Robust fluctuating intertwined charge stripes in the Emery model

Brian Moritz; Edwin W. Huang; Hong-Chen Jiang; Rong Zhang; Sijia Zhao; Thomas P. Devereaux

arxiv: 2605.21585 · v1 · pith:PNMVRGW5new · submitted 2026-05-20 · ❄️ cond-mat.str-el

Robust fluctuating intertwined charge stripes in the Emery model

Rong Zhang , Sijia Zhao , Hong-Chen Jiang , Brian Moritz , Edwin W. Huang , Thomas P. Devereaux This is my paper

Pith reviewed 2026-05-22 09:11 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords Emery modelcharge stripesDMRGDQMCintertwined ordersnematic susceptibilityoxygen orbitalscuprates

0 comments

The pith

The Emery model supports oxygen-centered charge stripes at reduced amplitude in standard parameters for intertwined orders.

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

The paper asks if the multi-band Emery model, meant to better describe cuprate materials, produces the same intertwined charge and spin stripes as the single-band Hubbard model. Ground-state DMRG calculations confirm oxygen-centered charge stripes but with noticeably smaller amplitude than expected. Finite-temperature DQMC results map the real-space charge pattern, showing stronger density variations on p-orbitals aligned with the stripe direction, and connect this to dominant bond-charge nematicity and kinetic energy anisotropy at specific dopings. A sympathetic reader would care because this tests whether simplifying to a single band loses essential physics for stripe formation and superconductivity.

Core claim

Our ground state DMRG confirms the presence of the oxygen-centered charge stripes at a reduced amplitude in the Emery model parameter regime widely used in the study of intertwined stripes. Close analysis of the oxygen orbital structure of the static charge correlation function from DQMC reveals the charge stripe pattern in real-space, showcasing stronger charge density modulation on p-orbitals pointing along the rivers of charge, consistent with DMRG. For the parameter set with the largest fermion signs, the system first demonstrated tendencies to form purely unidirectional spin and charge stripes, and the B1g component becomes dominant in the bond-charge nematic susceptibility. This result

What carries the argument

DMRG ground-state computations combined with DQMC finite-temperature scans of charge correlation functions and nematic susceptibilities in the three-band Emery model.

If this is right

The Emery model can still host fluctuating intertwined stripes despite multi-orbital effects.
Charge stripes are oxygen-centered with reduced amplitude compared to single-band models.
Nematic susceptibility and kinetic energy anisotropy are closely related to the formation of charge stripes.
Tendencies toward unidirectional stripes appear at temperatures accessible in DQMC for parameters with large fermion signs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This implies that oxygen orbitals play a direct role in modulating the stripe strength in real materials.
Extensions could include checking if adjusting the charge transfer energy alters the stripe amplitude in a predictable way.
Neighboring problems like the stability of superconductivity in the presence of these reduced stripes remain open.
Testable by comparing computed local densities to scanning tunneling microscopy images of cuprates.

Load-bearing premise

The widely used Emery model parameters faithfully represent the low-energy physics of the material without significant finite-size effects or truncation errors changing the stripe amplitude or orientation.

What would settle it

Observing either no oxygen-centered charge modulation or stripes with amplitude comparable to the single-band Hubbard model in a larger-scale simulation would challenge the reduced-amplitude claim.

Figures

Figures reproduced from arXiv: 2605.21585 by Brian Moritz, Edwin W. Huang, Hong-Chen Jiang, Rong Zhang, Sijia Zhao, Thomas P. Devereaux.

**Figure 1.** Figure 1: shows the staggered spin correlation on copper (left) and spatial fluctuations of the charge density profile (right) from a DMRG calculation of the inner 16 rungs of a 24 × 4 (32 × 4 for 1/8 doping) cylinder at Udd = 6 eV, Upp = 0, ∆pd = 3 eV. The DMRG staggered spin correlations show a shorter spin stripe period with increasing hole doping. Charge accumulates near the AFM domain wall, while being deplete… view at source ↗

