent Anderson impurity model (AIM): one interacting site ( operators) coupled to a non-interacting bath ( operators),

with in the single-band model. The bath parameters are tuned so the impurity Green’s function matches the local lattice one,

This is exact in infinite dimensions. Integrating out the bath gives the effective impurity action

where the Weiss dynamical field is the bare propagator of (the analogue of the classical mean field) and is the hybridization. From the interacting , the local self-energy follows from Dyson’s equation,

For the lattice, the interacting Green’s function is

The key infinite-dimensional fact is that the self-energy is purely local, so its -dependence drops and we identify . The local Green’s function then becomes

a closed self-consistent equation. With the non-interacting density of states ,

The DMFT loop

In practice DMFT iterates:

1. Guess the Weiss field .
2. Solve the impurity problem (an impurity solver) for .
3. Get the self-energy from Dyson, .
4. Compute from the self-consistency condition.
5. Rebuild the Weiss field, .
6. Repeat until (and ) stop changing.

Landau Fermi-liquid theory

Landau’s idea: the low-energy excitations of an interacting Fermi system are long-lived, weakly interacting quasiparticles resembling those of a free Fermi gas, obtained by adiabatically switching on interactions,

Near the Fermi surface the scattering phase space vanishes, so quasiparticles become long-lived (infinitely so at ). Formally, with the retarded function

(with so ). Expanding the real part at small , , and defining the quasiparticle weight and renormalized dispersion

the spectral function is a Lorentzian of weight ,

a stable quasiparticle with renormalized dispersion and weight . Equivalently, expanding the imaginary part instead, ,

and Cauchy’s identity again gives — the same result. This coherent peak emerging from an incoherent background distinguishes metallic ( finite, sharp quasiparticles) from insulating (incoherent/gapped) regimes.

The IPT solver

We use a simple, cheap impurity solver — Iterated Perturbation Theory (IPT) — computing the self-energy to second order in . The first-order terms are the Hartree shift (just a shift in the paramagnet) and the Fock term, which vanishes here since the on-site interaction only couples opposite spins. The second-order skeleton diagram,

uses the Weiss field as the bare propagator . With the spectral representation , the Fermi function , and the particle/hole parts , , analytic continuation gives

At half-filling, particle–hole symmetry reduces this to convolutions (),

and the real part follows from the Kramers–Kronig relations (since is causal). Scanning up and down through the DMFT loop will reveal hysteresis and phase coexistence — the first-order Mott transition.

Model and lattice choice: the Bethe lattice

We take the single-band Hubbard model at half-filling in infinite dimensions (), where single-site DMFT is exact (non-local self-energy components vanish as ). For concreteness we use the Bethe lattice, whose non-interacting density of states is the semicircle

with the hopping rescaled so stays finite. On the Bethe lattice the self-consistency integral collapses to the simple closed form

The Bethe lattice idealizes real 3D compounds with large but finite coordination — e.g. the corundum oxide (sixfold connectivity), routinely modelled this way with a quantitatively reasonable description of its pressure-driven Mott transition. With the method in hand, the next topic runs the loop and maps the transition.