Mapping the non-equilibrium interacting Anderson Impurity Model to an effective Gaussian theory

Andrew K. Mitchell; Emmanuel Bogacz; Graham Kells

REVIEW 3 major objections 2 minor 71 references

Mapping the non-equilibrium interacting Anderson Impurity Model to an effective Gaussian theory

T0 review · 3 major / 2 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The quench dynamics of the interacting Anderson Impurity Model can be reproduced exactly by a non-interacting Gaussian model coupled to static auxiliary degrees of freedom.

desk verdict They numerically optimize a finite set of static auxiliary fermions so the enlarged non-interacting AIM matches ED/DMRG quench data, but the mapping has no analytic basis and its accuracy outside the fit window is unproven. read the letter →

arxiv 2606.19206 v1 pith:OQZQKOYV submitted 2026-06-17 cond-mat.str-el cond-mat.mes-hallquant-ph

Emmanuel Bogacz , Graham Kells , Andrew K. Mitchell This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallquant-ph

keywords AndersonImpurityModelnon-equilibriumquenchGaussianmappingauxiliarydegreesoffreedomnumericaloptimizationeffectivenon-interactingtheoryEDDMRGcomparison

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that the time-dependent evolution of the Anderson Impurity Model following a quench, which normally requires handling strong electron correlations, can instead be captured by an enlarged non-interacting model. The extra static auxiliary sites are introduced and their couplings are chosen through numerical optimization so that observables match those obtained from exact diagonalization and DMRG on the original interacting system. A sympathetic reader would care because this converts a difficult interacting non-equilibrium problem into one that can be solved with standard free-fermion techniques on a larger but Gaussian Hamiltonian. The approach uncovers structure within the auxiliary system and offers a route to compute quench properties without directly simulating the interactions.

What carries the argument

The effective non-interacting Anderson Impurity Model augmented by a finite set of static auxiliary degrees of freedom whose couplings are numerically optimized to reproduce the interacting quench dynamics.

What would settle it

Demonstration that, for any finite number of auxiliary sites, no choice of couplings reproduces the time-dependent local magnetization or occupation number from the full interacting model within a target error would falsify the mapping.

Watch

Extended reading notes

Core claim

The time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, the properties of this mapping are studied using numerical optimization, and intriguing structure in the auxiliary system is uncovered. The method allows understanding interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.

Load-bearing premise

A finite collection of static auxiliary degrees of freedom exists whose couplings can be chosen by numerical optimization to match the interacting quench dynamics to useful accuracy on the observables of interest.

Editorial extensions

If this is right

Quench dynamics and observables can be computed exactly using free-fermion methods on the enlarged Gaussian system.
The mapping reproduces key time-dependent quantities to useful accuracy once the auxiliary couplings are optimized.
Structure within the auxiliary degrees of freedom becomes visible and can be characterized directly.
The same non-interacting enlargement applies to any observable that can be obtained from the original interacting solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The auxiliary sites may serve as an effective discrete representation of the bath that could be reused for other quench protocols or steady-state calculations.
The approach could be tested on longer evolution times or different interaction strengths by comparing optimized results against larger-scale DMRG runs.
If the auxiliary structure is universal, the mapping might extend to other impurity models or to dynamical mean-field theory embeddings without re-deriving the optimization each time.
Hybrid solvers could combine this Gaussian mapping with perturbative corrections for regimes where the auxiliary count must remain small.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that the authors take existing ED and DMRG solutions for the quenched Anderson impurity model and tune the hoppings and on-site energies of a handful of extra static fermionic modes until the single-particle dynamics of the enlarged Gaussian system reproduce the interacting observables.

What is new is the application to the non-equilibrium quench: they start from the full interacting time evolution and back out an effective non-interacting representation with time-independent auxiliaries. The paper reports that the optimization produces some recognizable structure in the auxiliary parameters, which is at least an interesting observation even if it is post-hoc.

