A Scalable Configuration-Interaction Impurity Solver via Active Learning

Ara Go; Jeongmoo Lee

arxiv: 2604.10230 · v2 · submitted 2026-04-11 · ❄️ cond-mat.str-el

A Scalable Configuration-Interaction Impurity Solver via Active Learning

Jeongmoo Lee , Ara Go This is my paper

Pith reviewed 2026-05-11 00:50 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords active learningconfiguration interactionimpurity solverdynamical mean-field theoryHubbard modelSr2RuO4bath discretizationexact diagonalization

0 comments

The pith

Active learning selects the relevant determinants so configuration-interaction impurity solvers scale to larger baths with only weak cost growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces AL-ATCI, an active-learning version of adaptive-truncation configuration interaction, to identify the determinant manifold that matters for a correlated impurity state. By querying only a finite number N_query of candidates, the method keeps the active space manageable even when the full combinatorial space grows rapidly with added bath orbitals. A reader would care because this directly attacks the bath-discretization limit that has restricted exact-diagonalization and configuration-interaction solvers in dynamical mean-field theory. The authors show that, over the tested range, cost grows only weakly with bath size and that the spectra match exact-diagonalization results while reaching cluster sizes up to ten sites. They further demonstrate systematic convergence for a three-orbital Sr2RuO4 problem as bath orbitals increase from nine to eighteen.

Core claim

AL-ATCI uses active learning to locate the physically relevant subset of Slater determinants inside the rapidly expanding Hilbert space of an enlarged impurity problem; the query size N_query both controls the approximation and serves as an internal convergence parameter. Because the correlated state occupies only a small fraction of the full determinant space, the computational effort grows only weakly with bath size. In one-dimensional Hubbard-model benchmarks the resulting spectra and energies reproduce those of exact diagonalization, while cellular calculations become feasible up to ten-site clusters. For a rotationally invariant three-orbital Sr2RuO4 impurity the method yields converged

What carries the argument

AL-ATCI: an active-learning extension of adaptive-truncation configuration interaction that iteratively queries and retains only the determinant manifold relevant to the correlated state, with accuracy controlled by the finite query size N_query.

If this is right

AL-ATCI reproduces exact-diagonalization accuracy on one-dimensional Hubbard-model DMFT benchmarks.
Cellular DMFT calculations become practical for clusters as large as ten sites.
Dynamical quantities for a three-orbital Sr2RuO4 impurity converge systematically when bath orbitals are increased from nine to eighteen.
The relevant determinant manifold stays far smaller than the combinatorial space, so cost grows only weakly with bath size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same selection strategy could be tested on other many-body methods whose Hilbert spaces are dominated by a sparse set of important configurations.
If N_query proves reliable across a wider range of models, it could serve as a parameter-free convergence diagnostic for any determinant-based impurity solver.
The approach might be combined with existing bath-optimization techniques to further reduce the number of orbitals needed for a given accuracy.

Load-bearing premise

The active-learning procedure with a fixed query size N_query will always locate every determinant that materially affects the dynamical quantities, even as the bath grows larger.

What would settle it

A case in which increasing N_query after the reported convergence threshold still shifts the impurity spectral function or self-energy by more than numerical noise.

Figures

Figures reproduced from arXiv: 2604.10230 by Ara Go, Jeongmoo Lee.

**Figure 3.** Figure 3: FIG. 3. Spectral function from cellular DMFT for the one [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Convergence of AL-ATCI for the multi-orbital im [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original abstract

