Scalable Simulation of Strongly Correlated Electron-Phonon Systems via Non-Gaussian Matrix Product States

Siyuan Jiang; Tao Shi

Scalable Simulation of Strongly Correlated Electron-Phonon Systems via Non-Gaussian Matrix Product States

T0 review · 0 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Soft phonons pin stripes in doped cuprate-like models by two distinct retardation effects.

desk verdict Solid hybrid solver plus large-scale HH results that cleanly separate local vs non-local phonon retardation; soft-phonon caveats are real but already stress-tested in the paper. read the letter →

arxiv 2607.08176 v1 pith:FJCBNBYY submitted 2026-07-09 cond-mat.str-el

Siyuan Jiang , Tao Shi This is my paper · ORCID

classification cond-mat.str-el

keywords electron-phononHubbard-Holsteinnon-Gaussianstatesmatrixproductstripeorderphaseseparationretardationsoftphonons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a hybrid non-Gaussian matrix product state method that can treat strongly correlated electrons and phonons of unbounded Hilbert space at the large scales needed to resolve competing orders. Using that tool on generalized Hubbard–Holstein models, the authors show that soft phonons drive a strong tendency to phase separation in one dimension and, in two dimensions, open an intermediate metallic window at half filling. Upon doping they find that the same soft phonons stabilize stripe order in two different ways: a local retardation that pins charge while weakening the spin response in the doped antiferromagnet, and a non-local retardation that uses long-range phonon-mediated interactions to suppress phase separation and lock a novel bipolaronic stripe with a 16-site period in the doped charge-density-wave regime. The concrete claim is that these retardation mechanisms are the microscopic reason soft phonons favor stripes over phase separation, giving a systematic route to the electron–phonon interplay that is thought to matter for superconductivity.

What carries the argument

The non-Gaussian matrix product state (NGS-MPS) ansatz: a non-Gaussian transformation that entangles a bosonic Gaussian phonon state with an electronic matrix product state, followed by a self-consistent alternating optimization that yields an analytic effective electronic Hamiltonian containing both instantaneous and retarded phonon-mediated interactions.

What would settle it

On a 48 imes4 cylinder at 1/8 doping in the doped CDW regime, an independent method that can resolve large unit cells either finds a lower-energy macroscopic phase-separated state than the claimed bipolaronic stripe, or finds that the 16-site period collapses once bond dimension is pushed higher.

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Extended reading notes

Core claim

Soft phonons stabilize stripe phases in doped two-dimensional Hubbard–Holstein models through two distinct retardation effects: a local one that pins charge order and diminishes the spin response in the doped antiferromagnet, and a non-local one in which long-range phonon-mediated interactions suppress phase separation and stabilize a bipolaronic stripe of 16-site period in the doped charge-density-wave regime.

Load-bearing premise

The hybrid variational ansatz plus its self-consistent optimization remains faithful for soft phonons on large cylinders, where near-degenerate states and extended phonon clouds make local minima and insufficient bond dimension especially dangerous.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new piece is the NGS-MPS hybrid: non-Gaussian transformation for e-ph entanglement, Gaussian phonons, MPS for the electrons, plus a self-consistent loop that starts from pure NGS seeds. That combination actually reaches 1D chains of length 120 and 48 imes4 cylinders at soft frequencies, which pure NGSED and standard DMRG with phonon cutoffs cannot. The physics payoff is concrete: macroscopic PS in soft-phonon 1D HH (confirmed by Maxwell construction under the restricted ansatz), a half-filled metallic window on 48 imes4 between AFM and CDW, and—most useful—the distinction between local retardation that pins the fully filled stripe when doping the AFM and non-local V that arrests PS into a spontaneous 16 imes2 bipolaronic stripe when doping the CDW.

What works: the effective electronic Hamiltonian (Eqs. 6, 12) is derived analytically, so the local pinning terms (18a–c) and the long-range V profile in Fig. 6b are not post-hoc stories. Benchmarks against NGSED on V_nn and 4 imes4 observables, energy consistency with LBO, and phase-diagram features that line up with existing QMC/VMC are all present. Truncation is held at 10^{-5}, multiple seeds are compared, and the paper itself flags when the bosonic sector can lose positive-definiteness.

Soft spots are real but proportional. Soft phonons (ω ≤ t) on 48 imes4 still sit near dense near-degeneracies and extended clouds; the ansatz can diverge or trap if D or the seed is poor. Finite-width effects (even-odd spin gap, plaquette pairing) mean the metallic window will shrink toward the 2D limit. Neither issue is hidden, and neither overturns the local/non-local narrative that is the paper’s strongest claim.

This is for people who actually run or need large-scale e-ph numerics on cuprate-relevant models. The math and citation pattern look solid; self-cites are to the prior NGS machinery that is being extended, not circular. I would send it to referees. Worth reading and, for anyone working on stripes or Holstein physics, worth citing.

Referee Report

0 major / 5 minor

Summary. The manuscript introduces a hybrid non-Gaussian matrix-product-state (NGS-MPS) variational method for strongly correlated electron-phonon systems. The ansatz (Eq. 3) applies a non-Gaussian transformation that encodes e-ph entanglement, after which residual phonons are treated as a bosonic Gaussian state and the electrons by an MPS; a self-consistent workflow of NGS flows and MPS sweeps is used to minimize the energy. The method is applied to generalized Hubbard-Holstein models. In 1D it recovers the phonon-mediated nearest-neighbor attraction of prior NGSED work and, on larger systems, finds a pronounced tendency to phase separation. In 2D it maps the half-filled phase diagram (metallic intermediate phase between AFM and CDW) and, at 1/8 doping, shows that soft phonons stabilize fully filled stripes via a local retardation effect in the doped AFM and a novel bipolaronic stripe (16 imes2 unit cell) via non-local phonon-mediated interactions that suppress phase separation in the doped CDW regime. Analytic effective electronic Hamiltonians (Eqs. 6, 12) are used to interpret both mechanisms.

