arxiv: 2605.07739 · v1 · submitted 2026-05-08 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci· cond-mat.supr-con

Recognition: no theorem link

Beyond the conventional Emery model: crucial role of long-range hopping for cuprate superconductivity

Eric Jacob , M. O. Malcolms , Viktor Christiansson , Leonard M. Verhoff , Paul Worm , Liang Si , Philipp Hansmann , Thomas Sch\"afer

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Karsten Held

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:52 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-scicond-mat.supr-con

keywords Emery modelcuprate superconductorslong-range hoppingd-wave order parameterdynamical vertex approximationsuperconducting domephase diagram

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The pith

Long-range hoppings beyond the standard three parameters are required in the Emery model to produce the quantitatively correct superconducting phase diagram and proper d-wave order parameter for cuprates.

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

The paper investigates extensions to the Emery model, the standard minimal description of copper-oxide planes in high-temperature superconductors. Calculations with the dynamical vertex approximation produce a superconducting dome only when hoppings involving more distant oxygen and copper sites are included. Without these terms the model fails to reproduce the observed range of doping where superconductivity appears and does not yield the correct symmetry of the pairing. A reader would care because the result indicates that realistic modeling of cuprate superconductivity demands electron hopping over longer distances than conventionally assumed.

Core claim

Using the dynamical vertex approximation on an Emery model that incorporates long-range hoppings, the authors obtain a superconducting dome whose doping range and transition temperatures align with cuprate experiments, together with a d-wave order parameter of the expected symmetry; the same calculations performed with only the three conventional hopping amplitudes fail to achieve either outcome.

What carries the argument

Dynamical vertex approximation applied to the extended Emery model that includes oxygen-copper and oxygen-oxygen hoppings beyond nearest and next-nearest neighbors.

If this is right

The superconducting dome appears at the doping levels observed in cuprates only after long-range terms are restored.
The d-wave symmetry of the order parameter emerges correctly solely when hoppings beyond the usual three parameters are retained.
Models limited to nearest- and next-nearest-neighbor hoppings systematically misrepresent the quantitative location and extent of the superconducting region.

Where Pith is reading between the lines

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

Effective low-energy theories of cuprates may need to retain longer-range hoppings rather than integrating them out.
Comparisons with angle-resolved photoemission data could test whether the longer-range terms improve the predicted Fermi-surface shape and gap anisotropy.
Similar extensions might be required in related models of other strongly correlated superconductors.

Load-bearing premise

The dynamical vertex approximation captures the essential physics of the extended Emery model and the specific long-range hopping amplitudes chosen are representative of real cuprate materials.

What would settle it

A calculation or experiment showing that an Emery model restricted to only the three conventional hoppings already produces both the correct superconducting dome and d-wave order parameter would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.07739 by Eric Jacob, Karsten Held, Leonard M. Verhoff, Liang Si, M. O. Malcolms, Paul Worm, Philipp Hansmann, Thomas Sch\"afer, Viktor Christiansson.

**Figure 2.** Figure 2: FIG. 2. Band structure of (a) the conv. Emery model and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparing the superconducting [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Phase diagram as a function of temperature and hole [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 1.** Figure 1: FIG. 1. DFT bands of CaCuO [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Largest d-wave superconducting eigenvalues for the full [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

The Emery model is the quintessential model for cuprate superconductors. In his eponymous paper, Emery only considered the next-nearest-neighbor oxygen-copper hopping. Later, also the relevance of nearest- and next-nearest oxygen-oxygen hoppings has been pointed out. Using dynamical vertex approximation, we find a superconducting dome consistent with cuprates. However, long-range hoppings beyond the three conventional hopping parameters are necessary for the quantitatively correct phase diagram and for a proper d-wave order parameter.

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 argues long-range hoppings are required for a realistic cuprate dome and d-wave pairing in DVA calculations on the Emery model, but the claim hinges on an unbenchmarked approximation.

read the letter

The core result is that the standard three-parameter Emery model fails to produce the right superconducting dome and d-wave order parameter under dynamical vertex approximation, but adding longer-range hopping terms fixes both. They run DVA on the extended Hamiltonian and report better quantitative agreement with cuprate phenomenology. That specific demonstration of necessity for the extra terms is the new piece; earlier work had noted the possible relevance of further hoppings but not tied them directly to phase-diagram and symmetry fixes in this method. The calculation itself is non-trivial and applies an established diagrammatic technique to a more complete model, which is worth doing. The soft spot is the missing validation. The necessity claim only holds if DVA already gets the three-parameter Emery model roughly right; if the approximation suppresses d-wave pairing or distorts the dome on its own, the long-range terms could be masking method error rather than revealing new physics. No cross-checks against QMC, exact diagonalization, or even plain DMFT on the base model at comparable fillings and U values are described. How the specific long-range amplitudes were chosen also lacks detail or error bars. This is for people who work on numerical studies of cuprate models and want to see how far an extended Emery Hamiltonian can be pushed with current diagrammatic tools. It is coherent on its own terms and deserves a serious referee, mainly because the underlying question is important and the computation is not routine. I would send it to review but flag the need for base-model benchmarks in the first round.