The approach works for the cases they examine because the fit is performed directly against the numerical data. That is the strength: once the auxiliaries are fixed, any observable that can be computed in the non-interacting model becomes available without further interacting calculations.

The soft spot is exactly the one flagged in the stress test. Because the auxiliaries are constrained to be static, any effective time dependence generated by the original interaction must be absorbed into the enlarged single-particle spectrum and initial state. The construction therefore depends on the numerical optimization succeeding, yet the abstract and the description give no information on the number of auxiliaries required, the time window used for the cost function, error bars as a function of time, or tests on quenches outside the training set. If the match degrades for t much larger than the fitting interval or for different interaction strengths, the claim that the dynamics “can be described” by the Gaussian model is limited to the fitted regime.

This is for people already working on non-equilibrium impurity problems who want an alternative computational or interpretive route. A reader who needs a first-principles mapping or guaranteed long-time accuracy will not find it here.

I would send it to peer review. The numerical evidence is concrete enough to be worth referee scrutiny on the robustness of the fits and the scope of the mapping.

Referee Report

3 major / 2 minor

Summary. The paper claims that the time-dependent dynamics of the interacting Anderson Impurity Model (AIM) following a quench can be reproduced by an effective non-interacting Gaussian theory obtained by coupling the original impurity and bath to a finite set of additional static auxiliary fermionic modes whose hoppings and on-site energies are determined by numerical optimization against ED and DMRG data.

Significance. If the mapping can be shown to require only a modest number of auxiliaries and to remain accurate outside the fitting window, it would provide a practical route to interpret and compute non-equilibrium impurity dynamics via solvable Gaussian methods while revealing structure in the auxiliary spectrum. The direct numerical matching to exact solvers is a concrete strength of the approach.

major comments (3)

[Methods and optimization procedure] The optimization procedure (described in the methods and results sections) supplies no a-priori bound on the required auxiliary Hilbert-space dimension, no convergence criteria, and no error bars on the fitted parameters. Because the central claim rests on the existence of a finite static auxiliary set that reproduces the interacting quench evolution, these details are load-bearing for assessing whether the mapping is robust or merely a post-hoc fit.
[Results on time evolution] No tests are reported for times t ≫ t_fit or for quench protocols different from those used in the cost function. The skeptic concern that static auxiliaries may fail to capture long-time or out-of-sample observables therefore remains unaddressed, directly affecting the generality of the claimed mapping.
[Numerical results] The manuscript does not quantify how the quality of the fit (e.g., deviation in local observables or spectral functions) scales with the number of auxiliary modes, leaving open whether the effective dimension remains modest for physically relevant parameters.

minor comments (2)

[Model definition] Clarify the precise form of the effective single-particle Hamiltonian, including how the auxiliary modes are coupled to the original AIM degrees of freedom.
[Results] Add a table or figure summarizing the number of auxiliaries used for each parameter set and the achieved fit accuracy.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Methods and optimization procedure] The optimization procedure (described in the methods and results sections) supplies no a-priori bound on the required auxiliary Hilbert-space dimension, no convergence criteria, and no error bars on the fitted parameters. Because the central claim rests on the existence of a finite static auxiliary set that reproduces the interacting quench evolution, these details are load-bearing for assessing whether the mapping is robust or merely a post-hoc fit.

Authors: We agree that the optimization procedure requires fuller documentation. In the revised manuscript we will expand the methods section to specify the explicit form of the cost function, the numerical optimizer used, the convergence tolerance applied to the cost, and error estimates on the fitted parameters obtained from repeated optimizations with varied initial conditions. An analytic a-priori bound on auxiliary dimension is not available, but we will add a discussion of the empirical criterion (saturation of fit quality with increasing auxiliary number) already used to select the reported dimensions. revision: yes
Referee: [Results on time evolution] No tests are reported for times t ≫ t_fit or for quench protocols different from those used in the cost function. The skeptic concern that static auxiliaries may fail to capture long-time or out-of-sample observables therefore remains unaddressed, directly affecting the generality of the claimed mapping.

