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Tensor cross interpolation approach for quantum impurity problems based on the weak-coupling expansion

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Tensor cross interpolation makes quantum impurity weak-coupling expansions tractable to 40th order.

desk verdict A credible, well-benchmarked first application of TCI to equilibrium weak-coupling impurity problems, with the low-rank assumption empirical rather than proven and the sign-problem claim a bit overstated, but clearly worth refereeing. read the letter →

arxiv 2501.12643 v2 pith:7CBWWRLJ submitted 2025-01-22 cond-mat.str-el

Shuta Matsuura , Hiroshi Shinaoka , Philipp Werner , Naoto Tsuji This is my paper · ORCID
classification cond-mat.str-el PACS 71.27.+a71.30.+h02.70.-c
keywords tensorcrossinterpolationweak-couplingexpansionquantumimpurityproblemdynamicalmean-fieldtheoryMotttransitiontrainfreeenergyGreen'sfunction
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 aims to show that tensor cross interpolation (TCI) can make the weak-coupling expansion of quantum impurity problems practical, evaluating the perturbative integrals up to 40th order instead of being limited to low orders. The core claim is that after a variable transformation from the time-ordered simplex to a hypercube, the integrand—a product of two determinants—has a low-rank tensor-train structure that the TCI algorithm can discover from a small number of samples. If this is true, impurity solvers no longer need Monte Carlo sampling: they are sign-problem-free and can compute the partition function, and hence the free energy, directly. Benchmarks on an exactly solvable impurity model and on DMFT for the Hubbard model support the claim, reproducing the metal-to-Mott-insulator crossover and the first-order Mott transition with accuracy comparable to continuous-time quantum Monte Carlo.

What carries the argument

The machinery is the tensor cross interpolation (TCI) algorithm, an active-learning scheme that approximates a high-dimensional tensor by a tensor train (matrix product state) built from a few selected pivot slices instead of the full tensor. The paper combines TCI with Gauss-Kronrod quadrature: the simplex domain of the imaginary-time integrals is mapped to a hypercube by the bijection $h^{a,b}_n$, whose Jacobian is separable, and the Green's function's $\tau$-dependence is handled by adding $\tau$ as an extra tensor leg and splitting the simplex at $\tau$ into $n+1$ regions so the integrand is continuous. The load-bearing step is that TCI finds a low-rank tensor-train representation of $\tilde P$ and $\tilde Q^k_\sigma$ from a polynomial number of determinant evaluations, reducing an $n$-fold integral to a product of one-dimensional sums with $\mathrm{O}(nd\chi^2)$ cost.

What would settle it

A direct test would be to run the TCI solver at $\beta=30/t$ and $U=5t$ (where the paper reports unphysical positive $G_{\rm loc}(\tau)$ even with $\chi=200$) and measure the bond dimension needed to drive $G_{\rm loc}(\tau)$ below zero and to match a CT-QMC reference to $10^{-4}$; if no plateau in the error-versus-$\chi$ curve appears within feasible bond dimensions, the low-rank premise fails in that regime.

Watch

Extended reading notes

Core claim

The central discovery is that the integrands in the weak-coupling expansion, $\tilde P(v_1,\dots,v_n)=P(h^{0,\beta}_n(v))J_{h^{0,\beta}_n}(v)$ for the partition function and the analogous $\tilde Q^k_\sigma(v;\tau)$ for Green's functions, are low-rank when viewed as tensors, and that the rank stays small enough for TCI to capture the integral to high precision. The paper demonstrates this by the fast convergence of the computed Green's function with respect to the tensor-train bond dimension $\chi$, and by agreement with exact solutions at order $n_{\max}=40$, where errors of order $10^{-5}$ are reached with $\chi=200$. In DMFT, the solver with $\chi\simeq200$ reproduces CT-HYB and CT-AUX results at the level of $10^{-4}$ and captures both the crossover at $\beta=16/t$ and the first-order transition with hysteresis at $\beta=20/t$.

Load-bearing premise

The whole method rests on the assumption, supported only by numerical evidence, that the transformed integrands are well approximated by a tensor train with small bond dimension; this assumption is already seen to fail in the paper at $\beta=30/t$, $U=5t$, where the computed Green's function becomes slightly positive.