Significance. If the numerical results hold, the work supplies a scalable, systematically benchmarked tool that reaches system sizes previously inaccessible to NGSED while remaining competitive with pure fermionic MPS. The concrete physical claims—soft-phonon stabilization of stripes by local versus non-local retardation, and the spontaneous appearance of a large-unit-cell bipolaronic stripe—are of direct interest for cuprate phenomenology and for the broader theory of intertwined orders. Strengths include quantitative benchmarks against NGSED (Fig. 1, Appendix B.2), independent Maxwell-construction confirmation of 1D phase separation (Appendix B.1), analytic derivation of the effective interactions used for interpretation, and explicit control of truncation error and multi-seed validation. These features make the central narrative falsifiable and reusable by other groups.

minor comments (5)

In Sec. II B the switch from imaginary-time evolution to DMRG is described only qualitatively; a short statement of the practical criterion (e.g., energy change or truncation-error threshold) used to decide the switch would improve reproducibility.
Fig. 1 caption and panels (b),(d) note the absence of data points at large λ as the onset of phase separation; a brief quantitative criterion (energy divergence, density variance, or compressibility sign) would make the identification unambiguous.
Table II reports renormalized hoppings along the u=2λ line; the corresponding bare values or the explicit definition of the central bulk site would help the reader verify the polaron-mass trend discussed in the text.
Appendix A.4, Eq. (A25): the approximation 〈PS〉_{0} ≈ 〈PS〉 is stated to leading order; a one-sentence estimate of the neglected O((t̃/ΔE)^{4}) correction for the parameters of Fig. 5 would strengthen the comparison with the full NGS-MPS double occupancy.
A few typographical inconsistencies appear (e.g., “NGS/uni2010ED”, “L = 80 system,,” double commas). A final proofreading pass would remove them.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: variational energy minimization of a derived effective Hamiltonian yields the phases; self-citations supply method machinery but do not force the PS/stripe conclusions.

full rationale

The central claims (1D PS tendency; half-filled metallic phase between AFM and CDW; soft-phonon stabilization of fully-filled and bipolaronic stripes via local vs non-local retardation) are obtained by numerical minimization of the variational energy of the NGS-MPS ansatz (Eq. 3) on large systems, not by algebraic identity with the inputs. The effective electronic Hamiltonian (Eqs. 6, 12) and the decomposition into V_inst and V_ret follow analytically from the non-Gaussian transformation and Gaussian average; they are used only for post-hoc microscopic interpretation of the already-computed charge/spin profiles and correlation lengths (Figs. 5–6, Eqs. 18). Benchmarks against NGSED (Appendix B.2), LBO, Maxwell construction (Appendix B.1), and literature QMC/VMC supply independent checks. Self-citations to prior NGS papers (Shi et al.) introduce the transformation and EoMs but are accompanied by full re-derivations in Appendix A; they do not uniquely force the phase diagram or the 16 imes2 bipolaronic stripe. No parameter is fitted to the target observables and then re-predicted; no uniqueness theorem is imported to exclude alternatives. Truncation-error and multi-seed validation criteria (Sec. II B) further keep the workflow from being tautological. Score 1 reflects only the ordinary presence of method self-citations that are not load-bearing for the physics claims.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central claims rest on a standard variational principle applied to a specific hybrid ansatz, on the Hubbard–Holstein Hamiltonian with diagonal e-ph coupling, and on numerical controls (bond dimension, truncation error). No new physical particles or forces are postulated; the bipolaronic stripe is an emergent numerical state, not an invented entity. Free parameters are model and numerical knobs, not fits that define the target observables.

free parameters (3)

MPS bond dimension D and truncation-error threshold (~10^{-5})
Controls accuracy of the electronic state; results are extrapolated or checked for convergence, but residual finite-D bias remains possible near critical points.
Phonon frequency ω, coupling λ, Hubbard U (and range r_g of g_lj)
Model parameters chosen to match experimental or prior theoretical regimes (e.g., ω/t=0.12–5, u=8); they define the physical setting rather than being fitted to produce the stripe claim.
Imaginary-time step δτ and NGS–MPS iteration schedule
Numerical hyperparameters of the self-consistent loop; affect convergence path and risk of local minima.

assumptions (5)

standard math McLachlan variational principle / projected imaginary-time evolution on the NGS tangent space yields the ground-state flow (Eqs. 11a–c).
Standard for Gaussian and non-Gaussian variational manifolds; invoked in Sec. II B and Appendix A 2.
domain assumption Ground state respects time-reversal symmetry so Δ_p=0 and off-diagonal Γ blocks vanish.
Used to reduce bosonic parameters (Sec. II A); plausible for the TRS Hamiltonian but excludes possible TRS-broken states.
domain assumption Diagonal Holstein-type e-ph coupling (Eq. 2) and Einstein or finite-range phonons suffice for the physics of interest.
Stated scope of the paper (Sec. II, IV); off-diagonal coupling is left for future work.
domain assumption 4-leg cylinders with OBC along the axis capture the essential 2D competition among AFM, CDW, metal, and stripe orders.
Standard quasi-2D DMRG setting; paper itself discusses finite-width caveats (Haldane gap, plaquette pairing, λ_ρ→0 as L_y→∞).
domain assumption Area-law entanglement of the electronic ground state makes MPS an efficient representation after the non-Gaussian dressing.
Underlying justification for the hybrid ansatz (Sec. II A).

invented entities (2)

NGS-MPS hybrid variational ansatz (Eq. 3) independent evidence
purpose: Compactly encode non-local e-ph entanglement while treating residual phonons as Gaussian and electrons as MPS.
Methodological construct, not a new physical degree of freedom; independent evidence is the match to NGSED/LBO benchmarks.
Bipolaronic stripe phase with 16×2 unit cell
purpose: Emergent ground-state order found upon doping the CDW parent; claimed to be stabilized by long-range phonon-mediated interactions.
Numerical discovery within the model, not postulated a priori; falsifiable by other methods or larger systems.

