What holes in superconductors reveal about superconductivity

J. E. Hirsch

REVIEW 4 major objections 4 minor 1 cited by

What holes in superconductors reveal about superconductivity

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A type I superconductor with an interior hole cannot expel the field from it, the paper argues—so the Meissner effect needs physics BCS lacks.

desk verdict A sharp, readable thought experiment whose thermodynamic bookkeeping is neat, but whose universal no-expulsion claim overreaches the derivation and contradicts the paper's own caveats. read the letter →

arxiv 2506.07361 v1 pith:SJMV2EW6 submitted 2025-06-09 cond-mat.supr-con

J. E. Hirsch This is my paper

classification cond-mat.supr-con

keywords MeissnereffecttypeIsuperconductormagneticfluxtrappingPoyntingtheoremholesuperconductivityBCStheoryfirst-orderphasetransitionradialchargeflow

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a type I superconductor—one that, when pure, expels all magnetic field from its bulk—cannot expel the field from a hole in its interior when it is cooled through its transition in a magnetic field. Instead the system ends up in a higher-energy state with the field trapped in the hole and normal regions letting field lines escape, no matter how small the hole is. The reason offered is that expelling a magnetic field is a local dynamical process requiring radial charge flow, the outward motion of charge as electron orbits expand, and an empty hole cannot host that process. If true, this would mean the Meissner effect—the defining expulsion of field on entering the superconducting state—cannot be explained by the standard BCS theory, whose Hamiltonians contain no radial charge flow. The paper supplies thermodynamic and Poynting-theorem arguments for the impossibility and proposes a cooling experiment that could settle it.

What carries the argument

The argument rests on two coupled pieces of machinery. The first is energy bookkeeping at constant applied field: the free-energy difference $F_n - F_s = H_c(T)^2/8\pi$ per unit volume does not cover the work $W = H_c(T_1)^2 V/4\pi$ returned to the power source when flux is expelled from the whole volume, leaving a deficit proportional to the hole volume $V_h$. The second is Poynting's theorem applied to the hole's lateral surface: with the Faraday field $\mathbf{E} = -(1/2\pi r c)(\partial\phi/\partial t)\,\hat{\theta}$, the electromagnetic-energy outflow can balance the decrease of field energy inside the hole only if the electric field does negative work on charges there, but a vacuum hole contains no charges. The paper argues the only way to supply that missing term is the process of hole superconductivity: conduction electrons expand their orbits from microscopic radius to mesoscopic radius $2\lambda_L$, moving radially outward in the magnetic field, so the Lorentz force gives the Meissner current its azimuthal momentum and the expanding orbits generate the 'Meissner pressure' that overcomes the opposing 'Maxwell pressure' of the field.

What would settle it

Cool an ultra-pure type I superconducting cylinder with a small interior hole of radius below $\sqrt{\xi\lambda_L}$ slowly through its transition in a fixed magnetic field and measure the flux through the hole: the paper predicts the initial flux remains trapped and part of the material stays normal, so observing complete expulsion from the hole, or a threshold hole size below which flux is fully expelled, would falsify the central claim. A second decisive check would be measuring whether the boundary field at the hole decreases during expulsion, which would remove the Poynting no-go assumption.

Watch

Extended reading notes

Core claim

The central claim is that the normal-to-superconducting transition in a field is not an ordinary reversible first-order transition when the body contains a cavity. Cooling to just below $T_1$ with applied field $H = H_c(T_1)$, the work that must be returned to the constant-current source that maintains the field is $H_c^2 V/4\pi$, while the condensing material can supply at most $(H_c^2/8\pi)(V - V_h)$; the missing energy $H_c^2 V_h/8\pi$ is exactly the condensation energy of the hole's volume. Poynting's theorem applied to the hole's surface reaches the same conclusion: the induced Faraday field drives electromagnetic energy outward, and the balance can close only if charges inside the hole do negative work on the field—there are none. The author therefore concludes that superconducting domains growing around the hole produce a current that sustains the field in the hole, leaving normal flux tubes above and below it, and that this is what the theory of hole superconductivity, in which field expulsion is driven by kinetic-energy lowering and radial charge flow, predicts.

