Real-Space DMFT: Inhomogeneity & Correlations

• Real-space DMFT is an extension of DMFT that assigns a local, site-dependent self-energy, capturing dynamic local correlations and spatial inhomogeneity.
• It maps each inequivalent site to an independent impurity problem, enabling detailed analysis of surfaces, interfaces, and nanostructures.
• The method retains key DMFT features like temporal fluctuations while relaxing translational symmetry to reveal Mott physics and quasiparticle renormalization.

Real-space dynamical mean-field theory (R-DMFT) is the inhomogeneous extension of dynamical mean-field theory in which the irreducible self-energy remains local in real space but is allowed to vary from site to site,

$$\Sigma_{ij}(i\omega_n)=\delta_{ij}\Sigma_i(i\omega_n).$$

It retains the central DMFT replacement of a static mean field by a dynamical environment, so temporal quantum fluctuations, quasiparticle renormalization, Hubbard bands, and Mott physics remain accessible, while translational invariance is relaxed. In practice, each inequivalent site or layer is mapped to its own self-consistent impurity problem, which makes the framework natural for surfaces, interfaces, multilayers, impurities, disorder realizations, traps, nanostructures, and nonequilibrium heterostructures (Vollhardt, 2010, Vollhardt et al., 2011, Miranda et al., 2011).

1. Conceptual definition and position within the DMFT family

The conceptual starting point is the distinction between correlations and static mean-field factorization. In correlated systems one cannot in general replace objects such as $$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$ by $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$ without destroying the physics of large-$U$ local moments and the Mott transition. DMFT addresses this by replacing the static Hartree picture with a dynamical bath, so the self-energy is frequency dependent even when it is spatially local.

Standard single-site DMFT combines two logically distinct statements. The first is locality of the self-energy,

$\Sigma_{ij}(\omega)=0\qquad (i\neq j),$

and the second is homogeneity of the local component,

$$\Sigma_{ii}(\omega)=\Sigma(\omega)\qquad \text{independent of }i.$$

R-DMFT keeps the first statement and abandons the second. The result is a site-resolved local self-energy, which preserves the dynamical treatment of local correlations while allowing the local environment to vary across the system.

This places R-DMFT between homogeneous single-site DMFT and cluster extensions. A concise classification is

$$\Sigma_{ij}(\omega)=\delta_{ij}\Sigma(\omega)\quad \text{(single-site DMFT)},$$

$$\Sigma_{ij}(\omega)=\delta_{ij}\Sigma_i(\omega)\quad \text{(R-DMFT)},$$

while cluster DMFT allows short-range nonlocal self-energy structure within a finite cluster. The distinction matters because R-DMFT restores spatial inhomogeneity but not nonlocal self-energy effects. A separate terminological caution is that “real-space renormalized DMFT” (rr-DMFT) is not conventional site-resolved R-DMFT; it is a real-space renormalization scheme built on top of cluster DMFT, especially CDMFT (Vollhardt, 2010, Vollhardt et al., 2011, Kubota et al., 2016).

2. Formal foundations: from high-dimensional DMFT to site-resolved self-consistency

The formal basis of R-DMFT is the large-dimension or large-coordination construction of DMFT. For the Hubbard model,

$$\hat H=\sum_{ij,\sigma} t_{ij}\hat c^\dagger_{i\sigma}\hat c_{j\sigma}+U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow},$$

the hopping must be scaled as

$$t\to \frac{t^*}{\sqrt d}\qquad\text{or}\qquad t\to \frac{t^*}{\sqrt Z},$$

so that kinetic and interaction energies remain comparable. In this limit, irreducible self-energy diagrams with spatially separated external vertices vanish, yielding

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$0

Vollhardt’s notes do not formulate R-DMFT by name, but this locality proof already isolates the ingredient that survives when translational invariance is broken (Vollhardt, 2010).

Once the self-energy is assumed site-diagonal, the lattice problem is solved from a real-space matrix Dyson equation. In the presence of an external site potential $$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$1,

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$2

with

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$3

Equivalently,

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$4

for the simplest Hubbard form without an explicit impurity potential. The local Green’s function on site $$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$5 is the diagonal element

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$6

Each site is then mapped to an effective impurity problem with action

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$7

where the site-dependent Weiss field obeys

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$8

or, in hybridization form,

$$\langle \hat n_{i\uparrow}\hat n_{i\downarrow}\rangle$$9

The self-consistency condition is imposed site by site: $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$0 This site-based impurity mapping is not a secondary reformulation; it is the natural real-space reading of the cavity construction already present in standard DMFT (Vollhardt, 2010, Byczuk et al., 2018).

