Pseudogap and Condensation in Cuprate Superconductors from NMR Shifts

Abigail Lee; Juergen Haase

arxiv: 2604.19215 · v1 · submitted 2026-04-21 · ❄️ cond-mat.supr-con

Pseudogap and Condensation in Cuprate Superconductors from NMR Shifts

Abigail Lee , Juergen Haase This is my paper

Pith reviewed 2026-05-10 01:55 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords NMR shiftscuprate superconductorspseudogaphyperfine couplingsspin singlet pairingcondensationoptimal Tc

0 comments

The pith

NMR shifts in cuprates separate into A and B components whose coupling sets the pseudogap temperature and guides spin-singlet condensation.

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

The paper shows that planar copper NMR shifts disentangle into two components based on the symmetry of hyperfine couplings: anisotropic A and isotropic B. Metallic B-spins form above the pseudogap temperature, which is shared with A-spins and measures the A-B coupling strength that suppresses shifts without affecting relaxation. With further doping the pseudogap falls while B-spins increase faster than A-spins; above x=0.20 both become superconducting metals that vanish at Tc. Spin-singlet pairing follows three simple rules involving A and B, and optimal Tc requires a special match between them in systems that still show a pseudogap. A sympathetic reader cares because this supplies a coherent, data-driven picture of the pseudogap and its link to superconductivity across many cuprates using only symmetry arguments and existing NMR measurements.

Core claim

Based only on symmetry of the two Cu hyperfine couplings, an anisotropic A_alpha and isotropic B, the Cu shifts are disentangled, and two different shift components emerge. Upon doping the cuprates, metallic B-spins are created above the pseudogap temperature which is shared with metallic A-spins. Further doping decreases the pseudogap temperature and increases the B-spin, but less so the A-spin. The apparent linear rate of increase in density of states of the B-spin with doping increases nearly threefold above about x=0.20, where the pseudogap has disappeared and A and B turn into superconducting metals, i.e. they disappear rapidly at Tc. The pseudogap temperature is a measure of the A-B, A

What carries the argument

Disentanglement of Cu NMR shifts into anisotropic A_alpha and isotropic B hyperfine components that reveal independent doping behaviors and their coupling.

If this is right

Pseudogap temperature falls with doping as B-spins increase faster than A-spins.
Above x approximately 0.20 both A and B components become superconducting metals that vanish rapidly at Tc.
Optimal Tc requires a special match between A and B and occurs in systems that retain a pseudogap.
Highest Tc values are not carried by the shifts but appear in nuclear relaxation rates and charge sharing between planar Cu and O.

Where Pith is reading between the lines

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

If A-B coupling sets the pseudogap scale, modest structural changes that alter this coupling could raise Tc at fixed doping.
The three condensation rules might supply a minimal template for pairing in other doped Mott insulators.
Combined shift and relaxation measurements could rank candidate cuprate compositions by predicted Tc before synthesis.

Load-bearing premise

The two Cu hyperfine couplings can be cleanly separated into a single anisotropic A_alpha and isotropic B form that remains valid for all cuprates and doping levels.

What would settle it

NMR shift data from one more cuprate compound that cannot be decomposed into consistent A and B components matching the reported doping trends or where pseudogap temperature fails to track the inferred A-B coupling strength.

Figures

Figures reproduced from arXiv: 2604.19215 by Abigail Lee, Juergen Haase.

**Figure 2.** Figure 2: Two Cu shift components, from two hyperfine coefficients. (a,b) Sketch of how the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: (a) High-temperature BχB for single-, bi-, and multilayer materials plotted against doping. Inset: High-temperature −A∥χA plotted against doping (note the lack of clear doping dependence). (b) High-temperature BχB plotted against high-temperature −A∥χA for a number of different families. BχB is never smaller than −A∥χA. A few families have been connected with dashed lines as guide to the eye to show their … view at source ↗

**Figure 4.** Figure 4: Temperature-implicit plot of −A∥χA(T) vs BχB(T). Dashed lines are guides to the eye to indicate the typical slopes according to (3) to (5), see main text for details. At high doping levels, both components are nearly indistinguishable from superconducting metals (the B-component is significantly larger than the A-component), i.e., both show a high-temperature plateau above Tc and rapidly disappear below Tc… view at source ↗

**Figure 1.** Figure 1: How the formation of Fermi surface arcs/and or pockets in the underdoped regime [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 5.** Figure 5: High-temperature spin components and Tc. (a) Given the dependence of Tc and BχB on doping, it is not surprising that the evolution of BχB traces out the superconducting dome. (b) A relationship between Tc and −A∥χA is not readily apparent. [27] relate to the overall change in DOS and the pseudogap [28, 29] remains to be seen. As expected, the high-temperature B-spin draws the superconducting dome when plot… view at source ↗

**Figure 6.** Figure 6: Relative isotropic shift suppression squared vs maximum in 1/ [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

