Functional interpolation expansion for nonequilibrium correlated impurities

This directory tree contains the dataset for the publication included here (publication.pdf):

D. Werner and E. Arrigoni, "Functional interpolation expansion for nonequilibrium correlated impurities", Phys. Rev. Research 7, L022044 (2025). DOI: 10.1103/PhysRevResearch.7.L022044

The paper introduces a functional interpolation (FI) approach within the auxiliary master equation approach (AMEA) to efficiently solve nonequilibrium correlated quantum impurity problems. The physical hybridization function of the impurity, Delta_phys, is approximated by an auxiliary bath with a small number Nb of sites (typically 6-8), giving a residual error epsilon = Delta_phys - Delta_aux. The FI method estimates the linear-order correction in epsilon by solving the impurity problem for a small batch of N_T (typically 2-4) auxiliary baths with suitably chosen parameters, and combining the resulting self-energies (Eqs. (5)-(9) of the paper). The approach is tested on the Anderson impurity model in and out of equilibrium, and then applied, within DMFT, to the Hubbard model on the Bethe lattice (equilibrium metal/insulator and a nonequilibrium photodoped steady state).

Units: unless noted otherwise, all energies (U, T, V, t, omega, ...) are given in units of Gamma, defined by -Im(Delta_phys^R(w=0)) = Gamma, i.e. minus the imaginary part of the retarded physical hybridization function at zero frequency at the start of each calculation.

Reproducibility: every result file carries a "metadata" block with a "git_commit" hash of the solver code that produced it, together with a timestamp and the name of the driver script ("script_file"). No separate parameter schema file is included in this dataset; the "input" block stored in every JSON file (see below) self-documents the numerical/physical parameters used for that particular run.

The dataset is organized in two top-level directories, corresponding to different parts of the paper:

"Taylor_analyse" -> benchmark of the FI method itself on the single-orbital Anderson impurity model, in and out of equilibrium. This is the data underlying Fig. 1 of the paper, as well as the hyperparameter benchmark (choice of k, N_T, weighted vs. unweighted auxiliary-bath fits) discussed in the Results section and in the Supplemental Material.

"compare_dmft" -> application of the method as an impurity solver within nonequilibrium DMFT for the Hubbard model on the infinite-dimensional Bethe lattice: equilibrium calculations close to the Mott metal-insulator transition (Fig. 2), and a nonequilibrium photodoped quasi-steady state benchmarked against numerically exact inchworm QMC data (Fig. 3).

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"Taylor_analyse"
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Each subfolder corresponds to one physical setup (a bias voltage sweep, or a single point) for the Anderson impurity model. The subfolder name encodes the run parameters, as far as they could be identified:

"Nb_<n>"      -> number of bath sites of the auxiliary bath (also recoverable from "input"."solver"."modifiable"."N_sys" = Nb + 1 in the JSON files below)
"T_<x>"       -> physical temperature of the bath(s)
"N_<n>"       -> N_T, the number of auxiliary-bath realizations combined by the FI expansion; for a given folder this equals the number of "solver_output_*.json" files inside its "solver" subfolder (e.g. "N_2" -> solver_output_0.json and solver_output_1.json)
"nexc_<n>"    -> N_exc, a truncation parameter of the many-body solver (number of excitations kept in the auxiliary-bath Hilbert space; a numerical accuracy/cost knob, not a physical parameter)
"normal"      -> plain AMEA calculation without the FI correction (N_T = 1, a single auxiliary bath)
"Taylor"      -> FI-corrected calculation (see caveat below on naming)
"plus_eps"/"minus_eps" -> FI calculations built around a "desired" hybridization Delta_phys + k*epsilon with k on the positive/negative side, used to test different choices of k (cf. Eq. (3)-(4) and footnote 61 of the paper)
"orbital_Taylor_..._compare_weighed_..." -> comparison between an unweighted fit and a fit using a frequency-dependent weight function W(w) (emphasizing, e.g., low-energy regions) when constructing the initial auxiliary bath, cf. the "modification of this procedure" paragraph of the paper
"V_<x>"       -> subfolders named "V_0.0", "V_0.5", ... contain the bias-voltage sweep; V is the bias voltage applied across the two leads of the nonequilibrium Anderson impurity model (V = 0 is the equilibrium case)