Editorial extensions

If this is right

  • The weak-coupling TCI solver evaluates impurity Green's functions and partition functions without stochastic sampling, so it is free of the sign problem that limits CT-QMC in some multi-orbital, cluster, spin-orbit, and nonequilibrium setups.
  • In DMFT on the Bethe lattice it reproduces both the metal-to-insulator crossover at $\beta=16/t$ and the first-order Mott transition with coexisting solutions and hysteresis at $\beta=20/t$, matching CT-HYB and CT-AUX to roughly $10^{-4}$.
  • Because it computes $Z/Z_0$ directly, observables such as the doublon number and the lattice free energy are obtained almost without extra cost, quantities QMC accesses only with difficulty.
  • The computational cost grows polynomially, roughly $\mathrm{O}(\chi^2 d\, n_{\max}^6)$, so reaching order 40 is feasible with bond dimension $\chi\approx200$ and small-scale parallelization.
  • The same approach is expected to extend to multi-site cluster impurity problems, where strong-coupling TCI struggles with exponentially large local Hilbert spaces.

Reading between the lines

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

  • Editorial inference: if the low-rank property survives when site or orbital indices are included as tensor legs, the same TCI machinery could solve multi-site cluster and multi-orbital problems in regimes where CT-QMC has a sign problem; the paper only offers this as a future possibility.
  • Editorial inference: the observed failure at $\beta=30/t$, $U=5t$ suggests the required bond dimension grows with $\beta U$, so an environment-aware TCI or a different simplex-to-hypercube map (both mentioned as future work) may be needed before the solver reaches the deep-Mott regime.
  • Editorial inference: TCI's deterministic quadrature errors could make it useful as a reference for validating CT-QMC codes on benchmark impurity models, since it provides independent free-energy estimates.
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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 paper presents a deterministic impurity solver based on tensor cross interpolation (TCI) applied to the weak-coupling expansion. The high-dimensional integrals over the simplex are mapped to hypercubes, discontinuities are handled by splitting the time-ordered domain into k sectors, and a tensor-train representation of each integrand is constructed by TCI. The solver is benchmarked on an exactly solvable impurity model and then used as a DMFT impurity solver for the Hubbard model on the Bethe lattice, where it reproduces the metal-insulator crossover, the first-order Mott transition with hysteresis, the doublon number, and the lattice free energy. The central numerical claims are supported by external comparisons with exact solutions and CT-HYB/CT-AUX results.

Significance. If the numerical results are correct, the method offers a deterministic, sign-problem-free (in principle) alternative to CT-QMC for impurity models, with the additional capability of computing the free energy directly. The paper includes strong external benchmarks: errors near 1e-5 for the exactly solvable model, agreement with CT-HYB and CT-AUX at the 1e-4 level in DMFT, and a correctly located first-order Mott transition. The method has no fitted parameters in the physical results, and the authors are candid about known failure modes. The main open question is the scaling of the required tensor-train rank with perturbation order, which is the pillar on which the efficiency claim rests.