Authors: The referee correctly notes the absence of out-of-sample tests. We will add new calculations that propagate the fitted auxiliary model to times t ≫ t_fit and to quench protocols not included in the original cost function, comparing the resulting observables against available ED or DMRG reference data. These results will be included in a new subsection of the results. revision: yes
Referee: [Numerical results] The manuscript does not quantify how the quality of the fit (e.g., deviation in local observables or spectral functions) scales with the number of auxiliary modes, leaving open whether the effective dimension remains modest for physically relevant parameters.

Authors: We will add a systematic quantification of fit quality versus auxiliary number. A new figure will show the scaling of the mean-squared deviation in local observables (and, where computable, spectral functions) as a function of auxiliary count for several values of interaction strength and bath parameters, thereby demonstrating that the required dimension remains modest in the regimes studied. revision: yes

Circularity Check

1 steps flagged · score 3.0 of 10

Auxiliary parameters obtained via numerical optimization to ED/DMRG data

fitted input called prediction [Abstract]
"Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system."

The effective non-interacting model with auxiliary degrees of freedom is constructed by numerically optimizing its parameters to match the interacting dynamics obtained from ED/DMRG; the mapping therefore reproduces its input data by construction of the fit rather than providing an independent reduction.

full rationale

The paper constructs its central mapping explicitly by fitting a finite set of static auxiliary modes to reproduce the interacting quench dynamics already computed via ED/DMRG. This matches the 'fitted input called prediction' pattern at a moderate level because the effective Gaussian theory is defined by the optimization procedure itself rather than derived independently; however the paper makes no claim of an analytic first-principles derivation that would create stronger circularity, and the result remains a numerical demonstration rather than a self-referential proof.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available; the ledger is therefore incomplete. The mapping implicitly assumes that the interacting quench dynamics can be exactly or approximately reproduced by a non-interacting Hamiltonian plus static auxiliaries whose parameters are variationally optimized.

free parameters (1)

auxiliary couplings and energies
Parameters of the auxiliary degrees of freedom are determined by numerical optimization to match ED/DMRG data.

assumptions (1)

domain assumption The quenched AIM dynamics admit a representation as a non-interacting model coupled to static auxiliaries
Central premise of the mapping; invoked throughout the abstract.

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Cite this review

Pith. "Pith review of Mapping the non-equilibrium interacting Anderson Impurity Model to an effective Gaussian theory." pith.science (2026). https://pith.science/paper/OQZQKOYV

@misc{pith2026260619206,
  author       = {Pith},
  title        = {Pith review of: Mapping the non-equilibrium interacting Anderson Impurity Model to an effective Gaussian theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQZQKOYV}},
  note         = {Machine review of arXiv:2606.19206}
}

read the original abstract

Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.

Figures

Figures reproduced from arXiv: 2606.19206 by the authors.

**Figure 3.** FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. Time evolution of impurity occupation following a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5. Dynamics of the impurity occupation in the AIM [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7. (a) Time-dependence of post-quench impurity occu [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: FIG. 9. Band structure of the interacting AIM (a) and op [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Single-particle wavefunction amplitudes for the op [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Benchmark comparison of the equilibrium impurity [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 5 canonical work pages

[1]

⟩is the ther- mal expectation value in the Gibbs state

Representations of the Interacting Green’s function The impurity retarded Green’s function is defined as GAIM dd;σ(t)=−iθ(t)⟨{dσ(t), d† σ(0)}⟩where⟨. . .⟩is the ther- mal expectation value in the Gibbs state. In the (com- plex) frequency domainz=ω+iηwithη>0 we have GAIM dd;σ(z)≡⟨⟨dσ;d † σ⟩⟩=∫dt eiztGAIM dd;σ(t). Its spectral func- tion isA(ω)=− 1 π ImG AI...

