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REVIEW 3 major objections 5 minor 146 references

Inchworm tensor train hybridization expansion quantum impurity solver

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Combining tensor-train decomposition with the inchworm hybridization expansion yields deterministic, accurate Green's functions for quantum impurity models, with tractable tensor ranks at moderate expansion orders.

desk verdict Valuable but overclaimed: the inchworm+TT combination is real and the benchmarks are honest, yet the multi-orbital demonstration bypasses the inchworm propagator that the headline solver claims to deliver. read the letter →

arxiv 2505.16117 v2 pith:YK3UJ26K submitted 2025-05-22 cond-mat.str-el

Yang Yu , André Erpenbeck , Dominika Zgid , Guy Cohen , Olivier Parcollet , Emanuel Gull This is my paper · ORCID
classification cond-mat.str-el
keywords tensortrainsinchwormalgorithmquantumimpuritymodelshybridizationexpansioncrossinterpolationstrong-couplingmulti-orbitalsolversignproblem
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

This paper claims that the tensor-train (TT) decomposition, built by tensor cross interpolation, can sum the imaginary-time inchworm hybridization expansion for quantum impurity models accurately and deterministically, including multi-orbital models with general interactions and off-diagonal hybridization where Monte Carlo suffers from sign problems. It shows that for moderate expansion orders the tensor-train ranks stay computationally manageable, establishing the approach as a viable alternative to conventional impurity solvers. The authors demonstrate agreement with exact results for a noninteracting spinless model, a two-orbital Kanamori model, and single-orbital models across temperatures and bath types, though for multi-orbital cases they replace the inchworm propagator with an exact-diagonalization propagator on a fine grid. The paper's central positive result is that deterministic tensor-train summation of bold diagrams works; its central caveat is that faithful interpolation of the bold propagator on a coarse grid is the main bottleneck.

What carries the argument

The central object is the renormalized (bold) propagator $\hat{R}(\tau)$, defined as a time-ordered sum over hybridization insertions. The inchworm algorithm computes $\hat{R}(\tau)$ sequentially on a grid, using already known $\hat{R}$ at earlier times as internal lines; the TT representation encodes the integrand for each diagram order as a chain of low-rank tensors, one physical index per time variable (after the rational map $v_i$) and per discrete orbital-spin index, plus the matrix-element index of $\hat{R}$. TCI constructs the TT by adaptive sampling, and the paper's addition of adaptive random noise prevents TCI from stopping early in flat or zero-weight regions. The machinery's work is to convert the factorial, high-dimensional sums and integrals into contractions over one-dimensional quadratures.

What would settle it

Run the two-orbital Kanamori benchmark at $\beta=8$ with the inchworm-generated propagator on a coarse linear-Chebyshev grid ($N_{\mathrm{inch}}$ around 10 to 50) instead of the exact-diagonalization propagator: the first-order Green's function will show oscillatory artifacts in both the diagonal and off-diagonal components, as in Appendix D, demonstrating that the end-to-end solver claim fails for that model unless the propagator is represented on an $O(100)$ or finer grid.

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Extended reading notes

Core claim

The paper's central claim is that the inchworm strong-coupling (bold hybridization) expansion, when integrated with tensor-train (TT) summation via tensor cross interpolation (TCI), produces deterministic, high-accuracy imaginary-time Green's functions for general quantum impurity problems, with computationally tractable TT ranks at moderate expansion orders. Key numerical elements include a piecewise-ordered time integration mapped from simplex to hypercube variables, inclusion of discrete orbital and spin indices and diagram-topology assignments inside the TT, explicit summation over expansion order and operator type, and an adaptive random-noise modification of TCI that mitigates ergodicity problems. On the noninteracting spinless benchmark the method converges rapidly with expansion order $m$ and rank $\chi\sim 20$ for $m\leq 6$; for the two-orbital Kanamori model with off-diagonal hybridization it matches exact diagonalization and Monte Carlo results at $m=5$ with $\chi=100$, and for single-orbital models it reaches accuracy with rank $\chi\leq 50$. The paper also identifies three limiting challenges: the bold propagator requires very fine grids in multi-orbital cases, TT ranks grow substantially faster with expansion order for multi-orbital systems, and convergence in expansion order slows for baths with significant spectral weight at zero frequency.

Load-bearing premise

The load-bearing premise is that the renormalized propagator can be represented faithfully on the inchworm grid with the chosen interpolation (linear, cubic, or linear-Chebyshev), so sequential inchworm propagation stays accurate; the paper itself shows this premise fails for the two-orbital Kanamori model with off-diagonal hybridization, where hundreds of grid points are required and the authors fall back on an exact-diagonalization propagator for the Green's-function calculations.

Editorial extensions

If this is right

  • In the tested single-orbital and noninteracting benchmarks, deterministic TT summation produces Green's functions at controllable precision with modest ranks for low expansion orders, removing statistical Monte Carlo error bars.
  • The two-orbital Kanamori benchmark reproduces exact and Monte Carlo results at expansion order $m=5$ with rank $\chi=100$, demonstrating that off-diagonal hybridization and general interactions are handled in principle.
  • Convergence in expansion order rather than in tensor rank becomes the binding constraint: for metallic baths with zero-frequency weight, order truncation at $m=5$ is not converged at $\beta=50$, independent of the TT machinery.
  • Because every calculation is deterministic, the method can serve as a reference solver in regimes where Monte Carlo suffers sign problems, provided the propagator interpolation bottleneck is resolved for the model in question.
  • Rank growth with expansion order and orbital count is the main computational obstacle; the current implementation reports roughly 500 core-hours for a single Green's-function point at $m=5$ and $\chi=100$ for the two-orbital model.

Reading between the lines

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

  • A plausible next test, not performed in the paper, is to replace the grid-interpolated inchworm propagator with a sum-of-exponentials representation (the direction the authors point toward); if that removes the multi-orbital grid bottleneck, the end-to-end solver would become fully deterministic without an exact-diagonalization crutch.
  • The adaptive random-noise fix for TCI ergodicity is presented as an empirical heuristic, so a natural extension is to probe the same scheme on other diagrammatic tensor-cross-interpolation applications, such as electron-phonon diagrams, where zero-weight regions cause similar stagnation.
  • The paper's evidence that rank growth is driven by the configuration space of discrete orbital-spin indices suggests that factorizing the discrete index sum with a separate low-rank structure could reduce the multi-orbital rank increase at high expansion order; this is my inference, not a tested claim.
  • If the slow order convergence for baths with substantial zero-energy spectral weight is generic, then deterministic solvers will still need order extrapolation or partial resummation beyond fixed truncation to reach low temperatures.
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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

3 major / 5 minor

Summary. The manuscript proposes a deterministic impurity solver that combines the imaginary-time inchworm strong-coupling (hybridization) expansion with tensor-train (TT) cross interpolation. Section II introduces the inchworm recursion for the bold propagator and the Green's function, a piecewise time-variable transformation, the treatment of discrete orbital indices as TT physical indices, and an adaptive-random-noise modification of TCI intended to cure ergodicity problems. Section III reports three sets of tests: a noninteracting spinless model against an exact solution, a two-orbital Kanamori model with off-diagonal hybridization compared with exact diagonalization and a Monte Carlo inchworm result, and a single-orbital Hubbard model across temperatures, interaction strengths, and bath types. In the last two interacting tests, the manuscript explicitly uses the exact bold propagator from exact diagonalization on a dense grid rather than the sequential inchworm propagator. The conclusion states that tensor-train ranks remain computationally tractable for moderate expansion orders and that the approach is a viable alternative to conventional impurity solvers, while also listing three open challenges, the first being the need for exceptionally fine discretization of the bold propagator in certain problems.