Load-bearing premise

The load-bearing assumption is that the magnetic field at the hole's boundary stays fixed at the transition value while the field inside the hole is expelled; if the boundary field could instead fall as a screening current develops, the energy and Poynting balances would close without any current inside the hole, and the paper's reason to exclude that—that condensation energy is unavailable while field remains inside—is an unproved dynamical assertion.

Editorial extensions

If this is right

Field-cooled type I superconductors with interior holes would retain trapped flux in every hole, with normal regions around it, even for holes far smaller than the coherence length and the London penetration depth.
Energy lowering alone would not explain the Meissner effect; a valid microscopic theory would have to supply the dynamical mechanism by which the field is pushed out.
BCS-type Hamiltonians would be incomplete in a specific, testable way: they omit radial charge flow, so they cannot describe how the Meissner current is generated, maintained, and stopped.
The transition between normal and superconducting states in a field would not be fully reversible when holes are present, and the reverse transition would generate entropy.
The conventional and hole-superconductivity pictures make opposite predictions for flux expulsion from holes, so a controlled field-cooling experiment can distinguish the two theories.

Reading between the lines

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

Beyond the paper: if holes trap flux, then tiny metallurgical voids, cracks, or inclusions would act as flux-trapping defects in real superconductors, giving a microstructure-based explanation for why field-cooled diamagnetic signals are systematically weaker than zero-field-cooled signals in type I samples.
Beyond the paper: the Poynting no-go condition suggests a general electrodynamic rule—magnetic field cannot vanish from a source-free region through purely external screening currents—which could be probed in inductive analogue systems where a cavity field is screened by external coils.
Beyond the paper: the claim that trapping is independent of hole size could be sharpened into a quantitative prediction: ultra-pure single-crystal type I samples with fabricated holes of decreasing radius should show a size threshold set by $\sqrt{\xi\lambda_L}$ below which complete expulsion either does or does not occur, depending on whether disorder converts the sample to type II behavior.

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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

Colleague,

The paper is at its best as a thought experiment: it points to a real energy-bookkeeping puzzle for expelling field from an interior cavity, and it honestly states where its own arguments stop. The thermodynamic accounting in Sec II is internally consistent as a formal exercise, and the proposed experiment is a legitimate discriminator. That said, the central claim—that a type I superconductor can never expel flux from an interior hole, no matter the hole size or cooling protocol—does not follow from the arguments given.

The weakest load-bearing step is in Sec IV. The Poynting no-go assumes the field at the hole boundary stays pinned at H_c(T1) throughout expulsion. If that field decays with the field inside the hole, the Poynting outflow matches the field-energy decrease exactly and Eq. (20) reduces to 0=0. The paper's reason to exclude this—that a screening current at the outer surface needs condensation energy from the interior that is unavailable while field remains inside—is an unproved dynamical assertion, not a theorem of BCS or GL. The author effectively concedes this at the end of Sec IV by saying he cannot rule out other conventional processes.

There is also an internal tension with Sec III. The paper admits that for deep supercooling, energetic arguments cannot rule out complete expulsion, and that the Joule-heat plausibility argument is not rigorous. Yet Sec V asserts the field remains trapped 'no matter how the cooling takes place.' The conclusion overstates what the analysis delivers.

One more soft spot: the Goodman et al. experiment on flux quantization in Sn and In cylinders is dismissed as 'probably disordered' and therefore type II. Those are classic type I elements; the dismissal needs stronger support than a guess. That experiment, if taken at face value, already shows complete expulsion for sub-half-flux-quantum initial flux, which cuts against the atomic-dimension claim.

What is genuinely new here is the energy balance framing and the invitation to test it. The paper is clearly written, the author identifies his own caveats, and the falsifiability of the proposal is a real virtue. But as a proof against BCS, it is not there. The radial-charge-flow mechanism cited to the author's own prior work is the interpretation, not an independent derivation.

Who is this for? Someone working on the Meissner effect or flux trapping who wants a sharp, contestable claim to test. It deserves a serious referee—this is not a desk reject—but the referee should insist the central claim be reframed as a conjecture/experimental proposal, with the Poynting boundary condition justified or removed, and the Goodman dismissal addressed. I'd take it to reading group for the discussion value.