major comments (2)
  1. [III.A, Fig. 2(d)] The statement in Sec. III.A that 'the integrand in the weak-coupling expansion has a low-rank structure' is not established at the highest order tested. For nmax=40, the error at tau=beta/2 decreases monotonically with chi and shows no plateau up to chi=200, so the rank needed to represent the n=40 integrand is not demonstrated to be small. Since the cost estimate in Sec. III.A scales as O(chi^2 d n^6), a rank that grows with n would limit the method to lower orders than the abstract implies. Please provide a scaling analysis (e.g., the minimal chi required to reach a target error as a function of n for n=10,...,40, or the bond-dimension profile of the tensor train at n=40) and qualify the abstract's 'naturally have a low-rank structure' accordingly.
  2. [III.C and Eq. (51)] The rough estimate (51) is used both to choose nmax and to draw the 'works stably' boundary in Fig. 4. This estimate controls only the truncation error of the perturbative series, not the convergence of the TCI rank or the cancellation among k-sectors. The observed failure at beta=30/t, U=5t (slightly positive Gloc despite chi=200) illustrates that accuracy is also limited by rank convergence, which Eq. (51) does not capture. I recommend stating explicitly in Sec. III.C that Eq. (51) is a necessary but not sufficient condition for accuracy, and, if feasible, reporting the bond-dimension requirement at the failure point.
minor comments (5)
  1. [Eq. (38)] The summation in Eq. (38) starts at n=1, omitting the n=0 term (which equals unity); either start at n=0 or add a sentence explaining that the n=0 contribution has been separated.
  2. [II.E.3] In the comparison of the three summation methods, the phrase 'approximately n+1 times smaller' should be clarified, e.g., as 'by a factor of about n+1' or 'roughly 1/(n+1) of the cost'; the current wording is ambiguous.
  3. [References] Reference [39] is typeset as 'N´u˜nez-Fern´andezet al.' with a missing space before 'et al.'; please fix the formatting.
  4. [Abstract and Sec. III.C] The abstract's claim of being 'free from the sign problem' is later qualified in Sec. III.C by the cancellation between positive and negative contributions in the k-sum, which can require larger bond dimensions. Adding a sentence to the abstract or Introduction with this qualification would prevent overreading of the claim.
  5. [Fig. 2] In Fig. 2(a) and (b), the TCI and exact curves are nearly indistinguishable; adding explicit error insets or distinct markers at the plotted points would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: external benchmarks, no fitted parameters relabeled as predictions, and the low-rank claim is an empirical finding rather than a definitional or self-citation input.

full rationale

The paper's derivation chain is self-contained on the points that matter for circularity. The weak-coupling expansion formulas (Sec. II.A) are standard; the TCI evaluation (Sec. II.C) is an external numerical algorithm; the variable transformation with separable Jacobian (Appendix A) is an explicit mathematical identity, not a fit. The central numerical claim, that P(h(v)) and Q_k(v;tau) have low tensor-train rank after the simplex-to-hypercube map (Sec. III.A), is supported by direct comparison against the exact solution of the Falicov-Kimball impurity model (Eq. (52)) and by convergence plots, not by fitting any parameter of the target data. No result is defined in terms of the quantity it claims to predict: Gloc(tau) is computed by summing the truncated series and compared with independent CT-HYB and CT-AUX data (Sec. III.B), and the free energy follows from Z/Z0 by explicit formulas (Eqs. (59)-(60)). The nmax estimate of Eq. (51) is a stated Poisson/Gaussian heuristic used to choose the truncation order, not a fit to the benchmark values. The documented failure at beta=30/t, U=5t (Sec. III.B.3), where Gloc(tau) becomes slightly positive, is an acknowledged accuracy limitation and does not indicate that a fitted input was relabeled as a prediction. Self-citations (e.g., SparseIR.jl [44-47] and the TCI library [39]) are tool citations; although H. Shinaoka is a coauthor of several cited references [39,44-47,62-64], none is invoked as an unverified uniqueness theorem or as the sole justification for the method's central claim, and the low-rank property is explicitly tested rather than imported. Overall, the paper's inputs and predictions are distinct, so there is no circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No physical free parameters are fitted. Two computational cutoffs, nmax and chi, are chosen by hand and affect accuracy. The central assumption, low-rank separability of the integrand, is numerical rather than proven. No new physical entities are postulated.

free parameters (2)
  • Maximum perturbation order nmax = up to 40
    Truncation cutoff chosen by heuristic estimate Eq. (51); not fitted to target data but affects accuracy and convergence.
  • Bond dimension chi = 50 to 200
    TCI rank chosen from convergence checks; not fitted to target data but controls precision, and is central to the low-rank assumption.
assumptions (4)
  • standard math Wick's theorem and the determinant structure of the weak-coupling expansion, Eqs. (4)-(8)
    Standard many-body perturbation theory used as the starting point; not introduced by this paper.
  • domain assumption Infinite-coordination Bethe lattice DMFT mapping with local self-energy, Eq. (14)
    Standard DMFT assumption used to compare with known Mott transition results; not specific to TCI.
  • ad hoc to paper The integrand after the variable transform has low-rank tensor-train structure at small bond dimension
    Empirically observed in Sec. III.A and heuristically motivated by near-constant Weiss fields, but not proven; breaks at beta=30, U=5 as described in Sec. III.C.
  • domain assumption The Gaussian/Poisson estimate Eq. (51) is sufficient for choosing nmax
    Heuristic truncation rule validated by numerical convergence checks, but not a proven bound.