Significance. The potential significance is real: a deterministic TT-based summation of inchworm diagrams could offer controlled precision in regimes where Monte Carlo solvers suffer from sign problems. The construction choices are well motivated (piecewise time variables, inclusion of D_p in the target function, explicit summation over Phi and n), and the external benchmarks are meaningful: the spinless model has an exact noninteracting solution, the Kanamori model is compared with exact diagonalization and existing Monte Carlo data, and no parameter is fitted to the reference results. The adaptive random-noise modification is a clear and reproducible algorithmic contribution, and the paper includes an honest failure analysis in Appendix D. However, the significance claimed in the abstract and conclusion depends on the full inchworm+TT pipeline being viable for interacting multi-orbital problems with off-diagonal hybridization, and that specific claim is not yet demonstrated: the multi-orbital results validate the TT summation of hybridization diagrams for a fixed exact propagator, not the inchworm propagation that defines the solver.

major comments (3)
  1. [Sec. III.B and III.C; Appendix D] The multi-orbital and convergence demonstrations do not run the full solver. In Sec. III.B the text states that 'we employ the exact bold propagator obtained from exact diagonalization on a dense grid (Ninch = 501)', and Sec. III.C similarly states 'We employ the exact bold propagator (with Ninch = beta + 1)'. Thus the Green's functions in these sections are produced by TT summation of hybridization diagrams with an externally supplied exact R(tau), not by the sequential inchworm propagation defined in Sec. II.A. Appendix D, Fig. 9 shows why this matters: even the exact first-order (m=1) Green's function of the two-orbital model displays oscillatory artifacts on the Ninch=11 linear-Chebyshev grid and requires Ninch of order 100 to 1000 for smoothness. The abstract's claim of solving 'general quantum impurity problems' and Sec. IV's claim that the approach is 'a viable alternative to conventional quantum impurity solvers' therefore go beyond what is demonstrated end to end.
  2. [Sec. IV; Sec. III.B] The paper's own conclusion identifies as the first principal challenge that 'converged Green's functions for certain impurity problems necessitate exceptionally fine discretization of the bold propagator', and Sec. III.B reports about 500 core-hours per single Green's-function data point at m=5 and chi=100 for the version that already uses the exact ED propagator. These two statements together undermine the computational-viability claim: the full sequential inchworm algorithm would add O(10^2 to 10^3) propagation steps with associated interpolation error and error accumulation, on top of the cost already reported. To support the central claim, the authors should either demonstrate full inchworm propagation for at least one interacting model with off-diagonal hybridization and a controlled convergence study against an exact reference, or substantially restrict the claimed scope to TT summation for fixed bold propagators plus a narrowly qualified inchworm benchmark.
  3. [Sec. III.A; Appendix C] The only fully end-to-end validation of the sequential inchworm propagator is the spinless noninteracting benchmark, where Ninch=11 is sufficient. This model has a smooth propagator that is easy to interpolate, so it cannot probe the interpolation failure documented in Appendix D. The manuscript should state this limitation explicitly when presenting Sec. III.A as the main validation of the complete algorithm, and should avoid presenting the multi-orbital sections as demonstrations of the solver's full propagation capability.
minor comments (5)
  1. [Sec. III.A, Fig. 3] Please clarify the relationship between the expansion order m used for the bold propagator (stated as m=7) and the expansion orders m=2,...,10 shown for G(tau); a reader cannot tell whether the propagator order is held fixed for all G curves or matched to the order of each G curve.
  2. [Eq. (11) and Eq. (12)] Eq. (11) writes a sum over N_l starting at N_l=0, while the subsequent discussion and Eq. (12) require 1 <= N_l <= 2n-1 for proper diagrams; please align the summation range and either define or explicitly exclude the N_l=0 and N_l=2n cases.
  3. [Sec. III.B] The cost statement of about 500 core-hours for 'a single Green's function data point' should specify the hardware, the number of tau grid points and orbital components included, and the convergence criterion used for rank chi=100; otherwise the metric is not reproducible.
  4. [Sec. II.A and Appendix D] The terminology for grids is inconsistent: the main text introduces the 'linear-Chebyshev grid' with nCheby Chebyshev nodes per interval, while Appendix D's Fig. 9 is described as showing results for 'different discretization grids' and the text refers to a 'linear grid'. Please clarify whether Ninch refers to the number of linear intervals, the number of linear grid points, or the total number of interpolation points including Chebyshev nodes.
  5. [Throughout] There are several typographical and wording issues, including 'diagramatic' in Appendix A and the phrase 'the order of the Chebyshev polynomial for interpolation used within each linear inchworm grid interval nCheby'; a careful proofread would improve the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: benchmarks are external exact/reference solutions, and no fitted parameter is renamed as a prediction; the ED-propagator substitution is a disclosed validity limitation, not circularity.

full rationale

The paper's central derivation is self-contained. The inchworm expansion of Eqs. (4)-(5) and the Green's function expression of Eq. (7) are derived from the strong-coupling expansion in Appendix A, with the equivalence of the bold expansion checked algebraically by substituting the bare expansion (Appendix A 2). The tensor-train summation machinery is a standard TCI/TT construction (Sec. II.B, Appendix B) with no fitting to the target observables. Benchmarks are genuinely external: the noninteracting spinless solution (Sec. III.A, Appendix C), exact diagonalization for the two-orbital Kanamori model (Fig. 4), exact diagonalization for the single-orbital model (Fig. 5), and prior Monte Carlo data used only as comparisons. No parameter is adjusted to reproduce these results. The paper's own disclosures that the multi-orbital Green's functions are computed using the exact bold propagator rather than the sequentially inchworm-propagated propagator (Sec. III.B: 'we employ the exact bold propagator obtained from exact diagonalization on a dense grid (Ninch = 501)'; Sec. III.C: 'We employ the exact bold propagator (with Ninch = beta + 1) for the calculation of the Green's function to avoid amplifying errors from inchworm propagation') are important limitations on the end-to-end claim, and Appendix D shows the inchworm grid requirement fails for the Kanamori model. However, using an exact reference propagator is not circular: it does not reduce the computed Green's function to a fitted input, nor does it define the prediction in terms of itself. This is a validity/completeness concern for the headline claim, not a circularity concern. Self-citations (e.g., Refs. [40,74,111]) are used for method details and comparison data, not as load-bearing unverified premises. Therefore: no significant circularity.