Referee Report

4 major / 4 minor

Summary. The paper argues that a type I superconducting body containing an interior hole cannot fully expel an applied magnetic field when cooled into the superconducting state, and that the system will necessarily remain partly normal with trapped flux (Fig. 1b). The argument proceeds from a thermodynamic energy balance (Sec. II), a discussion of supercooling (Sec. III), and a Poynting-theorem no-go result (Sec. IV). The paper concludes that magnetic-field expulsion requires physical ingredients absent from BCS Hamiltonians, namely the radial charge flow of the author's hole-superconductivity theory, and that this explains why the Meissner effect cannot be accounted for within conventional theory.

Significance. If the central claim were established, it would constitute a major challenge to the conventional BCS-based understanding of the Meissner effect and would provide a sharp experimental discriminator between BCS and hole superconductivity. The thermodynamic energy bookkeeping in Sec. II is a clean and internally consistent formal argument for the special case of a reversible transition infinitesimally below T1, and the paper usefully frames a falsifiable experimental prediction for holes of various sizes. However, the broader universal claim—that flux is trapped 'irrespective of dimensions' and 'no matter how the cooling takes place'—is not established. The Poynting no-go theorem in Sec. IV rests on an unproved assumption about the boundary field, and the paper itself concedes that other conventional processes cannot be ruled out. As a result, the paper's main conclusion currently depends on the very theory it aims to support.

major comments (4)

[Sec. IV, Eqs. (17)-(20)] The no-go result is derived under the explicit assumption that the magnetic field at the hole surface remains pinned at H = H_c(T1) while the interior field is expelled. If the boundary field instead decays together with the field in the hole, the Poynting outflow is reduced and Eq. (20) becomes an identity with a zero current term; no charges inside the hole are then required. The author's rebuttal, that a screening current at the outer cylinder boundary cannot draw on condensation energy while field remains in the interior, is an unproved dynamical assertion rather than a theorem of BCS or GL theory. This assumption is load-bearing, and the paper itself concedes in the same section that other processes cannot be ruled out within the conventional theory. Consequently, the claimed universal impossibility of flux expulsion is not established.
[Sec. III and Sec. V] The thermodynamic argument in Sec. II applies only to cooling infinitesimally below T1. In Sec. III the author states that for strong supercooling to T2 << T_h, energetic arguments alone cannot rule out complete expulsion, and that he is 'unable to rule this possibility out theoretically based on thermodynamic arguments.' The conclusion in Sec. V that flux remains trapped 'no matter how the cooling takes place' therefore goes beyond what the paper proves. Either the universal claim must be retracted or a new argument must be supplied for the supercooled case.
[Sec. VIII] The claim that flux is trapped for holes 'down to atomic dimensions' is in tension with the Goodman et al. experiments cited as Ref. [31], which report complete flux expulsion for initial fluxes below about half a flux quantum. The paper dismisses these experiments by assuming the samples were disordered type II superconductors, but no evidence is offered for that assumption. Since the paper's claim concerns arbitrarily small holes in clean type I materials, this unsupported dismissal leaves a direct experimental counterexample unaddressed.
[Sec. VII] The central interpretation is circular as presented. The absence of radial charge flow inside a hole is invoked as the mechanism that prevents flux expulsion, but radial charge flow is a postulate of the author's hole-superconductivity theory, not a consequence of the thermodynamic or Poynting arguments. Since the Poynting argument fails, the claimed independent support for hole superconductivity reduces to an assumption of that theory. To be convincing, the paper would need either to derive the holed-sample behavior from a conventional BCS/GL dynamics or to provide a direct experimental test that cleanly distinguishes the two theories.

minor comments (4)

[Fig. 2 caption] The caption says the current I is given by Eq. (1), but the current is defined in Eq. (2); Eq. (1) is the free-energy difference.
[Sec. IV] The text contains a typo: 'Pointing vector' should be 'Poynting vector.'
[Sec. IV, Eq. (19)] Equation (19) is dimensionally unclear as printed; the left-hand side appears to contain a spurious factor 1/(πr0^2) or a missing volume factor. Please rewrite the equation so that the per-volume normalization is explicit.
[Sec. IV, discussion of boundary-field decay] The paragraph beginning 'This would happen if...' presents a plausible physical process and then dismisses it using the unproved condensation-energy claim. The distinction between an explicit assumption and a derived theorem should be stated more carefully, perhaps by labeling the boundary-field hypothesis as an axiom.

