Two-Level DMFT for Correlated Systems

• Two-Level DMFT is a reduced-space framework that confines the impurity model to a minimal Hilbert space, ensuring computational tractability in simulating strongly correlated systems.
• It employs self-consistent moment matching and exact double-counting corrections to accurately capture Mott transitions, local moment formation, and spectral properties.
• The approach is versatile, serving as a conceptual tool and practical solver in quantum simulations and ab initio embedding for molecules and solids.

Two-Level Dynamical Mean-Field Theory (DMFT) is a controlled truncation of conventional DMFT in which the effective quantum impurity model is confined to a minimal Hilbert space, typically consisting of one or two impurity sites and a single bath level. This construction allows for a fully self-consistent dynamical treatment of strong correlation physics within a computationally tractable framework. Two-level DMFT is employed both as a conceptual tool (e.g., for analytic insight into Mott transitions or moment formation) and as a practical solver for simplified models, with recent demonstrations on quantum hardware. The approach further enables the formulation of fully ab initio embedding schemes (such as LDA+DMFT), with analytically defined double-counting corrections, applicable to finite systems like molecules and, with screening, to realistic solids.

1. Anderson Impurity Model and Two-Level Truncation

The foundation of two-level DMFT is the mapping of a lattice model with local interactions—most notably the single-band Hubbard model—onto an Anderson impurity model (AIM) with a dramatically reduced set of degrees of freedom. In the standard case, the AIM Hamiltonian contains an interacting impurity site coupled to a non-interacting bath:

$H = H_{\rm imp} + H_{\rm bath} + H_{\rm mix}$

with

$$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$

where $c_{0\sigma}^\dagger$ creates a spin-$\sigma$ electron on the impurity, $a_{l\sigma}^\dagger$ on bath level $l$, $U$ is the on-site Coulomb repulsion, and $\epsilon_l, V_l$ are the bath parameters. In the two-site (one bath level, $l=1$) approximation, the problem reduces to only four fermionic modes. This reduction is exact for certain moments of the density of states and allows implementation on small quantum devices or as a controlled analytic model (Rungger et al., 2019).

2. DMFT Self-Consistency and Moment-Matching Criteria

The key principle of DMFT is the identification of the local lattice Green’s function $G_{\rm loc}(\omega)$ with the impurity Green’s function $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$0. The bath parameters $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$1 are iteratively updated to enforce self-consistency. In the two-site DMFT formulation (Potthoff), the bath is determined by matching the first two moments of the local density of states:

• The impurity filling equals the lattice filling: $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$2.
• The hybridization $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$3 matches the quasiparticle weight $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$4 of the system, with

$$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$5

For the Bethe lattice with a semi-circular density of states, the hybridization function satisfies $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$6, where $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$7 is the hopping amplitude ($$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$8 for half-bandwidth $$H_{\rm imp} = U n_{0\uparrow} n_{0\downarrow},\qquad H_{\rm bath} = \sum_{l,\sigma}\epsilon_l a_{l\sigma}^\dagger a_{l\sigma},\qquad H_{\rm mix} = \sum_{l,\sigma}V_l(c_{0\sigma}^\dagger a_{l\sigma} + a_{l\sigma}^\dagger c_{0\sigma})$$9). The iteration is continued until convergence of bath parameters and the impurity self-energy (Rungger et al., 2019).

3. Implementation: Classical, Quantum, and Ab Initio Approaches

Two-level DMFT admits various implementation strategies:

• Quantum Simulation: The small Hilbert space allows mapping the problem onto quantum circuits via Jordan–Wigner transformation (four qubits for the two-site case). The Variational Quantum Eigensolver (VQE) is used as the impurity solver, targeting both ground and excited states. Matrix elements relevant for the Lehmann spectral representation of $c_{0\sigma}^\dagger$0 can be accessed by overlap circuits on current NISQ devices. This methodology has been demonstrated on superconducting and trapped-ion quantum hardware, achieving DMFT self-consistency despite realistic quantum noise, and recovering the expected impurity-bath hybridization and Kondo/side-peak structure in the local density of states (Rungger et al., 2019).
• LDA+DMFT with Exact Double Counting: The two-level paradigm is used for finite quantum systems (e.g., H$c_{0\sigma}^\dagger$1 molecule), where the correlated subspace is defined by localized orbitals constructed from the bonding/antibonding combinations of molecular one-electron states. The exact double-counting correction is identified as the static Hartree plus exchange-correlation energy of the local density in this subspace. The procedure yields total energies and excitation spectra in excellent agreement with exact quantum chemistry benchmarks (Lee et al., 2014).