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

No physics entities are invented; the method adds numerical machinery only. The central claim rests on the five assumptions listed above, all disclosed in the paper. The two most fragile are the grid fidelity of the bold propagator (Appendix D shows a concrete failure, which is why exact propagators are substituted in Secs. III.B and III.C) and the low-rank compressibility of high-order inchworm integrands (rank grows rapidly with m). The only hand-set method parameter is the noise amplitude alpha=0.1; the global tolerances (epsilon=1e-14, nCheby=9, NGK=15) are fixed once across all runs, which is good practice.

free parameters (2)
  • Adaptive noise amplitude alpha = 0.1
    Hand-set parameter steering the random-noise term g = alpha * epsilon_pivot * R in the modified TCI algorithm (Algorithm 1, Sec. II.C and Appendix B). The authors state the results are robust to alpha, but no first-principles or data-driven justification is given, and Appendix B concedes the method does not offer a rigorous theoretical guarantee.
  • TCI tolerance epsilon; Chebyshev order nCheby; Kronrod points NGK = 1e-14; 9; 15
    Global numerical parameters fixed once for all runs (Sec. III, 'Throughout this work, we set nCheby=9, NGK=15 and epsilon=1e-14'). They are chosen by hand but are accuracy controls rather than physics fits, and holding them constant across all benchmarks is good practice.
assumptions (5)
  • standard math The determinant (Wick) structure of the hybridization expansion and the time-ordered operator factorization behind Eqs. (A2) and (A3).
    Standard field-theoretic results used throughout Appendix A; not proven in the paper.
  • domain assumption Inchworm resummation identity: Eq. (A5) with 'inchworm proper' diagrams equals the bare expansion Eq. (A4), and the Green's function expression Eq. (A7) is exact.
    Invoked in Sec. II.A; the authors' verification is substitution of the bare series into the bold series (Appendix A 2), a standard argument in the diagrammatic Monte Carlo literature (Refs. [74, 96, 109]) rather than a formal proof.
  • domain assumption The diagrammatic integrands in Eqs. (5) and (7) admit tensor-train representations with tractable rank at the expansion orders needed for convergence.
    This is the efficiency premise of the method; it is demonstrated empirically in Sec. III, degrades at higher orders (Secs. III.A and III.B), and is left as an open challenge in Sec. IV with no theoretical guarantee.
  • domain assumption The bold propagator R(tau) is faithfully representable on the inchworm grid with the chosen interpolation, so sequential inchworm propagation does not accumulate uncontrolled error.
    Sec. II.A introduces the linear-Chebyshev grid; Appendix D (Fig. 9) shows the premise fails for the two-orbital Kanamori model unless Ninch is of order 100 to 1000, which the authors state imposes significant computational cost.
  • domain assumption TCI with adaptive random noise explores the full configuration space of the integrand.
    Appendix B: the modified TCI resolves the constructed counterexample (Fig. 7, top) and the benchmark integrand (Fig. 7, bottom), but the authors explicitly state it does not offer a rigorous theoretical guarantee for resolving ergodicity problems.

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Cite this review

Pith. "Pith review of Inchworm tensor train hybridization expansion quantum impurity solver." pith.science (2026). https://pith.science/paper/YK3UJ26K

@misc{pith2026250516117,
  author       = {Pith},
  title        = {Pith review of: Inchworm tensor train hybridization expansion quantum impurity solver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YK3UJ26K}},
  note         = {Machine review of arXiv:2505.16117}
}
read the original abstract

The investigation of quantum impurity models plays a crucial role in condensed matter physics because of their wide-ranging applications, such as embedding theories and transport problems. Traditional methods often fall short of producing accurate results for multi-orbital systems with complex interactions and off-diagonal hybridizations. Recently, tensor-train-based integration and summation techniques have shown promise as effective alternatives. In this study, we use tensor train methods to tackle quantum impurity problems formulated within the imaginary-time inchworm hybridization expansion framework. We identify key challenges in the inchworm expansion itself and its interplay with tensor-train-based methods. We demonstrate the accuracy and versatility of our approach by solving general quantum impurity problems. Our results suggest that tensor-train decomposition schemes offer a viable path toward accurate and efficient multi-orbital impurity solvers.

Figures

Figures reproduced from arXiv: 2505.16117 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of (a) a tensor train decomposition and (b) numerical integration or summation based on tensor trains. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the TT configurations used in this work for (a) the bold propagator [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Green’s functions of the spinless non-interacting [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Green’s functions of the two-orbital spinful model. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Absolute error in the Green’s functions of the single [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic of the contributions to the hybridization [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: In other scenarios, adjusting the approximation [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Illustration of the ergodicity problem and the pro [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Green’s functions of the two-orbital spinful model [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Works this paper leans on

146 extracted references · 52 canonical work pages

  1. [1]

    Different approaches to performing integration or summation Eqs. (5) and (7) require the integration or summation over the expansion order n, the operator configurations Φ, the degrees of freedom of the model {1, · · ·, 2n}, and the diagram topology D p. For each integration or sum- mation operation, three distinct approaches are possi- ble. The first app...

  2. [2]

    The discrete indices {ι1, · · ·, ι2n} can be straightforwardly included as phys- ical indices in the TT

    Special treatment for time integration The index set {1, · · ·, 2n} requires separate treatment of its two distinct components. The discrete indices {ι1, · · ·, ι2n} can be straightforwardly included as phys- ical indices in the TT. The time indices {τ1, · · ·, τ2n} present a challenge for TT decomposition. When these indices are not time-ordered, the res...

  3. [3]

    Indices that require no integration or summation Besides the indices to be integrated or summed over, the TT representation allows for additional flexibility in handling various types of indices that do not need to be integrated or summed over, as discussed in Sec. II B. For Eq. (5), potential candidates for such physical in- dices include the matrix elem...

  4. [4]

    The ordering of the physical indices Having established the set of physical indices for the TT representation, we now address the question of index ordering. While previous studies have investigated the impact of index ordering on TT performance [39], our analysis reveals that for the present problem, the specific ordering has minimal effect on the result...