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Cite this review

Pith. "Pith review of Tensor cross interpolation approach for quantum impurity problems based on the weak-coupling expansion." pith.science (2026). https://pith.science/paper/7CBWWRLJ

@misc{pith2026250112643,
  author       = {Pith},
  title        = {Pith review of: Tensor cross interpolation approach for quantum impurity problems based on the weak-coupling expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CBWWRLJ}},
  note         = {Machine review of arXiv:2501.12643}
}
read the original abstract

We apply the tensor cross interpolation (TCI) algorithm to solve equilibrium quantum impurity problems with high precision based on the weak-coupling expansion. The TCI algorithm, a kind of active learning method, factorizes high-dimensional integrals that appear in the perturbative expansion into a product of low-dimensional ones, enabling us to evaluate higher-order terms efficiently. This method is free from the sign problem which quantum Monte Carlo methods sometimes suffer from, and allows one to directly calculate the free energy. We benchmark the TCI impurity solver on an exactly solvable impurity model, and find good agreement with the exact solutions. We also incorporate the TCI impurity solver into the dynamical mean-field theory to solve the Hubbard model, and show that the metal-to-Mott insulator transition is correctly described with comparable accuracy to the Monte Carlo methods. Behind the effectiveness of the TCI approach for quantum impurity problems lies the fact that the integrands in the weak-coupling expansion naturally have a low-rank structure in the tensor-train representation.

Figures

Figures reproduced from arXiv: 2501.12643 by the authors.

Figure 1
Figure 1. FIG. 1. Tensor-train decomposition of a tensor [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Results of the weak-coupling TCI solver for the exactly solvable impurity model with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Results of the weak-coupling TCI solver for the exactly solvable impurity model with constant hybridization functions [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. DMFT phase diagram for the Hubbard model on [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. DMFT results for the half-filled Hubbard model on the Bethe lattice at [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. DMFT results obtained by the weak-coupling TCI impurity solver for the half-filled Hubbard model on the Bethe [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Forward citations

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Reference graph

Works this paper leans on

76 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [1]

    Let us consider an M × N matrix A

    Matrix cross interpolation We first introduce the matrix cross interpolation (CI) [34–39], which forms a basis of the TCI approximation. Let us consider an M × N matrix A. We define the set of the row and column indices ofA as I = {1, 2, · · ·, M} and J = {1, 2, · · ·, N}, respectively, and write the subset of I, J with size χ (≤ rank A) as I = {i1, i2, ·...

    work page

  2. [2]

    Let us consider a tensor A with n legs labeled by σ1, σ2, · · ·, σn

    Tensor cross interpolation The tensor cross interpolation (TCI) is an extension of the matrix CI that approximates a tensor by a product of its low-dimensional slices [34–39]. Let us consider a tensor A with n legs labeled by σ1, σ2, · · ·, σn. We as- sume that the index σℓ (1 ≤ ℓ ≤ n) takes values from the set Sℓ = {1, 2, · · ·, dℓ}, which means that A i...

    work page

  3. [3]

    (10) and (11)

    Evaluating high-dimensional integrals by TCI To evaluate the Green’s function of the impurity model, we have to perform the high-dimensional inte- 5 grals appearing in Eqs. (10) and (11). In this work, we evaluate these integrals by combining Gauss–Kronrod (GK) quadrature and the tensor-train decomposition generated by the TCI algorithm following the appr...

    work page

  4. [4]

    (32)], the one in the Green’s function [Eq

    Treatment of the non-integrated variableτ Unlike the integral appearing in the partition function [Eq. (32)], the one in the Green’s function [Eq. (39)] in- volves the variable τ , which is not integrated. There are (at least) two ways to deal with it. The first one is to get the tensor-train approximation of Qσ(τ1, · · ·, τn; τ ) at each sampling point τ...

    work page

  5. [5]

