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REVIEW 2 major objections 5 minor 109 references

Tensor network algorithm to solve polaron impurity problems

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims an extended tensor-network path-integral method solves polaron impurity problems in real time with only two controllable sources of error.

desk verdict This paper extends GTEMPO to electron-phonon impurity problems with a clean scalar-reweighting trick, and needs only a tempered real-time baseline claim and code release to be a solid contribution. read the letter →

arxiv 2507.05580 v2 pith:UZDPWMYS submitted 2025-07-08 quant-ph

Ruofan Chen , Lei Gu , Chu Guo This is my paper · ORCID
classification quant-ph PACS 71.38.-k
keywords polaronimpurityproblemtensornetworkGrassmanntime-evolvingmatrixproductoperatorFeynman-VernoninfluencefunctionalpathintegralKeldyshcontourKadanoffreal-timeGreen'sfunctions
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 aims to solve the polaron impurity problem, an old and still largely unsolved model in which an impurity is coupled to both an electron bath and a phonon bath, on imaginary-time, Keldysh, and L-shaped Kadanoff contours. Its central claim is that a tensor-network representation of the impurity path integral, called extended GTEMPO, makes both baths analytically integrable through the Feynman-Vernon influence functional, so the simulation has no bath discretization error. The method is claimed to handle non-diagonal impurity couplings and to compute multi-time correlations beyond single-particle Green's functions from the same compressed object. Extensive benchmarks against analytic solutions, exact diagonalization, and continuous-time quantum Monte Carlo support the claim that the only remaining error sources are time-step discretization and matrix-product-state bond truncation.

What carries the argument

The load-bearing object is the augmented density tensor $A[\bar a a] = K[\bar a a] I_{\mathrm{el}}[\bar a a] I_{\mathrm{ph}}[\bar a a]$ discretized on a contour and represented as a Grassmann MPS. The key identity is the componentwise action of the phonon influence functional: after expanding in Fock states, $I_{\mathrm{ph}}[\hat n]$ multiplies each component $K[\bar a a;\bar n n] = \eta[\bar n n]\langle \bar n_0|\hat U_{\mathrm{imp}}|n_{M-1}\rangle \cdots \langle \bar n_1|\hat U_{\mathrm{imp}}|n_0\rangle I_{\mathrm{ph}}[n]$, so the phonon IF can be built as a normal MPS in the Fock basis with the partial-IF algorithm and applied before the electron IF. This ordering avoids the invalid replacement $\hat n \to \bar a a$ at finite $\delta\tau$, which would introduce an extra first-order error.

What would settle it

A conclusive test would be a strong-coupling continuous-bath polaron calculation with a large bond dimension, comparing the real-time greater Green's function against an independent numerically exact short-time benchmark: if reducing $\delta t$ does not produce the linear error decrease reported here, or if the error at fixed $\delta t$ grows with final time $t$ instead of staying controlled, the claim that only two controllable error sources remain would be falsified. For Kadanoff-contour calculations, a direct check is to fix $\delta\tau$, lower $\delta t$, and see whether errors to the analytic independent-bosons solution stay bounded and decreasing as the paper's note about keeping the ratio roughly the same suggests.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a construction: the augmented density tensor for the polaron impurity problem, $A[\bar a a] = K[\bar a a] I_{\mathrm{el}}[\bar a a] I_{\mathrm{ph}}[\bar a a]$, can be built entirely as a Grassmann matrix product state. The obstacle is that the phonon influence functional is naturally a bosonic object in the Fock basis, while the electron influence functional is a Grassmann object in the coherent-state basis, and replacing the density operator $\hat n$ by $\bar a a$ at finite time step is only first-order accurate. The paper's resolution is to apply the phonon influence functional to the Fock-state components of the impurity propagator first, forming $K I_{\mathrm{ph}}$ by componentwise multiplication of a Grassmann MPS with a normal MPS, and only then multiply in the electron influence functional. With this ordering, all existing GTEMPO techniques apply, and the paper demonstrates accurate Matsubara, nonequilibrium, and equilibrium real-time Green's functions and density-density correlations, including full-fledged continuous-bath real-time calculations it states have not been done before.

Load-bearing premise

The central assumption is that the first-order splitting $e^{-\delta\tau H}\approx e^{-\delta\tau H_{\mathrm{imp}}}e^{-\delta\tau H_{\mathrm{el}}}e^{-\delta\tau H_{\mathrm{ph}}}$ makes the path integral accurate enough when the time step is small; the paper demonstrates convergence empirically, but does not bound this error a priori, so for strong coupling or long real times this discretization error could dominate.

Editorial extensions

If this is right

  • Real-time greater and lesser Green's functions and density-density correlations for polaron impurity models can be computed directly on the Keldysh or L-shaped Kadanoff contour, without analytic continuation.
  • Bath discretization error is absent by construction, so only the time step and the bond dimension need to be converged.
  • Non-diagonal impurity-bath couplings and generic impurity Hamiltonians with off-diagonal flavor tunneling are treated on the same footing as diagonal ones.
  • The same augmented density tensor yields single-particle Green's functions and higher-order multi-time correlations at comparable cost.
  • The method can serve as an impurity solver in dynamical mean-field-type calculations with retarded interactions, where the phonon bath is the source of the retarded interaction.