Finite-Hamiltonian impurity solvers provide direct real-frequency spectra and a natural route to enlarged impurity Hamiltonians, but their applicability is limited by the rapid Hilbert-space growth with the number of bath or other added one-particle orbitals. We introduce an active-learning extension of adaptive-truncation configuration interaction (AL-ATCI) that identifies the determinant manifold relevant to the correlated state. The approximation is systematically controlled by the query size N_query, which also provides an internal convergence parameter when no external benchmark is available. Over the benchmark range studied here, the computational cost grows only weakly with bath size, because enlarging the bath mainly expands the combinatorial determinant space rather than the physically relevant manifold. In dynamical mean-field-theory benchmarks for the one-dimensional Hubbard model, AL-ATCI reproduces exact-diagonalization accuracy and extends cellular calculations to clusters as large as N_c = 10. For a three-orbital rotationally invariant Sr2RuO4 impurity problem, we demonstrate systematic convergence of dynamical quantities and a highly compressed configuration space as N_b is increased from 9 to 18. These results substantially alleviate the bath-discretization bottleneck of exact-diagonalization- and configuration-interaction-based impurity solvers and make large-bath and enlarged-orbital calculations more practical.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

AL-ATCI keeps the determinant space manageable as bath size grows by selecting relevant configurations via active learning, with benchmarks showing ED-level accuracy on tested models.

read the letter

The paper's main advance is wrapping active learning around adaptive-truncation configuration interaction to pick out the physically relevant determinants for impurity problems. This keeps the computational cost from exploding with larger baths, since the combinatorial space grows fast but the important manifold stays small. They control the approximation through the query size N_query, which doubles as an internal convergence check when no exact benchmark exists. That setup is new for this class of solvers and directly targets the bath-size limit that has constrained exact and CI-based DMFT calculations. On the Hubbard chain they recover exact-diagonalization results, then push cellular calculations to clusters of size 10. For the three-orbital Sr2RuO4 impurity they show dynamical quantities converging as the bath increases from 9 to 18 orbitals while the configuration space remains highly compressed. Those benchmarks are the practical evidence that the approach works on realistic multi-orbital cases. The weak scaling with bath size is the clearest payoff for anyone trying to move beyond small baths. The soft spot is whether the learned manifold is complete enough for the Green's function and self-energy at finite frequencies. Even when ground-state energy or density matrix looks converged, small-weight determinants can still shift poles or residues. The paper reports systematic convergence for the dynamical quantities in the Sr2RuO4 example, so the tested cases appear to hold, but the acquisition function's sensitivity to virtual excitations that matter only at specific Matsubara frequencies would benefit from more explicit checks. No error bars or raw selection statistics are highlighted in the summary, which leaves the robustness a bit harder to judge without the full data tables. This work is aimed at condensed-matter groups running DMFT on multi-orbital or large-cluster problems where standard ED or CI hits the wall. Readers who need a scalable impurity solver with controllable truncation will find the benchmarks and implementation details useful. It deserves a serious referee because it supplies concrete scaling data and a working method for a known numerical bottleneck rather than just another formal proposal.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an active-learning extension (AL-ATCI) to adaptive-truncation configuration interaction for finite-Hamiltonian impurity solvers in DMFT. It claims that the query size N_query systematically controls the approximation by identifying the physically relevant determinant manifold, yielding only weak growth in computational cost with bath size N_b (because the manifold does not expand combinatorially), exact-diagonalization accuracy for the 1D Hubbard model, extension of cellular calculations to N_c=10, and systematic convergence of dynamical quantities for a three-orbital Sr2RuO4 impurity as N_b increases from 9 to 18.

Significance. If the active-learning selection is shown to be complete for dynamical quantities, the work would meaningfully extend the reach of configuration-interaction impurity solvers by reducing the effective Hilbert-space size, thereby addressing the bath-discretization bottleneck that currently limits ED- and CI-based DMFT calculations for multi-orbital and large-cluster problems.

major comments (2)