Caveat: the folder-naming convention is not perfectly uniform across all subfolders (this reflects the exploratory nature of the underlying calculations); e.g. one folder is named "orbital_Taylor_N_6_..." where the "N_6" refers to Nb = 6 (confirmed via N_sys = 7 in its JSON) rather than N_T (which is 4 there, from the number of solver_output_*.json files present). When in doubt, prefer the values found inside the "input" block of the JSON files, or contact the authors.

Each parameter-set folder (i.e., each "V_<x>" folder, or the top folder itself when there is no voltage sweep) contains:

"orbital_output_<timestamp>.json" -> the combined/final output of the FI procedure for that parameter set (Eq. (7) of the paper). Besides the "input" block (solver and orbital/model settings, including "Taylor_coulomb"."N_max_it" = the maximum N_T attempted and "max_condition_number" = the cutoff on the condition number of the matrix A of Eq. (8) beyond which an auxiliary bath is discarded), the "results" block is split by spin flavor ("up", "down") and contains:
    "w"                    -> frequency grid (in units of Gamma)
    "Delta_fit_R"/"Delta_fit_K" -> fitted physical hybridization function (retarded/Keldysh) that the auxiliary baths were fitted to
    "DR_aux_*"/"DK_aux_*"  -> hybridization functions of the individual auxiliary baths entering the combination, for the unweighted ("_norm"/"_normal"... naming), weighted ("_weighed"), and, where present, transposed/combined variants ("_T", "_nw", "_wn")
    "SR_el_real_*"/"SR_el_imag_*"/"SK_el_real_*"/"SK_el_imag_*" -> real/imaginary parts of the retarded (R) and Keldysh (K) self-energy, for each of the above combination schemes; the "_total" suffix is the final FI-combined self-energy of Eq. (7), i.e. the main physical result of the calculation
    "Delta_R_imag_*"/"Delta_K_imag_*" -> imaginary parts of the corresponding hybridization functions

"solver/solver_output_<n>.json" -> the n-th individual auxiliary-bath many-body calculation (n = 0, ..., N_T-1) that enters the FI combination above. For each spin flavor it contains the frequency grid "w", the interacting retarded/Keldysh Green's function ("GR_real", "GR_imag", "GK_real", "GK_imag") and self-energy ("SR_real", "SR_imag", "SK_real", "SK_imag") of that single auxiliary-bath realization, the correlated-site occupation "n_cc", and, at the end of the file, the double occupation "N_double" and the accuracy indicator "chi" together with the boolean flag "SS_converged" (whether the nonequilibrium steady state was reached).

"c-self.dat" -> two-column ASCII reference/cross-check data (frequency, value); kept here as an auxiliary comparison curve for this benchmark.

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"compare_dmft"
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"dmft_eq_data/insulator/" and "dmft_eq_data/metal/" -> equilibrium DMFT self-consistency runs for the Hubbard model on the Bethe lattice (Fig. 2), for a point in the insulating phase (U = 5.0) and one in the metallic phase (U = 4.4), both at T = 0.05. Subfolder names follow the pattern

    normal_results_QB_t_<t>_U_<U>_T_<T>_Nb_<Nb>   (plain AMEA, no FI correction)
    taylor_results_QB_t_<t>_U_<U>_T_<T>_Nb_<Nb>    (FI-corrected)

with "t" the nearest-neighbor hopping (sets the Bethe lattice half-bandwidth) and the other parameters as above. A trailing "_old" on a folder name marks a superseded run kept for reference.