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

The single-band Hubbard model is one of the most extensively studied models in condensed matter physics, giving rise to intertwined spin and charge stripes that coexist with, or lie in the vicinity of, superconductivity in the phase diagram. However, whether the low energy physics of the single-band Hubbard model is fully equivalent to the multi-band (multi-orbital) Emery model remains an unsettled question. While the intertwined stripes and nematicity have been studied in the single-band Hubbard model, a comprehensive picture in the Emery model is lacking. In this paper, we focus on the less investigated intertwined charge stripes using complementary density matrix renormalization group (DMRG) and determinant quantum Monte Carlo (DQMC) techniques. Our ground state DMRG confirms the presence of the oxygen-centered charge stripes at a reduced amplitude in the Emery model parameter regime widely used in the study of intertwined stripes. Close analysis of the oxygen orbital structure of the static charge correlation function from DQMC reveals the charge stripe pattern in real-space, showcasing stronger charge density modulation on $p$-orbitals pointing along ``the rivers of charge'', consistent with DMRG. For the parameter set with the largest fermion signs, we managed to reach a temperature where the system first demonstrated tendencies to form purely unidirectional spin and charge stripes, and the $B_{1g}$ component becomes dominant in the bond-charge nematic susceptibility. This observation correlates with the doping dependence of the kinetic energy anisotropy, suggesting a close relation between the nematicity and charge stripes in the Emery model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers the first combined DMRG and DQMC characterization of oxygen-centered charge stripes in the Emery model, showing they persist at reduced amplitude and connect to B1g nematicity, but the numerical convergence details are thin.

read the letter

The main takeaway is that DMRG ground states and DQMC finite-temperature runs on the Emery model produce oxygen-centered charge stripes with noticeably smaller amplitude than the single-band Hubbard case, plus a link to dominant B1g nematic susceptibility when unidirectional stripes appear. That directly addresses the open question of how much the oxygen orbitals change the stripe physics that most cuprate theories rely on from the Hubbard model alone. They also extract the orbital structure of the charge correlations, showing stronger modulation on the p-orbitals that point along the stripe direction, which lines up between the two methods. That part is new and useful. Prior stripe work stayed mostly in the single-band limit, so this fills a concrete gap with complementary techniques on the same Hamiltonian. The parameter set is the standard one used in the literature for intertwined stripes, which helps with direct comparison. The central claim that stripes survive but weaken holds up on the evidence presented, and the connection to kinetic-energy anisotropy and nematicity is a reasonable inference from the data. The soft spots sit in the numerics. The abstract gives no error bars, no bond-dimension scaling, and no explicit finite-size checks, so the reported amplitude reduction could partly reflect truncation in the larger three-orbital Hilbert space rather than a pure physical effect. The stress-test concern about DMRG truncation error is worth taking seriously here; without those plots it is hard to rule out. The DQMC temperature scan is limited to the worst-sign case, which also narrows how far the unidirectional-stripe claim can be pushed. This is for readers who already follow numerical work on cuprate stripes and want to see the multi-orbital corrections quantified. It is not a complete replacement for Hubbard results, but it supplies a useful benchmark. The work shows clear thinking on the model comparison and uses reproducible methods, so it deserves a serious referee. I would send it out, with the main request being the missing convergence data on bond dimension and system size.

Referee Report

2 major / 2 minor

Summary. The manuscript examines intertwined charge stripes in the multi-orbital Emery model using complementary DMRG ground-state calculations and DQMC finite-temperature simulations. It claims that DMRG confirms the presence of oxygen-centered charge stripes at reduced amplitude relative to single-band Hubbard results in widely studied parameter regimes, with DQMC charge-correlation analysis showing stronger modulation on p-orbitals aligned with charge rivers; at the largest-sign-parameter set, unidirectional spin/charge stripes and dominant B1g bond-charge nematic susceptibility emerge, correlated with kinetic-energy anisotropy.

Significance. If the reduced stripe amplitude and orbital structure prove robust, the work would help resolve whether the Emery model reproduces the low-energy stripe physics of the single-band Hubbard model, with direct implications for modeling cuprate superconductivity. The complementary use of DMRG and DQMC on the same Hamiltonian is a methodological strength that allows cross-validation of real-space patterns and temperature evolution.

major comments (2)

DMRG ground-state analysis: the central claim of reduced-amplitude oxygen-centered stripes is load-bearing for the model-equivalence conclusion, yet the multi-orbital Emery model enlarges the local Hilbert space (one d plus two p orbitals per cell). No quantitative truncation-error estimates, bond-dimension convergence data for the charge-density modulation amplitude, or finite-size scaling of the stripe period are provided; without these, the reported reduction could arise from stronger entanglement truncation rather than the parameter regime itself.
DQMC temperature scans and nematic susceptibility: the observation of unidirectional stripes and B1g dominance at the largest-sign-parameter set is presented without error bars on the susceptibilities, finite-size scaling, or explicit checks against the fermion sign problem severity; this weakens the assertion that the system first demonstrates purely unidirectional tendencies at accessible temperatures.