    (39) is defined over a simplex, which needs to be transformed into a hypercube

    Change of variables As in the case of the partition function, the integral in Eq. (39) is defined over a simplex, which needs to be transformed into a hypercube. Furthermore, since ˜Dσ n depends on Gσ(τ − τi) that has a discontinuous jump at τ = τi, the integrand Qσ(τ1, · · ·, τn; τ ) is also discontin- uous at τi = τ for 1 ≤ i ≤ n. To eliminate the disco...

    work page

  6. [6]

    Discrete summation over k In addition to the integral over v1, · · ·, vn, there ap- pears a discrete summation over k in Eq. (48). We have tried three ways to perform the discrete sum with respect to k. The first method is to get the tensor-train approxi- mation of Pn k=0 ˜Qσ k (v1, · · ·, vn; τ ) by applying the TCI algorithm and integrate it over a hype...

    work page

  7. [7]

    From the analysis of the exactly 10 FIG

    Analysis of the crossover region Figure 5(a)-(c) shows the results of the weak-coupling TCI solver for β = 16/t, which is slightly above the Mott transition endpoint. From the analysis of the exactly 10 FIG. 3. Results of the weak-coupling TCI solver for the exactly solvable impurity model with constant hybridization functions ∆↑(τ ) = −ε2/2 and ∆ ↓(τ ) =...

    work page

  8. [8]

    As the temperature is lowered, the required perturbation order nmax increases, making the calculation more chal- lenging

    Analysis of the Mott transition Next we show the results around the Mott transition in the Hubbard model on the Bethe lattice at β = 20/t. As the temperature is lowered, the required perturbation order nmax increases, making the calculation more chal- lenging. Our weak-coupling TCI solver can explore the temperature regime slightly below the critical endp...

    work page

Show all 76 references
  1. [9]

    Current limitations of the weak-coupling TCI solver To finish this section, let us discuss the limitations of the current weak-coupling TCI solver. From the analysis of the exactly solvable model and the comparison with CT-QMC in the DMFT calculations, we found that the weak-c...

  2. [10]

    To this end, we use a map f a,b n : S0,1 n → Sa,b n , [f a,b n (y)]i = a + (b − a)yi, (A1) where y = ( y1, · · ·, yn) ∈ S0,1 n

    T ransformation from Sa,b n to S0,1 n Let us consider changing the integral domain fromSa,b n into S0,1 n through a variable transformation. To this end, we use a map f a,b n : S0,1 n → Sa,b n , [f a,b n (y)]i = a + (b − a)yi, (A1) where y = ( y1, · · ·, yn) ∈ S0,1 n . This ma...

  3. [11]

    To this end, we use a map gn : [0, 1]n → S0,1 n , [gn(z)]i = 1 − i−1Y j=1 (1 − zj), (A5) where z = (z1, · · ·, zn) ∈ [0, 1]n

    T ransformation from S0,1 n to [0, 1]n Let us consider changing the integral domain from the simplex S0,1 n to the unit hypercube [0 , 1]n. To this end, we use a map gn : [0, 1]n → S0,1 n , [gn(z)]i = 1 − i−1Y j=1 (1 − zj), (A5) where z = (z1, · · ·, zn) ∈ [0, 1]n. This transf...

  4. [12]

    (A9) The Jacobian of this map is given by Jha,b n (z) = Jf a,b n (gn(z))Jgn (z) = (b − a)n(1 − z1)n−1 · · ·(1 − zn−1)

    T ransformation from Sa,b n to [0, 1]n By combining the two maps, f a,b n and gn, we can con- struct a bijective map from [0 , 1]n to Sa,b n , ha,b n = f a,b n ◦ gn. (A9) The Jacobian of this map is given by Jha,b n (z) = Jf a,b n (gn(z))Jgn (z) = (b − a)n(1 − z1)n−1 · · ·(1 −...

