The Vortex Phase Diagram of Rotating Superfluid $^3$He-B

J. A. Sauls; J. J. Wiman; Robert C. Regan

REVIEW 3 major objections 5 minor 2 cited by

The Vortex Phase Diagram of Rotating Superfluid $^3$He-B

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper reports the first calculation of the pressure–temperature–field phase diagram for vortex phases in rotating superfluid 3He-B, and identifies the observed cooling and warming transitions with a metastability limit and an…

desk verdict First quantitative vortex phase diagram for rotating 3He-B, with a genuine metastability line, but the strong-coupling GL extrapolation is stretched below its validity and the paper admits the supercooling line misses the low-pressure data. read the letter →

arxiv 1908.04190 v2 pith:JNNXJNHX submitted 2019-08-12 cond-mat.supr-con

Robert C. Regan , J. J. Wiman , J. A. Sauls This is my paper · ORCID

classification cond-mat.supr-con

keywords superfluid3He-BvortexphasediagramA-coreD-corestrong-couplingGinzburg-Landautheorysupercoolingtransitiongyromagneticeffect

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 seeks to establish that the vortex states of rotating superfluid 3He-B form a first-order phase diagram with two stable phases, and that the experimentally observed hysteresis can be understood quantitatively. In zero field, a transition line separates a high-pressure, high-temperature phase of axially symmetric vortices whose cores contain the A and β phases from a low-pressure, low-temperature phase of doubly structured, axially asymmetric vortices. The paper argues that the high-temperature phase is metastable over a wide region and can supercool to a lower limit, below which it is globally unstable. If the calculations are correct, the observed transition on cooling marks the metastability limit, while the warming transition marks the true equilibrium line. This matters because it gives a parameter-free, material-property-based account of a phase transition between different vortex cores, a phenomenon unique to unconventional superfluids.

What carries the argument

The central object is the strong-coupling Ginzburg-Landau free energy with temperature-dependent fourth-order parameters, Eq. (12), calibrated by bulk thermodynamic data so that the bulk A and B phase boundaries are reproduced over the full pressure range. The vortex order parameter is expanded in the angular-momentum basis, where each component carries an integer phase winding N; the A-core vortex is carried by amplitudes with zero phase winding, C0+ and C+0, which are favored by strong-coupling energies, while the D-core vortex is driven by the dissociation of N=2 winding components C0− and C−0 into pairs of N=1 vortices. The competition between the gain in condensation energy from this dissociation and the loss of strong-coupling core energy is the mechanism that sets the transition temperature T_v(p,H) and the metastability limit T*_v(p,H).

What would settle it

A pressure-sweep measurement of the vortex transition on cooling at H=284 G that reaches pressures below the point where the paper's metastability line terminates would test the theory; the paper's own comparison shows its A-core metastability region ends at higher pressure than the measured cooling transitions, so observing a cooling transition at those lower pressures would indicate a stabilizing mechanism missing from the strong-coupling Ginzburg-Landau functional.

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Extended reading notes

Core claim

The central claim is that the vortex phase diagram of rotating 3He-B can be calculated from a strong-coupling Ginzburg-Landau functional whose fourth-order coefficients carry both weak-coupling and strong-coupling contributions, with the strong-coupling part scaled by T/Tc. Using this functional, the paper finds two globally stable vortex phases: the A-core phase, with the chiral A phase and non-unitary β phase filling the vortex core, and the D-core phase, whose core spontaneously breaks axial rotation symmetry and develops a double-core structure. In zero field the first-order boundary T_v(p) between these phases terminates on the bulk transition line at a triple point; the A-core phase is metastable below that line and supercools to a global-instability line T*_v(p,H). For magnetic fields H≳60 G parallel to the rotation axis, the A-core phase is stabilized over an extended region down to low pressure. The authors compare their calculated lines with experimental transitions and find that the cooling transitions track the supercooling line while a warming transition tracks the equilibrium line, thereby giving a unified interpretation of the observed hysteresis.

Load-bearing premise

The calculation assumes that strong-coupling corrections measured near the superfluid transition, scaled linearly with temperature, remain accurate at the much lower temperatures and inside the strongly distorted vortex cores where the D-core phase is stable; if that extrapolation fails, the predicted phase boundaries shift.

Editorial extensions

If this is right

The observed first-order vortex transition in rotating 3He-B is explained as an equilibrium transition on warming and a supercooling transition on cooling, resolving the hysteresis seen in NMR experiments.
In zero field, the A-core vortex phase exists only in a window near the bulk transition at high pressure, ending at a triple point on T_c(p), so no A-core phase should be found at lower pressures in zero field.
An external field H∥Ω with H≳60 G widens the A-core stability region and can open a window of A-core stability at pressures below the zero-field triple point.
At the first-order transition the vortex magnetization jumps discontinuously, giving a magnetic signature that should be observable in the gyromagnetic NMR shift.
Below the metastability line the A-core phase is globally unstable, so no amount of supercooling can preserve it; the observed sharp drop in the NMR signal at cooling is the signature of this global instability.