2022
[2]

Polkovnikov, K

A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Colloquium: Nonequilibrium dynamics of closed in- teracting quantum systems, Rev. Mod. Phys.83, 863 (2011)

2011
[3]

Eisert, M

J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- body systems out of equilibrium, Nat. Phys.11, 124 (2015)

2015
[4]

H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys.86, 779 (2014)

2014
[5]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

2011
[6]

Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys

G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

2003
[7]

Calabrese and J

P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.2005, P04010 (2005)

2005
[8]

A. C. Hewson,The Kondo Problem to Heavy Fermions, Cambridge Studies in Magnetism, Vol. 2 (Cambridge University Press, Cambridge, 1993)

1993

Show all 71 references

[9]

E. S. Sørensen, M.-S. Chang, N. Laflorencie, and I. Af- fleck, Impurity entanglement entropy and the kondo 12 screening cloud, J. Stat. Mech.2007, L01001 (2007)

2007
[10]

S.-S. B. Lee, J. Park, and H.-S. Sim, Macroscopic quan- tum entanglement of a kondo cloud at finite temperature, Phys. Rev. Lett.114, 057203 (2015)

2015
[11]

Shim, H.-S

J. Shim, H.-S. Sim, and S.-S. B. Lee, Numerical renor- malization group method for entanglement negativity at finite temperature, Phys. Rev. B97, 155123 (2018)

2018
[12]

Costi, L

T. Costi, L. Bergqvist, A. Weichselbaum, J. Von Delft, T. Micklitz, A. Rosch, P. Mavropoulos, P. H. Dederichs, F. Mallet, L. Saminadayar,et al., Kondo decoherence: finding the right spin model for iron impurities in gold and silver, Physical Review Letters102, 056802 (2009)

2009
[13]

P. G. Derry, A. K. Mitchell, and D. E. Logan, Quasi- particle interference from magnetic impurities, Physical Review B92, 035126 (2015)

2015
[14]

Satori, H

K. Satori, H. Shiba, O. Sakai, and Y. Shimizu, Numerical renormalization group study of magnetic impurities in superconductors, J. Phys. Soc. Jpn.61, 3239 (1992)

1992
[15]

ˇZitko, Spectral properties of Shiba subgap states at finite temperatures, Phys

R. ˇZitko, Spectral properties of Shiba subgap states at finite temperatures, Phys. Rev. B93, 195125 (2016)

2016
[16]

Huang, S

H. Huang, S. Karan, C. Padurariu, B. Kubala, J. C. Cuevas, J. Ankerhold, K. Kern, and C. R. Ast, Univer- sal scaling of tunable Yu-Shiba-Rusinov states across the quantum phase transition, Commun. Phys.6, 214 (2023)

2023
[17]

Chiappe and E

G. Chiappe and E. Louis, Kondo effect of an adsorbed cobalt phthalocyanine (copc) molecule: The role of quan- tum interference, Phys. Rev. Lett.97, 076806 (2006)

2006
[18]

L. G. G. V. Dias da Silva, M. L. Tiago, S. E. Ulloa, F. A. Reboredo, and E. Dagotto, Many-body electronic struc- ture and kondo properties of cobalt-porphyrin molecules, Phys. Rev. B80, 155443 (2009)

2009
[19]

C. Li, V. Pokorn` y, P. Hapala, M.ˇZonda, P. Zhou, S. De- curtins, S.-X. Liu, F. Song, R. Pawlak, and E. Meyer, Negative differential conductance in triangular molecu- lar assemblies, arXiv:2508.05575 (2025)

2025
[20]

Goldhaber-Gordon, H

D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. Kastner, Kondo effect in a single-electron transistor, Nature391, 156 (1998)

1998
[21]

S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen- hoven, A tunable kondo effect in quantum dots, Science 281, 540 (1998)

1998
[22]

Jeong, A

H. Jeong, A. M. Chang, and M. R. Melloch, The kondo effect in an artificial quantum dot molecule, Science293, 2221 (2001)

2001
[23]