  5. [13]

    Kondo, Resistance minimum in dilute magnetic alloys, Prog

    J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32, 37 (1964)

    1964

  6. [14]

    A. M. Clogston, B. T. Matthias, M. Peter, H. J. Williams, E. Corenzwit, and R. C. Sherwood, Local Magnetic Mo- ment Associated with an Iron Atom Dissolved in Various Transition Metal Alloys, Phys. Rev. 125, 541 (1962)

    1962

  7. [15]

    M. P. Sarachik, E. Corenzwit, and L. D. Longinotti, Re- sistivity of Mo-Nb and Mo-Re Alloys Containing 1% Fe, Phys. Rev. 135, A1041 (1964)

    1964

  8. [16]

    Y. Li, R. Deng, W. Lin, Y. Tian, H. Peng, J. Yi, B. Yao, and T. Wu, Electrostatic tuning of Kondo effect in a rare-earth-doped wide-band-gap oxide, Phys. Rev. B 87, 155151 (2013)

    2013

  9. [17]

    Goldhaber-Gordon, H

    D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A. Kastner, Kondo effect in a single-electron transistor, Nature 391, 156 (1998)

    1998

  10. [18]

    S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen- hoven, A Tunable Kondo Effect in Quantum Dots, Sci- 15 ence 281, 540 (1998)

    1998

  11. [19]

    I. V. Borzenets, J. Shim, J. Chen, A. Ludwig, A. Wieck, S. Tarucha, H.-S. Sim, and M. Yamamoto, Observation of the Kondo Screening Cloud of Micron Lengths, Nature 579, 210 (2020)

    2020

  12. [20]

    A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. Rey, Two-orbital SU( N ) magnetism with ultracold alkaline-earth atoms, Nat. Phys. 6, 289 (2010)

    2010

  13. [21]

    Riegger, N

    L. Riegger, N. Darkwah Oppong, M. H¨ ofer, D. R. Fer- nandes, I. Bloch, and S. F¨ olling, Localized Magnetic Mo- ments with Tunable Spin Exchange in a Gas of Ultracold Fermions, Phys. Rev. Lett. 120, 143601 (2018)

    2018

  14. [22]

    K. Ono, Y. Amano, T. Higomoto, Y. Saito, and Y. Taka- hashi, Observation of spin-exchange dynamics between itinerant and localized 176Yb atoms, Phys. Rev. A 103, L041303 (2021)

    2021

  15. [23]

    Georges and G

    A. Georges and G. Kotliar, Hubbard model in infinite dimensions, Phys. Rev. B 45, 6479 (1992)

    1992

  16. [24]

    Georges, G

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

    1996

  17. [25]

    Kotliar, S

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

    2006

  18. [26]

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

    2014

  19. [27]

    Sun and G

    Q. Sun and G. K.-L. Chan, Quantum Embedding Theo- ries, Acc. Chem. Res. 49, 2705 (2016)

    2016

  20. [28]

    Jarrell, Hubbard model in infinite dimensions: A quantum Monte Carlo study, Phys

    M. Jarrell, Hubbard model in infinite dimensions: A quantum Monte Carlo study, Phys. Rev. Lett. 69, 168 (1992)

    1992

  21. [29]

    X. Y. Zhang, M. J. Rozenberg, and G. Kotliar, Mott transition in the d = ∞ Hubbard model at zero temper- ature, Phys. Rev. Lett. 70, 1666 (1993)

    1993

  22. [30]

    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

  23. [31]

    Bl¨ umer, Metal-Insulator Transition and Optical Con- ductivity in High Dimensions (Shaker Verlag, Aachen, 2003)

    N. Bl¨ umer, Metal-Insulator Transition and Optical Con- ductivity in High Dimensions (Shaker Verlag, Aachen, 2003)

    2003

  24. [32]

    A. I. Lichtenstein and M. I. Katsnelson, Antiferromag- netism and d-wave superconductivity in cuprates: A clus- ter dynamical mean-field theory, Phys. Rev. B 62, R9283 (2000)

    2000

  25. [33]

    Maier, M

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

    2005

  26. [34]

    Haule and G

    K. Haule and G. Kotliar, Strongly correlated super- conductivity: A plaquette dynamical mean-field theory study, Phys. Rev. B 76, 104509 (2007)

    2007

  27. [35]

    Kitatani, N

    M. Kitatani, N. Tsuji, and H. Aoki, FLEX+DMFT ap- proach to the d-wave superconducting phase diagram of the two-dimensional Hubbard model, Phys. Rev. B 92, 085104 (2015)