Circularity Check

2 steps flagged · score 6.0 of 10

Poynting no-go presupposes the hole-boundary field is pinned; the only argument against a decaying boundary field asserts the conclusion, and the universal flux-trapping claim rests on the author's own prior papers.

self definitional [Sec. IV (Electromagnetic Energy Flow), paragraph after Eq. (20)]
"This would happen if, for example, the Meissner current expelling the magnetic field would develop at the boundary of the cylinder, so that the magnetic field would decrease uniformly inside the cylinder including the region of the hole and its surface. This is however impossible even within the conventional theory, since in order for the current at the boundary of the cylinder to overcome the Faraday counter-emf requires condensation energy from the interior, which is not available while any magnetic field is still in the interior."

Eq. (20) presupposes that the boundary field stays pinned at H_c(T1); if the boundary field decays with the interior field, the Poynting bookkeeping is an identity and no current inside the hole is needed. The paper excludes this by asserting that a boundary screening current would need condensation energy from the interior, 'which is not available while any magnetic field is still in the interior.' That assertion is exactly the disputed conclusion: a superconducting shell has its own local condensation energy to drive the screening current while field remains inside. Thus the escape from Eq. (20) is rejected by a premise equivalent to the claim being proved. The text then concedes 'we cannot rule out that within the conventional theory there could be other processes.'
self citation load bearing [Sec. I (Introduction), paragraph beginning 'Instead, within the alternative theory...']
"Within that theory it is clear that the required dynamic processes are impeded by the presence of the cavity [10, 11] so that the transition shown in Fig. 1 (a) will not take place. The system cooled into the superconducting state cannot expel the entire magnetic field as shown in Fig. 1 (a), but rather will end up in the higher energy state shown in Fig. 1 (b), with part of the system remaining in the normal phase, irrespective of the dimensions of the hole."

Refs [10,11] are prior papers by the same author that already assert the hole-flux-trapping conclusion. The independent arguments in this manuscript do not close the gap: Sec. III ends 'we are unable to rule this possibility out theoretically based on thermodynamic arguments,' and Sec. IV ends 'we cannot rule out that within the conventional theory there could be other processes.' With those concessions, the universal 'irrespective of the dimensions of the hole' claim has no support outside the self-cited version of the author's own theory. Citing that theory's own conclusion to support the theory is load-bearing self-citation.

full rationale

The paper contains a genuinely independent thermodynamic calculation (Sec. II) showing that at T just below T1 the free energy released by the superconducting material volume V-V_h is less than the work needed to remove flux from the hole volume V_h; that is a substantive argument. But the paper itself closes the energetic gap on deeper supercooling (Eq. (9)) and states it cannot rule out complete expulsion thermodynamically. The Poynting theorem argument (Sec. IV) is also substantive but conditional: it holds only if the magnetic field at the hole boundary is assumed pinned at H_c(T1) throughout the process. The paper's only reason for rejecting the alternative decaying-boundary-field case is an assertion that condensation energy is unavailable while field remains in the interior, which is the same conclusion in different words. The paper's own final sentence in Sec. IV concedes that other conventional processes cannot be ruled out. Once those independent arguments are seen to be conditional or conceded, the universal claim that holes always trap flux regardless of size is supported only by the author's prior papers [10,11], which already contained that conclusion. Hence the central prediction is partially circular: it is a theory-internal prediction whose independent derivations fail at their decisive step, plus a self-citation chain. I do not score 8+ because the thermodynamic identity and the Poynting identity are real calculations with independent content, and the hole theory's mechanism (orbit expansion/radial charge flow) is not merely a renaming of the prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted in this paper; H_c, lambda_L, and V_h are inputs. The argument leans on the author's prior orbit-expansion mechanism and on a boundary-field assumption in the Poynting no-go. No new particles or entities are introduced.

