Dynamical mean-field self-consistency

Let zc denote the number of nearest neighbors and scale a bond hopping as a fixed energy t star divided by the square root of zc. The sum of squared hopping amplitudes then remains finite as coordination grows. Keeping hopping fixed instead would make the kinetic-energy scale grow and would define a different singular limit. The interaction U stays local and unscaled in the usual single-band Hubbard construction. This limit is a property of a family of models, not a claim that a cubic material literally has infinitely many neighbors.

For appropriate lattices and local interactions, irreducible self-energy becomes local in that limit while retaining its full energy dependence. Locality does not make every correlation function local: propagation through the lattice still produces momentum-dependent Green functions and response structure. Single-site DMFT retains nontrivial local charge and spin dynamics. A purely static Hartree shift is therefore a much stronger approximation and cannot reproduce the general dynamical bath or the separated coherent and Hubbard-band spectral features.

Assume a local self-energy and use it in the lattice momentum sum, or equivalently in an integral over the noninteracting density of states. This gives the local lattice Green function. Independently define an interacting impurity with the same on-site interaction and an adjustable hybridization function describing exchange with a noninteracting bath. Solving the impurity yields its Green function and self-energy. Self-consistency requires the impurity Green function to equal the local lattice one.

The bath is often written through a noninteracting impurity Green function, called the Weiss function. Its inverse contains the hybridization, and subtracting the interacting impurity inverse gives the self-energy. A practical iteration updates these functions, solves an interacting impurity problem and repeats until suitable residuals converge. A converged numerical loop alone does not certify physical uniqueness, thermodynamic stability or spectral accuracy. Coexisting solutions and solver discretization, sampling, continuation and truncation errors need separate checks.

For an infinite-coordination Bethe lattice with the semicircular noninteracting density of states of half-bandwidth twice t star, self-consistency simplifies: hybridization equals squared t star times the local Green function. That convenient relation is lattice specific. A different density of states generally requires the full integral, and inserting the same shortcut for a square lattice changes the problem. The noninteracting limit provides a clean branch and units check before using an interacting impurity solver.

In finite-dimensional correlated metals, a single-site approximation omits momentum-dependent self-energy effects, short-range singlets and some collective fluctuations. Cluster or diagrammatic extensions can recover selected spatial structure, at further cost and with their own checks. A Mott boundary must specify the density of states, temperature, magnetic constraints and solver. The optimization lab only illustrates convergence of one elementary iteration; its controls are not U, filling, bath discretization or an impurity solver and cannot validate a DMFT phase diagram.