Reading between the lines

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

The same competition between zero-winding core amplitudes and dissociation of higher-winding amplitudes might govern vortex core transitions in other spin-triplet or p-wave superfluids, including neutron-star matter, where the strong-coupling parameters would be different.
The paper's admitted discrepancy—its metastability region ends at higher pressure than the measured cooling transitions—suggests that an additional stabilizing mechanism for the A-core phase exists below p_cv, possibly requiring the full quasiclassical strong-coupling functional rather than the GL extrapolation.
A quantitative prediction of the axial mass-current anomaly in D-core vortices, for example the transit time of ions carried along the core, would provide a direct experimental test of the double-core structure beyond the phase diagram itself.

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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

Paul — quick take on Regan, Wiman, Sauls, arXiv:1908.04190.

The real news is that they actually compute the full p-T-H vortex phase diagram for rotating 3He-B — equilibrium A-core/D-core boundary, the zero-field triple point on T_c, and the supercooling line T*_v where the A-core phase becomes globally unstable. That is new. Thuneberg and Salomaa-Volovik identified the vortices in the 80s, and Wiman/Sauls had the strong-coupling GL functional, but nobody had put the stability map together over the full pressure range, including the metastability region. The solver is benchmarked against Kasamatsu et al., and the same β parameters reproduce the bulk A-B transition and heat capacity jumps, so the vortex phase diagram is effectively parameter-free once those inputs are fixed. That is solid work.

The soft spot is the one the stress-test names. Eq. 12 is an ansatz: β_i(p,T) = β_wc_i(p) + (T/T_c) β_sc_i(p), with β_sc fixed by bulk fits near T_c. The paper uses it down to 0.25 T_c and inside strongly distorted vortex cores, with no error estimate. The paper itself admits that the calculated T*_v does not extend to the low pressures where the cooling transitions are observed, and says it cannot resolve the origin of the discrepancy. That is not a minor cosmetic issue: the central identification of the observed cooling transition as T*_v is only established for pressures above about 20 bar, not over the full experimental range. The qualitative story — a metastable A-core phase that supercools to a global instability — is supported, and the one warming data point at 29.3 bar matches T_v nicely. But the quantitative reach of the prediction is narrower than the abstract suggests.

Also worth noting: the T-linear suppression of strong coupling is motivated by phase-space arguments, not derived from the microscopic theory at low T. A quasi-classical check at one or two low-pressure points would have been the obvious way to bound the error. They did not do it, so the reader has no handle on how wrong the extrapolation might be.

Who benefits: anyone working on superfluid 3He vortices, NMR signatures, and the neutron-star analogue literature. It deserves a serious referee — the calculation is non-trivial, the solver is benchmarked, and the claims are concrete. I would send it to peer review but ask the authors to quantify the GL truncation error and address the low-pressure T*_v mismatch honestly, perhaps with a quasi-classical calculation or a clear discussion of why the discrepancy is expected.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a strong-coupling Ginzburg-Landau theory for vortex phases in rotating superfluid 3He-B, using temperature-dependent fourth-order coefficients beta_i(p,T) = beta_i^wc(p)+(T/Tc) beta_i^sc(p) (Eq. 12). The authors solve the Euler-Lagrange equations numerically with an L-BFGS solver on a 60xi x 60xi cell for the o-vortex, A-core vortex, and D-core vortex, and compute free energies across pressure and temperature. They report a zero-field first-order transition line T_v(p) between D-core and A-core vortex phases that terminates at a triple point on T_c(p), a field-shifted equilibrium line for H=284 G, and a metastability (supercooling) line T*_v(p,H) below which the A-core vortex is globally unstable. The central interpretive claim is that the experimentally observed cooling transition is T*_v and the warming transition is T_v, with quantitative agreement over most of the measured pressure range and an admitted discrepancy at low pressures.