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Cite this review

Pith. "Pith review of Scalable Simulation of Strongly Correlated Electron-Phonon Systems via Non-Gaussian Matrix Product States." pith.science (2026). https://pith.science/paper/FJCBNBYY

@misc{pith2026260708176,
  author       = {Pith},
  title        = {Pith review of: Scalable Simulation of Strongly Correlated Electron-Phonon Systems via Non-Gaussian Matrix Product States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJCBNBYY}},
  note         = {Machine review of arXiv:2607.08176}
}

read the original abstract

We investigate strongly correlated electron-phonon (e-ph) systems via a non-Gaussian matrix product state method. By combining non-Gaussian states with matrix product states, our method efficiently characterizes the intractable entanglement between strongly correlated electrons and phononic modes of unbounded Hilbert space, enabling scalable simulations across broad parameter regimes. In one-dimensional generalized Hubbard--Holstein (HH) models, we identify a pronounced tendency toward phase separation (PS), an instability relevant to recent angle-resolved photoemission spectroscopy observations on doped cuprate chain. In two-dimensional HH models, we construct the phase diagram at half-filling featuring a metallic phase emerging from the competition between non-local phonon-mediated attraction and local Hubbard repulsion. Upon doping, we elucidate the role of soft phonons in stabilizing stripe phases. In the antiferromagnet, the stabilization of the fully filled stripe is attributed to a local retardation effect, wherein the charge order is pinned by phonons, leading to a diminished response to spin fluctuations. In the doped charge-density-wave regime, a novel bipolaronic stripe phase with an enlarged unit cell is stabilized via a non-local retardation effect, where long-range phonon-mediated interactions suppress PS. Our work establishes a systematic route to decoding the e-ph interplay that is crucial for superconductivity.

Figures

Figures reproduced from arXiv: 2607.08176 by the authors.

**Figure 2.** FIG. 2. Ground-state properties of the generalized Hubbard–Holstein model on an [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Phase diagram of the half-filled Hubbard–Holstein [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. Correlation functions and Luttinger exponents for the half-filled Hubbard–Holstein model on a 48 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Ground-state properties of the fully filled stripe phase in the 1 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Stabilization of the bipolaronic stripe phase via long [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Microscopic decomposition of the double occupancy for the fully filled stripe phase in the 1 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Energy per hole [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 10.** Figure 10: FIG. 10. Supplemental results for the 4-leg half-filled [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 9.** Figure 9: FIG. 9. Benchmarks of the NGS-MPS method against NGS [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 11.** Figure 11: FIG. 11. Impact of geometric frustration on the half-filled Hubbard–Holstein model at [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

150 extracted references · 150 canonical work pages

[1]

Half-filled case For the half-filled HH model atω= 5t, we present the phase diagram in Fig. 3. The anti-adiabatic limit pro- vides a theoretical baseline (dotted line,u= 2λ). In this limit, the system maps onto an effective Hubbard model (U→U−2λt), predicting a quantum phase transition atu= 2λ. This line separates the (π, π)-AFM phase foru >2λ[104] from d...

work page
[2]

arrested

1 8 hole doped case Upon doping the half-filled parent compound top= 1/8, the system enters the regime of intertwined orders TABLE II. Renormalized polaronic hopping amplitudes for a central bulk site along theu= 2λline. (u, λ) ˜tc,c+ ˆα a (0,0) (2,1) (4,2) (8,4) ˆα= ˆx 1.000 0.930 0.862 0.730 ˆα= ˆy 1.000 0.929 0.861 0.730 a Here, ˜tc,c+ ˆαdenotes the dr...

work page
[3]

Brief Introduction to Fermionic Gaussian MPS This section outlines the construction of the MPS rep- resentation for fermionic Gaussian states [71, 136], which we employ to generate robust initial seeds for the opti- mization workflow. A fermionic Gaussian state [71] is defined as|Ψ f ⟩=U f |0⟩f, where|0⟩ f is the vacuum and Uf is a unitary transformation ...

work page
[4]

The derivation is based on the McLachlan variational principle [141]

Derivation of EoMs This appendix details the derivation of the EoMs for the NGS parameters. The derivation is based on the McLachlan variational principle [141]. We project ITE, dτ |Ψ(τ)⟩=−(H− ⟨H⟩)|Ψ(τ)⟩,(A3) onto the tangent space of the variational manifold [71, 72]: ⟨Vν|dτ ξµ|Vµ⟩=−⟨V ν|δH|Ψ⟩,(A4) where|V µ⟩=Q Ψ∂µ|Ψ⟩= (1− |Ψ⟩⟨Ψ|) ∂|Ψ⟩ ∂ξ µ is the tangen...

work page
[5]

(3)] are denoted by⟨· · · ⟩, while those with respect to the electronic wave function|Ψ e⟩are denoted by⟨· · · ⟩ e

Expectation values on the full variational wavefunction As mentioned in the main text, expectation values with respect to the full variational state [Eq. (3)] are denoted by⟨· · · ⟩, while those with respect to the electronic wave function|Ψ e⟩are denoted by⟨· · · ⟩ e. We express key ob- servables⟨· · · ⟩in terms of⟨· · · ⟩ e and the associated dress- ing...

work page
[6]

We estimate the residual double occupancy by applying second-order perturbation theory to the effective electronic Hamiltonian derived in Eq

Perturbative Estimation on the Double Occupancy In the regime of strong effective repulsion (U−2λt≫ t), the ground state is dominated by configurations with- out double occupancy. We estimate the residual double occupancy by applying second-order perturbation theory to the effective electronic Hamiltonian derived in Eq. (6). The first-order correction to ...

work page
[7]