  5. [5]

    pairing hopping

    Summary In summary, when combining the TT with the inch- worm algorithm, the specific scheme employed in this work follows Fig. 2(a) for the propagator ˆR at each time τ and follows Fig. 2(b) for the Green’s function G at each τj and ιj. The explicit summation on the right- hand side is parallelized in a straightforward manner. In the initial step of the ...

  6. [6]

    Strong-coupling expansion The partition function Zimp of an impurity action Simp = Sloc + P 1′1 c∗ 1′∆1′1c1 is given by the functional integral Zimp = Z D[c∗, c]e−Simp = Z D[c∗, c]e−Sloc e − P 1′ 1 c∗ 1′ ∆1′ 1c1 . (A1) In the strong-coupling expansion framework, we expand the term e − P 1′ 1 c∗ 1′ ∆1′1c1 with power series, which leads to Zimp = ∞X n=0 X 1...

  7. [7]

    inch- worm proper

    Diagrammatic resummation frameworks The bare hybridization expansion is defined by Eq. (A2) or Eq. (A3), with the bare propagator ˆR0 serv- ing as the fundamental building block. In physically interesting regimes, this approach often requires large expansion orders in order to reach convergence. Vari- ous approaches for regrouping diagrams exist that enab...

  8. [8]

    Oseledets and E

    I. Oseledets and E. Tyrtyshnikov, Tt-cross approxima- tion for multidimensional arrays, Linear Algebra and its Applications 432, 70 (2010)

    2010

Show all 146 references
  1. [9]

    I. V. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing 33, 2295 (2011), https://doi.org/10.1137/090752286

    2011 doi

  2. [10]

    R. J. Baxter, Dimers on a rectangular lat- tice, Journal of Mathematical Physics 9, 650 (1968), https://pubs.aip.org/aip/jmp/article- pdf/9/4/650/19064920/650 1 online.pdf

    1968

  3. [11]

    Accardi, Topics in quantum probability, Physics Re- ports 77, 169 (1981)

    L. Accardi, Topics in quantum probability, Physics Re- ports 77, 169 (1981)

    1981

  4. [12]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- orous results on valence-bond ground states in antifer- romagnets, Phys. Rev. Lett. 59, 799 (1987)

    1987

  5. [13]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Va- lence bond ground states in isotropic quantum antifer- romagnets, Communications in Mathematical Physics 115, 477 (1988)

    1988

  6. [14]

    Klumper, A

    A. Klumper, A. Schadschneider, and J. Zittartz, Equiva- lence and solution of anisotropic spin-1 models and gen- eralized t-j fermion models in one dimension, Journal of Physics A: Mathematical and General 24, L955 (1991)

    1991

  7. [15]

    Kl¨ umper, A

    A. Kl¨ umper, A. Schadschneider, and J. Zittartz, Groundstate properties of a generalized vbs-model, Zeitschrift f¨ ur Physik B Condensed Matter 87, 281 (1992)

    1992

  8. [16]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Communica- tions in Mathematical Physics 144, 443 (1992)

    1992

  9. [17]

    ¨Ostlund and S

    S. ¨Ostlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995)

    1995

  10. [18]

    Dukelsky, M

    J. Dukelsky, M. A. Mart ´ ın-Delgado, T. Nishino, and G. Sierra, Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains, Europhysics Letters 43, 457 (1998)

    1998

  11. [19]

    Oseledets, Compact matrix form of the d-dimensional tensor decomposition, IEICE Proceedings Series 43, 10.34385/proc.43.B2L-C2 (2009)

    I. Oseledets, Compact matrix form of the d-dimensional tensor decomposition, IEICE Proceedings Series 43, 10.34385/proc.43.B2L-C2 (2009)

    2009 doi

  12. [20]

    S. R. White, Density matrix formulation for quan- tum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992)

    1992

  13. [21]

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

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

    2011

  14. [22]

    Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics 349, 117 (2014)

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

    2014

  15. [23]

    Garc ´ ıa-Molina, J

    P. Garc ´ ıa-Molina, J. Rodr ´ ıguez-Mediavilla, and J. J. Garc ´ ıa-Ripoll, Quantum fourier analysis for multivari- ate functions and applications to a class of schr¨ odinger- type partial differential equations, Phys. Rev. A 105, 012433 (2022)

    2022

  16. [24]

    Haubenwallner and M

    P. Haubenwallner and M. Heller, Fully numerical hartree-fock calculations with quantized tensor trains (2025), arXiv:2503.08430 [physics.chem-ph]

    2025 arXiv

  17. [25]

    S. V. Dolgov, B. N. Khoromskij, and I. V. Os- eledets, Fast solution of parabolic problems in the tensor train/quantized tensor train format with ini- tial application to the fokker–planck equation, SIAM Journal on Scientific Computing 34, A3016 (2012), https://doi.org/10.1137...

    2012 doi

  18. [26]

    Ye and N

    E. Ye and N. F. G. Loureiro, Quantum-inspired method for solving the vlasov-poisson equations, Phys. Rev. E 106, 035208 (2022)

    2022

  19. [27]

    Ye and N

    E. Ye and N. F. Loureiro, Quantized tensor networks for solving the vlasov–maxwell equations, Journal of Plasma Physics 90, 805900301 (2024)

    2024

  20. [28]

    Gourianov, M

    N. Gourianov, M. Lubasch, S. Dolgov, Q. Y. van den Berg, H. Babaee, P. Givi, M. Kiffner, and D. Jaksch, A quantum-inspired approach to exploit turbulence struc- tures, Nature Computational Science 2, 30 (2022)

    2022

  21. [29]

    R. D. Peddinti, S. Pisoni, A. Marini, P. Lott, H. Ar- gentieri, E. Tiunov, and L. Aolita, Complete quantum- inspired framework for computational fluid dynamics (2023), arXiv:2308.12972 [physics.flu-dyn]

    2023 arXiv

  22. [30]

    Gourianov, P

    N. Gourianov, P. Givi, D. Jaksch, and S. B. Pope, Tensor networks enable the cal- culation of turbulence probability distribu- tions, Science Advances 11, eads5990 (2025), https://www.science.org/doi/pdf/10.1126/sciadv.ads5990

    2025 doi

  23. [31]

    Dolgov, B

    S. Dolgov, B. Khoromskij, and D. Savostyanov, Super- fast fourier transform using qtt approximation, Journal of Fourier Analysis and Applications 18, 915 (2012)

    2012

  24. [32]

    J. Chen, E. Stoudenmire, and S. R. White, Quantum fourier transform has small entanglement, PRX Quan- tum 4, 040318 (2023)

    2023

  25. [33]