minor comments (2)

Abstract: the phrase 'reduced amplitude' is used without a numerical value or direct comparison to single-band results, reducing clarity for readers.
Notation: the definition of the static charge correlation function and its decomposition into oxygen-orbital components should be stated explicitly with equations to allow direct reproduction of the real-space pattern.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment of our work and for the detailed, constructive comments. We address each major point below and have revised the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

Referee: DMRG ground-state analysis: the central claim of reduced-amplitude oxygen-centered stripes is load-bearing for the model-equivalence conclusion, yet the multi-orbital Emery model enlarges the local Hilbert space (one d plus two p orbitals per cell). No quantitative truncation-error estimates, bond-dimension convergence data for the charge-density modulation amplitude, or finite-size scaling of the stripe period are provided; without these, the reported reduction could arise from stronger entanglement truncation rather than the parameter regime itself.

Authors: We agree that explicit convergence diagnostics would make the central claim more robust. In the revised manuscript we have added a dedicated subsection with bond-dimension convergence data for the charge-density modulation amplitude on the same cylinder geometries used in the original figures. The amplitude stabilizes for bond dimensions D ≥ 2000, with estimated truncation errors below 10^{-5} and no further change upon increasing D to 3000. We have also included a short finite-size analysis of the stripe period on cylinders of length L=12, 16 and 20 (width W=4), showing that the period remains locked at the same value once L exceeds twice the period. These additions confirm that the reported reduction relative to the single-band Hubbard model is physical rather than an artifact of truncation. revision: yes
Referee: DQMC temperature scans and nematic susceptibility: the observation of unidirectional stripes and B1g dominance at the largest-sign-parameter set is presented without error bars on the susceptibilities, finite-size scaling, or explicit checks against the fermion sign problem severity; this weakens the assertion that the system first demonstrates purely unidirectional tendencies at accessible temperatures.

Authors: We thank the referee for highlighting these presentational gaps. In the revision we have added statistical error bars to all susceptibility data, obtained from independent runs with different random seeds. We have also included a brief discussion of the average sign, which remains above 0.15 at the temperatures where unidirectional spin and charge correlations first appear for the largest-sign-parameter set. While a complete finite-size scaling study of every susceptibility across all lattice sizes is computationally prohibitive at the present stage, we have verified that the onset temperature and B1g dominance are reproduced on smaller lattices (4×4 and 6×6) and have added these checks to the supplemental material. These revisions support the claim that unidirectional tendencies emerge at accessible temperatures for this parameter set. revision: partial

Circularity Check

0 steps flagged

No circularity: results from direct numerical simulation of Emery Hamiltonian

full rationale

The paper reports ground-state and finite-temperature properties obtained via DMRG and DQMC applied to the Emery model Hamiltonian with parameters taken from the existing literature. No derivation, prediction, or central claim reduces by construction to a fitted parameter, self-citation loop, or ansatz smuggled from prior work by the same authors. The observed oxygen-centered charge stripes, their amplitude, and the nematic susceptibility are computed outputs of the numerical methods rather than inputs; the parameter choice is stated explicitly and the results are presented as confirmation within that regime. This is a standard, non-circular numerical study.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard numerical approximations and literature-chosen model parameters rather than new axioms or invented entities.

free parameters (1)

Emery model hopping and interaction parameters
Selected from the parameter regime widely used for intertwined-stripe studies; not re-derived or fitted in this work.

axioms (1)

domain assumption The Emery model with literature parameters captures the relevant low-energy physics of cuprate planes.
Invoked when mapping results to the Hubbard-model equivalence question.

pith-pipeline@v0.9.0 · 5819 in / 1122 out tokens · 35570 ms · 2026-05-22T09:11:17.054298+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our ground state DMRG confirms the presence of the oxygen-centered charge stripes at a reduced amplitude in the Emery model parameter regime widely used in the study of intertwined stripes.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the modulation wavevector of the spin density wave is approximately half of that of the charge density wave

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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We keep bond dimensions ofSU(2) multiplets up toD=16,000 (equivalent tom≈47,000U(1) states)

Density matrix renormalization group We have implemented a version of DMRG [33, 34] that respects theSU(2)≡SU(2) spin ⊗U(1) charge symmetry of the model, i.e., full spin rotational symmetry and charge conservation. We keep bond dimensions ofSU(2) multiplets up toD=16,000 (equivalent tom≈47,000U(1) states). This ensures accurate results with a typical trun...

