DMFT Self-Consistency: Lattice to Impurity Mapping

Recommended resources

The freely available resources this lesson will direct you to.

G. Kotliar - Towards a Predictive Theory of Strongly-Correlated Electron Materialswww.youtube.com
Dynamical Mean-Field Theory: An Introduction - Flatiron Instituteindico.flatironinstitute.org
4 The Foundations of Dynamical Mean-Field Theorywww.cond-mat.de

Hello! Welcome back.

In our last lesson, we explored the Mott metal-insulator transition, a cornerstone of strong-correlation physics. We saw that the competition between electron hopping ( $t$ ) and on-site repulsion ( $U$ ) could transform a would-be metal into an insulator. We discussed how experimental probes like photoemission and optical spectroscopy reveal the unique signatures of this transition, such as the emergence of a sharp quasiparticle peak at the Fermi level accompanied by broader Hubbard bands. Throughout that discussion, we repeatedly mentioned that Dynamical Mean-Field Theory (DMFT) is the modern theoretical framework that successfully captures this complex phenomenology.

Today, we will open the "black box" of DMFT. Our goal is to understand its fundamental conceptual leap and formulate the iterative procedure that lies at its heart. We will see how DMFT brilliantly simplifies an intractable many-body lattice problem by mapping it onto a solvable quantum impurity problem, embedded within a self-consistently determined environment. This lesson is theoretical, but it provides the essential machinery for producing the very spectral functions and physical insights we discussed last time.

1. The Core Idea: Beyond Simple Mean-Field Theory

You are likely familiar with mean-field theories from other areas of physics, like the Weiss theory of magnetism. There, the complex interactions of a single spin with all its neighbors are replaced by an interaction with a static, average "mean field". This works well when fluctuations are small.

However, in a strongly correlated electron system, this static approximation fails. An electron's environment is not static; it is highly dynamic. An electron hopping away from a site and returning later is influenced by the quantum fluctuations that occurred in its absence. A simple mean field cannot capture this "memory" or the dynamics of the environment.

Dynamical Mean-Field Theory addresses this by replacing the environment of a single lattice site not with a static field, but with a dynamical effective medium or "bath". This bath has a frequency-dependent response that encodes the complex quantum fluctuations of the rest of the lattice.

The central approximation that makes this possible is to assume the electron self-energy $Σ$ , which contains all the effects of interactions, is local. This means it is a function of frequency ( $ω$ ) but is independent of momentum ( $k$ ):

Σ (k, i ω_{n}) \approx Σ (i ω_{n})

This approximation becomes exact in the formal limit of infinite spatial dimensions ( $d \to \infty$ ) or infinite lattice coordination. While this limit may seem unphysical, it captures the essential local physics of strong correlations remarkably well, much like how mean-field theories become exact for infinitely-ranged interactions.

To get a feel for the core concepts, let's watch a segment from a lecture by Prof. Gabriel Kotliar, one of the pioneers of DMFT.

G. Kotliar - Towards a Predictive Theory of Strongly-Correlated Electron Materials

In this clip from 'Towards a Predictive Theory of Strongly-Correlated Electron Materials,' Prof. Kotliar explains the foundational idea of DMFT: reducing the lattice problem to a single-site quantum impurity problem within a self-consistent medium.

Please watch from 21:55 to 25:27. Focus on the analogy to classical mean-field theory and how it is extended to the quantum case by introducing a dynamical 'bath' or 'Weiss field'. Pay attention to his description of the self-consistency condition.

As Prof. Kotliar explains, the result of this local approximation is the mapping of the original lattice problem (like the Hubbard model) onto a single-site quantum impurity model. This is a problem of a single interacting site coupled to a non-interacting bath of electrons. A common example of such a model is the Single-Impurity Anderson Model (SIAM).

Dynamical Mean-Field Theory (DMFT) and the Hubbard Model — This image illustrates the core idea of DMFT. The complex Hubbard model on a lattice (left) is mapped to a more tractable problem of a single quantum impurity (right). This impurity represents one site of the original lattice, now coupled to an effective medium or 'bath' that represents the rest of the lattice. The properties of this bath are determined self-consistently.