[Abstract and Sr2RuO4 results] Abstract and Sr2RuO4 benchmarks: the central claim that AL-ATCI reproduces ED-level accuracy for dynamical quantities while compressing the configuration space rests on the assumption that finite-N_query active learning never omits low-weight determinants that shift poles, residues, or the self-energy at finite frequencies. The presented results show convergence of dynamical quantities but supply no error bars, no quantitative truncation-error estimates for the Green's function, and no direct comparison to full ED (where feasible for smaller N_b) that would confirm the manifold is complete for spectra rather than only for the ground-state energy or density matrix.
[Methods (active-learning section)] Methods description of the active-learning procedure: N_query is introduced as an external control parameter that also serves as an internal convergence diagnostic, yet no equation or pseudocode specifies the acquisition function or uncertainty estimator. Without this, it is impossible to assess whether the selection criterion is insensitive to virtual excitations that contribute only at specific Matsubara or real frequencies, which directly affects the reliability of the reported weak scaling and spectral accuracy.

minor comments (2)

The abstract states that the cost 'grows only weakly with bath size' and that the configuration space is 'highly compressed,' but the main text should include a quantitative plot or table of the retained determinant fraction versus N_b to make the compression claim concrete.
Figure captions and axis labels for the dynamical-quantity convergence plots should explicitly state the value of N_query used and whether the curves are for fixed or increasing N_query.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the recognition that AL-ATCI could meaningfully extend the reach of configuration-interaction impurity solvers. We address each major comment below and will incorporate revisions to strengthen the quantitative support for our claims.

read point-by-point responses

Referee: [Abstract and Sr2RuO4 results] Abstract and Sr2RuO4 benchmarks: the central claim that AL-ATCI reproduces ED-level accuracy for dynamical quantities while compressing the configuration space rests on the assumption that finite-N_query active learning never omits low-weight determinants that shift poles, residues, or the self-energy at finite frequencies. The presented results show convergence of dynamical quantities but supply no error bars, no quantitative truncation-error estimates for the Green's function, and no direct comparison to full ED (where feasible for smaller N_b) that would confirm the manifold is complete for spectra rather than only for the ground-state energy or density matrix.

Authors: We agree that additional quantitative error analysis would strengthen the presentation of dynamical accuracy. For the 1D Hubbard model we already provide direct ED comparisons up to N_c=10 that include both energies and spectral functions. For the Sr2RuO4 impurity, full ED becomes prohibitive beyond N_b=9, but we will add in revision: (i) a direct ED versus AL-ATCI comparison at N_b=9 to quantify spectral truncation error, and (ii) error bars on the self-energy and Green's function obtained from the variation across N_query values. These additions will explicitly demonstrate that the selected manifold captures the relevant poles and residues for the reported dynamical quantities. revision: yes
Referee: [Methods (active-learning section)] Methods description of the active-learning procedure: N_query is introduced as an external control parameter that also serves as an internal convergence diagnostic, yet no equation or pseudocode specifies the acquisition function or uncertainty estimator. Without this, it is impossible to assess whether the selection criterion is insensitive to virtual excitations that contribute only at specific Matsubara or real frequencies, which directly affects the reliability of the reported weak scaling and spectral accuracy.

Authors: We acknowledge that the current methods section does not provide an explicit equation or pseudocode for the acquisition function. In the revision we will insert a dedicated subsection that defines the uncertainty estimator (based on the variance of CI coefficients across an ensemble of trial wavefunctions) and the acquisition function used to rank determinants. We will also include pseudocode for the iterative selection loop, clarifying how N_query controls inclusion of both ground-state and frequency-relevant virtual excitations. This will make the weak-scaling claim and spectral reliability fully reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; N_query is external control and scaling claims rest on benchmarks

full rationale

The paper presents AL-ATCI as an active-learning extension of adaptive-truncation CI, with N_query as an explicit external convergence parameter that controls the determinant manifold size. The central observation—that computational cost grows only weakly with bath size because the physically relevant manifold remains compressed—is reported as an empirical outcome from explicit benchmarks (1D Hubbard model and Sr2RuO4 impurity) rather than a mathematical identity or self-fit. No equation in the provided text reduces the reported accuracy, compression ratio, or weak scaling to a redefinition of the input data or to a parameter fitted to the same observables being predicted. Self-citations, if present, are not invoked to justify uniqueness or to close a derivation loop. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the assumption that active learning can isolate a physically relevant subset of determinants whose size grows much more slowly than the full Hilbert space.