Reading between the lines

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

  • If the claimed accuracy holds, the technique should provide reference real-time data for continuum-bath polaron models where imaginary-time quantum Monte Carlo data must be analytically continued; a natural check is to compare its real-time results with those obtained by real-time diagrammatic Monte Carlo at short times.
  • The first-order Trotter/QuAPI splitting is the likely bottleneck: the paper shows linear convergence in $\delta t$ and notes that $\delta\tau$ and $\delta t$ must be decreased together on Kadanoff contours, so a higher-order splitting or an error estimate for the splitting would be the natural next step.
  • Because the phonon influence functional is built once as a normal MPS in the Fock basis, the same $I_{\mathrm{ph}}$ tensor may be reusable across contours and initial states, which would make thermal and quench calculations share the most expensive part of the construction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript generalizes the GTEMPO method to polaron impurity problems, where an impurity is coupled to both a fermionic electron bath and a bosonic phonon bath. The central technical step is the representation of the phonon influence functional as a normal bosonic MPS and its subsequent use as a scalar reweighting of the Grassmann tensor components of the bare impurity dynamics, which avoids the inaccurate replacement of the density operator by Grassmann bilinears in the discretized influence functional. The method is formulated on the imaginary, Keldysh, and Kadanoff contours. The paper benchmarks the method against analytic solutions of the independent bosons model, exact diagonalization of a two-bath toy model, and CTQMC for continuous-bath imaginary-time calculations. It also presents real-time Keldysh results for a full-fledged continuous-bath model, with convergence checks in bond dimension and time step.

Significance. If the method performs as claimed, it fills a clear gap: it provides real-time and multi-time impurity correlation functions for polaron problems with continuous baths, free of bath discretization error and sign problem, with essentially two controllable hyperparameters (time step and bond dimension). The central derivation (Eqs. 40-46) is clear and appears sound. The benchmarks against analytic, ED, and CTQMC references are appropriate and give strong support for the imaginary-time and the discrete-bath real-time results. The real-time continuous-bath results in Sec. V.B are the least independently verified part; because they are promoted in the abstract as a potential 'benchmarking baseline,' the lack of an independent reference is a load-bearing gap that should be addressed.

major comments (2)
  1. [Sec. V.B (Figs. 11-14)] The real-time Keldysh results for the full-fledged model are validated only by self-convergence with respect to the bond dimension χ and time step δt, with the δt-convergence measured relative to a baseline at δt=0.0125 rather than against an independent solution. The abstract and Sec. VI present these results as a 'benchmarking baseline,' which is not yet supported by the evidence. The ED benchmarks in Sec. IV use delta-function spectral functions, and the authors argue that this is a harder case because the influence functional does not decay; nevertheless, those benchmarks do not directly establish the absolute accuracy of the continuous-bath real-time observables, where the first-order Trotter error in Eq. (36) is not bounded a priori and could in principle be larger for strong coupling. To support the central claim of accurate real-time polaron simulations, the authors should add an independent check—for example, a limiting case that reduces to a known solvable problem (such as α=0 or g=0), a comparison against an alternative real-time method in a restricted regime, or a careful statement that the results are unvalidated predictions rather than a benchmarking baseline.
  2. [Sec. VI and Eq. (27)] The statement that 'the only two sources of errors in our method are the first-order discretization error of the impurity path integral... and the MPS bond truncation error' is somewhat imprecise: the discretization error also includes the QuAPI approximation of the influence-functional integrals (Eq. 27), and its magnitude for the continuous-bath Keldysh case is not quantified independently. Because the convergence plots in Sec. V.B compare one extended GTEMPO result against another (rather than against an exact or independent result), they demonstrate internal consistency but not absolute error control. The authors should either provide an external error scale for the real-time results or clarify that the claimed error control is only verified relative to the method's own convergence.
minor comments (5)
  1. [Figs. 9 and 10] The inset axis labels in the figure panels read 'E δt', but the convergence variable analyzed in those insets is the imaginary-time step δτ; the figures should be relabeled to avoid confusion.
  2. [Eq. (43)] The first factor in the expanded propagator is written as '⟨ak+1| ˆU ′ imp fN −1⟩' and should be typeset with an explicit ket, e.g., '⟨a_{k+1}| \hat{U}'_{imp}|f_{N-1}\rangle'.
  3. [Abstract] The sentence 'The polaron problem is a very old problem... but still remain largely unsolved today' has a subject-verb disagreement; 'remain' should be 'remains'.
  4. [Sec. III, first paragraph] The statement that 'for this model the choice of δτ is irrelevant for extended GTEMPO' would benefit from a one-sentence reminder that QuAPI is exact here because the relevant paths are piecewise constant on the time grid, as shown in Appendix A.
  5. [Sec. IV, first sentence] The phrase 'In the next we will consider' should be 'In this section we consider' or 'Next, we consider'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the phonon-IF reweighting step is derived from the path integral, and the central accuracy claims are benchmarked against independent methods (analytic, ED, CTQMC).