    2015

  28. [36]

    Tanaskovi´ c, V

    D. Tanaskovi´ c, V. Dobrosavljevi´ c, E. Abrahams and G. Kotliar, Disorder Screening in Strongly Correlated Sys- tems, Phys. Rev. Lett. 91, 066603 (2003)

    2003

  29. [37]

    Byczuk, W

    K. Byczuk, W. Hofstetter, and D. Vollhardt, Mott- Hubbard Transition versus Anderson Localization in Cor- related Electron Systems with Disorder, Phys. Rev. Lett. 94, 056404 (2005)

    2005

  30. [38]

    Byczuk, W

    K. Byczuk, W. Hofstetter, and D. Vollhardt, Anderson localization vs. Mott-Hubbard metal-insulator transition in disordered, interacting lattice fermion systems, Int. J. Mod. Phys. B 24, 1727 (2010)

    2010

  31. [39]

    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

  32. [40]

    Werner, A

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

    2006

  33. [41]

    E. Gull, P. Werner, A. Millis, and M. Troyer, Perfor- mance analysis of continuous-time solvers for quantum impurity models, Phys. Rev. B 76, 235123 (2007)

    2007

  34. [42]

    E. Gull, P. Werner, O. Parcollet, and M. Troyer, Continuous-time auxiliary-field Monte Carlo for quantum impurity models, Europhys. Lett. 82, 57003 (2008)

    2008

  35. [43]

    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

  36. [44]

    Maˇ cek, P

    M. Maˇ cek, P. T. Dumitrescu, C. Bertrand, B. Triggs, O. Parcollet, and X. Waintal, Quantum Quasi-Monte Carlo Technique for Many-Body Perturbative Expan- sions, Phys. Rev. Lett. 125, 047702 (2020)

    2020

  37. [45]

    Bertrand, D

    C. Bertrand, D. Bauernfeind, P. T. Dumitrescu, M. Maˇ cek, X. Waintal, and O. Parcollet, Quantum quasi Monte Carlo algorithm for out-of-equilibrium Green functions at long times, Phys. Rev. B103, 155104 (2021)

    2021

  38. [46]

    Oseledets and E

    I. Oseledets and E. Tyrtyshnikov, TT-cross approxima- tion for multidimensional arrays, Linear Algebra and its Applications 432(1), 70 (2010)

    2010

  39. [47]

    I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33(5), 2295 (2011)

    2011

  40. [48]

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

    2014

  41. [49]

    Dolgov and D

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

    2020

  42. [50]

    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, Learning Feynman Diagrams with Tensor Trains, Phys. Rev. X12, 041018 (2022)

    2022

  43. [51]

    N´ u˜ nez-Fern´ andez, M

    Y. N´ u˜ nez-Fern´ andez, M. K. Ritter, M. Jeannin, Jheng- Wei Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, Jan von Delft, H. Shinaoka, and X. Waintal, Learning tensor networks with tensor cross interpolation: new al- gorithms and libraries, SciPost Phys. 18, 104 (2025)

    2025

  44. [52]

    Erpenbeck, W.-T

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

    2023

  45. [53]

    Eckstein, Solving impurity models in the non-equilibrium steady state with tensor trains, arXiv:2410.19707 (2024)

    M. Eckstein, Solving impurity models in the non-equilibrium steady state with tensor trains, arXiv:2410.19707 (2024)

    2024 arXiv

  46. [54]

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

    2025

  47. [55]

    Metzner and D

    W. Metzner and D. Vollhardt, Correlated Lattice Fermions in d = ∞ Dimensions, Phys. Rev. Lett. 62, 324 (1989). 16

    1989

  48. [56]

    Shinaoka, J

    H. Shinaoka, J. Otsuki, M. Ohzeki, and K. Yoshimi, Compressing Green’s function using intermediate repre- sentation between imaginary-time and real-frequency do- mains, Phys. Rev. B 96, 035147 (2017)

    2017

  49. [57]

    J. Li, M. Wallerberger, N. Chikano, C.-N. Yeh, E. Gull, and H. Shinaoka, Sparse sampling approach to efficient ab initio calculations at finite temperature, Phys. Rev. B 101, 035144 (2020)