2(a), the 1D generalized HH model exhibits PS over the doping level from 0% to 15%

Maxwell construction of 1D generalized HH model with NGS-MPS ∗ As reported in Fig. 2(a), the 1D generalized HH model exhibits PS over the doping level from 0% to 15%. Here, we independently verify the PS via a Maxwell construc- tion [84]. Under PBC, we enforce a uniform electron density and calculate the ground-state energy using the restricted NGS-MPS∗ a...

work page
[8]

Benchmarks against NGSED in 2D torus Figure 9 benchmarks the NGS-MPS method against NGS-ED on a 4×4 torus. We examine various physi- cal observables, including pairing correlations, structure factors, and local moments: P0 = 1 N X i,j ⟨c† i↑c† i↓cj↓cj↑⟩,(B1a) Sσ(q) = 1 N X i,j ⟨Si ·S j⟩eiq·(i−j),(B1b) Sρ(q) = 1 N X i,j ⟨ninj⟩eiq·(i−j),(B1c) ⟨m2 z⟩= 1 N X ...

work page

Show all 150 references

[9]

III B 1, we analyze pair-pair correlations in the intermediate metallic regime of the half-filled HH model atω= 5ton 4-leg cylinders

Finite-width effects for 2D Half-filled HH model withω= 5t In Sec. III B 1, we analyze pair-pair correlations in the intermediate metallic regime of the half-filled HH model atω= 5ton 4-leg cylinders. Neither the ex- tendeds-wave channel [Eq. (17c)] nor thed x2−y2-wave channel...
[10]

J. Lee, K. Fujita, K. McElroy, J. A. Slezak, M. Wang, Y. Aiura, H. Bando, M. Ishikado, T. Masui, J.-X. Zhu, A. V. Balatsky, H. Eisaki, S. Uchida, and J. C. Davis, Nature442, 546 (2006)

2006
[11]

J. L. Tallon, R. S. Islam, J. Storey, G. V. M. Williams, and J. R. Cooper, Physical Review Letters94, 237002 (2005)

2005
[12]

Lanzara, P

A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fuji- mori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.-X. Shen, Nature412, 510 (2001)

2001
[13]

T. Cuk, D. H. Lu, X. J. Zhou, Z.-X. Shen, T. P. Dev- ereaux, and N. Nagaosa, physica status solidi (b)242, 11 (2005)

2005
[14]

K. M. Shen, F. Ronning, D. H. Lu, W. S. Lee, N. J. C. Ingle, W. Meevasana, F. Baumberger, A. Damascelli, N. P. Armitage, L. L. Miller, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, and Z.-X. Shen, Physical Review Letters93, 267002 (2004)

2004
[15]

Y. He, M. Hashimoto, D. Song, S.-D. Chen, J. He, I. M. Vishik, B. Moritz, D.-H. Lee, N. Nagaosa, J. Zaanen, T. P. Devereaux, Y. Yoshida, H. Eisaki, D. H. Lu, and Z.-X. Shen, Science362, 62 (2018)

2018
[16]

Thomsen, M

C. Thomsen, M. Cardona, B. Gegenheimer, R. Liu, and A. Simon, Physical Review B37, 9860 (1988)

1988
[17]

Reznik, L

D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita, K. Yamada, G. D. Gu, and J. M. Tranquada, Nature440, 1170 (2006)

2006
[18]

Z. Chen, Y. Wang, S. N. Rebec, T. Jia, M. Hashimoto, D. Lu, B. Moritz, R. G. Moore, T. P. Devereaux, and Z.-X. Shen, Science373, 1235 (2021)

2021
[19]

Y. Wang, Z. Chen, T. Shi, B. Moritz, Z.-X. Shen, and T. P. Devereaux, Physical Review Letters127, 197003 (2021)

2021
[20]

Qu, B.-B

D.-W. Qu, B.-B. Chen, H.-C. Jiang, Y. Wang, and W. Li, Communications Physics5, 257 (2022)

2022
[21]

Y. Wang, I. Esterlis, T. Shi, J. I. Cirac, and E. Demler, Physical Review Research2, 043258 (2020)

2020
[22]

T. Tang, B. Moritz, C. Peng, Z.-X. Shen, and T. De- vereaux, Traces of Electron-Phonon Coupling in One- Dimensional Cuprates (2022)

2022
[23]

P.-Y. Zhao, K. Ding, and S. Yang, Physical Review Re- search5, 023026 (2023)

2023
[24]

Thomas, D

J. Thomas, D. Banerjee, A. Nocera, and S. Johnston, Physical Review X15, 021030 (2025)

2025
[25]

C. Chen, X. Chen, W. Tang, Z. Li, S. Wang, S. Ding, Z. Kang, C. Jozwiak, A. Bostwick, E. Rotenberg, M. Hashimoto, D. Lu, J. P. C. Ruff, S. G. Louie, R. J. Birgeneau, Y. Chen, Y. Wang, and Y. He, Physical Re- view Research5, 043089 (2023)

2023
[26]

Cai, Z.-X

X. Cai, Z.-X. Li, and H. Yao, Physical Review B112, 144517 (2025)

2025
[27]

Wang, Y.-F

H.-X. Wang, Y.-F. Jiang, and H. Yao, Science Bulletin 70, 2260 (2025)

2025
[28]

Fausti, R

D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Science331, 189 (2011)

2011
[29]

W. Hu, S. Kaiser, D. Nicoletti, C. R. Hunt, I. Gierz, M. C. Hoffmann, M. Le Tacon, T. Loew, B. Keimer, and A. Cavalleri, Nature Materials13, 705 (2014)

2014
[30]

Nicoletti, D

D. Nicoletti, D. Fu, O. Mehio, S. Moore, A. S. Disa, G. D. Gu, and A. Cavalleri, Physical Review Letters 121, 267003 (2018)

2018
[31]

K. A. Cremin, J. Zhang, C. C. Homes, G. D. Gu, Z. Sun, M. M. Fogler, A. J. Millis, D. N. Basov, and R. D. Averitt, Proceedings of the National Academy of Sci- ences116, 19875 (2019)

2019
[32]

Buzzi, D

M. Buzzi, D. Nicoletti, M. Fechner, N. Tancogne- Dejean, M. A. Sentef, A. Georges, T. Biesner, E. Uykur, M. Dressel, A. Henderson, T. Siegrist, J. A. Schlueter, K. Miyagawa, K. Kanoda, M.-S. Nam, A. Ardavan, J. Coulthard, J. Tindall, F. Schlawin, D. Jaksch, and A. Cavalleri, P...