free parameters (1)

N_query
Query size that controls truncation and serves as internal convergence parameter; its value is chosen by the user and affects which determinants are retained.

axioms (1)

domain assumption The physically relevant determinant manifold for the correlated impurity state can be identified by repeated active-learning queries without exhaustive enumeration.
This is the core premise that allows weak scaling with bath size.

pith-pipeline@v0.9.0 · 5514 in / 1399 out tokens · 52149 ms · 2026-05-11T00:50:59.050245+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

AL-ATCI addresses this by introducing a targeted selection step before Hamiltonian construction and diagonalization... Only the top N_query configurations are then retained for Hamiltonian construction and diagonalization
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the computational cost grows only weakly with bath size, because enlarging the bath mainly expands the combinatorial determinant space rather than the physically relevant manifold

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

[1]

Single-site DMFT (a) captures local correlations but misses spatial fluctuations. AsN c grows (b)–(d), the spectral weight redistributes and the dominant fea- tures align more closely with the Bethe-ansatz spinon and holon branches, reflecting the improved treatment 4 210 212 214 216 218 Nquery 10 7 10 5 10 3 (a) 210 212 214 216 218 Nquery 10 4 10 3 10 2 ...

work page 2023
[2]

Georges, G

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys.68, 13 (1996)

work page 1996
[3]

Kotliar, S

G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys.78, 865 (2006)

work page 2006
[4]

Werner, A

P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett.97, 076405 (2006)

work page 2006
[5]

E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys.83, 349 (2011)

work page 2011
[6]

Eidelstein, E

E. Eidelstein, E. Gull, and G. Cohen, Phys. Rev. Lett. 124, 206405 (2020)

work page 2020
[7]

Erpenbeck, T

A. Erpenbeck, T. Blommel, L. Zhang, W.-T. Lin, G. Co- hen, and E. Gull, The Journal of Chemical Physics161, 094104 (2024)

work page 2024
[8]

K. G. Wilson, Rev. Mod. Phys.47, 773 (1975)

work page 1975
[9]

Bulla, T

R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008)

work page 2008
[10]

K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Phys. Rev. Lett.115, 136401 (2015)

work page 2015
[11]

F. B. Kugler, M. Zingl, H. U. R. Strand, S.-S. B. Lee, J. von Delft, and A. Georges, Phys. Rev. Lett.124, 016401 (2020)

work page 2020
[12]

Gleis, S.-S

A. Gleis, S.-S. B. Lee, G. Kotliar, and J. von Delft, Phys. Rev. Lett.134, 106501 (2025)

work page 2025
[13]

Caffarel and W

M. Caffarel and W. Krauth, Phys. Rev. Lett.72, 1545 (1994)

work page 1994
[14]

Capone, L

M. Capone, L. de’ Medici, and A. Georges, Phys. Rev. B 76, 245116 (2007)

work page 2007
[15]

Lu and M

Y. Lu and M. W. Haverkort, The European Physical Journal Special Topics226, 2549 (2017)

work page 2017
[16]

Zgid and G

D. Zgid and G. K.-L. Chan, J. Chem. Phys.134, 094115 (2011)

work page 2011
[17]

D. Zgid, E. Gull, and G. K.-L. Chan, Phys. Rev. B86, 165128 (2012)

work page 2012
[18]

Lin and A

C. Lin and A. A. Demkov, Phys. Rev. B88, 035123 (2013)

work page 2013
[19]

Y. Lu, M. H¨ oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B90, 085102 (2014)

work page 2014
[20]

Go and A

A. Go and A. J. Millis, Phys. Rev. Lett.114, 016402 (2015)

work page 2015
[21]

Go and A

A. Go and A. J. Millis, Phys. Rev. B96, 085139 (2017)

work page 2017
[22]

Mejuto-Zaera, N

C. Mejuto-Zaera, N. M. Tubman, and K. B. Whaley, Phys. Rev. B100, 125165 (2019)

work page 2019
[23]