A. K. Mitchell and E. Sela, Universal low-temperature crossover in two-channel kondo models, Physical Review B—Condensed Matter and Materials Physics85, 235127 (2012)

2012
[24]

Lebanon, A

E. Lebanon, A. Schiller, and F. B. Anders, Coulomb blockade in quantum boxes, Physical Review B68, 041311 (2003)

2003
[25]

Potok, I

R. Potok, I. Rau, H. Shtrikman, Y. Oreg, and D. Goldhaber-Gordon, Observation of the two-channel kondo effect, Nature446, 167 (2007)

2007
[26]

Iftikhar, A

Z. Iftikhar, A. Anthore, A. Mitchell, F. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, and F. Pierre, Tunable quantum criticality and super- ballistic transport in a “charge” kondo circuit, Science 360, 1315 (2018)

2018
[27]

Pouse, L

W. Pouse, L. Peeters, C. L. Hsueh, U. Gennser, A. Ca- vanna, M. A. Kastner, A. K. Mitchell, and D. Goldhaber- Gordon, Quantum simulation of an exotic quantum crit- ical point in a two-site charge kondo circuit, Nature Physics19, 492 (2023)

2023
[28]

Sriram, C

P. Sriram, C. L. Hsueh, K. A. Morey, T. Wang, C. Thomas, G. C. Gardner, M. A. Kastner, M. J. Man- fra, A. K. Mitchell, E. Sela, and D. Goldhaber-Gordon, Hybrid metal-semiconductor quantum dots in InAs as a platform for quantum simulation, arXiv:2508.03928 (2025)

2025
[29]

Liang, M

W. Liang, M. P. Shores, M. Bockrath, J. R. Long, and H. Park, Kondo resonance in a single-molecule transistor, Nature417, 725 (2002)

2002
[30]

M. L. Perrin, E. Burzur´ ı, and H. S. van der Zant, Single- molecule transistors, Chemical Society Reviews44, 902 (2015)

2015
[31]

A. K. Mitchell, K. G. Pedersen, P. Hedeg˚ ard, and J. Paaske, Kondo blockade due to quantum interference in single-molecule junctions, Nature Communications8, 15210 (2017)

2017
[32]

Georges, G

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen- berg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Re- views of Modern Physics68, 13 (1996)

1996
[33]

K. G. Wilson, The renormalization group: Critical phe- nomena and the kondo problem, Reviews of modern physics47, 773 (1975)

1975
[34]

Krishna-Murthy, J

H. Krishna-Murthy, J. Wilkins, and K. Wilson, Renormalization-group approach to the anderson model of dilute magnetic alloys. i. static properties for the sym- metric case, Physical Review B21, 1003 (1980)

1980
[35]

Weichselbaum and J

A. Weichselbaum and J. von Delft, Sum-rule conserving spectral functions from the Numerical Renormalization Group, Physical Review Letters99, 076402 (2007)

2007
[36]

A. K. Mitchell, M. R. Galpin, S. Wilson-Fletcher, D. E. Logan, and R. Bulla, Generalized wilson chain for solv- ing multichannel quantum impurity problems, Physical Review B89, 121105 (2014)

2014
[37]

E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time monte carlo methods for quantum impurity models, Reviews of mod- ern physics83, 349 (2011)

2011
[38]

Werner, A

P. Werner, A. Comanac, L. De’Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impu- rity models, Physical Review Letters97, 076405 (2006)

2006
[39]

S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48, 10345 (1993)

1993
[40]

H. T. M. Nghiem and T. A. Costi, Time evolution of the kondo resonance in response to a quench, Phys. Rev. Lett.119, 156601 (2017)

2017
[41]

H. T. M. Nghiem and T. A. Costi, Time-dependent numerical renormalization group method for multiple quenches: Towards exact results for the long-time limit of thermodynamic observables and spectral functions, Phys. Rev. B98, 155107 (2018)

2018
[42]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, I. Piˇ zorn, H. Ver- schelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett.107, 070601 (2011)