Significance. The result, if quantitatively reliable, is significant: it is the first calculation of the p-T-H vortex phase diagram for 3He-B, and it is parameter-free in the sense that the strong-coupling beta parameters are fixed by prior fits to bulk properties, not by vortex data. The numerical solver is benchmarked against independent results by Kasamatsu et al., and the bulk A-B transition line is reproduced. The proposed identification of the cooling transition as a global instability of the A-core vortex and the warming transition as the equilibrium transition is physically plausible and gives a natural explanation of the observed hysteresis. The strength of the paper is that it converts a phenomenological picture into concrete, falsifiable predictions for T_v(p,H) and T*_v(p,H); the weakness is the quantitative reliance on an extrapolated strong-coupling ansatz without error estimates.

major comments (3)

[Equilibrium & Metastability Transitions] The paper states, 'We are not able to resolve the origin of the discrepancy in the minimum pressure for the metastable A-core phase within the strong-coupling GL theory.' Because the central claim is that the experimentally observed cooling transitions are T*_v(p,H), the fact that the calculated T*_v terminates at higher pressure than the low-pressure cooling data in Fig. 1 means the identification is not established over the full pressure range. The authors should either quantify the pressure mismatch, identify a physical mechanism (e.g., non-axial field components, cell texture, or nucleation effects) that shifts T*_v downward, or narrow the claim to the pressure range where agreement holds.
[Strong-Coupling Theory, Eq. (12)] The linear temperature scaling beta_i(p,T) = beta_i^wc(p) + (T/Tc) beta_i^sc(p) is motivated by phase-space arguments near T_c, but the stable D-core region and the computed transition lines use solutions at T = 0.25-0.55 T_c, as shown in Figs. 2 and 5. No estimate is provided for the error from truncating the GL expansion at fourth order or from assuming the same linear T/Tc scaling for the strongly distorted order-parameter gradients inside vortex cores. Since T_v and T*_v are determined by small free-energy differences between two vortex phases, this unquantified extrapolation is load-bearing; a sensitivity analysis with respect to variations of beta_i^sc within the uncertainties of Ref. [14] would materially strengthen the paper.
[Equilibrium & Metastability Transitions] The identification of the warming transition as the equilibrium line T_v rests on a single experimental point at p = 29.3 bar, with the authors noting an uncertainty between T_v = 1.81 mK and 1.85 mK. The case for the hysteresis interpretation would be stronger if additional warming-transition data, or an analysis of the NMR merging criterion, were presented.

minor comments (5)

[Fig. 1 caption] The legend contains duplicate entries for 'TV (p, H) - Expt., On Warming, H = 284G'; please remove the duplicate.
[Magnetic Susceptibility] The text 'The increase in TV relative to the zero-field transition for H & 60mK' should read 'H & 60G'; the unit mK is inconsistent with the magnetic field context.
[Table II] The particle density at p = 22.0 bar (22.96 nm^-3) is not monotonic with the neighboring values and is likely a typographical error; please verify.
[Throughout] Cross-references to sections appear as placeholders (e.g., 'in Sec. '); please supply the actual section numbers.
[Equilibrium & Metastability Transitions] The sentence 'This is indicated on the pressure-temperature phase diagram for p = 29.3bar the transition on cooling occurs at T*_V = 1.43mK' is missing punctuation and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the vortex phase diagram is obtained by solving the strong-coupling GL equations with beta-parameters fixed by bulk thermodynamic data, then compared, not fitted, to experiment.

full rationale

The derivation chain starts from the GL functional (Eqs. 1-4) with material parameters fixed by weak-coupling BCS theory and strong-coupling corrections (Eq. 12), whose coefficients are taken from Ref. [14]. Those coefficients are constrained by bulk normal-state and heat-capacity data: the paper states that the strong-coupling corrections 'reproduce the heat capacity jumps for the A and B transitions over the full pressure range.' These inputs do not depend on the vortex transition data. The paper then solves the Euler-Lagrange equations (Eq. 14) for competing vortex ansatze, computes their free energies, and reads off TV and T*V as genuine outputs. The A-core/D-core comparison is not equivalent to the input condition beta_A = beta_B: the vortex free energies involve gradient terms, core structure, and field distortions. The experimental cooling/warming transitions are used only for comparison, not as fitting targets; the paper explicitly reports a low-pressure discrepancy ('We are not able to resolve the origin of the discrepancy in the minimum pressure for the metastable A-core phase'), which is evidence that the comparison is not being forced. The benchmark against Kasamatsu et al. (Fig. 12) is independent code-level support. The self-citations [13,14] do not raise the circularity score because their content is externally constrained by bulk thermodynamic data and the numerical benchmark, and no equation in the paper reduces the predicted vortex transition line to a fit of that line. The linear T/Tc scaling in Eq. 12 is an input assumption with a phase-space motivation, but it is not derived from or fitted to the vortex phase diagram; it is an accuracy risk rather than a circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The core physical inputs are standard GL material parameters from microscopic theory plus measured normal-state properties; the only fitted quantities are the strong-coupling beta_sc coefficients. The linear T/T_c scaling and the isolated-vortex cell are explicit modeling assumptions. No new entities are introduced.

free parameters (1)

Strong-coupling corrections beta_sc_i(p) = Set of 19 values (Table II), e.g. at p=0: [-0.0098, -0.0419, -0.0132, -0.0047, -0.0899] in units of |beta_wc_1|; at…
Determined in prior work [14] by fitting the microscopic strong-coupling theory to normal-state thermodynamic and transport data and to the heat capacity jumps of bulk A and B phases. These coefficients set the relative stability of A-phase vs B-phase material, which drives the A-core vs D-core vortex balance.