The electronic properties of the high-temperature superconducting cuprates are encoded in complex sets of NMR data, but without microscopic theory, reliable NMR phenomenologies are in demand. Early analyses of NMR could only focus on very few materials and discovered spin singlet pairing and the enigmatic pseudogap. However, a coherent phenomenology of shift and relaxation could not be established, as incoming data from other cuprates complicated the picture. Today, due to work of many groups worldwide, planar copper and oxygen NMR data are available for most cuprates. Here, based only on symmetry of the two Cu hyperfine couplings, an anisotropic $A_\alpha$ and isotropic $B$, the Cu shifts are disentangled, and two different shift components emerge. Upon doping the cuprates, metallic B-spins are created above the pseudogap temperature which is shared with metallic A-spins. Further doping decreases the pseudogap temperature and increases the B-spin, but less so the A-spin. The apparent linear rate of increase in density of states of the B-spin with doping increases nearly threefold above about $x=0.20$, where the pseudogap has disappeared and A and B turn into superconducting metals, i.e. they disappear rapidly at $T_\mathrm{c}$. The pseudogap temperature is a measure of the coupling between A and B, which suppresses the shifts but not nuclear relaxation. Spin singlet pairing involves A and B according to three simple rules for condensation which will be discussed. The optimal $T_\mathrm{c}$ demands a special match between A and B and involves systems with a pseudogap. However, the highest $T_\mathrm{c}$ of all cuprates is not encoded in the shift, but rather in nuclear relaxation and charge sharing between planar Cu and O. Relations to other probes are discussed.

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 splits Cu NMR shifts into two components via fixed hyperfine symmetry and ties the pseudogap to their coupling, but the separation assumes those forms never change with doping.

read the letter

The central claim is that symmetry lets them pull the observed Cu shifts apart into an anisotropic A part and an isotropic B part, then read the pseudogap temperature as the strength of A-B coupling that suppresses shifts without affecting relaxation. They also give three rules for how spin-singlet condensation works with these two components and note that optimal Tc needs a particular A-B balance while the absolute highest Tc tracks relaxation and Cu-O charge sharing instead. This is applied to data from most cuprate families and a wide doping range, including the point near x=0.20 where the pseudogap ends and both components turn metallic and drop at Tc. That broader compilation is the useful part; earlier NMR work stayed with only a few materials, so seeing the same patterns hold across families gives a clearer phenomenology than before. The density-of-states increase for the B component and its jump above x=0.20 are concrete observations that follow from the separation. The soft spot is the starting assumption that the same A_alpha and B hyperfine forms stay valid at every doping level and in every cuprate. If orbital character or transferred fields shift with doping, the linear combination used to isolate A and B will mix the two, and the claimed distinction between their behaviors collapses. The abstract does not show explicit tests that the forms are doping-independent, so the disentanglement could be partly circular with the input data rather than an independent extraction. The three condensation rules also need to be checked against whether they are fitted to the same shifts used to define T*. This is for readers who already work with cuprate NMR or pseudogap models and want a single framework that organizes the shift data. Someone outside that niche will find the symmetry step hard to follow without the prior literature. It deserves a serious referee because the dataset is large and the claims are specific enough to be tested against other probes, even though the hyperfine invariance needs direct scrutiny in review. I would send it out rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper claims to develop a coherent NMR phenomenology for cuprates by disentangling Cu shifts into two components (A-spins and B-spins) using symmetry of anisotropic A_α and isotropic B hyperfine couplings. It identifies the pseudogap temperature as a measure of A-B coupling suppressing shifts (but not relaxation), describes doping evolution of B-spin DOS with a change at x≈0.20, and proposes three rules for spin-singlet condensation where optimal Tc requires a special A-B match in pseudogap systems. Highest Tc relates to relaxation and Cu-O charge sharing.

Significance. This could be significant as it unifies data from many cuprates without microscopic theory, offering simple rules for condensation and explaining pseudogap as inter-component coupling. It gives credit to extensive experimental data. If the separation is robust, it provides a new framework for interpreting NMR in superconductors. However, its impact hinges on whether the hyperfine assumption holds.

major comments (1)

[Shift disentanglement (based on symmetry arguments)] The disentanglement into independent A and B components assumes that the hyperfine form factors A_α (anisotropic) and B (isotropic) are doping-independent and the same for all cuprates. This is load-bearing for identifying metallic B-spins above T*, the suppression by coupling, and the condensation rules. If covalency or transferred hyperfine fields vary with doping (as suggested by changes at x≈0.20), the linear combination used to isolate components would mix A and B, making the claimed distinction an artifact rather than physical. No quantitative check against doping-dependent hyperfine literature is referenced.

minor comments (2)

The abstract refers to 'three simple rules for condensation which will be discussed' without specifying their location or providing equations; this makes it difficult to evaluate their derivation from the shift data.
Notation for A_α and B should be defined explicitly with reference to standard hyperfine Hamiltonian early in the text for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below.

read point-by-point responses

Referee: The disentanglement into independent A and B components assumes that the hyperfine form factors A_α (anisotropic) and B (isotropic) are doping-independent and the same for all cuprates. This is load-bearing for identifying metallic B-spins above T*, the suppression by coupling, and the condensation rules. If covalency or transferred hyperfine fields vary with doping (as suggested by changes at x≈0.20), the linear combination used to isolate components would mix A and B, making the claimed distinction an artifact rather than physical. No quantitative check against doping-dependent hyperfine literature is referenced.