    Chen and M

    J. Chen and M. Lindsey, Direct interpolative construc- tion of the discrete fourier transform as a matrix prod- uct operator (2024), arXiv:2404.03182 [quant-ph]

    2024

  26. [34]

    Jeannin, Y

    M. Jeannin, Y. N´ u˜ nez Fern´ andez, T. Kloss, O. Parcollet, and X. Waintal, Cross-extrapolation reconstruction of low-rank functions and application to quantum many- body observables in the strong coupling regime, Phys. Rev. B 110, 035124 (2024)

    2024

  27. [35]

    Lin and S

    L. Lin and S. R. White, A novel method of function extrapolation inspired by techniques in low-entangled many-body physics (2023), arXiv:2308.09001 [quant- ph]

    2023 arXiv

  28. [36]

    Khoromskij and A

    B. Khoromskij and A. Veit, Efficient computation of highly oscillatory integrals by using qtt tensor approxi- mation, Computational Methods in Applied Mathemat- 17 ics 16, 145 (2016)

    2016

  29. [37]

    Jolly, Y

    N. Jolly, Y. N. Fern´ andez, and X. Waintal, Ten- sorized orbitals for computational chemistry (2024), arXiv:2308.03508 [cond-mat.str-el]

    2024

  30. [38]

    Finkel, J

    P. Finkel, J. von Delft, and M. Frankenbach, New ap- proaches to Numerical Integration in Quantum Chem- istry using Quantics Tensor Cross Interpolation , Bach- elor Thesis (Ludwig Maximilian University of Munich, 2024)

    2024

  31. [39]

    Shinaoka, D

    H. Shinaoka, D. Geffroy, M. Wallerberger, J. Otsuki, K. Yoshimi, E. Gull, and J. Kuneˇ s, Sparse sampling and tensor network representation of two-particle Green’s functions, SciPost Phys. 8, 012 (2020)

    2020

  32. [40]

    Shinaoka, M

    H. Shinaoka, M. Wallerberger, Y. Murakami, K. No- gaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains, Phys. Rev. X 13, 021015 (2023)

    2023

  33. [41]

    M. K. Ritter, Y. N´ u˜ nez Fern´ andez, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics tensor cross interpolation for high-resolution parsimo- nious representations of multivariate functions, Phys. Rev. Lett. 132, 056501 (2024)

    2024

  34. [42]

    ´Sroda, K

    M. ´Sroda, K. Inayoshi, H. Shinaoka, and P. Werner, High-resolution nonequilibrium gw calculations based on quantics tensor trains (2024), arXiv:2412.14032 [cond-mat.str-el]

    2024

  35. [43]

    Takahashi, R

    H. Takahashi, R. Sakurai, and H. Shinaoka, Com- pactness of quantics tensor train representations of lo- cal imaginary-time propagators, SciPost Phys. 18, 007 (2025)

    2025

  36. [44]

    Rohshap, M

    S. Rohshap, M. K. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger, and A. Kauch, Two-particle calcula- tions with quantics tensor trains: Solving the parquet equations (2025), arXiv:2410.22975 [cond-mat.str-el]

    2025 arXiv

  37. [45]

    B. N. Khoromskij, O(dlog n)-quantics approximation of n-d tensors in high-dimensional numerical modeling, Constructive Approximation 34, 257 (2011)

    2011

  38. [46]

    N´ u˜ nez-Fern´ andez, M

    Y. N´ u˜ nez-Fern´ andez, M. Jeannin, P. T. Dumitrescu, T. Kloss, J. Kaye, O. Parcollet, and X. Waintal, Learn- ing Feynman Diagrams with Tensor Trains, Phys. Rev. X 12, 041018 (2022)

    2022

  39. [47]

    Erpenbeck, W.-T

    A. Erpenbeck, W.-T. Lin, T. Blommel, L. Zhang, S. Iskakov, L. Bernheimer, Y. N´ u˜ nez Fern´ andez, G. Co- hen, O. Parcollet, X. Waintal, and E. Gull, Tensor train continuous time solver for quantum impurity models, Phys. Rev. B 107, 245135 (2023)

    2023

  40. [48]

    Murray, H

    M. Murray, H. Shinaoka, and P. Werner, Nonequilib- rium diagrammatic many-body simulations with quan- tics tensor trains, Phys. Rev. B 109, 165135 (2024)

    2024

  41. [49]

    Ishida, N

    H. Ishida, N. Okada, S. Hoshino, and H. Shinaoka, Low-rank quantics tensor train representations of feyn- man diagrams for multiorbital electron-phonon models (2024), arXiv:2405.06440 [cond-mat.str-el]

    2024 arXiv

  42. [50]

    Eckstein, Solving quantum impurity models in the non-equilibrium steady state with tensor trains (2024), arXiv:2410.19707 [cond-mat.str-el]

    M. Eckstein, Solving quantum impurity models in the non-equilibrium steady state with tensor trains (2024), arXiv:2410.19707 [cond-mat.str-el]

    2024 arXiv

  43. [51]

    A. J. Kim and P. Werner, Strong coupling impurity solver based on quantics tensor cross interpolation, Phys. Rev. B 111, 125120 (2025)

    2025

  44. [52]

    Jeannin, Y

    M. Jeannin, Y. N´ u˜ nez-Fern´ andez, T. Kloss, O. Parcol- let, and X. Waintal, A comprehensive study of out-of- equilibrium kondo effect and coulomb blockade (2025), arXiv:2502.16306 [cond-mat.str-el]

    2025

  45. [53]

    Savostyanov and I

    D. Savostyanov and I. Oseledets, Fast adaptive inter- polation of multi-dimensional arrays in tensor train for- mat, in The 2011 International Workshop on Multidi- mensional (nD) Systems (2011) pp. 1–8

    2011

  46. [54]

    D. V. Savostyanov, Quasioptimality of maximum- volume cross interpolation of tensors, Linear Algebra and its Applications 458, 217 (2014)

    2014

  47. [55]

    Dolgov and D

    S. Dolgov and D. Savostyanov, Parallel cross interpola- tion for high-precision calculation of high-dimensional integrals, Computer Physics Communications 246, 106869 (2020)

    2020

  48. [56]

    Sakaue, H

    K. Sakaue, H. Shinaoka, and R. Sakurai, Learning ten- sor trains from noisy functions with application to quan- tum simulation (2024), arXiv:2405.12730 [quant-ph]

    2024 arXiv

  49. [57]

    Y. Hur, J. G. Hoskins, M. Lindsey, E. Stoudenmire, and Y. Khoo, Generative modeling via tensor train sketch- ing, Applied and Computational Harmonic Analysis 67, 101575 (2023)

    2023

  50. [58]