2. Formulating the DMFT Self-Consistent Loop

The power of DMFT lies in determining the properties of this effective medium self-consistently. The bath that the impurity feels must be equivalent to the medium created by the impurity itself when viewed as part of the lattice. This requirement leads to a closed set of equations that are solved iteratively until convergence.

The following text provides a more formal derivation of the self-consistency equations. We will walk through the steps together.

4 The Foundations of Dynamical Mean-Field Theory

The document 'The Foundations of Dynamical Mean-Field Theory' by Marcus Kollar provides a rigorous yet clear derivation of the DMFT equations. We will use it to construct the self-consistent loop step-by-step.

Please read sections 3.5 ('Local self-energy', p. 12-13), 4.2 ('Mapping onto effective impurity models', p. 14), and 4.3 ('Dynamical mean-field equations', p. 15). Focus on how the assumption of a local self-energy (Eq. 43) leads to the three core DMFT equations (DMFT-1, DMFT-2, DMFT-3).

Let's break down the iterative loop described in the text and visualized in the diagram below. We will use Matsubara frequencies $i ω_{n}$ . The main quantities are the local self-energy $Σ (i ω_{n})$ , the local lattice Green's function $G_{l oc} (i ω_{n})$ , and the effective medium's Green's function, or Weiss field, $G_{0} (i ω_{n})$ .

LDA+DMFT Self-Consistency Loop — A flowchart illustrating the iterative DMFT cycle. While this diagram is for LDA+DMFT (a method for real materials), the core loop involving the self-energy $\Sigma$, the Green's function $G$, the impurity solver, and the self-consistency condition (S.C.C) is exactly what we are discussing for the model Hubbard Hamiltonian.

Here is the step-by-step procedure:

1. Start with a guess for the self-energy $Σ (i ω_{n})$ . A common starting point is $Σ (i ω_{n}) = 0$ (the non-interacting limit).

2. Calculate the local lattice Green's function $G_{l oc} (i ω_{n})$ .
Using the current self-energy, compute the full lattice Green's function $G (k, i ω_{n})$ and then integrate over all momentum $k$ to find the local part.

G (k, i ω_{n}) = \frac{1}{i ω _{n} + μ - ϵ ( k ) - Σ ( i ω _{n} )}

G_{l oc} (i ω_{n}) = k \sum G (k, i ω_{n}) = \int d ϵ \frac{ρ _{0} ( ϵ )}{i ω _{n} + μ - ϵ - Σ ( i ω _{n} )}

Here, $ϵ (k)$ is the non-interacting band dispersion and $ρ_{0} (ϵ)$ is the corresponding non-interacting density of states (DOS). This is equation (DMFT-3) from the Kollar notes.

3. Determine the Weiss field $G_{0} (i ω_{n})$ .
The Weiss field defines the effective medium for our impurity problem. It is the non-interacting Green's function of the impurity model. It is defined such that if we were to solve the impurity Dyson equation with our current self-energy, we would get back our current local Green's function. This allows us to extract it via algebraic rearrangement:

G_{0} (i ω_{n})^{- 1} = G_{l oc} (i ω_{n})^{- 1} + Σ (i ω_{n})

This step is a rearrangement of equation (DMFT-2) from the notes.

4. Solve the quantum impurity problem.
This is the most computationally demanding step. We now have a fully defined quantum impurity problem: a single interacting site described by an action whose non-interacting part is governed by $G_{0}$ . We must solve this model to find the new, fully interacting impurity Green's function, which we'll call $G_{im p} (i ω_{n})$ .