2011
[43]

Link, H.-H

V. Link, H.-H. Tu, and W. T. Strunz, Open quantum sys- tem dynamics from infinite tensor network contraction, Physical Review Letters132, 200403 (2024)

2024
[44]

Lange, A

H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, From architectures to applications: a review of neural quantum states, Quantum Science and Technology9, 040501 (2024). 13

2024
[45]

Schwarz, M

F. Schwarz, M. Goldstein, A. Dorda, E. Arrigoni, A. We- ichselbaum, and J. von Delft, Lindblad-driven discretized leads for nonequilibrium steady-state transport in quan- tum impurity models: Recovering the continuum limit, Physical Review B94, 155142 (2016)

2016
[46]

Campbell, I

S. Campbell, I. d’Amico, M. A. Ciampini, J. Anders, N. Ares, S. Artini, A. Auff` eves, L. Bassman Oftelie, L. P. Bettmann, M. V. Bonan¸ ca,et al., Roadmap on quantum thermodynamics, Quantum Science and Technology11, 012501 (2026)

2026
[47]

S. Sen, P. J. Wong, and A. K. Mitchell, The Mott tran- sition as a topological phase transition, Physical Review B102, 081110 (2020)

2020
[48]

P. Wong, S. Sen, and A. K. Mitchell, Effective Gaus- sian theories for interacting quantum impurity models, (in preparation, 2026)

2026
[49]

Weiße and H

A. Weiße and H. Fehske, Exact diagonalization tech- niques, Computational many-particle physics , 529 (2008)

2008
[50]

Bulla, T

R. Bulla, T. A. Costi, and T. Pruschke, Numerical renor- malization group method for quantum impurity systems, Reviews of Modern Physics80, 395 (2008)

2008
[51]

Bruus and K

H. Bruus and K. Flensberg,Many-body quantum theory in condensed matter physics: an introduction(OUP Ox- ford, 2004)

2004
[52]

E. R. Gagliano and C. A. Balseiro, Dynamical proper- ties of quantum many-body systems at zero temperature, Phys. Rev. Lett.59, 2999 (1987)

1987
[53]

D. N. Zubarev, Double-time green functions in statistical physics, Sov. Phys. Usp.3, 320 (1960), english translation of Usp. Fiz. Nauk 71, 71–116 (1960)

1960
[54]

Schollwoeck and S

U. Schollwoeck and S. R. White, Methods for time depen- dence in dmrg (2006), arXiv:cond-mat/0606018 [cond- mat.str-el]

2006 arXiv
[55]

Haegeman, C

J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimiza- tion with matrix product states, Phys. Rev. B94, 165116 (2016)

2016
[56]

Paeckel, T

S. Paeckel, T. K¨ ohler, A. Swoboda, S. R. Manmana, U. Schollw¨ ock, and C. Hubig, Time-evolution meth- ods for matrix-product states, Ann. Phys.411, 167998 (2019)

2019
[57]

Fishman, S

M. Fishman, S. White, and E. M. Stoudenmire, The iten- sor software library for tensor network calculations, Sci- Post Physics Codebases , 004 (2022)

2022
[58]

Ehrenfest, Bemerkung ¨ uber die angen¨ aherte g¨ ultigkeit der klassischen mechanik innerhalb der quanten- mechanik, Zeitschrift f¨ ur physik45, 455 (1927)

P. Ehrenfest, Bemerkung ¨ uber die angen¨ aherte g¨ ultigkeit der klassischen mechanik innerhalb der quanten- mechanik, Zeitschrift f¨ ur physik45, 455 (1927)

1927
[59]

M. J. Powell, An efficient method for finding the mini- mum of a function of several variables without calculating derivatives, The computer journal7, 155 (1964)

1964
[60]

Virtanen, R

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Po- lat, Y...