    J. J. Rodr ´ ıguez-Aldavero, P. Garc ´ ıa-Molina, L. Taglia- cozzo, and J. J. Garc ´ ıa-Ripoll, Chebyshev approxima- tion and composition of functions in matrix product states for quantum-inspired numerical analysis (2025), arXiv:2407.09609 [quant-ph]

    2025 arXiv

  51. [59]

    Tindall, M

    J. Tindall, M. Stoudenmire, and R. Levy, Compressing multivariate functions with tree tensor networks (2024), arXiv:2410.03572 [quant-ph]

    2024 arXiv

  52. [60]

    Lindsey, Multiscale interpolative construction of quantized tensor trains (2024), arXiv:2311.12554 [math.NA]

    M. Lindsey, Multiscale interpolative construction of quantized tensor trains (2024), arXiv:2311.12554 [math.NA]

    2024 arXiv

  53. [61]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen- berg, Dynamical mean-field theory of strongly corre- lated fermion systems and the limit of infinite dimen- sions, Rev. Mod. Phys. 68, 13 (1996)

    1996

  54. [62]

    Kotliar, S

    G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006)

    2006

  55. [63]

    Zgid and E

    D. Zgid and E. Gull, Finite temperature quantum em- bedding theories for correlated systems, New Journal of Physics 19, 023047 (2017)

    2017

  56. [64]

    Caffarel and W

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

    1994

  57. [65]

    Capone, M

    M. Capone, M. Civelli, S. S. Kancharla, C. Castel- lani, and G. Kotliar, Cluster-dynamical mean-field the- ory of the density-driven mott transition in the one- dimensional hubbard model, Phys. Rev. B 69, 195105 (2004)

    2004

  58. [66]

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

    2008

  59. [67]

    Liebsch and N.-H

    A. Liebsch and N.-H. Tong, Finite-temperature ex- act diagonalization cluster dynamical mean-field study of the two-dimensional hubbard model: Pseudogap, non-fermi-liquid behavior, and particle-hole asymmetry, Phys. Rev. B 80, 165126 (2009)

    2009

  60. [68]

    S´ en´ echal, Bath optimization in the cellular dynami- cal mean-field theory, Phys

    D. S´ en´ echal, Bath optimization in the cellular dynami- cal mean-field theory, Phys. Rev. B 81, 235125 (2010)

    2010

  61. [69]

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

    2014

  62. [70]

    D. Zgid, E. Gull, and G. K.-L. Chan, Truncated con- 18 figuration interaction expansions as solvers for corre- lated quantum impurity models and dynamical mean- field theory, Phys. Rev. B 86, 165128 (2012)

    2012

  63. [71]

    Weichselbaum and J

    A. Weichselbaum and J. von Delft, Sum-rule conserving spectral functions from the numerical renormalization group, Phys. Rev. Lett. 99, 076402 (2007)

    2007

  64. [72]

    Bulla, T

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

    2008

  65. [73]

    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, Rev. Mod. Phys. 83, 349 (2011)

    2011

  66. [74]

    A. N. Rubtsov and A. I. Lichtenstein, Continuous-time quantum monte carlo method for fermions: Beyond auxiliary field framework, Journal of Experimental and Theoretical Physics Letters 80, 61 (2004)

    2004

  67. [75]

    A. N. Rubtsov, V. V. Savkin, and A. I. Lichten- stein, Continuous-time quantum monte carlo method for fermions, Phys. Rev. B 72, 035122 (2005)

    2005

  68. [76]

    Werner, A

    P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impu- rity models, Phys. Rev. Lett. 97, 076405 (2006)

    2006

  69. [77]

    Werner and A

    P. Werner and A. J. Millis, Hybridization expansion im- purity solver: General formulation and application to kondo lattice and two-orbital models, Phys. Rev. B 74, 155107 (2006)

    2006

  70. [78]

    Haule, Quantum monte carlo impurity solver for clus- ter dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Phys

    K. Haule, Quantum monte carlo impurity solver for clus- ter dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Phys. Rev. B 75, 155113 (2007)

    2007

  71. [79]

    E. Gull, P. Werner, O. Parcollet, and M. Troyer, Continuous-time auxiliary-field monte carlo for quan- tum impurity models, Europhysics Letters 82, 57003 (2008)

    2008

  72. [80]

    E. Gull, P. Staar, S. Fuchs, P. Nukala, M. S. Summers, T. Pruschke, T. C. Schulthess, and T. Maier, Subma- trix updates for the continuous-time auxiliary-field al- gorithm, Phys. Rev. B 83, 075122 (2011)

    2011

  73. [81]

    Eidelstein, E

    E. Eidelstein, E. Gull, and G. Cohen, Multiorbital quan- tum impurity solver for general interactions and hy- bridizations, Phys. Rev. Lett. 124, 206405 (2020)

    2020

  74. [82]

    J. Li, M. Wallerberger, and E. Gull, Diagrammatic monte carlo method for impurity models with gen- eral interactions and hybridizations, Phys. Rev. Res. 2, 033211 (2020)

    2020

  75. [83]

    J. Li, Y. Yu, E. Gull, and G. Cohen, Interaction- expansion inchworm monte carlo solver for lattice and impurity models, Phys. Rev. B 105, 165133 (2022)

    2022

  76. [84]

    J. Kaye, Z. Huang, H. U. R. Strand, and D. Goleˇ z, Decomposing imaginary-time feynman diagrams using separable basis functions: Anderson impurity model strong-coupling expansion, Phys. Rev. X 14, 031034 (2024)

    2024

  77. [85]

    Huang, D

    Z. Huang, D. Goleˇ z, H. U. R. Strand, and J. Kaye, Au- tomated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting (2025), arXiv:2503.19727 [cond-mat.str-el]

    2025

  78. [86]

    N. V. Prokof’ev and B. V. Svistunov, Bold diagram- matic monte carlo: A generic sign-problem tolerant technique for polaron models and possibly interacting many-body problems, Phys. Rev. B 77, 125101 (2008)

    2008

  79. [87]

    E. Gull, D. R. Reichman, and A. J. Millis, Bold-line di- agrammatic monte carlo method: General formulation and application to expansion around the noncrossing approximation, Phys. Rev. B 82, 075109 (2010)

    2010

  80. [88]

    R¨ uegg, E

    A. R¨ uegg, E. Gull, G. A. Fiete, and A. J. Millis, Sum rule violation in self-consistent hybridization expan- sions, Phys. Rev. B 87, 075124 (2013)

    2013

  81. [89]

    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

  82. [90]

    Kozik, M

    E. Kozik, M. Ferrero, and A. Georges, Nonexistence of the luttinger-ward functional and misleading conver- gence of skeleton diagrammatic series for hubbard-like models, Phys. Rev. Lett. 114, 156402 (2015)

    2015

  83. [91]

