Real-Time Iteration Scheme for Dynamical Mean-Field Theory: A Framework for Near-Term Quantum Simulation

TL;DR

We address solving the DMFT self-consistency for strongly correlated systems using a real-time, time-domain approach compatible with near-term quantum hardware. The method maps the impurity problem onto a short one-dimensional chain and updates the hybridization $Δ^R(t)$ through real-time Green's functions, achieving stable convergence across U from 2 to 8. Despite a minimal 5-site bath and modest time resolution, the approach reproduces key spectral features including Hubbard bands and suppression of spectral weight at the Fermi level, signaling a metal-insulator transition. This framework provides a quantum-friendly DMFT solver that can be extended to non-equilibrium DMFT, Real-Space DMFT, and quantum solvers, potentially enabling more detailed spectral analyses beyond two-site approximations.

Abstract

We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.

Figures (4)

• Figure 1: Schematic of the effective Anderson impurity model mapped onto a linear chain geometry. The interacting impurity (Site 0) with Coulomb repulsion $U$ is coupled to a finite, non-interacting bath of $N_{bath} = 5$ sites. The chain is parameterized by the hopping amplitudes $t_i$ and on-site energies $\varepsilon_i$.
• Figure 2: Convergence analysis of the iterative fitting procedure across the metal-insulator crossover. The columns correspond to interaction strengths $U = 2, 4, 6,$ and $8$, respectively. Top Row: The convergence of the self-consistency loop, quantified by the root mean square error $\epsilon_{RMS}$ between consecutive hybridization functions [as defined in Eq. \ref{['eqn:RMS']}], plotted against the iteration number on a logarithmic scale. Bottom Row: The evolution of the five independent bath parameters $\{t_0, t_1, t_2, \epsilon_1, \epsilon_2\}$ as a function of iteration count. The parameters rapidly settle to their fixed-point values within approximately 20-30 iterations, confirming the stability of the method even in the strongly correlated Mott insulating phase ($U=8$).
• Figure 3: Interpolation of the converged impurity Green's function $G^R_{imp}(t)$ across the four interaction strengths ($U = 2, 4, 6, 8$) using cubic splines. Top Row: The imaginary part $\text{Im}[G^R(t)]$ plotted on a linear scale. The red dots represent the discrete time points from the solver, while the solid blue line shows the cubic spline interpolation. Bottom Row: The magnitude $|G^R(t)|$ plotted on a logarithmic scale to highlight the fitting quality at smaller amplitudes and later times. The evolution from slow decay at $U=2$ to rapid, sustained oscillations at $U=8$ reflects the emergence of high-energy Hubbard bands.
• Figure 4: Evolution of the single-particle spectral properties across the metal-insulator crossover for interaction strengths $U = 2, 4, 6, 8$. Top Row: The lattice spectral function $A_{lat}(\omega)$. The system transitions from a correlated metal ($U=2$) with a dominant central quasiparticle peak, through a coexistence regime ($U=4, 6$) characterized by a "three-peak structure", to a fully gapped Mott insulator ($U=8$) where spectral weight is transferred to the high-energy Hubbard bands. Bottom Row: The local self-energy $\Sigma(\omega)$, showing the real (blue) and imaginary (red) parts. In the metallic regimes ($U=2, 4$), $\text{Im}[\Sigma(\omega)]$ exhibits a minimum magnitude at $\omega=0$ indicating the presence of long-lived quasiparticles. In the Mott insulating limit ($U=8$), the real part develops a pole-like divergence and the imaginary part diverges at $\omega=0$.