full rationale

The paper's central derivation (Sec. II.D, Eqs. (37)-(46)) starts from the impurity path integral, inserts Fock-space resolutions of identity, and shows that the phonon influence functional multiplies each Grassmann component by a scalar Iph[n]. This is an algebraic identity derived from the path integral, not an ansatz, a fit, or a renamed known result; no parameter is adjusted to reproduce any target Green's function. The observable correlations are then defined as traces over the constructed augmented density tensor, so the 'predictions' are computed, not fitted. Benchmarks in Sec. III (independent bosons model vs analytic Lang-Firsov solutions), Sec. IV (toy model vs exact diagonalization), and Sec. V.A (continuous-bath imaginary-time vs CTQMC) use external methods whose assumptions do not include the present paper's outputs; convergence insets confirm that the only tunable hyperparameters are the bond dimension χ and the time steps δτ/δt. The full-fledged real-time Keldysh results (Sec. V.B) are explicitly stated to lack an independent benchmark ('there does not exist any reliable methods for benchmarking to our knowledge'), and the paper supports them only by self-convergence in χ and δt. This is a genuine validation gap, but not a circular reduction: the same underlying discretized path-integral equations were already validated on the imaginary axis and against ED for delta-function baths, and the real-time results are not obtained by fitting to the convergence targets. The manuscript cites prior GTEMPO works ([67]-[71], [93]) for algorithmic machinery, but the novel phonon-IF-to-Grassmann conversion is derived in this paper, and the central accuracy statements are checked against independent, non-self-cited benchmarks; no uniqueness theorem from prior work is invoked to forbid alternatives. Therefore no step meets the required standard of exhibiting Eq. X = Eq. Y by construction or a fitted parameter renamed as a prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities and fits no physical constants to target observables. The only hand-chosen quantities are the numerical convergence parameters δτ/δt and χ. The derivation relies on standard coherent state path integral machinery and on the Feynman-Vernon influence functional, plus a first-order Trotter/QuAPI discretization of the impurity propagators.

free parameters (2)
  • time step δτ (imaginary) / δt (real) = 0.0125 to 0.2
    Discretization step for the QuAPI path integral; controls first-order Trotter error. Chosen by hand for convergence; not fitted to data.
  • bond dimension χ = 100 to 800
    MPS bond dimension truncation parameter; controls truncation error. Chosen to achieve convergence; not fitted to target observables.
assumptions (5)
  • domain assumption The Feynman-Vernon influence functional exactly accounts for the effect of the non-interacting electron and phonon baths on the impurity after integrating them out (Eq. 6).
    The entire method rests on this standard result; the paper assumes the bath couplings are linear (electron hybridization) and bilinear (phonon density-phonon), with non-interacting baths, as specified in Eqs. (3)-(4), Sec. II A-B.
  • domain assumption First-order Trotter decomposition e^{-δτ H} ≈ e^{-δτ H_imp} e^{-δτ H_el} e^{-δτ H_ph} is valid for the impurity dynamics (Eq. 36).
    The paper inherits this from QuAPI; the claimed accuracy relies on this first-order error being small and controllable with δτ, which is verified empirically but not bounded a priori.
  • standard math Grassmann coherent states provide a faithful basis for fermionic impurity path integrals, with the standard coherent state resolution of identity (Eqs. 33-44).
    Standard coherent state path integral formalism, per Negele and Orland (Ref. 64).
  • standard math QuAPI discretization with piecewise-constant trajectories converges to the continuous path integral in the δτ→0 limit (Eq. 27).
    Standard QuAPI approximation (Refs. 89-90); also the basis of TEMPO.
  • standard math The phonon influence functional Iph[n] can be exactly written as a bosonic MPS using the partial-IF algorithm because the density operators n_j at different times commute (Eq. 28 and Algorithm 1).
    This is the TEMPO construction (Ref. 65); the factorization is exact since n_j Λ_jk n_k = Λ_jk n_j n_k for bosonic density operators.

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Pith. "Pith review of Tensor network algorithm to solve polaron impurity problems." pith.science (2026). https://pith.science/paper/UZDPWMYS

@misc{pith2026250705580,
  author       = {Pith},
  title        = {Pith review of: Tensor network algorithm to solve polaron impurity problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZDPWMYS}},
  note         = {Machine review of arXiv:2507.05580}
}
read the original abstract

The polaron problem is a very old problem in condensed matter physics that dates back to the thirties, but still remain largely unsolved today, especially when electron-electron interaction is taken into consideration. The presence of both electron-electron and electron-phonon interactions in the problem invalidates most existing numerical methods, either computationally too expensive or simply intractable. The continuous time quantum Monte Carlo (CTQMC) methods could tackle this problem, but are only effective in the imaginary-time axis. In this work we present a method based on tensor network and the path integral formalism to solve polaron impurity problems. As both the electron and phonon baths can be integrated out via the Feynman-Vernon influence functional in the path integral formalism, our method is free of bath discretization error. It can also flexibly work on the imaginary, Keldysh, and the L-shaped Kadanoff contour. In addition, our method can naturally resolve several long-existing challenges: (i) non-diagonal hybridization function; (ii) measuring multi-time correlations beyond the single particle Green's functions. We demonstrate the effectiveness and accuracy of our method with extensive numerical examples against analytic solutions, exact diagonalization and CTQMC. We also perform full-fledged real-time calculations that have never been done before to our knowledge, which could be a benchmarking baseline for future method developments.