Kitatani, S

M. Kitatani, S. Sakai, and R. Arita, Phys. Rev. B108, 195124 (2023)

work page 2023
[24]

Y. Wang, G. Fabbris, M. Dean, and G. Kotliar, Com- puter Physics Communications243, 151 (2019)

work page 2019
[25]

Hariki, M

A. Hariki, M. Winder, T. Uozumi, and J. Kuneˇ s, Phys. Rev. B101, 115130 (2020)

work page 2020
[26]

Higashi, M

K. Higashi, M. Winder, J. Kuneˇ s, and A. Hariki, Phys. Rev. X11, 041009 (2021)

work page 2021
[27]

J. P. Coe, J. Chem. Theory Comput.14, 5739 (2018)

work page 2018
[28]

Jeong, C

W. Jeong, C. A. Gaggioli, and L. Gagliardi, J. Chem. Theory Comput.17, 7518 (2021)

work page 2021
[29]

Herzog, P

B. Herzog, P. Thunstr¨ om, and O. Eriksson, npj Compu- tational Materials11, 373 (2025)

work page 2025
[30]

Bilous, L

P. Bilous, L. Thirion, H. Menke, M. W. Haverkort, A. P´ alffy, and P. Hansmann, Phys. Rev. B111, 035124 (2025)

work page 2025
[31]

Olsen, B

J. Olsen, B. O. Roos, P. Jørgensen, and H. J. A. Jensen, J. Chem. Phys.89, 2185 (1988)

work page 1988
[32]

Lanczos, J

C. Lanczos, J. Res. Natl. Bur. Stand.45, 255 (1950)

work page 1950
[33]

E. R. Davidson, J. Comput. Phys.17, 87 (1975)

work page 1975
[34]

See Supplemental Material at [URL will be inserted by publisher] for additional benchmark results

work page
[35]

Breiman, Mach

L. Breiman, Mach. Learn.45, 5 (2001)

work page 2001
[36]

Pedregosa, G

F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cour- napeau, M. Brucher, M. Perrot, and E. Duchesnay, J. Mach. Learn. Res.12, 2825 (2011)

work page 2011
[37]

Hubbard, Proc

J. Hubbard, Proc. R. Soc. Lond. A276, 238 (1963)

work page 1963
[38]

E. H. Lieb and F. Y. Wu, Phys. Rev. Lett.20, 1445 (1968)

work page 1968
[39]

K. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. Lett.77, 1390 (1996)

work page 1996
[40]

Capone, M

M. Capone, M. Civelli, S. S. Kancharla, C. Castellani, and G. Kotliar, Phys. Rev. B69, 195105 (2004)

work page 2004
[41]

Go and G

A. Go and G. S. Jeon, Journal of Physics: Condensed Matter21, 485602 (2009)

work page 2009
[42]

Kotliar, S

G. Kotliar, S. Y. Savrasov, G. P´ alsson, and G. Biroli, Phys. Rev. Lett.87, 186401 (2001)

work page 2001
[43]

Maier, M

T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys.77, 1027 (2005)

work page 2005
[44]

Mravlje, M

J. Mravlje, M. Aichhorn, T. Miyake, K. Haule, G. Kotliar, and A. Georges, Phys. Rev. Lett.106, 096401 (2011)

work page 2011
[45]

Georges, L

A. Georges, L. d. Medici, and J. Mravlje, Annual Review of Condensed Matter Physics4, 137 (2013)

work page 2013
[46]

Kanamori, Prog

J. Kanamori, Prog. Theor. Phys.30, 275 (1963)

work page 1963
[47]

H. J. Lee, C. H. Kim, and A. Go, Phys. Rev. B102, 195115 (2020)

work page 2020
[48]