G_{im p} (i ω_{n}) = ⟨ c (i ω_{n}) c^{†} (i ω_{n}) ⟩_{A [G_{0}]}

This is equation (DMFT-1). This problem is typically mapped onto a Single-Impurity Anderson Model (SIAM) and solved numerically using methods like:

Continuous-Time Quantum Monte Carlo (CT-QMC): A stochastic method, powerful at finite temperatures.
Numerical Renormalization Group (NRG): Excellent for low-energy properties and zero temperature.
Exact Diagonalization (ED): Solves the problem exactly for a small, finite-sized bath.

5. Calculate the new self-energy $Σ_{n e w} (i ω_{n})$ .
Using the solution from the impurity solver ( $G_{im p}$ ) and the Weiss field we constructed ( $G_{0}$ ), we can find the new self-energy for the impurity problem from its own Dyson equation:

Σ_{n e w} (i ω_{n}) = G_{0} (i ω_{n})^{- 1} - G_{im p} (i ω_{n})^{- 1}

6. Check for convergence.
The loop is closed. We compare the new self-energy, $Σ_{n e w} (i ω_{n})$ , with the one we started with in Step 1, $Σ (i ω_{n})$ .

If they are sufficiently close (e.g., $\int d ω ∣ Σ_{n e w} - Σ ∣^{2} < δ$ ), the solution is self-consistent, and we stop. The converged $G_{l oc} (i ω_{n})$ contains the full physical information about the local electronic properties.
If not, we update our guess for the self-energy (e.g., set $Σ \leftarrow Σ_{n e w}$ or a mix of old and new) and return to Step 2.

Test your understanding!

In the DMFT loop, what is the physical meaning of the self-consistency condition $G_{l oc} (ω) = G_{im p} (ω)$ ? Why is this the central consistency check?

Show answer

The self-consistency condition demands that the local environment of the lattice model is identical to the effective medium seen by the impurity.

$G_{l oc} (ω)$ represents the "true" local dynamics of an electron on a single site within the full, interacting lattice (under the DMFT approximation). It's what an experimental probe measuring local properties would see.
$G_{im p} (ω)$ is the Green's function of the auxiliary impurity model, which is constructed to be solvable.

The condition $G_{l oc} = G_{im p}$ ensures that the auxiliary problem we are solving is not just any impurity problem, but the specific one that correctly mimics the local physics of the original lattice. If they are equal, it means the effective bath ( $G_{0}$ ) we constructed correctly represents the influence of all other sites on our chosen "impurity" site.

Conclusion

Today, we have delved into the theoretical heart of Dynamical Mean-Field Theory. We've moved from the qualitative picture of the Mott transition to the quantitative machinery used to describe it.

Key Takeaways:

DMFT's core idea is to map an interacting lattice model onto a quantum impurity model coupled to a self-consistent effective medium.
This mapping is enabled by the approximation that the electron self-energy $Σ$ is local (momentum-independent), an assumption that is exact in the limit of infinite dimensions.
The DMFT self-consistent loop is an iterative algorithm to find the effective medium:
1. Assume a self-energy $Σ$ .
2. Calculate the local lattice Green's function $G_{l oc}$ .
3. Extract the effective medium (Weiss field) $G_{0}$ .
4. Solve the impurity problem defined by $G_{0}$ to find a new impurity Green's function $G_{im p}$ .
5. Calculate a new self-energy $Σ_{n e w}$ from $G_{im p}$ .
6. Repeat until $Σ_{n e w} = Σ$ .

The output of this computationally intensive loop is the converged local Green's function, a rich object that contains detailed information about the electronic structure.

Preview of the Next Lesson

Now that we understand the process for calculating the local Green's function with DMFT, what do we do with it? The most important physical quantity we can extract is the spectral function, $A (ω) = - \frac{1}{π} Im G_{l oc} (ω)$ , which is directly comparable to photoemission experiments.

In the next lesson, we will interpret typical DMFT spectral functions for a metal, a correlated metal, and a Mott insulator. We will see precisely how the features we discussed previously—the quasiparticle peak, the Hubbard bands, and the Mott gap—emerge from the DMFT solution as we vary the interaction strength $U$ , thereby completing the connection between the physics of the Mott transition and the theory of DMFT.