Figures

Figures reproduced from arXiv: 2507.05580 by the authors.

Figure 1
Figure 1. FIG. 1. Unified view of different contours that are commonly used [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Matsubara Green’s function [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figures from the paper (7 more)
Figure 6
Figure 6. Figure 6: FIG. 6. (a, c) The Matsubara Green’s function [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The non-equilibrium greater Green’s function [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (a,c) Matsubara Green’s function [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The non-equilibrium greater Green’s function [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The non-equilibrium greater Green’s function [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

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Works this paper leans on

109 extracted references · 72 canonical work pages

  1. [1]

    The Matsubara Green’s function We first consider the single-flavor case with impurity Hamiltonian in Eq.(48). In this case, the analytical solution of the independent bosons model is often derived in literatures by employing the Lang-Firsov canonical transformation [51, 96], and the Matsubara Green’s function is given as G(τ ) = −⟨T ˆa(τ )ˆa†⟩eq = ˜G0(τ )...

  2. [2]

    The equilibrium Green’s functions The analytic solutions for the real-time Green’s functions can also be obtained through the Lang-Firsov transformation. The concretely form for the equilibrium greater and lesser Green’s functions in the single-flavor case are Geq,>(t) = −ie−i˜ϵat[1 − ¯f (˜ϵa)]F (t); (A13) Geq,<(t) = ie−i˜ϵat ¯f (˜ϵa)F (t), (A14) where F ...

  3. [3]

    L. D. Landau, Electron motion in crystal lattices, Phys. Z. Sowjet. 3, 664 (1933)

    1933

  4. [4]

    L. D. Landau and S. I. Pekar, Effective mass of a polaron, Zh. Eksp. Teor. Fiz. 18, 419 (1948)

    1948

  5. [5]

    A. S. Alexandrov and J. T. Devreese, Advances in polaron physics, V ol. 159 (Springer, 2010)

    2010

  6. [6]

    J. L. Bredas and G. B. Street, Polarons, bipolarons, and soli- tons in conducting polymers, Accounts of Chemical Research 18, 309 (1985), https://doi.org/10.1021/ar00118a005

    work page doi:10.1021/ar00118a005 1985

  7. [7]

    Franchini, M

    C. Franchini, M. Reticcioli, M. Setvin, and U. Diebold, Po- larons in materials, Nature Reviews Materials 6, 560 (2021)

    2021

  8. [8]

    Thomas, D

    J. Thomas, D. BanerJee, A. Nocera, and S. Johnston, The- ory of electron-phonon interactions in extended correlated sys- tems probed by resonant inelastic x-ray scattering, Phys. Rev. X 15, 021030 (2025)

    2025

Show all 109 references
  1. [9]

    Grusdt, N

    F. Grusdt, N. Mostaan, E. Demler, and L. A. P. Ardila, Impuri- ties and polarons in bosonic quantum gases: a review on recent progress (2024), arXiv:2410.09413 [cond-mat.quant-gas]

    2024 arXiv

  2. [10]

    Lanzara, P

    A. Lanzara, P. Bogdanov, X. Zhou, S. Kellar, D. Feng, E. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, et al. , Evi- dence for ubiquitous strong electron–phonon coupling in high- temperature superconductors, Nature 412, 510 (2001)

    2001

  3. [11]

    A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Dou- ble exchange alone does not explain the resistivity of La1−x srx mno3, Phys. Rev. Lett. 74, 5144 (1995)

    1995

  4. [12]

    A. J. Millis, B. I. Shraiman, and R. Mueller, Dynamic jahn- teller effect and colossal magnetoresistance inla1-xsrxmno3, Phys. Rev. Lett. 77, 175 (1996)

    1996

  5. [13]

    A. J. Millis, Lattice effects in magnetoresistive manganese perovskites, Nature 392, 147 (1998)

    1998

  6. [14]

    Yamasaki, S

    Y . Yamasaki, S. Miyasaka, Y . Kaneko, J.-P. He, T. Arima, and Y . Tokura, Magnetic reversal of the ferroelectric polariza- tion in a multiferroic spinel oxide, Phys. Rev. Lett.96, 207204 (2006)

    2006

  7. [15]

    Werner and A

    P. Werner and A. J. Millis, Efficient dynamical mean field sim- ulation of the holstein-hubbard model, Phys. Rev. Lett. 99, 146404 (2007)

    2007

  8. [16]

    Werner and A

    P. Werner and A. J. Millis, Dynamical screening in correlated electron materials, Phys. Rev. Lett. 104, 146401 (2010)

    2010

  9. [17]