    2020

  50. [58]

    Shinaoka, N

    H. Shinaoka, N. Chikano, E. Gull, J. Li, T. Nomoto, J. Otsuki, M. Wallerberger, T. Wang, and K. Yoshimi, Ef- ficient ab initio many-body calculations based on sparse modeling of matsubara green’s function, SciPost Phys. Lect. Notes 63, 1 (2022)

    2022

  51. [59]

    Wallerberger, S

    M. Wallerberger, S. Badr, S. Hoshino, F. Kakizawa, T. Koretsune, Y. Nagai, K. Nogaki, T. Nomoto, H. Mori, J. Otsuki, S. Ozaki, R. Sakurai, C. Vogel, N. Witt, K. Yoshimi and H. Shinaoka, sparse-ir: optimal compression and sparse sampling of many-body propagators, Soft- wareX 21...

    2023

  52. [60]

    J. K. Freericks and V. Zlati´ c, Exact dynamical mean-field theory of the Falicov-Kimball model, Rev. Mod. Phys. 75, 1333 (2003)

    2003

  53. [61]

    Kotliar, E

    G. Kotliar, E. Lange, and M.J. Rozenberg, Landau The- ory of the Finite Temperature Mott Transition, Phys. Rev. Lett. 84, 5180 (2000)

    2000

  54. [62]

    Kotliar, Sahana Murthy, and M

    G. Kotliar, Sahana Murthy, and M. J. Rozenberg, Com- pressibility Divergence and the Finite Temperature Mott Transition, Phys. Rev. Lett. 89, 046401 (2002)

    2002

  55. [63]

    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

  56. [64]

    Capone, L

    M. Capone, L. de Medici, and A. Georges, Solving the Dynamical Mean-Field Theory at Very Low Tempera- tures Using the Lanczos Exact Diagonalization, Phys. Rev. B 76, 245116 (2007)

    2007

  57. [65]

    Liebsch and H

    A. Liebsch and H. Ishida, Temperature and bath size in exact diagonalization dynamical mean field theory, J. Phys.: Condens. Matter 24, 053201 (2012)

    2012

  58. [66]

    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. B 90, 115124 (2014)

    2014

  59. [67]

    Ganahl, P

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

    2014

  60. [68]

    F. A. Wolf, A. Go, I. P. McCulloch, A. J. Millis, and U. Schollw¨ ock, Imaginary-Time Matrix Product State Im- purity Solver for Dynamical Mean-Field Theory, Phys. Rev. X 5, 041032 (2015)

    2015

  61. [69]

    Thoenniss, A

    J. Thoenniss, A. Lerose, and D. A. Abanin, Nonequi- librium quantum impurity problems via matrix-product states in the temporal domain, Phys. Rev. B 107, 195101 (2023)

    2023

  62. [70]

    Thoenniss, M

    J. Thoenniss, M. Sonner, A. Lerose, and D. A. Abanin, Efficient method for quantum impurity problems out of equilibrium, Phys. Rev. B 107, L201115 (2023)

    2023

  63. [71]

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

    2024

  64. [72]

    Chen and C

    R. Chen and C. Guo, Solving equilibrium quantum impu- rity problems on the L-shaped Kadanoff-Baym contour, Phys. Rev. B 110, 165114 (2024)

    2024

  65. [73]

    Z. Sun, R. Chen, Z. Li, and C. Guo, Infi- nite Grassmann time-evolving matrix product opera- tors for non-equilibrium quantum impurity problems, arXiv:2412.04702 (2024)

    2024

  66. [74]

    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, arXiv:2405.06440 (2024)

    2024 arXiv

  67. [75]

    Shinaoka, M

    H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale Space- Time Ansatz for Correlation Functions of Quantum Sys- tems Based on Quantics Tensor Trains, Phys. Rev. X13, 021015 (2023)

    2023

  68. [76]

    M. K. Ritter, Y. N´ u˜ nez Fern´ andez, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal, Quantics Ten- sor Cross Interpolation for High-Resolution Parsimonious Representations of Multivariate Functions, Phys. Rev. Lett. 132, 056501 (2024)

    2024

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