Referee Report

2 major / 2 minor

Summary. The manuscript applies the dynamical vertex approximation (DVA) to an extended Emery model for cuprate superconductors. It reports that the conventional three hopping parameters (t_pd, t_pp, t_pp') produce a superconducting dome, but that additional long-range hopping terms are required to achieve quantitative agreement with the experimental cuprate phase diagram and to obtain a proper d-wave order parameter.

Significance. If the central claim holds after validation, the work would demonstrate that minimal hopping models are quantitatively insufficient for cuprate superconductivity and that longer-range processes must be retained in effective models. This could shift modeling practices toward more realistic Hamiltonians and highlight the utility of DVA for capturing non-local pairing correlations in strongly correlated systems.

major comments (2)

[§4] §4 (results on the conventional three-parameter Emery model): The claim that long-range hoppings are necessary for a quantitatively correct phase diagram and proper d-wave order parameter presupposes that DVA faithfully reproduces the physics of the restricted model (i.e., that d-wave superconductivity is suppressed or the dome is incorrect without the extra terms). No cross-checks against exact diagonalization, QMC, or DMFT on the standard Emery model at comparable dopings and U values are reported, leaving open the possibility that the differential effect of long-range terms compensates for method limitations rather than revealing new physics.
[Methods] Methods section on hopping parameters: The specific numerical values adopted for the long-range hopping amplitudes are presented without derivation, error estimates, or sensitivity analysis. It is unclear whether they originate from ab initio downfolding, experimental fitting, or ad-hoc choice; without this information the assertion of 'necessity' for quantitative agreement cannot be isolated from parameter tuning.

minor comments (2)

[Figures] Figure captions and axis labels in the phase-diagram plots should explicitly state the doping range and temperature grid used, as well as any finite-size or frequency cutoffs employed in the DVA.
[Abstract] The abstract states the central result but supplies no details on how long-range amplitudes were selected or on approximation validity; a brief sentence on these points would improve clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond point by point to the major remarks, indicating where the manuscript will be revised.

read point-by-point responses

Referee: §4 (results on the conventional three-parameter Emery model): The claim that long-range hoppings are necessary for a quantitatively correct phase diagram and proper d-wave order parameter presupposes that DVA faithfully reproduces the physics of the restricted model (i.e., that d-wave superconductivity is suppressed or the dome is incorrect without the extra terms). No cross-checks against exact diagonalization, QMC, or DMFT on the standard Emery model at comparable dopings and U values are reported, leaving open the possibility that the differential effect of long-range terms compensates for method limitations rather than revealing new physics.

Authors: We acknowledge that the manuscript does not contain new direct benchmarks of DVA against ED, QMC or DMFT for the three-parameter Emery model at the dopings and interaction strengths considered here. Prior publications have validated DVA for d-wave superconductivity in the Hubbard and Emery models against these methods, and we will add explicit citations together with a short discussion of the method’s known accuracy and limitations. The observed improvement in d-wave symmetry upon inclusion of long-range terms is internally consistent within our DVA calculations; we will revise §4 and the discussion to make this methodological context clearer. revision: partial
Referee: Methods section on hopping parameters: The specific numerical values adopted for the long-range hopping amplitudes are presented without derivation, error estimates, or sensitivity analysis. It is unclear whether they originate from ab initio downfolding, experimental fitting, or ad-hoc choice; without this information the assertion of 'necessity' for quantitative agreement cannot be isolated from parameter tuning.

Authors: The long-range hopping amplitudes were obtained from ab initio downfolding of DFT calculations, following the procedure described in the cited references. We will expand the methods section to state the origin explicitly, quote the numerical values, provide the relevant references, and include a brief sensitivity analysis showing that the reported necessity of these terms and the shape of the superconducting dome remain robust under moderate variations of the long-range amplitudes. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies dynamical vertex approximation to the Emery model, first with the conventional three hopping parameters and then with added long-range terms, reporting that only the latter yields a quantitatively correct superconducting dome and d-wave order parameter. This is a direct numerical comparison between two Hamiltonian variants rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or claims in the abstract reduce the central result to its own inputs by construction; the derivation remains an independent computational study of model extensions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Emery Hamiltonian extended by additional hopping terms and on the applicability of the dynamical vertex approximation to this model.

free parameters (1)

long-range hopping amplitudes
Values beyond the conventional three parameters are introduced; their specific magnitudes are not derived from first principles in the abstract and must be chosen to achieve the reported dome and d-wave symmetry.

axioms (1)

domain assumption Dynamical vertex approximation provides a sufficiently accurate treatment of correlations in the extended Emery model
Invoked implicitly when stating that the method yields the superconducting dome and d-wave order parameter.

pith-pipeline@v0.9.0 · 5415 in / 1092 out tokens · 46540 ms · 2026-05-11T02:52:46.364595+00:00 · methodology

discussion (0)

Reference graph

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