    Otsuki, Spin-boson coupling in continuous-time quantum monte carlo, Phys

    J. Otsuki, Spin-boson coupling in continuous-time quantum monte carlo, Phys. Rev. B 87, 125102 (2013)

    2013

  10. [18]

    Caffarel and W

    M. Caffarel and W. Krauth, Exact diagonalization approach to correlated fermions in infinite dimensions: Mott transition and superconductivity, Phys. Rev. Lett.72, 1545 (1994)

    1994

  11. [19]

    E. Koch, G. Sangiovanni, and O. Gunnarsson, Sum rules and bath parametrization for quantum cluster theories, Phys. Rev. B 78, 115102 (2008)

    2008

  12. [20]

    Granath and H

    M. Granath and H. U. R. Strand, Distributional exact diagonal- ization formalism for quantum impurity models, Phys. Rev. B 86, 115111 (2012)

    2012

  13. [21]

    Y . Lu, M. H¨oppner, O. Gunnarsson, and M. W. Haverkort, Ef- ficient real-frequency solver for dynamical mean-field theory, Phys. Rev. B 90, 085102 (2014)

    2014

  14. [22]

    Mejuto-Zaera, L

    C. Mejuto-Zaera, L. Zepeda-N ´u˜nez, M. Lindsey, N. Tubman, B. Whaley, and L. Lin, Efficient hybridization fitting for dy- namical mean-field theory via semi-definite relaxation, Phys. Rev. B 101, 035143 (2020)

    2020

  15. [23]

    Y . Lu, X. Cao, P. Hansmann, and M. W. Haverkort, Natural- orbital impurity solver and projection approach for green’s functions, Phys. Rev. B 100, 115134 (2019)

    2019

  16. [24]

    He and Z.-Y

    R.-Q. He and Z.-Y . Lu, Quantum renormalization groups based on natural orbitals, Phys. Rev. B 89, 085108 (2014)

    2014

  17. [25]

    R.-Q. He, J. Dai, and Z.-Y . Lu, Natural orbitals renormaliza- tion group approach to the two-impurity kondo critical point, Phys. Rev. B 91, 155140 (2015)

    2015

  18. [26]

    K. G. Wilson, The renormalization group: Critical phenomena and the kondo problem, Rev. Mod. Phys. 47, 773 (1975)

    1975

  19. [27]

    Bulla, Zero temperature metal-insulator transition in the infinite-dimensional hubbard model, Phys

    R. Bulla, Zero temperature metal-insulator transition in the infinite-dimensional hubbard model, Phys. Rev. Lett. 83, 136 (1999)

    1999

  20. [28]

    Bulla, T

    R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormaliza- tion group method for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008)

    2008

  21. [29]

    F. B. Anders, A numerical renormalization group approach to non-equilibrium green functions for quantum impurity mod- els, J. Phys. Condens. Matter 20, 195216 (2008)

    2008

  22. [30]

    ˇZitko and T

    R. ˇZitko and T. Pruschke, Energy resolution and discretization artifacts in the numerical renormalization group, Phys. Rev. B 79, 085106 (2009)

    2009

  23. [31]

    X. Deng, J. Mravlje, R. ˇZitko, M. Ferrero, G. Kotliar, and A. Georges, How bad metals turn good: Spectroscopic signa- tures of resilient quasiparticles, Phys. Rev. Lett. 110, 086401 (2013)

    2013

  24. [32]

    K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Dynamical mean-field theory plus numeri- cal renormalization-group study of spin-orbital separation in a three-band hund metal, Phys. Rev. Lett. 115, 136401 (2015)

    2015

  25. [33]

    S.-S. B. Lee and A. Weichselbaum, Adaptive broadening to improve spectral resolution in the numerical renormalization group, Phys. Rev. B 94, 235127 (2016)

    2016

  26. [34]

    S.-S. B. Lee, J. von Delft, and A. Weichselbaum, Doublon- holon origin of the subpeaks at the hubbard band edges, Phys. Rev. Lett. 119, 236402 (2017)

    2017

  27. [35]

    P. S. Cornaglia, H. Ness, and D. R. Grempel, Many-body ef- fects on the transport properties of single-molecule devices, Phys. Rev. Lett. 93, 147201 (2004)

    2004

  28. [36]

    Paaske and K

    J. Paaske and K. Flensberg, Vibrational sidebands and the kondo effect in molecular transistors, Phys. Rev. Lett. 94, 176801 (2005)

    2005

  29. [37]

    P. S. Cornaglia, D. R. Grempel, and H. Ness, Quantum trans- port through a deformable molecular transistor, Phys. Rev. B 71, 075320 (2005)

    2005

  30. [38]

    M. A. Laakso, D. M. Kennes, S. G. Jakobs, and V . Meden, Functional renormalization group study of the ander- son–holstein model, New Journal of Physics 16, 023007 (2014)

    2014

  31. [39]

    F. A. Wolf, I. P. McCulloch, O. Parcollet, and U. Schollw¨ock, Chebyshev matrix product state impurity solver for dynamical mean-field theory, Phys. Rev. B90, 115124 (2014)