2020
[33]

S. J. Zhang, Z. X. Wang, D. Wu, Q. M. Liu, L. Y. Shi, T. Lin, S. L. Li, P. C. Dai, T. Dong, and N. L. Wang, Physical Review B98, 224507 (2018)

2018
[34]

S. J. Zhang, Z. X. Wang, L. Y. Shi, T. Lin, M. Y. Zhang, G. D. Gu, T. Dong, and N. L. Wang, Physical Review B98, 020506 (2018)

2018
[35]

Sun and A

Z. Sun and A. J. Millis, Physical Review X10, 021028 (2020)

2020
[36]

Boschini, E

F. Boschini, E. H. Da Silva Neto, E. Razzoli, M. Zonno, S. Peli, R. P. Day, M. Michiardi, M. Schneider, B. Zwartsenberg, P. Nigge, R. D. Zhong, J. Schneeloch, G. D. Gu, S. Zhdanovich, A. K. Mills, G. Levy, D. J. Jones, C. Giannetti, and A. Damascelli, Nature Mate- rials17, 416 (2018)

2018
[37]

Lemonik and A

Y. Lemonik and A. Mitra, Physical Review B100, 094503 (2019)

2019
[38]

A. A. Patel and A. Eberlein, Physical Review B93, 195139 (2016)

2016
[39]

M. H. Michael, A. Von Hoegen, M. Fechner, M. F¨ orst, A. Cavalleri, and E. Demler, Physical Review B102, 174505 (2020)

2020
[40]

J. J. Lee, F. T. Schmitt, R. G. Moore, S. Johnston, Y.-T. Cui, W. Li, M. Yi, Z. K. Liu, M. Hashimoto, Y. Zhang, D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen, Nature515, 245 (2014). 21

2014
[41]

Xiang, F

Y.-Y. Xiang, F. Wang, D. Wang, Q.-H. Wang, and D.- H. Lee, Physical Review B86, 134508 (2012)

2012
[42]

Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee, Science Bul- letin61, 925 (2016)

2016
[43]

Suzuki, T

T. Suzuki, T. Someya, T. Hashimoto, S. Michi- mae, M. Watanabe, M. Fujisawa, T. Kanai, N. Ishii, J. Itatani, S. Kasahara, Y. Matsuda, T. Shibauchi, K. Okazaki, and S. Shin, Communications Physics2, 115 (2019)

2019
[44]

E. F. Talantsev and V. V. Chistyakov, Letters on Ma- terials14, 262 (2024)

2024
[45]

J. Zhan, Y. Gu, X. Wu, and J. Hu, Physical Review Letters134, 136002 (2025)

2025
[46]

Y. Li, Y. Cao, L. Liu, P. Peng, H. Lin, C. Pei, M. Zhang, H. Wu, X. Du, W. Zhao, K. Zhai, X. Zhang, J. Zhao, M. Lin, P. Tan, Y. Qi, G. Li, H. Guo, L. Yang, and L. Yang, Science Bulletin70, 180 (2025)

2025
[47]

Dagotto, Reviews of Modern Physics66, 763 (1994)

E. Dagotto, Reviews of Modern Physics66, 763 (1994)

1994
[48]

Keimer, S

B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature518, 179 (2015)

2015
[49]

M. Qin, T. Sch¨ afer, S. Andergassen, P. Corboz, and E. Gull, Annual Review of Condensed Matter Physics 13, 275 (2022)

2022
[50]

D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, Annual Review of Condensed Matter Physics13, 239 (2022)

2022
[51]

J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bu- lik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Fer- rero, T. M. Henderson, C. A. Jim´ enez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. V. Prokof’ev, M. Qin, G. E. Scuseria, H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupits...

2015
[52]

Kozik, M

E. Kozik, M. Ferrero, and A. Georges, Physical Review Letters114, 156402 (2015)

2015
[53]

Gunnarsson, G

O. Gunnarsson, G. Rohringer, T. Sch¨ afer, G. Sangio- vanni, and A. Toschi, Physical Review Letters119, 056402 (2017)

2017
[54]

Reitner, P

M. Reitner, P. Chalupa, L. Del Re, D. Springer, S. Ciuchi, G. Sangiovanni, and A. Toschi, Physical Re- view Letters125, 196403 (2020)

2020
[55]

A. S. Darmawan, Y. Nomura, Y. Yamaji, and M. Imada, Physical Review B98, 205132 (2018)

2018
[56]

Zheng, C.-M

B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Science358, 1155 (2017)

2017
[57]

Chang and S

C.-C. Chang and S. Zhang, Physical Review Letters 104, 116402 (2010)

2010
[58]

Corboz, S

P. Corboz, S. R. White, G. Vidal, and M. Troyer, Phys- ical Review B84, 041108 (2011)

2011
[59]

E. W. Huang, C. B. Mendl, H.-C. Jiang, B. Moritz, and T. P. Devereaux, npj Quantum Materials3, 22 (2018)

2018
[60]

Xu, C.-M

H. Xu, C.-M. Chung, M. Qin, U. Schollw¨ ock, S. R. White, and S. Zhang, Science384, eadh7691 (2024)

2024
[61]

Kokalj, Physical Review B95, 041110 (2017)