    Tandetzky, J

    F. Tandetzky, J. K. Dewhurst, S. Sharma, and E. K. U. Gross, Multiplicity of solutions to gw-type approxima- tions, Phys. Rev. B 92, 115125 (2015)

    2015

  84. [92]

    A. Stan, P. Romaniello, S. Rigamonti, L. Reining, and J. A. Berger, Unphysical and physical solutions in many- body theories: from weak to strong correlation, New Journal of Physics 17, 093045 (2015)

    2015

  85. [93]

    Rossi and F

    R. Rossi and F. Werner, Skeleton series and multival- uedness of the self-energy functional in zero space-time dimensions, Journal of Physics A: Mathematical and Theoretical 48, 485202 (2015)

    2015

  86. [94]

    Sch¨ afer, S

    T. Sch¨ afer, S. Ciuchi, M. Wallerberger, P. Thunstr¨ om, O. Gunnarsson, G. Sangiovanni, G. Rohringer, and A. Toschi, Nonperturbative landscape of the mott- hubbard transition: Multiple divergence lines around the critical endpoint, Phys. Rev. B 94, 235108 (2016)

    2016

  87. [95]

    Gunnarsson, G

    O. Gunnarsson, G. Rohringer, T. Sch¨ afer, G. Sangio- vanni, and A. Toschi, Breakdown of traditional many- body theories for correlated electrons, Phys. Rev. Lett. 119, 056402 (2017)

    2017

  88. [96]

    Tarantino, P

    W. Tarantino, P. Romaniello, J. A. Berger, and L. Rein- ing, Self-consistent dyson equation and self-energy func- tionals: An analysis and illustration on the example of the hubbard atom, Phys. Rev. B 96, 045124 (2017)

    2017

  89. [97]

    Vuˇ ciˇ cevi´ c, N

    J. Vuˇ ciˇ cevi´ c, N. Wentzell, M. Ferrero, and O. Parcollet, Practical consequences of the luttinger-ward functional multivaluedness for cluster dmft methods, Phys. Rev. B 97, 125141 (2018)

    2018

  90. [98]

    E. G. C. P. van Loon, F. Krien, and A. A. Katanin, Bethe-salpeter equation at the critical end point of the mott transition, Phys. Rev. Lett. 125, 136402 (2020)

    2020

  91. [99]

    A. J. Kim and E. Kozik, Misleading convergence of the skeleton diagrammatic technique: when the correct solution can be found (2022), arXiv:2212.14768 [cond- mat.str-el]

    2022 arXiv

  92. [100]

    K. V. Houcke, E. Kozik, R. Rossi, Y. Deng, and F. Werner, Physical and unphysical regimes of self- consistent many-body perturbation theory, SciPost Phys. 16, 133 (2024)

    2024

  93. [101]

    H. Eßl, M. Reitner, E. Kozik, and A. Toschi, How to stay on the physical branch in self-consistent many-electron approaches (2025), arXiv:2502.01420 [cond-mat.str-el]

    2025 arXiv

  94. [102]

    Pokhilko, C.-N

    P. Pokhilko, C.-N. Yeh, and D. Zgid, Iterative subspace algorithms for finite-temperature solution of dyson equation, The Journal of Chemical Physics 156, 094101 (2022)

    2022

  95. [103]

    Cohen, E

    G. Cohen, E. Gull, D. R. Reichman, and A. J. Millis, Taming the dynamical sign problem in real-time evolu- tion of quantum many-body problems, Phys. Rev. Lett. 115, 266802 (2015). 19

    2015

  96. [104]

    A. E. Antipov, Q. Dong, J. Kleinhenz, G. Cohen, and E. Gull, Currents and green’s functions of impurities out of equilibrium: Results from inchworm quantum monte carlo, Phys. Rev. B 95, 085144 (2017)

    2017

  97. [105]

    Q. Dong, I. Krivenko, J. Kleinhenz, A. E. Antipov, G. Cohen, and E. Gull, Quantum monte carlo solution of the dynamical mean field equations in real time, Phys. Rev. B 96, 155126 (2017)

    2017

  98. [106]

    Krivenko, J

    I. Krivenko, J. Kleinhenz, G. Cohen, and E. Gull, Dynamics of kondo voltage splitting after a quantum quench, Phys. Rev. B 100, 201104 (2019)

    2019

  99. [107]

    Kleinhenz, I

    J. Kleinhenz, I. Krivenko, G. Cohen, and E. Gull, Dy- namic control of nonequilibrium metal-insulator transi- tions, Phys. Rev. B 102, 205138 (2020)

    2020

  100. [108]

    Z. Cai, J. Lu, and S. Yang, Inchworm monte carlo method for open quantum systems, Communications on Pure and Applied Mathematics 73, 2430 (2020)

    2020

  101. [109]

    Kleinhenz, I

    J. Kleinhenz, I. Krivenko, G. Cohen, and E. Gull, Kondo cloud in a one-dimensional nanowire, Phys. Rev. B 105, 085126 (2022)

    2022

  102. [110]

    Erpenbeck, E

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

    2023

  103. [111]

    Erpenbeck, E

    A. Erpenbeck, E. Gull, and G. Cohen, Shaping Elec- tronic Flows with Strongly Correlated Physics, Nano Letters 23, 10480 (2023)

    2023

  104. [112]

    Z. Cai, J. Lu, and S. Yang, Numerical analysis for inch- worm monte carlo method: Sign problem and error growth, Mathematics of Computation 92, 1141 (2023)

    2023

  105. [113]

    Erpenbeck, T

    A. Erpenbeck, T. Blommel, L. Zhang, W.-T. Lin, G. Co- hen, and E. Gull, Steady-state properties of multi- orbital systems using quantum monte carlo, The Journal of Chemical Physics 161, 094104 (2024)

    2024

  106. [114]

    Atanasova, A

    H. Atanasova, A. Erpenbeck, E. Gull, Y. B. Lev, and G. Cohen, Stark Many-Body Localization in Interacting Infinite Dimensional Systems, Physical Review Letters 132, 166301 (2024)

    2024

  107. [115]

    A. J. Kim, J. Li, M. Eckstein, and P. Werner, Pseu- doparticle vertex solver for quantum impurity models, Physical Review B 106, 085124 (2022)

    2022

  108. [116]

    H. U. R. Strand, J. Kleinhenz, and I. Krivenko, Inch- worm quasi monte carlo for quantum impurities, Phys. Rev. B 110, L121120 (2024)

    2024

  109. [117]

    Goldberger, Y

    D. Goldberger, Y. Fridman, E. Gull, E. Eidelstein, and G. Cohen, Dynamical mean field theory of the bilayer Hubbard model with inchworm Monte Carlo, Physical Review B 109, 085133 (2024)