assumptions (5)

ad hoc to paper The transition to superconductivity is driven by lowering of kinetic energy, with electrons expanding from Fermi-scale orbits to radius 2 lambda_L.
Central to Sec VII; this is the author's hole-superconductivity postulate, cited to refs [23-25], not derived here.
ad hoc to paper Expulsion of magnetic field requires radial charge flow; without it, field cannot be expelled.
Invoked in Secs V, VII, and IX as the only mechanism; asserted rather than proved.
ad hoc to paper The magnetic field at the hole boundary stays at H_c(T1) throughout the expulsion process in the Poynting argument.
Used in Eqs (17)-(20); the alternative of decreasing boundary field is dismissed with an unproved dynamical claim.
domain assumption The heat reservoir cannot supply the missing condensation energy without violating the second law.
Sec II assumes no entropy decrease from the heat bath; standard thermodynamic assumption, cited to ref [13] but not adopted.
domain assumption Conventional BCS theory predicts full flux expulsion from a body with holes because it only minimizes free energy.
Secs I and IX; this is a claim about the standard literature, not demonstrated; fluxoid quantization in multiply connected superconductors is not discussed.

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Cite this review

Pith. "Pith review of What holes in superconductors reveal about superconductivity." pith.science (2026). https://pith.science/paper/SJMV2EW6

@misc{pith2026250607361,
  author       = {Pith},
  title        = {Pith review of: What holes in superconductors reveal about superconductivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJMV2EW6}},
  note         = {Machine review of arXiv:2506.07361}
}

read the original abstract

We consider a type I superconducting body that contains one or more holes in its interior that undergoes a transition between normal and superconducting states in the presence of a magnetic field. We argue that unlike other thermodynamic systems that undergo first order phase transitions the system cannot reach its equilibrium thermodynamic state, and that this sheds new light on the physics of the Meissner effect. How the Meissner effect occurs has not been addressed within the conventional theory of superconductivity, BCS. The situation considered in this paper indicates that expulsion of magnetic field requires physical elements absent from Hamiltonians assumed to describe superconductors within BCS theory. These physical elements are essential components of the alternative theory of hole superconductivity.

Figures

Figures reproduced from arXiv: 2506.07361 by the authors.

**Figure 2.** FIG. 2: Critical field versus temperature for a type I super [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3: Initial stage of flux expulsion that could lead to the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: A superconducting simply connected body with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5: Cross section of cylinder of Fig. 4 as seen from [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7: Processes involved in the expansion of a supercon [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 6.** Figure 6: FIG. 6: Cross section of cylinder of Fig. 4 as seen from the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 4.** Figure 4: An electron expanding its orbit in the presence of a magnetic field B perpendicular to the orbit acquires an azimuthal velocity that generates a magnetic field opposite to the applied field. Note that 2λL orbits are closely related to Slater’s orbits hypothesized in 19…

**Figure 10.** Figure 10: FIG. 10: Growth of a domain, in more detail, as seen from [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11: Graphic explanation for why phase coherence is [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On a recent explanation of the dynamics of the Meissner effect within the conventional theory of superconductivity
cond-mat.supr-con 2025-12 conditional novelty 3.0 of 10

A reply to Markos-Hlubina arguing their time-dependent London treatment only assumes the superconducting fraction and thus leaves Hirsch's objections unanswered, plus a proposed cavity experiment to arbitrate.

Reference graph

Works this paper leans on

40 extracted references · 39 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [31]

J. E. Hirsch, ‘Charge Expulsion, Spin Meissner Effect, and Charge Inhomogeneity in Superconductors’, J. Sup. Nov. Mag.22, 131 (2009)

work page 2009

[2] [1]

How does the current acquire its azimuthal momen- tum when a domain is born?

work page

[3] [2]

How does the condensation energy become kinetic energy of the supercurrent?

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[4] [3]

How is the azimuthal momentum of the supercur- rent compensated so that angular momentum con- servation is not violated?

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[5] [4]

How does a domain grow outward overcoming the Maxwell pressure that wants to shrink it?

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[6] [5]

How does the azimuthal clockwise current ini- tially overcome the counterclockwise Faraday elec- tric field that eventually stops it?

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[7] [6]

The conventional theory has not addressed any of these questions

How does all of the above happens without energy dissipation to ensure the reversibility of the pro- cess? And of course the same questions in reverse exist and need to be answered about the reverse process, the su- perconductor to normal transition in a magnetic field [11]. The conventional theory has not addressed any of these questions. The answers acc...

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[8] [7]

Superconductivity begins with H

J. E. Hirsch, “Superconductivity begins with H”, World Scientific, Singapore, 2020

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