A steady-state study of the nonequilibrium properties of realistic materials: Application of the mixed-configuration approximation

TL;DR

This work introduces and extends the mixed-configuration approximation (MCA) within dynamical mean-field theory (DMFT) to treat nonequilibrium steady states of multiorbital systems using the auxiliary master equation approach (AMEA) as the impurity solver. By partitioning orbitals into a target and configuration set, MCA expresses the target’s Green's function as a weighted sum over independent single-impurity problems, with weights given by joint configuration probabilities computed recursively. Application to bulk and layered SrVO3 shows that MCA-AMEA captures qualitative nonequilibrium charge redistribution and bias-induced orbital polarization, while displaying quantitative limitations related to orbital degeneracy and DMFT self-consistency drift, especially for degenerate t_{2g} manifolds. The approach provides a computationally efficient pathway to explore nonequilibrium multiorbital physics in realistic materials, with potential improvements via cumulant-based extensions to address degeneracy and causality concerns.

Abstract

We present the mixed-configuration approximation (MCA) based on the auxiliary master equation approach impurity solver to study multiorbital correlated systems under equilibrium and nonequilibrium conditions within dynamical mean-field theory (DMFT). We benchmark the method for bulk and layered SrVO$_3$ in equilibrium and apply it to a prototypical nonequilibrium geometry in which a voltage bias is applied perpendicular to the layer via reservoirs held at different chemical potentials. For bulk SrVO$_3$, MCA reproduces the metallic state at moderate interaction strengths, but it overestimates the weight of the lower band relative to quantum Monte Carlo (QMC) and fork tensor product state (FTPS) solvers. With respect to QMC and FTPS, MCA yields an earlier metal-to-insulator transition as the electron-electron interaction is increased. In layered SrVO$_3$ at equilibrium, MCA partially captures the orbital polarization in favor of the in-plane $xy$ orbital, although not as strong as in the DMFT-converged results obtained with QMC. However, when performing a one-shot impurity calculation initialized with the DFMT-QMC results, MCA yields orbital occupations which show a stronger charge polarization in favor of orbital $xy$. This suggests that our approach can be used to study multiorbital impurity problems when the focus is to assess properties without performing the full DMFT self-consistent loop. Finally, under applied bias, we observe a pronounced redistribution of orbital occupations, demonstrating that the method captures bias-driven orbital charge transfer in realistic materials in nonequilibrium conditions.

Figures (4)

• Figure 1: Equilibrium spectra of bulk SrVO$_3$ obtained for selected values of $U$ and $J=0.7 \, \mathrm{eV}$. Results obtained within the (a) FTPS (figure adapted from https://doi.org/10.1103/PhysRevX.7.031013) and (b) MCA-AMEA. The inset in (b) magnifies the region around the Fermi level, while the dashed vertical lines denote the position of the chemical potential.
• Figure 2: DMFT equilibrium results for a free-standing monolayer of SrVO$_3$ at $1/6$ filling. (a) Imaginary part of the Matsubara GFs per spin for orbital $d_{xy}$ obtained with QMC and MCA-AMEA (The AMEA calculation is carried out in real frequencies, so the Matsubara data are obtained by the usual Laplace-type transform.). (b) Same for orbitals $d_{yz}$ and $d_{zx}$. (c) Orbital spectra obtained with QMC and MCA-AMEA. MCA-AMEA yields occupations $n_{d_{xy}} = 0.307$ and $n_{d_{yz}/d_{zx}} = 0.117$, hinting at charge polarization. By contrast, QMC predicts an almost fully polarized state with $n_{d_{xy}} = 0.462$ and $n_{d_{yz}/d_{zx}} = 0.019$ at numerically exact sixth-filling. Here $U = 5.5\, \mathrm{eV}$, $J = 0.75\, \mathrm{eV}$ and $T = 0.025\, \mathrm{eV}$.
• Figure 3: Comparison of the one‐shot MCA‐AMEA results against fully self‐consistent DMFT obtained with QMC. (a) Orbital-resolved retarded hybridization function (solid) from the converged DMFT run within QMC used to initialize the one‐shot MCA‐AMEA calculation. The dashed line shows the reconstructed auxiliary hybridization function $\Delta^{\text{R}}_{\text{aux}}$ necessary for the MCA-AMEA impurity solver. The inset shows the corresponding Keldysh components. (b) Imaginary part of the orbital Matsubara GFs. (c) Corresponding spectra to (b). MCA‐AMEA yields orbital occupations (per spin) $n_{d_{xy}} = 0.372$ and $n_{d_{yz}/d_{zx}} = 0.078$, hinting at a tendency toward a charge polarization in favor of orbital $d_{xy}$. By contrast, the occupations in the converged QMC are $n_{d_{xy}} = 0.462$ and $n_{d_{yz}/d_{zx}} = 0.019$, with numerically exact $1/6$ filling. Temperature is set to $T = 0.025\ \mathrm{eV}$, and the chemical potential is set to the last value from the DMFT‐converged result obtained with QMC, $\mu= 2.09\ \mathrm{eV}.$
• Figure 4: Nonequilibrium DMFT results obtained with the MCA-AMEA by starting from the polarized state in Fig. \ref{['fig:fig3']}(c). (a) Nonequilibrium steady-state current as function of applied bias $\Phi$. (b) Orbital occupations as function of the applied bias. (c) Orbital spectra for $\Phi = 2 \mathrm{eV}$. (d) Same as (c) for $\Phi = 3.5 \mathrm{eV}$. Here $U = 5.5\, \mathrm{eV}$, $J = 0.75\, \mathrm{eV}$ and $T = 0.025\ \mathrm{eV}$.