J. Kokalj, Physical Review B95, 041110 (2017)

2017
[62]

E. W. Huang, R. Sheppard, B. Moritz, and T. P. Dev- ereaux, Science366, 987 (2019)

2019
[63]

Jiang and T

H.-C. Jiang and T. P. Devereaux, Science365, 1424 (2019)

2019
[64]

Q. Li, Y. Gao, Y.-Y. He, Y. Qi, B.-B. Chen, and W. Li, Physical Review Letters130, 226502 (2023)

2023
[65]

Blankenbecler, D

R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Physical Review D24, 2278 (1981)

1981
[66]

E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, International Journal of Modern Physics C16, 1319 (2005)

2005
[67]

Troyer and U.-J

M. Troyer and U.-J. Wiese, Physical Review Letters94, 170201 (2005)

2005
[68]

Weber, F

M. Weber, F. F. Assaad, and M. Hohenadler, Physical Review Letters119, 097401 (2017)

2017
[69]

Or´ us, Nature Reviews Physics1, 538 (2019)

R. Or´ us, Nature Reviews Physics1, 538 (2019)

2019
[70]

J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Ver- straete, Reviews of Modern Physics93, 045003 (2021)

2021
[71]

Schollwoeck, Annals of Physics326, 96 (2011), arXiv:1008.3477 [cond-mat]

U. Schollwoeck, Annals of Physics326, 96 (2011), arXiv:1008.3477 [cond-mat]

2011 arXiv
[72]

Verstraete and J

F. Verstraete and J. I. Cirac, Renormalization algo- rithms for Quantum-Many Body Systems in two and higher dimensions (2004), arXiv:cond-mat/0407066

2004 arXiv
[73]

S. R. White, Physical Review Letters69, 2863 (1992)

1992
[74]

Zhang, E

C. Zhang, E. Jeckelmann, and S. R. White, Physical Review Letters80, 2661 (1998)

1998
[75]

Koch and C

O. Koch and C. Lubich, SIAM Journal on Matrix Anal- ysis and Applications29, 434 (2007)

2007
[76]

Evenbly, Physical Review B98, 085155 (2018)

G. Evenbly, Physical Review B98, 085155 (2018)

2018
[77]

C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Physical Review Letters108, 160401 (2012)

2012
[78]

Stolpp, T

J. Stolpp, T. K¨ ohler, S. R. Manmana, E. Jeckelmann, F. Heidrich-Meisner, and S. Paeckel, Computer Physics Communications269, 108106 (2021)

2021
[79]

Kn¨ orzer, T

J. Kn¨ orzer, T. Shi, E. Demler, and J. I. Cirac, Physical Review Letters128, 120404 (2022)

2022
[80]

T. Shi, E. Demler, and J. Ignacio Cirac, Annals of Physics390, 245 (2018)

2018
[81]

Hackl, T

L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and I. Cirac, SciPost Physics9, 048 (2020)

2020
[82]

Kan´ asz-Nagy, Y

M. Kan´ asz-Nagy, Y. Ashida, T. Shi, C. u. u. u. u. P. m. c. Moca, T. N. Ikeda, S. F¨ olling, J. I. Cirac, G. Zar´ and, and E. A. Demler, Phys. Rev. B97, 155156 (2018)

2018
[83]

Ashida, T

Y. Ashida, T. Shi, M. C. Ba˜ nuls, J. I. Cirac, and E. Demler, Physical Review B98, 024103 (2018)

2018
[84]

Ashida, T

Y. Ashida, T. Shi, M. C. Ba˜ nuls, J. I. Cirac, and E. Demler, Physical Review Letters121, 026805 (2018)

2018
[85]

Ashida, T

Y. Ashida, T. Shi, R. Schmidt, H. R. Sadeghpour, J. I. Cirac, and E. Demler, Physical Review A100, 043618 (2019)

2019
[86]

Ashida, T

Y. Ashida, T. Shi, R. Schmidt, H. R. Sadeghpour, J. I. Cirac, and E. Demler, Physical Review Letters123, 183001 (2019)

2019
[87]

P. E. Dolgirev, Y.-F. Qu, M. B. Zvonarev, T. Shi, and E. Demler, Physical Review X11, 041015 (2021)

2021
[88]

Y.-F. Qu, P. E. Dolgirev, E. Demler, and T. Shi, Effi- cient variational approach to the Fermi polaron prob- lem in two dimensions, both in and out of equilibrium (2022)

2022
[89]

Z.-Y. Wei, T. Shi, J. I. Cirac, and E. A. Demler, Kondo impurity in an attractive Fermi-Hubbard bath: Equilib- rium and dynamics (2025)

2025
[90]

T. Shi, J. I. Cirac, and E. Demler, Physical Review Re- search2, 033379 (2020)

2020
[91]

P. Sala, T. Shi, S. K¨ uhn, M. C. Ba˜ nuls, E. Demler, and J. I. Cirac, Physical Review D98, 034505 (2018)

2018
[92]

P. M. Schindler, T. Guaita, T. Shi, E. Demler, and J. I. Cirac, Physical Review Letters129, 220401 (2022)

2022
[93]

T. Shi, E. Demler, and J. I. Cirac, Physical Review Let- ters125, 180602 (2020)

2020
[94]

Y. Wang, T. Shi, and C.-C. Chen, Physical Review X 22 11, 041028 (2021)

2021
[95]

N. C. Costa, K. Seki, S. Yunoki, and S. Sorella, Com- munications Physics3, 80 (2020)

2020
[96]

Weber and M

M. Weber and M. Hohenadler, Physical Review B98, 085405 (2018)

2018
[97]

E. A. Nowadnick, S. Johnston, B. Moritz, R. T. Scalet- tar, and T. P. Devereaux, Physical Review Letters109, 246404 (2012)

2012
[98]