    2014

  32. [40]

    Ganahl, P

    M. Ganahl, P. Thunstr ¨om, F. Verstraete, K. Held, and H. G. Evertz, Chebyshev expansion for impurity models using ma- trix product states, Phys. Rev. B 90, 045144 (2014)

    2014

  33. [41]

    Ganahl, M

    M. Ganahl, M. Aichhorn, H. G. Evertz, P. Thunstr¨om, K. Held, and F. Verstraete, Efficient dmft impurity solver using real- time dynamics with matrix product states, Phys. Rev. B 92, 155132 (2015)

    2015

  34. [42]

    F. A. Wolf, A. Go, I. P. McCulloch, A. J. Millis, and U. Schollw ¨ock, Imaginary-time matrix product state impu- rity solver for dynamical mean-field theory, Phys. Rev. X 5, 041032 (2015)

    2015

  35. [43]

    D. J. Garc ´ıa, K. Hallberg, and M. J. Rozenberg, Dynami- cal mean field theory with the density matrix renormalization 18 group, Phys. Rev. Lett. 93, 246403 (2004)

    2004

  36. [44]

    Nishimoto, F

    S. Nishimoto, F. Gebhard, and E. Jeckelmann, Dynamical mean-field theory calculation with the dynamical density- matrix renormalization group, Physica B Condens. Matter 378-380, 283 (2006)

    2006

  37. [45]

    Weichselbaum, F

    A. Weichselbaum, F. Verstraete, U. Schollw ¨ock, J. I. Cirac, and J. von Delft, Variational matrix-product-state approach to quantum impurity models, Phys. Rev. B 80, 165117 (2009)

    2009

  38. [46]

    Bauernfeind, M

    D. Bauernfeind, M. Zingl, R. Triebl, M. Aichhorn, and H. G. Evertz, Fork tensor-product states: Efficient multiorbital real- time dmft solver, Phys. Rev. X7, 031013 (2017)

    2017

  39. [47]

    Werner, J

    D. Werner, J. Lotze, and E. Arrigoni, Configuration interaction based nonequilibrium steady state impurity solver, Phys. Rev. B 107, 075119 (2023)

    2023

  40. [48]

    Kohn and G

    L. Kohn and G. E. Santoro, Efficient mapping for anderson impurity problems with matrix product states, Phys. Rev. B 104, 014303 (2021)

    2021

  41. [49]

    Kohn and G

    L. Kohn and G. E. Santoro, Quench dynamics of the ander- son impurity model at finite temperature using matrix product states: entanglement and bath dynamics, J. Stat. Mech. Theory Exp. 2022, 063102 (2022)

    2022

  42. [50]

    E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time monte carlo meth- ods for quantum impurity models, Rev. Mod. Phys. 83, 349 (2011)

    2011

  43. [51]

    N. V . Prokof’ev and B. V . Svistunov, Polaron problem by di- agrammatic quantum monte carlo, Phys. Rev. Lett. 81, 2514 (1998)

    1998

  44. [52]

    Hafermann, P

    H. Hafermann, P. Werner, and E. Gull, Efficient implemen- tation of the continuous-time hybridization expansion quan- tum impurity solver, Computer Physics Communications 184, 1280 (2013)

    2013

  45. [53]

    I. G. Lang and Y . A. Firsov, Kinetic theory of semiconductors with low mobility, Soviet Physics JETP 16, 1301 (1963)

    1963

  46. [54]

    Hafermann, Self-energy and vertex functions from hybridization-expansion continuous-time quantum monte carlo for impurity models with retarded interaction, Phys

    H. Hafermann, Self-energy and vertex functions from hybridization-expansion continuous-time quantum monte carlo for impurity models with retarded interaction, Phys. Rev. B 89, 235128 (2014)

    2014

  47. [55]

    L. Gu, J. Luo, R. Wu, and G. Zhao, Extended hybridization ex- pansion solver for impurity models with retarded interactions, arXiv:2504.02274 (2025)

    2025 arXiv

  48. [56]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations, Phys. Rev. Lett. 94, 170201 (2005)

    2005

  49. [57]

    Fei, C.-N

    J. Fei, C.-N. Yeh, and E. Gull, Nevanlinna analytical continu- ation, Phys. Rev. Lett. 126, 056402 (2021)

    2021

  50. [58]

    Cohen, D

    G. Cohen, D. R. Reichman, A. J. Millis, and E. Gull, Green’s functions from real-time bold-line monte carlo, Phys. Rev. B 89, 115139 (2014)

    2014

  51. [59]

    Cohen, E

    G. Cohen, E. Gull, D. R. Reichman, and A. J. Millis, Green’s functions from real-time bold-line monte carlo calculations: Spectral properties of the nonequilibrium anderson impurity model, Phys. Rev. Lett. 112, 146802 (2014)

    2014

  52. [60]

    Cohen, E

    G. Cohen, E. Gull, D. R. Reichman, and A. J. Millis, Taming the dynamical sign problem in real-time evolution of quantum many-body problems, Phys. Rev. Lett. 115, 266802 (2015)

    2015

  53. [61]