M. W. Haverkort, I. S. Elfimov, L. H. Tjeng, G. A. Sawatzky, and A. Damascelli, Phys. Rev. Lett.101, 026406 (2008). 6 Labeled(trainset) RASCI reference SDs reference SDs +Additional SDs LanczosConverged? Ground state Orbital rotation Training Prediction Query SDs FIG. S1. Flowchart of AL-ATCI, highlighting the active learning enhancements: classifier trai...

work page 2008

[1] [1]

Single-site DMFT (a) captures local correlations but misses spatial fluctuations. AsN c grows (b)–(d), the spectral weight redistributes and the dominant fea- tures align more closely with the Bethe-ansatz spinon and holon branches, reflecting the improved treatment 4 210 212 214 216 218 Nquery 10 7 10 5 10 3 (a) 210 212 214 216 218 Nquery 10 4 10 3 10 2 ...

work page 2023

[2] [2]

Georges, G

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys.68, 13 (1996)

work page 1996

[3] [3]

Kotliar, S

G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys.78, 865 (2006)

work page 2006

[4] [4]

Werner, A

P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett.97, 076405 (2006)

work page 2006

[5] [5]

E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys.83, 349 (2011)

work page 2011

[6] [6]

Eidelstein, E

E. Eidelstein, E. Gull, and G. Cohen, Phys. Rev. Lett. 124, 206405 (2020)

work page 2020

[7] [7]

Erpenbeck, T

A. Erpenbeck, T. Blommel, L. Zhang, W.-T. Lin, G. Co- hen, and E. Gull, The Journal of Chemical Physics161, 094104 (2024)

work page 2024

[8] [8]

K. G. Wilson, Rev. Mod. Phys.47, 773 (1975)

work page 1975

[9] [9]

Bulla, T

R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008)

work page 2008

[10] [10]

K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Phys. Rev. Lett.115, 136401 (2015)

work page 2015

[11] [11]

F. B. Kugler, M. Zingl, H. U. R. Strand, S.-S. B. Lee, J. von Delft, and A. Georges, Phys. Rev. Lett.124, 016401 (2020)

work page 2020

[12] [12]

Gleis, S.-S

A. Gleis, S.-S. B. Lee, G. Kotliar, and J. von Delft, Phys. Rev. Lett.134, 106501 (2025)

work page 2025

[13] [13]

Caffarel and W

M. Caffarel and W. Krauth, Phys. Rev. Lett.72, 1545 (1994)

work page 1994

[14] [14]

Capone, L

M. Capone, L. de’ Medici, and A. Georges, Phys. Rev. B 76, 245116 (2007)

work page 2007

[15] [15]

Lu and M

Y. Lu and M. W. Haverkort, The European Physical Journal Special Topics226, 2549 (2017)

work page 2017

[16] [16]

Zgid and G

D. Zgid and G. K.-L. Chan, J. Chem. Phys.134, 094115 (2011)

work page 2011

[17] [17]

D. Zgid, E. Gull, and G. K.-L. Chan, Phys. Rev. B86, 165128 (2012)

work page 2012

[18] [18]

Lin and A

C. Lin and A. A. Demkov, Phys. Rev. B88, 035123 (2013)

work page 2013

[19] [19]

Y. Lu, M. H¨ oppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B90, 085102 (2014)

work page 2014

[20] [20]

Go and A

A. Go and A. J. Millis, Phys. Rev. Lett.114, 016402 (2015)

work page 2015

[21] [21]

Go and A

A. Go and A. J. Millis, Phys. Rev. B96, 085139 (2017)

work page 2017

[22] [22]

Mejuto-Zaera, N

C. Mejuto-Zaera, N. M. Tubman, and K. B. Whaley, Phys. Rev. B100, 125165 (2019)

work page 2019

[23] [23]

Kitatani, S

M. Kitatani, S. Sakai, and R. Arita, Phys. Rev. B108, 195124 (2023)

work page 2023

[24] [24]

Y. Wang, G. Fabbris, M. Dean, and G. Kotliar, Com- puter Physics Communications243, 151 (2019)

work page 2019

[25] [25]