Ohgoe and M

T. Ohgoe and M. Imada, Phys. Rev. Lett.119, 197001 (2017)

2017
[99]

Holstein, Annals of Physics8, 325 (1959)

T. Holstein, Annals of Physics8, 325 (1959)

1959
[100]

A. S. Alexandrov,Polarons in Advanced Materials, Springer Series in Materials Science No. 103 (Canopus Publishing Limited, Dordrecht, 2007)

2007
[101]

Zoli, Physical Review B71, 184308 (2005)

M. Zoli, Physical Review B71, 184308 (2005)

2005
[102]

P. E. Spencer, J. H. Samson, P. E. Kornilovitch, and A. S. Alexandrov, Physical Review B71, 184310 (2005)

2005
[103]

Pasquale Calabrese and John Cardy, Journal of Statis- tical Mechanics: Theory and Experiment2004, P06002 (2004)

2004
[104]

M. B. Hastings, Journal of Statistical Mechanics: The- ory and Experiment2007, P08024 (2007)

2007
[105]

Eisert, M

J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics82, 277 (2010)

2010
[106]

Verstraete and J

F. Verstraete and J. I. Cirac, Physical Review B73, 094423 (2006)

2006
[107]

Haegeman, C

J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Physical Review B94, 165116 (2016)

2016
[108]

T. Tang, D. Jost, B. Moritz, and T. P. Devereaux, Phys- ical Review B110, 165118 (2024)

2024
[109]

I. G. Lang and Yu. A. Firsov, Soviet Journal of Exper- imental and Theoretical Physics16, 1301 (1963)

1963
[110]

Devos and J

L. Devos and J. Haegeman, arXiv 10.48550/arXiv.2508.10076 (2025)

2025 doi arXiv
[111]

Chung, M

C.-M. Chung, M. Qin, S. Zhang, U. Schollw¨ ock, S. R. White, and The Simons Collaboration on the Many- Electron Problem, Physical Review B102, 041106 (2020)

2020
[112]

Cini and G

M. Cini and G. Stefanucci, Solid State Communications 117, 451 (2001)

2001
[113]

C. N. Yang, Physical Review Letters63, 2144 (1989)

1989
[114]

Zhang, Physical Review Letters65, 120 (1990)

S. Zhang, Physical Review Letters65, 120 (1990)

1990
[115]

Karakuzu, L

S. Karakuzu, L. F. Tocchio, S. Sorella, and F. Becca, Physical Review B96, 205145 (2017)

2017
[116]

Stoudenmire and S

E. Stoudenmire and S. R. White, Annual Review of Condensed Matter Physics3, 111 (2012)

2012
[117]

Pollmann, S

F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Physical Review Letters102, 255701 (2009)

2009
[118]

Haldane, Physics Letters A93, 464 (1983)

F. Haldane, Physics Letters A93, 464 (1983)

1983
[119]

F. D. M. Haldane, Physical Review Letters50, 1153 (1983)

1983
[120]

Chakravarty, Physical Review Letters77, 4446 (1996)

S. Chakravarty, Physical Review Letters77, 4446 (1996)

1996
[121]

Hohenadler and G

M. Hohenadler and G. G. Batrouni, Physical Review B 100, 165114 (2019)

2019
[122]

Fradkin, S

E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Re- views of Modern Physics87, 457 (2015)

2015
[123]

J. M. Tranquada, Advances in Physics69, 437 (2020)

2020
[124]

Jiang, J

Y.-F. Jiang, J. Zaanen, T. P. Devereaux, and H.-C. Jiang, Physical Review Research2, 033073 (2020)

2020
[125]

Zachar, Physical Review B65, 174411 (2002)

O. Zachar, Physical Review B65, 174411 (2002)

2002
[126]

Zaanen, Journal of Physics and Chemistry of Solids 59, 1769 (1998)

J. Zaanen, Journal of Physics and Chemistry of Solids 59, 1769 (1998)

1998
[127]

Proville and S

L. Proville and S. Aubry, Physica D: Nonlinear Phe- nomena113, 307 (1998)

1998
[128]

Proville and S

L. Proville and S. Aubry, The European Physical Jour- nal B11, 41 (1999)

1999
[129]

K. A. Chao, J. Spa lek, and A. M. Ole´ s, Physical Review B18, 3453 (1978)

1978
[130]

Micnas, J

R. Micnas, J. Ranninger, and S. Robaszkiewicz, Reviews of Modern Physics62, 113 (1990)

1990
[131]

Emery and S

V. Emery and S. Kivelson, Physica C: Superconductiv- ity209, 597 (1993)

1993
[132]

Karakuzu, A

S. Karakuzu, A. Tanjaroon Ly, P. Mai, J. Neuhaus, T. A. Maier, and S. Johnston, Communications Physics 5, 311 (2022)

2022
[133]

The choice of ∆Ne = 2 preserves the system in the spin-singlet sector and thereby mitigates finite-size effects

Here we calculate it using a central second-order finite- difference scheme,∂ 2 ν f∼E(N e + 2) +E(N e −2)− 2E(Ne). The choice of ∆Ne = 2 preserves the system in the spin-singlet sector and thereby mitigates finite-size effects
[134]

Bak, Reports on Progress in Physics45, 587 (1982)

P. Bak, Reports on Progress in Physics45, 587 (1982)

1982
[135]

H. J. Schulz, Physical Review B22, 5274 (1980)

1980
[136]

N. C. Costa, T. Blommel, W.-T. Chiu, G. Batrouni, and R. T. Scalettar, Physical Review Letters120, 187003 (2018)

2018
[137]

Pavarini, I

E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Physical Review Letters87, 047003 (2001)

2001
[138]

H. Q. Lin and J. E. Hirsch, Physical Review B35, 3359 (1987)

1987
[139]

Zhang, Z

W. Zhang, Z. Zeng, and T. Shi, Physical Review A111, 043317 (2025)

2025
[140]