4. Algorithmic Workflow and Self-Consistency Loop

A generic workflow for two-level DMFT consists of the following steps:

1. Initialization: Compute the non-interacting one-particle Hamiltonian and determine the projectors for the correlated subspace (e.g., localized atomic/molecular orbitals).
2. Starting Self-Energy: Set the initial self-energy $c_{0\sigma}^\dagger$2.
3. Lattice Green’s Function: Calculate the interacting lattice Green’s function (either in $c_{0\sigma}^\dagger$3-space or real space).
4. Projection: Extract the local correlated subspace Green’s function via projection.
5. Weiss Field / Hybridization: Construct the impurity Weiss field $c_{0\sigma}^\dagger$4 and/or hybridization $c_{0\sigma}^\dagger$5.
6. Impurity Solver: Solve the two-level impurity problem to obtain $c_{0\sigma}^\dagger$6 and update $c_{0\sigma}^\dagger$7.
7. Double-Counting Correction (if ab initio): Apply exact double-counting functional to remove overlapping static correlations.
8. Charge and Self-Energy Update: Update the total charge density or self-energy, and check for convergence.
9. Iteration: Repeat until convergence criteria are met in $c_{0\sigma}^\dagger$8 and/or total energy.

Resource requirements for quantum implementation are modest: four qubits, tens of single-qubit rotations and CNOTs, $c_{0\sigma}^\dagger$9 VQE iterations for the ground state, with additional excited-state optimization steps and 7–10 DMFT iterations for self-consistency (Rungger et al., 2019).

5. Numerical Benchmarks and Spectral Properties

Benchmark calculations for H$\sigma$0 using two-level DMFT combined with LDA and exact double-counting reveal:

• Total ground-state energies accurate within $\sigma$1 of quantum chemical reference data across all interatomic separations, outperforming both standalone HF and LDA (Lee et al., 2014).
• Correlation energies capturing the formation of local moments and proper dissociation behavior.
• Spectral functions (density of states) yielding improved highest occupied and lowest unoccupied molecular orbital positions, sharply reducing errors in ionization energy and electron affinity.
• In quantum hardware demonstrations, convergence to expected bath parameters and Kondo/side-band features is achieved to within several percent accuracy, even under realistic noise conditions (Rungger et al., 2019).

6. Extension to Solids and Screening

The methodology generalizes to periodic solids by incorporating screening effects. The bare Coulomb interaction is replaced by a screened form $\sigma$2, and the LDA exchange-correlation is adjusted to $\sigma$3. Screening parameters can be computed using constrained-LDA or constrained-RPA. The exact intersection ("double-counting") between LDA and DMFT remains well-defined, enabling parameter-free ab initio calculations for realistic materials (Lee et al., 2014). The algorithmic framework is unchanged; only the form of $\sigma$4 and the correlation functionals adapt to screening.

7. Applications, Limitations, and Outlook

Two-level DMFT provides a tractable platform for developing and benchmarking DMFT-based algorithms, quantum circuit implementations, and ab initio embedding schemes. While the truncation sacrifices exactness for high-energy and nonlocal features, the approach successfully captures essential dynamical correlation and moment-formation physics, as confirmed by comparisons to exact results for small molecules and to analytical DMFT solutions for model systems. The formalism is foundational for quantum-classical hybrid algorithms and offers a rigorous test-bed for error mitigation and circuit reduction techniques in quantum simulation.

By enabling exact double-counting corrections and explicit dynamical self-consistency in a minimal model, two-level DMFT forms a bridge between conceptual analytic studies, quantum algorithm development, and realistic materials simulations (Rungger et al., 2019, Lee et al., 2014).