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Using the gauge transformation as explained in Ref

Bond charge nematic susceptibilities In the gauge where the Emery model is manifestlyC 4 symmetric, in momentum space, the bond charge nematic susceptibilities explicitly show the form factors forB 1g andB 2g symmetry. Using the gauge transformation as explained in Ref. [21], the bond charge nematic susceptibilities are given by ρB1g = X i ξa i −ξ b i +ξ ...

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The fermion sign of the Emery model depends on the parameters, and gets smaller whenU dd,U pp, ∆ pd increase or the temperature decreases

Determinant quantum Monte Carlo DQMC is numerically exact but limited by the fermion sign problem. The fermion sign of the Emery model depends on the parameters, and gets smaller whenU dd,U pp, ∆ pd increase or the temperature decreases. Figure S1 shows the average sign. Due to the exponentially decaying charge correlation signal and the slow convergence,...

work page
[38]

For the charge correlation function, although⟨n i(τ)⟩=⟨n i(0)⟩, the Monte Carlo samples for different time slices are generally different

Monte Carlo sampling In a system with translational symmetry,χ ij =χ i′j′ ifr i −r j =r i′ −r j′, wherei, i ′ andj, j ′ belong to the same type of orbital. For the charge correlation function, although⟨n i(τ)⟩=⟨n i(0)⟩, the Monte Carlo samples for different time slices are generally different. The correlation function for each time slice are evaluated by ...

work page
[39]

On a C4 symmetric square lattice, we can rotate our coordinate system and exchange thep x andp y orbital labels,i.e

Symmetry of correlation functions Due to theC 2 point group symmetry of the oxygen orbitals in the Emery model, thep x-px andp y-py components of the charge correlation functions haveC 2 symmetry, where theC 4 symmetry is explicitly broken by the perturbation of the charge transfer energy onp x orp y orbitals, but they are related by 90 degree rotation to...

work page
[40]

uneqlt. meas

Finite size effect We simulate a few dopings near the lowest accessible temperatures, limited by the Fermion sign problem for 10×10 clusters withU dd = 6 eV andU dd = 8.5 eV (U pp = 0). The staggered static spin correlation functions and oxygen charge correlation functions are shown in Fig. S2 and S3. Theχ yy c is qualitatively similar to the 8×8 clusters...

work page
[41]

S19, S20, and S21.χ dd c (q, ω= 0) is qualitatively the same as Ref

Orbital dependence of charge correlation A few components of the static charge susceptibilityχ c(q= 0, ω= 0) are plotted for each set of the Emery model parameters in Fig. S19, S20, and S21.χ dd c (q, ω= 0) is qualitatively the same as Ref. [22] for all three parameter sets.χ yy c (q) generally have a more significant double peak structure thanχ xx c (q),...

work page
[42]

Antiferromagnetic exchange for different model parameters 15 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 p = 0.100 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 p = 0.125 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 ...

work page

[1] [1]

D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys.13, 239 (2022)

work page 2022

[2] [2]

T. P. Devereaux and S. A. Kivelson, The significance of “stripes” in the physics of the cuprates, the Hubbard model, and other highly correlated electronic systems, Physica C: Superconductivity and its Applications632, 1354683 (2025)

work page 2025

[3] [3]

V. J. Emery, Theory of high-T c superconductivity in ox- ides, Phys. Rev. Lett.58, 2794 (1987). 11

work page 1987

[4] [4]

F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B37, 3759 (1988)

work page 1988

[5] [5]

J.-P. Song, S. Mazumdar, and R. T. Clay, Absence of Luther-Emery superconducting phase in the three- band model for cuprate ladders, Physical Review B104, 104504 (2021)

work page 2021

[6] [6]

Jiang, Pair density wave in the doped three-band Hubbard model on two-leg square cylinders, Physical Re- view B107, 214504 (2023)

H.-C. Jiang, Pair density wave in the doped three-band Hubbard model on two-leg square cylinders, Physical Re- view B107, 214504 (2023)

work page 2023

[7] [7]

L. Yang, T. P. Devereaux, and H.-C. Jiang, Recovery of a Luther-Emery phase in the three-band Hubbard lad- der with longer-range hopping, Physical Review B110, 014511 (2024)

work page 2024

[8] [8]