Authors: The separation into A and B components is grounded in the distinct symmetry properties of the hyperfine couplings (anisotropic on-site A_α versus isotropic transferred B), as originally established by Mila and Rice and validated across multiple cuprate families in the experimental literature. These symmetry-based distinctions remain robust even if modest doping variations in covalency exist, because the extracted components display consistent doping trends and cross-material patterns that would be disrupted by significant mixing. The feature at x≈0.20 reflects a change in the B-spin density of states rather than a hyperfine discontinuity. We nevertheless agree that an explicit discussion of possible doping dependence would strengthen the presentation. In revision we will add references to key studies on the doping evolution of Cu hyperfine constants (showing variations are small relative to the observed shift changes) together with a brief quantitative estimate of the impact any such variation would have on the linear combinations. This directly addresses the concern that the distinction could be an artifact. revision: partial

Circularity Check

0 steps flagged

No significant circularity; phenomenological analysis under explicit symmetry assumptions

full rationale

The paper starts from an explicit symmetry-based assumption that the two Cu hyperfine couplings can be written as a fixed anisotropic A_α plus isotropic B form factor pair, then uses this to linearly disentangle observed shifts into two components. All subsequent statements (doping evolution of B-spin density of states, T* as a measure of A-B coupling, three condensation rules, optimal-Tc matching condition) are direct interpretations or re-descriptions of the disentangled data sets. No equation or claim is shown to reduce by construction to a parameter fitted from the same shifts, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The derivation chain therefore remains self-contained once the initial hyperfine-form assumption is granted; it does not loop back to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The analysis rests on the assumption that hyperfine couplings are strictly separable into one anisotropic and one isotropic term for all cuprates; no new particles or forces are introduced, but the two spin components are treated as distinct entities whose coupling is inferred from the data.

axioms (1)

domain assumption Planar copper hyperfine couplings consist of an anisotropic A_alpha term and an isotropic B term whose symmetry allows clean separation of observed shifts.
Invoked in the first sentence of the abstract as the basis for disentangling the Cu shifts.

invented entities (2)

A-spins no independent evidence
purpose: Direction-dependent spin component extracted from anisotropic hyperfine coupling.
Introduced to explain one part of the disentangled shift data.
B-spins no independent evidence
purpose: Isotropic spin component extracted from isotropic hyperfine coupling.
Introduced to explain the second part of the disentangled shift data and its doping dependence.

pith-pipeline@v0.9.0 · 5633 in / 1590 out tokens · 42051 ms · 2026-05-10T01:55:45.567041+00:00 · methodology

discussion (0)

Reference graph

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NMR. For example, planar O and Y show a superconducting metal for YBa 2Cu3O7 (just beyond optimal doping aty= 0.96). This means that aboveT c the shifts are temperature independent, and the relaxation is proportional to temperature, connected by the Korringa relation. A missing Hebel-Slichter peak just belowT c points to unconventional supercon- ductivity...

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ACKNOWLEDGMENTS We acknowledge financial support from Leipzig University, in particular Roger Gl¨ aser, and stimulating discussions with Nabeel Aslam

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For example, planar O and Y show a superconducting metal for YBa 2Cu3O7 (just beyond optimal doping aty= 0.96)

NMR. For example, planar O and Y show a superconducting metal for YBa 2Cu3O7 (just beyond optimal doping aty= 0.96). This means that aboveT c the shifts are temperature independent, and the relaxation is proportional to temperature, connected by the Korringa relation. A missing Hebel-Slichter peak just belowT c points to unconventional supercon- ductivity...

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As expected, the high-temperature B-spin draws the superconducting dome when plotted againstT c, as Fig

relate to the overall change in DOS and the pseudogap [28, 29] remains to be seen. As expected, the high-temperature B-spin draws the superconducting dome when plotted againstT c, as Fig. 5 shows. The variation in the A-component might be related to family dependent corrugation of the plane [30, 31]. The size of the pseudogap can be determined, from the c...

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ACKNOWLEDGMENTS We acknowledge financial support from Leipzig University, in particular Roger Gl¨ aser, and stimulating discussions with Nabeel Aslam

to other probes’ findings in greater detail. ACKNOWLEDGMENTS We acknowledge financial support from Leipzig University, in particular Roger Gl¨ aser, and stimulating discussions with Nabeel Aslam. Author contributions A.L. performed all data analysis. A.L. and J.H. contributed equally to preparing the manuscript. J.H. had the overall leadership. Competing ...

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