3. Broken translational symmetry, disorder, and the scope of site dependence

The immediate motivation for R-DMFT is the inadequacy of a single self-energy $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$1 when local environments differ. The explicit targets named in the supplied literature include thin films, multilayers, nanostructures, surfaces, interfaces, heterostructures, impurities, defects, disorder realizations, and cold atoms in traps. In this class of problems, the physically decisive approximation is not momentum independence as such, but locality in the site basis.

The disordered-systems review makes this point especially sharply. Homogeneous DMFT and CPA-like averaging retain a common bath and can miss Anderson localization, broad distributions of local observables, rare-region physics, electronic Griffiths phases, and the strong spatial fluctuations of the local hybridization. Statistical DMFT replaces the common bath by site-dependent impurity actions,

$$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$2

with local but site-dependent self-energy

$$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$3

and a full lattice resolvent

$$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$4

The review explicitly states that the same microscopic formulation is applicable “in any spatially non-uniform situation, whether random or not,” so the formal gap between disorder-oriented statDMFT and deterministic R-DMFT is mainly one of application and emphasis: site profiles in the latter, distributions over sites or realizations in the former (Miranda et al., 2011).

This broader landscape also clarifies neighboring extensions. Typical Medium Theory compresses the inhomogeneous problem into a single “typical” medium defined from the geometric average of the local density of states and is therefore not fully site resolved. Extended DMFT introduces self-consistent bosonic fields for retarded density or spin interactions and for glassy order; it enriches the dynamical content of the impurity problem rather than the spatial resolution of the fermionic self-energy. These methods are complementary rather than interchangeable with R-DMFT.

4. Analytical reformulations and equilibrium impurity physics

A formal development within R-DMFT is the scattering-theory or $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$5-matrix formulation for an inhomogeneous Hubbard system with external potential $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$6. The self-energy is decomposed into a homogeneous background $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$7 and an inhomogeneous correction $$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$8, so the full lattice Green’s function can be written relative to a homogeneous interacting host,

$$\langle \hat n_{i\uparrow}\rangle\langle \hat n_{i\downarrow}\rangle$$9

with effective dynamical scattering potential

$U$0

In this language the inhomogeneity is not merely a bare defect potential; it is the bare perturbation plus the interaction-induced self-energy rearrangement. The same reformulation yields a Friedel sum rule for interacting lattice fermions at $U$1,

$U$2

where $U$3 is the screening charge and $U$4 is a matrix of scattering phase shifts defined algebraically through the many-body $U$5-matrix. The derivation assumes a Fermi-liquid state, particle-number conservation, and zero temperature (Byczuk et al., 2018).

A more direct equilibrium application is the study of Friedel oscillations around a local impurity in the Hubbard model. Full site-dependent R-DMFT solved with CT-QMC at finite temperature, together with a homogeneous self-energy approximation at $U$6 using NRG, shows that in the Fermi-liquid phase the amplitude of Friedel oscillations and the screening charge decrease with increasing interaction and track the quasiparticle renormalization factor

$U$7

Inside the Mott-insulating regime the asymptotic oscillations disappear, but a residual screening charge remains finite because the density near the impurity site still differs from the homogeneous background. The homogeneous self-energy approximation used there,

$U$8

is an explicit approximation to full R-DMFT rather than an identity (Chatterjee et al., 2018).

5. Layered and nonequilibrium formulations

A major branch of R-DMFT concerns systems that are translationally invariant within layers but inhomogeneous along one spatial direction. In equilibrium this already covers thin films, multilayers, interfaces, and heterostructures; in nonequilibrium it leads naturally to mixed real-space/momentum formulations on the Keldysh contour.