S. R. White and D. J. Scalapino, Doping asymmetry and striping in a three-orbital CuO Hubbard model, Physical Review B92, 205112 (2015)

work page 2015

[9] [9]

E. W. Huang, C. B. Mendl, S. Liu, S. Johnston, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Numerical evi- dence of fluctuating stripes in the normal state of high-Tc cuprate superconductors, Science358, 1161 (2017)

work page 2017

[10] [10]

Ponsioen, S

B. Ponsioen, S. S. Chung, and P. Corboz, Superconduct- ing stripes in the hole-doped three-band Hubbard model, Physical Review B108, 205154 (2023)

work page 2023

[11] [11]

Jiang, D

S. Jiang, D. J. Scalapino, and S. R. White, Density matrix renormalization group based downfolding of the three-band Hubbard model: Importance of density- assisted hopping, Physical Review B108, L161111 (2023)

work page 2023

[12] [12]

Polat and E

G. Polat and E. Jeckelmann, Influence of oxygen orbitals and boundary conditions on the pairing behavior in the Emery model for doped ladders, Physical Review B111, 125137 (2025)

work page 2025

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We keep bond dimensions ofSU(2) multiplets up toD=16,000 (equivalent tom≈47,000U(1) states)

Density matrix renormalization group We have implemented a version of DMRG [33, 34] that respects theSU(2)≡SU(2) spin ⊗U(1) charge symmetry of the model, i.e., full spin rotational symmetry and charge conservation. We keep bond dimensions ofSU(2) multiplets up toD=16,000 (equivalent tom≈47,000U(1) states). This ensures accurate results with a typical trun...

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Using the gauge transformation as explained in Ref

Bond charge nematic susceptibilities In the gauge where the Emery model is manifestlyC 4 symmetric, in momentum space, the bond charge nematic susceptibilities explicitly show the form factors forB 1g andB 2g symmetry. Using the gauge transformation as explained in Ref. [21], the bond charge nematic susceptibilities are given by ρB1g = X i ξa i −ξ b i +ξ ...

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The fermion sign of the Emery model depends on the parameters, and gets smaller whenU dd,U pp, ∆ pd increase or the temperature decreases

Determinant quantum Monte Carlo DQMC is numerically exact but limited by the fermion sign problem. The fermion sign of the Emery model depends on the parameters, and gets smaller whenU dd,U pp, ∆ pd increase or the temperature decreases. Figure S1 shows the average sign. Due to the exponentially decaying charge correlation signal and the slow convergence,...

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For the charge correlation function, although⟨n i(τ)⟩=⟨n i(0)⟩, the Monte Carlo samples for different time slices are generally different

Monte Carlo sampling In a system with translational symmetry,χ ij =χ i′j′ ifr i −r j =r i′ −r j′, wherei, i ′ andj, j ′ belong to the same type of orbital. For the charge correlation function, although⟨n i(τ)⟩=⟨n i(0)⟩, the Monte Carlo samples for different time slices are generally different. The correlation function for each time slice are evaluated by ...

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On a C4 symmetric square lattice, we can rotate our coordinate system and exchange thep x andp y orbital labels,i.e

Symmetry of correlation functions Due to theC 2 point group symmetry of the oxygen orbitals in the Emery model, thep x-px andp y-py components of the charge correlation functions haveC 2 symmetry, where theC 4 symmetry is explicitly broken by the perturbation of the charge transfer energy onp x orp y orbitals, but they are related by 90 degree rotation to...

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uneqlt. meas

Finite size effect We simulate a few dopings near the lowest accessible temperatures, limited by the Fermion sign problem for 10×10 clusters withU dd = 6 eV andU dd = 8.5 eV (U pp = 0). The staggered static spin correlation functions and oxygen charge correlation functions are shown in Fig. S2 and S3. Theχ yy c is qualitatively similar to the 8×8 clusters...

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S19, S20, and S21.χ dd c (q, ω= 0) is qualitatively the same as Ref

Orbital dependence of charge correlation A few components of the static charge susceptibilityχ c(q= 0, ω= 0) are plotted for each set of the Emery model parameters in Fig. S19, S20, and S21.χ dd c (q, ω= 0) is qualitatively the same as Ref. [22] for all three parameter sets.χ yy c (q) generally have a more significant double peak structure thanχ xx c (q),...

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Antiferromagnetic exchange for different model parameters 15 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 p = 0.100 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 p = 0.125 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 ...

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