Y.-F. Qu, M. Stefanini, T. Shi, T. Esslinger, S. Gopalakrishnan, J. Marino, and E. Demler, Physi- cal Review B111, 155113 (2025)

2025
[141]

Bender, P

J. Bender, P. Emonts, and J. I. Cirac, Physical Review Research5, 043128 (2023)

2023
[142]

Carleo and M

G. Carleo and M. Troyer, Science355, 602 (2017)

2017
[143]

Brockt, F

C. Brockt, F. Dorfner, L. Vidmar, F. Heidrich-Meisner, and E. Jeckelmann, Physical Review B92, 241106 (2015)

2015
[144]

M. T. Fishman and S. R. White, Physical Review B92, 075132 (2015), arXiv:1504.07701 [cond-mat]

2015 arXiv
[145]

Fishman, S

M. Fishman, S. R. White, and E. M. Stoudenmire, Sci- Post Phys. Codebases , 4 (2022)

2022
[146]

A. H. Al-Mohy and N. J. Higham, SIAM Jour- nal on Scientific Computing33, 488 (2011), https://doi.org/10.1137/100788860

2011 doi
[147]

Jin, R.-Y

H.-K. Jin, R.-Y. Sun, Y. Zhou, and H.-H. Tu, Physical Review B105, L081101 (2022)

2022
[148]

G. F. Bertsch and L. M. Robledo, Physical Review Let- ters108, 042505 (2012)

2012
[149]

McLachlan, Molecular Physics8, 39 (1964), https://doi.org/10.1080/00268976400100041

A. McLachlan, Molecular Physics8, 39 (1964), https://doi.org/10.1080/00268976400100041

1964 doi
[150]

Z.-X. Li, M. L. Cohen, and D.-H. Lee, Physical Review B100, 245105 (2019)

2019

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.

[1] [1]

Half-filled case For the half-filled HH model atω= 5t, we present the phase diagram in Fig. 3. The anti-adiabatic limit pro- vides a theoretical baseline (dotted line,u= 2λ). In this limit, the system maps onto an effective Hubbard model (U→U−2λt), predicting a quantum phase transition atu= 2λ. This line separates the (π, π)-AFM phase foru >2λ[104] from d...

work page

[2] [2]

arrested

1 8 hole doped case Upon doping the half-filled parent compound top= 1/8, the system enters the regime of intertwined orders TABLE II. Renormalized polaronic hopping amplitudes for a central bulk site along theu= 2λline. (u, λ) ˜tc,c+ ˆα a (0,0) (2,1) (4,2) (8,4) ˆα= ˆx 1.000 0.930 0.862 0.730 ˆα= ˆy 1.000 0.929 0.861 0.730 a Here, ˜tc,c+ ˆαdenotes the dr...

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[3] [3]

Brief Introduction to Fermionic Gaussian MPS This section outlines the construction of the MPS rep- resentation for fermionic Gaussian states [71, 136], which we employ to generate robust initial seeds for the opti- mization workflow. A fermionic Gaussian state [71] is defined as|Ψ f ⟩=U f |0⟩f, where|0⟩ f is the vacuum and Uf is a unitary transformation ...

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[4] [4]

The derivation is based on the McLachlan variational principle [141]

Derivation of EoMs This appendix details the derivation of the EoMs for the NGS parameters. The derivation is based on the McLachlan variational principle [141]. We project ITE, dτ |Ψ(τ)⟩=−(H− ⟨H⟩)|Ψ(τ)⟩,(A3) onto the tangent space of the variational manifold [71, 72]: ⟨Vν|dτ ξµ|Vµ⟩=−⟨V ν|δH|Ψ⟩,(A4) where|V µ⟩=Q Ψ∂µ|Ψ⟩= (1− |Ψ⟩⟨Ψ|) ∂|Ψ⟩ ∂ξ µ is the tangen...

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[5] [5]

(3)] are denoted by⟨· · · ⟩, while those with respect to the electronic wave function|Ψ e⟩are denoted by⟨· · · ⟩ e

Expectation values on the full variational wavefunction As mentioned in the main text, expectation values with respect to the full variational state [Eq. (3)] are denoted by⟨· · · ⟩, while those with respect to the electronic wave function|Ψ e⟩are denoted by⟨· · · ⟩ e. We express key ob- servables⟨· · · ⟩in terms of⟨· · · ⟩ e and the associated dress- ing...

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[6] [6]

We estimate the residual double occupancy by applying second-order perturbation theory to the effective electronic Hamiltonian derived in Eq

Perturbative Estimation on the Double Occupancy In the regime of strong effective repulsion (U−2λt≫ t), the ground state is dominated by configurations with- out double occupancy. We estimate the residual double occupancy by applying second-order perturbation theory to the effective electronic Hamiltonian derived in Eq. (6). The first-order correction to ...

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[7] [7]

2(a), the 1D generalized HH model exhibits PS over the doping level from 0% to 15%

Maxwell construction of 1D generalized HH model with NGS-MPS ∗ As reported in Fig. 2(a), the 1D generalized HH model exhibits PS over the doping level from 0% to 15%. Here, we independently verify the PS via a Maxwell construc- tion [84]. Under PBC, we enforce a uniform electron density and calculate the ground-state energy using the restricted NGS-MPS∗ a...

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[8] [8]

Benchmarks against NGSED in 2D torus Figure 9 benchmarks the NGS-MPS method against NGS-ED on a 4×4 torus. We examine various physi- cal observables, including pairing correlations, structure factors, and local moments: P0 = 1 N X i,j ⟨c† i↑c† i↓cj↓cj↑⟩,(B1a) Sσ(q) = 1 N X i,j ⟨Si ·S j⟩eiq·(i−j),(B1b) Sρ(q) = 1 N X i,j ⟨ninj⟩eiq·(i−j),(B1c) ⟨m2 z⟩= 1 N X ...

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