    H.-T. Chen, G. Cohen, and D. R. Reichman, Inchworm Monte Carlo for exact non-adiabatic dynamics. I. Theory and algo- rithms, J. Chem. Phys. 146, 054105 (2017)

    2017

  54. [62]

    H.-T. Chen, G. Cohen, and D. R. Reichman, Inchworm Monte Carlo for exact non-adiabatic dynamics. II. Benchmarks and comparison with established methods, J. Chem. Phys. 146, 054106 (2017)

    2017

  55. [63]

    Bertrand, S

    C. Bertrand, S. Florens, O. Parcollet, and X. Waintal, Recon- structing nonequilibrium regimes of quantum many-body sys- tems from the analytical structure of perturbative expansions, Phys. Rev. X 9, 041008 (2019)

    2019

  56. [64]

    Erpenbeck, E

    A. Erpenbeck, E. Gull, and G. Cohen, Quantum monte carlo method in the steady state, Phys. Rev. Lett. 130, 186301 (2023)

    2023

  57. [65]

    R. P. Feynman and F. L. Vernon, The theory of a general quan- tum system interacting with a linear dissipative system, Ann. Phys. 24, 118 (1963)

    1963

  58. [66]

    J. W. Negele and H. Orland, Quantum Many-Particle Systems (Westview Press, 1998)

    1998

  59. [67]

    Strathearn, P

    A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-markovian quantum dynamics using time-evolving matrix product operators, Nat. Commun. 9, 3322 (2018)

    2018

  60. [68]

    Thoenniss, M

    J. Thoenniss, M. Sonner, A. Lerose, and D. A. Abanin, Effi- cient method for quantum impurity problems out of equilib- rium, Phys. Rev. B 107, L201115 (2023)

    2023

  61. [69]

    R. Chen, X. Xu, and C. Guo, Grassmann time-evolving matrix product operators for quantum impurity models, Phys. Rev. B 109, 045140 (2024)

    2024

  62. [70]

    R. Chen, X. Xu, and C. Guo, Grassmann time-evolving matrix product operators for equilibrium quantum impurity problems, New J. Phys. 26, 013019 (2024)

    2024

  63. [71]

    R. Chen, X. Xu, and C. Guo, Real-time impurity solver using grassmann time-evolving matrix product operators, Phys. Rev. B 109, 165113 (2024)

    2024

  64. [72]

    Guo and R

    C. Guo and R. Chen, Efficient construction of the Feynman- Vernon influence functional as matrix product states, SciPost Phys. Core 7, 063 (2024)

    2024

  65. [73]

    Chen and C

    R. Chen and C. Guo, Solving equilibrium quantum impurity problems on the l-shaped kadanoff-baym contour, Phys. Rev. B 110, 165114 (2024)

    2024

  66. [74]

    Guo and R

    C. Guo and R. Chen, Infinite grassmann time-evolving matrix product operator method in the steady state, Phys. Rev. B110, 045106 (2024)

    2024

  67. [75]

    Guo and R

    C. Guo and R. Chen, Infinite grassmann time-evolving ma- trix product operator method for zero-temperature equilib- rium quantum impurity problems, Phys. Rev. B 110, 165119 (2024)

    2024

  68. [76]

    Z. Sun, R. Chen, Z. Li, and C. Guo, Infinite grassmann time- evolving matrix product operators for non-equilibrium quan- tum impurity problems, arXiv:2412.04702 (2024)

    2024

  69. [77]

    M. R. Jørgensen and F. A. Pollock, Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-markovian path integrals, Phys. Rev. Lett. 123, 240602 (2019)

    2019

  70. [78]

    Popovic, M

    M. Popovic, M. T. Mitchison, A. Strathearn, B. W. Lovett, J. Goold, and P. R. Eastham, Quantum heat statistics with time-evolving matrix product operators, PRX Quantum 2, 020338 (2021)

    2021

  71. [79]

    G. E. Fux, E. P. Butler, P. R. Eastham, B. W. Lovett, and J. Keeling, Efficient exploration of hamiltonian parameter space for optimal control of non-markovian open quantum systems, Phys. Rev. Lett. 126, 200401 (2021)

    2021

  72. [80]

    Gribben, A

    D. Gribben, A. Strathearn, G. E. Fux, P. Kirton, and B. W. Lovett, Using the environment to understand non-markovian open quantum systems, Quantum 6, 847 (2021)

    2021

  73. [81]

    Otterpohl, P

    F. Otterpohl, P. Nalbach, and M. Thorwart, Hidden phase of the spin-boson model, Phys. Rev. Lett. 129, 120406 (2022)

    2022

  74. [82]

    Gribben, D

    D. Gribben, D. M. Rouse, J. Iles-Smith, A. Strathearn, H. Maguire, P. Kirton, A. Nazir, E. M. Gauger, and B. W. Lovett, Exact dynamics of nonadditive environments in non- markovian open quantum systems, PRX Quantum 3, 010321 19 (2022)

    2022

  75. [83]