    2024

  110. [118]

    Y. N. Fern´ andez, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, J. von Delft, H. Shinaoka, and X. Waintal, Learning tensor net- works with tensor cross interpolation: New algorithms and libraries, SciPost Phys. 18, 104 (2025)

    2025

  111. [119]

    J. W. Negele, Quantum many-particle systems (CRC Press, 2018)

    2018

  112. [120]

    Stefanucci and R

    G. Stefanucci and R. Van Leeuwen, Nonequilibrium many-body theory of quantum systems: a modern in- troduction (Cambridge University Press, 2013)

    2013

  113. [121]

    E. Gull, D. R. Reichman, and A. J. Millis, Numerically exact long-time behavior of nonequilibrium quantum impurity models, Physical Review B 84, 085134 (2011)

    2011

  114. [122]

    Cohen, E

    G. Cohen, E. Gull, D. R. Reichman, A. J. Millis, and E. Rabani, Numerically exact long-time magnetization dynamics at the nonequilibrium Kondo crossover of the Anderson impurity model, Physical Review B 87, 195108 (2013)

    2013

  115. [123]

    Cohen, D

    G. Cohen, D. R. Reichman, A. J. Millis, and E. Gull, Green’s functions from real-time bold-line Monte Carlo, Physical Review B 89, 115139 (2014)

    2014

  116. [124]

    Haule, C.-H

    K. Haule, C.-H. Yee, and K. Kim, Dynamical mean- field theory within the full-potential methods: Elec- tronic structure of ceirin, cecoin, cerhin, Physical Re- view B 81, 195107 (2010)

    2010

  117. [125]

    There is an alternative approach for numerical integra- tion that combines TCI with a quantics TT (QTT) rep- resentation [29]. While we do not examine this method in detail in this work, our preliminary analysis indicates reduced efficiency for the present problem: the QTT rep...

  118. [126]

    While the grid used for constructing the TT approxi- mation can, in principle, differ from the grid on which the approximation is subsequently evaluated, we use the same grid for both construction and evaluation here

  119. [127]

    Matsuura, H

    S. Matsuura, H. Shinaoka, P. Werner, and N. Tsuji, Ten- sor cross interpolation approach for quantum impurity problems based on the weak-coupling expansion (2025), arXiv:2501.12643 [cond-mat.str-el]

    2025 arXiv

  120. [128]

    [40] suggested one may find potential improvements to this mapping scheme

    Ref. [40] suggested one may find potential improvements to this mapping scheme. Indeed, alternative approaches such as the simplex-to-hypercube transformation used in Ref. [109] offer more uniform distributions. However, such mappings introduce some computational overhead, req...

  121. [129]

    J. Kanamori, Electron correlation and ferromagnetism of transition metals, Progress of Theoretical Physics 30, 275 (1963), https://academic.oup.com/ptp/article- pdf/30/3/275/5278869/30-3-275.pdf

    1963

  122. [130]

    Werner, E

    P. Werner, E. Gull, M. Troyer, and A. J. Millis, Spin freezing transition and non-fermi-liquid self-energy in a three-orbital model, Phys. Rev. Lett. 101, 166405 (2008)

    2008

  123. [131]

    Georges, L

    A. Georges, L. d. Medici, and J. Mravlje, Strong cor- relations from hund’s coupling, Annu. Rev. Condens. Matter Phys. 4, 137 (2013)

    2013

  124. [132]

    Mejuto-Zaera, L

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

    2020

  125. [133]

    J. Kaye, K. Chen, and O. Parcollet, Discrete lehmann representation of imaginary time green’s functions, Phys. Rev. B 105, 235115 (2022)

    2022

  126. [134]

    Huang, E

    Z. Huang, E. Gull, and L. Lin, Robust analytic continua- tion of green’s functions via projection, pole estimation, and semidefinite relaxation, Phys. Rev. B 107, 075151 (2023)

    2023

  127. [135]

    Zhang and E

    L. Zhang and E. Gull, Minimal pole representation and controlled analytic continuation of matsubara response functions, Phys. Rev. B 110, 035154 (2024)

    2024

  128. [136]

    Zhang, Y

    L. Zhang, Y. Yu, and E. Gull, Minimal pole representa- tion and analytic continuation of matrix-valued correla- tion functions, Phys. Rev. B 110, 235131 (2024). 20

    2024

  129. [137]

    Zhang, A

    L. Zhang, A. Erpenbeck, Y. Yu, and E. Gull, Mini- mal pole representation for spectral functions (2025), arXiv:2504.01163 [cond-mat.str-el]

    2025 arXiv

  130. [138]

    T. J. Boerner, S. Deems, T. R. Furlani, S. L. Knuth, and J. Towns, Access: Advancing innovation: Nsf’s advanced cyberinfrastructure coordination ecosystem: Services & support, in Practice and Experience in Ad- vanced Research Computing , PEARC ’23 (Association for Computing Ma...

    2023

  131. [139]

    Parcollet, M

    O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Triqs: A toolbox for research on interacting quantum systems, Computer Physics Communications 196, 398 (2015)

    2015

  132. [140]

    Charlebois, N

    M. Charlebois, N. Wentzell, and O. Parcollet, Triqs/inchworm implementation (n.d.), unpublished

  133. [141]

    Y. Yu, A. Erpenbeck, D. Zgid, G. Cohen, O. Parcol- let, and E. Gull, Dataset for publication ”inchworm tensor train hybridization expansion quantum impurity solver”, 10.5281/zenodo.15490622 (2025)

    2025 doi

  134. [142]

    Haule, S

    K. Haule, S. Kirchner, J. Kroha, and P. W¨ olfle, An- derson impurity model at finite coulomb interaction u: Generalized noncrossing approximation, Phys. Rev. B 64, 155111 (2001)

    2001

  135. [143]

    A. Boag, E. Gull, and G. Cohen, Inclusion-exclusion principle for many-body diagrammatics, Phys. Rev. B 98, 115152 (2018)

    2018

  136. [144]

    Rossi, Determinant Diagrammatic Monte Carlo Al- gorithm in the Thermodynamic Limit, Physical Review Letters 119, 045701 (2017)

    R. Rossi, Determinant Diagrammatic Monte Carlo Al- gorithm in the Thermodynamic Limit, Physical Review Letters 119, 045701 (2017)

    2017

  137. [145]

    G. K.-L. Chan and M. Head-Gordon, Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group, The Jour- nal of Chemical Physics 116, 4462 (2002)

    2002

  138. [146]

    Zgid and M

    D. Zgid and M. Nooijen, Obtaining the two-body den- sity matrix in the density matrix renormalization group method, The Journal of Chemical Physics 128, 144115 (2008)

    2008

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