assumptions (4)

domain assumption Ginzburg-Landau expansion remains quantitatively valid in vortex cores at T/T_c down to 0.25
The D-core phase and transition lines are computed at T/T_c of 0.25 to 0.6 (Figs. 2 and 1), far below the formal GL validity window near T_c; higher-order terms are not estimated.
ad hoc to paper Linear temperature scaling of strong-coupling corrections, Eq. 12
beta_i = beta_wc_i + (T/T_c) beta_sc_i assumes strong-coupling corrections vanish linearly as T approaches 0; the vortex phase diagram depends on this extrapolation, and the paper does not derive it.
domain assumption Single-vortex unit cell (d_c=60xi) with phase-winding boundary conditions represents a vortex in the rotating array
Vortex spacing at Omega of about 1.7 rad/s is around 150 um, far larger than d_c of about 60xi; the calculation neglects vortex-vortex interactions and uses an isolated-vortex cell, with no reported convergence study versus cell size.
domain assumption Dipolar energy and large-scale n(r) texture do not affect vortex phase stability
Justified by energy scales of about 10^-4 mK and the choice d_c much less than xi_D, but textural effects can shift NMR signatures used in the experimental comparison and are explicitly excluded from the p-T-H diagram.

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Pith. "Pith review of The Vortex Phase Diagram of Rotating Superfluid $^3$He-B." pith.science (2026). https://pith.science/paper/JNNXJNHX

@misc{pith2026190804190,
  author       = {Pith},
  title        = {Pith review of: The Vortex Phase Diagram of Rotating Superfluid $^3$He-B},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNNXJNHX}},
  note         = {Machine review of arXiv:1908.04190}
}

abstract

We present the first theoretical calculation of the pressure-temperature-field phase diagram for the vortex phases of rotating superfluid $^3$He-B. Based on a strong-coupling extension of the Ginzburg-Landau theory that accounts for the relative stability of the bulk A and B phases of $^3$He at all pressures, we report calculations for the internal structure and free energies of distinct broken-symmetry vortices in rotating superfluid $^3$He-B. Theoretical results for the equilibrium vortex phase diagram in zero field and an external field of $H=284\,\mbox{G}$ parallel to the rotation axis, $\vec{H}\parallel\vec{\Omega}$, are reported, as well as the supercooling transition line, $T^{*}_ {v} (p,H)$. In zero field the vortex phases of $^3$He-B are separated by a first-order phase transition line $T_ {v} (p)$ that terminates on the bulk critical line $T_{c}(p)$ at a triple point. The low-pressure, low-temperature phase is characterized by an array of singly-quantized vortices that spontaneously breaks axial rotation symmetry, exhibits anisotropic vortex currents and an axial current anomaly (D-core phase). The high-pressure, high-temperature phase is characterized by vortices with both bulk A phase and $\beta$ phase in their cores (A-core phase). We show that this phase is metastable and supercools down to a minimum temperature, $T^{*}_ {v} (p,H)$, below which it is globally unstable to an array of D-core vortices. For $H\gtrsim 60\,\mbox{G}$ external magnetic fields aligned along the axis of rotation increase the region of stability of the A-core phase of rotating $^3$He-B, opening a window of stability down to low pressures. These results are compared with the experimentally reported phase transitions in rotating $^3$He-B.

Figures

Figures reproduced from arXiv: 1908.04190 by the authors.

**Figure 2.** FIG. 2. Left panel: The axially symmetric B-phase vortex (“o-vortex”) has a hard core with a node in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Amplitudes and phases of the components of the o-vortex [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Amplitudes and phases of the components of the A-core [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Amplitudes and phases of the components of the D-core [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Left: Axially symmetric current density of the A-core vortex for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Axial mass current of the D-core vortex at [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Magnetization profiles, [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: shows our results for the local susceptibilities of the A- and D-core vortices. The D-core vortex has the larger susceptibility, and thus we expect that the equilibrium vortex transition line to shift to higher temperatures with the application of a weak magetic field…

**Figure 10.** Figure 10: FIG. 10. Field evolution of the equilibrium vortex transition tem [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11. Rate of convergence of the iterative solution of the GL [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: FIG. 12. Free energies for the A-core and D-core vortex states [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Figure 7 shows the z-axis current for the D-core vortex for the same pressure and temperature as that for the vortex currents in Fig. 6, also in units of jc. The axial current density spans an area of order A≈ 100 ξ 2≈ 6.25 µm2. An idea for detection of the current anomaly along the z- axis is to inject electrons into rotating 3He-B from the outer, radial...

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work page 1999