Inside each such folder:

"dmft_output.json" -> the final, self-consistency-converged DMFT result.
"time_ev_step_<n>/DMFT_iteration_<m>/dmft_output.json" -> intermediate results of the DMFT self-consistency loop: "time_ev_step_<n>" is the n-th step of the time-local quasi-steady-state (quantum Boltzmann) propagation, and within it "DMFT_iteration_<m>" is the m-th DMFT self-consistency iteration (lattice <-> impurity loop) at that time step.
"time_ev_step_<n>/quantum_boltzmann_output.json" -> output of the quantum-Boltzmann propagation step connecting consecutive time steps (see "Model and method" of the paper and Ref. [40] therein).

The "dmft_output.json" files share the same overall JSON layout as the Taylor_analyse files ("input"/"results"/"metadata"), with "results" split by spin flavor and containing, for the lattice/physical quantities after DMFT self-consistency:
    "w"                              -> frequency grid (in units of Gamma)
    "GR_phys_real"/"GR_phys_imag"    -> real/imaginary part of the physical (lattice) retarded Green's function
    "GK_phys_imag"                   -> imaginary part of the physical Keldysh Green's function
    "SR_el_real"/"SR_el_imag"        -> real/imaginary part of the physical retarded self-energy
    "SK_el_imag"                     -> imaginary part of the physical Keldysh self-energy
    "N_exc", "E_kin", "E_pot", "G_occ" -> auxiliary-bath truncation level and the kinetic/potential energy and orbital occupation of the converged solution
The "input" block additionally carries the DMFT self-consistency settings ("input"."DMFT"."static": "dmft_conv" = convergence threshold, "mixing_parameter", "lattice_type": "Bethe") and the quantum-Boltzmann propagation settings ("input"."QB").

"inchworm_data/" -> nonequilibrium steady-state benchmark for a cold photodoped Mott insulator (Fig. 3), reproducing the setup of F. Kunzel et al., Phys. Rev. Lett. 132, 176501 (2024) [Ref. 39 of the paper]. Contains:

    "normal_results_QB_t_<t>_Nb_<Nb>_conv_<c>_single_dmft/" -> plain AMEA calculation (no FI correction)
    "taylor_results_QB_t_<t>_Nb_<Nb>_conv_<c>_single_dmft/"  -> FI-corrected calculation
    (same internal dmft_output.json / time_ev_step / DMFT_iteration layout as above; "single_dmft" indicates a single (non-self-consistent) impurity calculation reproducing the reference steady state rather than a full lattice self-consistency loop)

    "inchworm_SE.dat" -> reference self-energy from numerically exact inchworm QMC (extracted from Ref. 39), three columns: frequency w, and two further columns corresponding to the real and imaginary parts of the self-energy (used for the comparison shown in Fig. 3).
    "inchworm_GR.dat" -> reference retarded Green's function from the same inchworm QMC calculation, same three-column layout (frequency, real part, imaginary part).
    "OCA.dat" -> reference data from the one-crossing approximation (OCA), a lower-order hybridization-expansion impurity solver used as an additional comparison benchmark; two columns (frequency, value).

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General note on the JSON files
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All JSON result files in this dataset (dmft_output.json, orbital_output_*.json, solver_output_*.json, quantum_boltzmann_output.json) share the same top-level layout:

"input"    -> full numerical and physical configuration used to produce this file (solver tolerances/iteration counts, model parameters such as U, the auxiliary-bath size N_sys = Nb + 1, lattice/self-consistency settings, etc.). This block makes each file self-contained and is the authoritative source for run parameters when in doubt about a directory name.
"results"  -> the physical output, generally split by spin flavor "up"/"down" (the model is spin-symmetric SU(2) unless stated otherwise). Retarded (R) and Keldysh (K) components of Green's functions ("G..."), self-energies ("S..."), and hybridization functions ("Delta...") follow the Keldysh formalism of the paper (see Eq. (1) and footnote 48): the retarded component's real part is independent information (its imaginary part is Kramers-Kronig related in the "phys" case), while only the imaginary part of the Keldysh component is independent (its real part vanishes).
"metadata" -> "timestamp", "git_commit" (code revision used), and "script_file" (driver script) for full reproducibility.

This research was funded in part by the Austrian Science Fund (FWF) [Grant DOI:10.55776/P33165]