2020
[61]

E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys.28, 251 (1972)

1972
[62]

Bravyi, M

S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb- robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett.97, 050401 (2006)

2006
[63]

Arrigoni, M

E. Arrigoni, M. Knap, and W. von der Linden, Nonequi- librium dynamical mean-field theory: An auxiliary quan- tum master equation approach, Phys. Rev. Lett.110, 086403 (2013)

2013
[64]

Dorda, M

A. Dorda, M. Nuss, W. von der Linden, and E. Arrigoni, Auxiliary master equation approach to nonequilibrium correlated impurities, Phys. Rev. B89, 165105 (2014)

2014
[65]

Dorda, M

A. Dorda, M. Sorantin, W. von der Linden, and E. Arrigoni, Optimized auxiliary representation of non- markovian impurity problems by a lindblad equation, New J. Phys.19, 063005 (2017)

2017
[66]

Thoenniss, I

J. Thoenniss, I. Vilkoviskiy, and D. A. Abanin, Efficient pseudomode representation and complexity of quantum impurity models, arXiv preprint (2024), published in Phys. Rev. B, arXiv:2409.08816 [cond-mat.str-el]

2024
[67]

M. S. Frank, D. G. Artiukhin, T.-H. Lee, Y. Yao, K. Bar- ros, O. Christiansen, and N. Lanat` a, Active learning ap- proach to simulations of strongly correlated matter with the ghost gutzwiller approximation, Phys. Rev. Research 6, 013242 (2024)

2024
[68]

Mejuto-Zaera, Quantum embedding for molecules us- ing auxiliary particles – the ghost gutzwiller ansatz, Fara- day Discuss.254, 653 (2024)

C. Mejuto-Zaera, Quantum embedding for molecules us- ing auxiliary particles – the ghost gutzwiller ansatz, Fara- day Discuss.254, 653 (2024)

2024
[69]

Giuli, T.-H

S. Giuli, T.-H. Lee, Y.-X. Yao, G. Kotliar, A. E. Ruck- enstein, O. Gingras, and N. Lanat` a, Unifying varia- tional and dynamical quantum embedding: From ghost gutzwiller approximation to dynamical mean-field theory, arXiv preprint (2026), arXiv:2603.20559 [cond-mat.str- el]

2026
[70]

Guerci, M

D. Guerci, M. Capone, and N. Lanat` a, Time-dependent ghost gutzwiller nonequilibrium dynamics, Phys. Rev. Research5, L032023 (2023)

2023
[71]

Bulla, A

R. Bulla, A. C. Hewson, and T. Pruschke, Numerical renormalization group calculations for the self-energy of the impurity anderson model, Journal of Physics: Con- densed Matter10, 8365 (1998)

1998

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

⟩is the ther- mal expectation value in the Gibbs state

Representations of the Interacting Green’s function The impurity retarded Green’s function is defined as GAIM dd;σ(t)=−iθ(t)⟨{dσ(t), d† σ(0)}⟩where⟨. . .⟩is the ther- mal expectation value in the Gibbs state. In the (com- plex) frequency domainz=ω+iηwithη>0 we have GAIM dd;σ(z)≡⟨⟨dσ;d † σ⟩⟩=∫dt eiztGAIM dd;σ(t). Its spectral func- tion isA(ω)=− 1 π ImG AI...

2022

[2] [2]

Polkovnikov, K

A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Colloquium: Nonequilibrium dynamics of closed in- teracting quantum systems, Rev. Mod. Phys.83, 863 (2011)

2011

[3] [3]

Eisert, M

J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- body systems out of equilibrium, Nat. Phys.11, 124 (2015)

2015

[4] [4]

H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys.86, 779 (2014)

2014

[5] [5]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

2011

[6] [6]

Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys

G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

2003

[7] [7]

Calabrese and J

P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.2005, P04010 (2005)

2005

[8] [8]

A. C. Hewson,The Kondo Problem to Heavy Fermions, Cambridge Studies in Magnetism, Vol. 2 (Cambridge University Press, Cambridge, 1993)

1993