    N. Ng, G. Park, A. J. Millis, G. K.-L. Chan, and D. R. Reich- man, Real-time evolution of anderson impurity models via ten- sor network influence functionals, Phys. Rev. B 107, 125103 (2023)

    2023

  76. [84]

    G. Park, N. Ng, D. R. Reichman, and G. K.-L. Chan, Ten- sor network influence functionals in the continuous-time limit: Connections to quantum embedding, bath discretization, and higher-order time propagation, Phys. Rev. B 110, 045104 (2024)

    2024

  77. [85]

    F. F. Assaad and T. C. Lang, Diagrammatic determinantal quantum monte carlo methods: Projective schemes and ap- plications to the hubbard-holstein model, Phys. Rev. B 76, 035116 (2007)

    2007

  78. [86]

    L. V . Keldysh, Diagram technique for non-equilibrium pro- cesses, Soviet Physics JETP 20, 1018 (1965)

    1965

  79. [87]

    E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics V olume 10: Physical Kinetics (Elsevier, 1981)

    1981

  80. [88]

    L. P. Kadanoff and G. Baym, Quantum Statistical Mechnics (W. A. Benjamin, New York, 1962)

    1962

  81. [89]

    Chen, Path integral formalism for quantum open systems, Ann

    R. Chen, Path integral formalism for quantum open systems, Ann. Phys. 480, 170083 (2025)

    2025

  82. [90]

    Jordan and E

    P. Jordan and E. Wigner, ¨Uber das Paulische ¨Aquivalenzverbot, Z. Physik 47, 631 (1928)

    1928

  83. [91]

    D. E. Makarov and N. Makri, Path integrals for dissipative systems by tensor multiplication. condensed phase quantum dynamics for arbitrarily long time, Chem. Phys. Lett.221, 482 (1994)

    1994

  84. [92]

    Makri, Numerical path integral techniques for long time dynamics of quantum dissipative systems, J

    N. Makri, Numerical path integral techniques for long time dynamics of quantum dissipative systems, J. Math. Phys. 36, 2430 (1995)

    1995

  85. [93]

    Schollw ¨ock, The density-matrix renormalization group in the age of matrix product states, Ann

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

    2011

  86. [94]

    Or ´us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann

    R. Or ´us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014)

    2014

  87. [95]

    Z. Sun, R. Chen, Z. Li, and C. Guo, Scalable tensor network algorithm for quantum impurity problems, arXiv:2507.12722 (2025)

    2025

  88. [96]

    X. Xu, C. Guo, and R. Chen, Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations, J. Chem. Phys. 161, 151001 (2024)

    2024

  89. [97]

    Chen, Heat current in non-markovian open systems, New J

    R. Chen, Heat current in non-markovian open systems, New J. Phys. 25, 033035 (2023)

    2023

  90. [98]

    G. D. Mahan, Many-Particle Physics (Springer; 3nd edition, 2000)

    2000

  91. [99]

    M. Xu, Y . Yan, Q. Shi, J. Ankerhold, and J. T. Stockburger, Taming quantum noise for efficient low temperature simula- tions of open quantum systems, Phys. Rev. Lett. 129, 230601 (2022)

    2022

  92. [100]

    F. A. Wolf, I. P. McCulloch, and U. Schollw ¨ock, Solving nonequilibrium dynamical mean-field theory using matrix product states, Phys. Rev. B 90, 235131 (2014)

    2014

  93. [101]

    Holstein, Studies of polaron motion, Ann

    T. Holstein, Studies of polaron motion, Ann. Phys. 8, 325 (1959)

    1959

  94. [102]

    Holstein, Studies of polaron motion, Ann

    T. Holstein, Studies of polaron motion, Ann. Phys. 8, 343–389 (1959)

    1959

  95. [103]

    H.-T. Chen, G. Cohen, A. J. Millis, and D. R. Reichman, Anderson-holstein model in two flavors of the noncrossing ap- proximation, Phys. Rev. B 93, 174309 (2016)

    2016

  96. [104]

    A. M. Sengupta and A. Georges, Non-fermi-liquid behav- ior near a t=0 spin-glass transition, Phys. Rev. B 52, 10295 (1995)

    1995

  97. [105]

    Si and J

    Q. Si and J. L. Smith, Kosterlitz-thouless transition and short range spatial correlations in an extended hubbard model, Phys. Rev. Lett. 77, 3391 (1996)

    1996

  98. [106]

    J. L. Smith and Q. Si, Spatial correlations in dynamical mean- field theory, Phys. Rev. B61, 5184 (2000)

    2000

  99. [107]

    Chitra and G

    R. Chitra and G. Kotliar, Effect of long range coulomb interac- tions on the mott transition, Phys. Rev. Lett. 84, 3678 (2000)

    2000

  100. [108]

    Sun and G

    P. Sun and G. Kotliar, Extended dynamical mean-field theory and GW method, Phys. Rev. B 66, 085120 (2002)

    2002

  101. [109]

    Huang, T

    L. Huang, T. Ayral, S. Biermann, and P. Werner, Extended dynamical mean-field study of the hubbard model with long- range interactions, Phys. Rev. B 90, 195114 (2014)

    2014

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