For the multilayer Hubbard Hamiltonian, the nonequilibrium extension uses a layer-resolved local self-energy,

$U$9

and, after Fourier transformation in the in-plane coordinates, an $\Sigma_{ij}(\omega)=0\qquad (i\neq j),$0 Dyson equation in layer space for each transverse momentum. The local layer Green’s function is

$\Sigma_{ij}(\omega)=0\qquad (i\neq j),$1

The method introduces left and right embedding hybridizations for the rest of the layered chain and solves the resulting contour equations by a recursive “zipper algorithm.” In the implementation discussed in the supplied material, the computational effort and storage scale linearly with the number of layers, making calculations with up to 39 interacting layers feasible. Applications include doublon diffusion after photoexcitation, current and polarization build-up under a voltage bias, and photo-induced current in biased Mott multilayers (Eckstein et al., 2013).

A steady-state formulation adds self-consistent electrostatics. In the layered heterostructure work, the correlated region is embedded between semi-infinite metallic leads and the long-range Coulomb interaction is treated at Hartree level through a Poisson equation,

$\Sigma_{ij}(\omega)=0\qquad (i\neq j),$2

solved by the Newton–Raphson method. The R-DMFT impurity problems are then coupled to a layer-dependent electrostatic potential $\Sigma_{ij}(\omega)=0\qquad (i\neq j),$3, and the impurity solver is the auxiliary master equation approach (AMEA). The paper emphasizes that the applied electrostatic bias $\Sigma_{ij}(\omega)=0\qquad (i\neq j),$4 need not coincide with the chemical-potential difference $\Sigma_{ij}(\omega)=0\qquad (i\neq j),$5, and studies how the Hubbard interaction $\Sigma_{ij}(\omega)=0\qquad (i\neq j),$6, the lead–correlated-region coupling strength, and the applied voltage influence charge redistribution in the correlated region and the adjacent lead layers (Titvinidze et al., 2018).

6. Accuracy, failure modes, and neighboring generalizations

The clearest benchmark of R-DMFT as an approximation is the two-impurity Anderson model describing two correlated adatoms coupled to a noninteracting tight-binding chain. This problem isolates the competition between local Kondo screening and nonlocal indirect magnetic exchange. R-DMFT treats the single-adatom Kondo effect exactly in the relevant sense because the self-energy is local for a single impurity, but the RKKY interaction is incorporated only approximately because its many-body feedback would generate nonlocal self-energy components. The comparison with DMRG shows a sharply regime-dependent outcome: there is excellent agreement, including subtle details of the competition between RKKY exchange and the Kondo effect, in a broad intermediate regime, but also complete failure in parameter regions where nonlocal singlet formation dominates. The failure appears as a spurious symmetry-broken mean-field solution and diverging susceptibilities; reliable results are obtained when the self-consistent solution remains paramagnetic and Fermi-liquid-like, well away from this artificial instability (Titvinidze et al., 2012).

This benchmark reflects the general limitation of R-DMFT. The method is local in space and dynamical in time. It restores spatial inhomogeneity, but it does not restore nonlocal self-energy physics. Cluster DMFT and related cluster methods address the latter by enlarging the impurity to a finite cluster. The distinction is structural, not terminological: R-DMFT and cluster DMFT solve different deficiencies of homogeneous single-site DMFT. In the same vein, rr-DMFT, despite its name, is not conventional R-DMFT but a real-space renormalized solver for large-cluster CDMFT problems; its aim is to approximate large-cluster solutions with many small-cluster impurity solves rather than to describe site-resolved inhomogeneity directly (Kubota et al., 2016).

A further extension of the dynamical mean-field concept appears in first-principles Green-function embedding. Real-space dynamical mean-field embedding (RDMFE) treats an entire unit cell or supercell as a local embedded region, assumes a cell-local self-energy correction,

$\Sigma_{ij}(\omega)=0\qquad (i\neq j),$7

and couples the embedded Green’s function to the periodic environment through two nested Dyson equations. This is explicitly DMFT-structured and real-space in the sense of cell-locality, but it is not conventional model-Hamiltonian R-DMFT for inhomogeneous lattices (Chibani et al., 2015).

Taken together, the supplied literature presents R-DMFT as a local self-energy theory whose essential content is precise: it abandons translational homogeneity without abandoning the dynamical impurity mapping of DMFT. Its success is strongest where local temporal correlations dominate and where inequivalent local environments are the primary source of complexity. Its limitations appear whenever short-range or long-range nonlocal correlations, rather than spatial inhomogeneity alone, control the physics.