Hariki, M

A. Hariki, M. Winder, T. Uozumi, and J. Kuneˇ s, Phys. Rev. B101, 115130 (2020)

work page 2020

[26] [26]

Higashi, M

K. Higashi, M. Winder, J. Kuneˇ s, and A. Hariki, Phys. Rev. X11, 041009 (2021)

work page 2021

[27] [27]

J. P. Coe, J. Chem. Theory Comput.14, 5739 (2018)

work page 2018

[28] [28]

Jeong, C

W. Jeong, C. A. Gaggioli, and L. Gagliardi, J. Chem. Theory Comput.17, 7518 (2021)

work page 2021

[29] [29]

Herzog, P

B. Herzog, P. Thunstr¨ om, and O. Eriksson, npj Compu- tational Materials11, 373 (2025)

work page 2025

[30] [30]

Bilous, L

P. Bilous, L. Thirion, H. Menke, M. W. Haverkort, A. P´ alffy, and P. Hansmann, Phys. Rev. B111, 035124 (2025)

work page 2025

[31] [31]

Olsen, B

J. Olsen, B. O. Roos, P. Jørgensen, and H. J. A. Jensen, J. Chem. Phys.89, 2185 (1988)

work page 1988

[32] [32]

Lanczos, J

C. Lanczos, J. Res. Natl. Bur. Stand.45, 255 (1950)

work page 1950

[33] [33]

E. R. Davidson, J. Comput. Phys.17, 87 (1975)

work page 1975

[34] [34]

See Supplemental Material at [URL will be inserted by publisher] for additional benchmark results

work page

[35] [35]

Breiman, Mach

L. Breiman, Mach. Learn.45, 5 (2001)

work page 2001

[36] [36]

Pedregosa, G

F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cour- napeau, M. Brucher, M. Perrot, and E. Duchesnay, J. Mach. Learn. Res.12, 2825 (2011)

work page 2011

[37] [37]

Hubbard, Proc

J. Hubbard, Proc. R. Soc. Lond. A276, 238 (1963)

work page 1963

[38] [38]

E. H. Lieb and F. Y. Wu, Phys. Rev. Lett.20, 1445 (1968)

work page 1968

[39] [39]

K. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. Lett.77, 1390 (1996)

work page 1996

[40] [40]

Capone, M

M. Capone, M. Civelli, S. S. Kancharla, C. Castellani, and G. Kotliar, Phys. Rev. B69, 195105 (2004)

work page 2004

[41] [41]

Go and G

A. Go and G. S. Jeon, Journal of Physics: Condensed Matter21, 485602 (2009)

work page 2009

[42] [42]

Kotliar, S

G. Kotliar, S. Y. Savrasov, G. P´ alsson, and G. Biroli, Phys. Rev. Lett.87, 186401 (2001)

work page 2001

[43] [43]

Maier, M

T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys.77, 1027 (2005)

work page 2005

[44] [44]

Mravlje, M

J. Mravlje, M. Aichhorn, T. Miyake, K. Haule, G. Kotliar, and A. Georges, Phys. Rev. Lett.106, 096401 (2011)

work page 2011

[45] [45]

Georges, L

A. Georges, L. d. Medici, and J. Mravlje, Annual Review of Condensed Matter Physics4, 137 (2013)

work page 2013

[46] [46]

Kanamori, Prog

J. Kanamori, Prog. Theor. Phys.30, 275 (1963)

work page 1963

[47] [47]

H. J. Lee, C. H. Kim, and A. Go, Phys. Rev. B102, 195115 (2020)

work page 2020

[48] [48]

M. W. Haverkort, I. S. Elfimov, L. H. Tjeng, G. A. Sawatzky, and A. Damascelli, Phys. Rev. Lett.101, 026406 (2008). 6 Labeled(trainset) RASCI reference SDs reference SDs +Additional SDs LanczosConverged? Ground state Orbital rotation Training Prediction Query SDs FIG. S1. Flowchart of AL-ATCI, highlighting the active learning enhancements: classifier trai...

work page 2008