Ferromagnetic fluctuations and odd-parity pairing

Scope

Ferromagnetic fluctuations describe enhanced magnetic response near zero momentum transfer. They can support odd-parity pairing in specified models, but neither a large uniform susceptibility nor a large upper critical field identifies a superconducting spin state. Magnetic order, correlations, gap symmetry and the pairing interaction are separate questions. Fay and Appel calculated equal-spin p-state pairing from longitudinal fluctuations in an itinerant-electron model; this is a theoretical possibility, not a universal material classification. R1

Mathematical assumptions

For a simple effective exchange $V=-J\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2$ with $J>0$, the spin operator has eigenvalues $-3$ in the singlet and $+1$ in the triplet sector. The corresponding interaction eigenvalues are $+3J$ and $-J$. This independent two-spin illustration assumes a ferromagnetic effective interaction; it does not derive a material's exchange constant or its frequency dependence.

For even-frequency, single-orbital pairing, Fermi antisymmetry requires $\Delta(\mathbf k)=-\Delta^T(-\mathbf k)$. Writing a triplet matrix as $\Delta=(\mathbf d\cdot\boldsymbol\sigma)i\sigma_y$ gives a symmetric spin matrix, so $\mathbf d(-\mathbf k)=-\mathbf d(\mathbf k)$. Odd parity here follows from these assumptions. Strong spin–orbit coupling requires a suitable pseudospin description; additional orbital or frequency structure changes this elementary argument.

A second independent illustration uses a circular two-dimensional Fermi surface and an attractive angular kernel $V(\theta)=-g/(a-b\cos\theta)$, with $g>0$ an energy scale and dimensionless $a>b>0$. Its dimensionless Fourier coefficients are

$C_\ell=\int_0^{2\pi}\frac{d\theta}{2\pi}\frac{\cos(\ell\theta)}{a-b\cos\theta}=\frac{r^\ell}{\sqrt{a^2-b^2}},\qquad r=\frac{a-\sqrt{a^2-b^2}}{b},\quad \ell\geq0.$

The odd-parity projection $[V(\theta)-V(\theta+\pi)]/2$ retains odd angular harmonics. Since $0<r<1$, the strongest attractive odd harmonic in this example is $\ell=1$. Numerical quadrature checks the coefficients and projection. Crystal anisotropy, competing interactions and self-energy are absent: this example neither selects a universal p-wave state nor predicts a transition temperature.

Experimental criteria

UGe2 superconductivity was reported in a restricted pressure range within an itinerant-ferromagnetic setting. URhGe supplied ambient-pressure thermodynamic evidence for coexistence; UCoGe supplied ambient-pressure superconducting and ferromagnetic signatures. These primary abstracts support the historical observations, while microscopic volume overlap and a particular gap representation require more than proximity in a phase diagram. Corrected original links replace two inherited citation errors. R2 R3 R5

Lévy and colleagues reported a second superconducting pocket in URhGe around spin-axis reorientation near 12 T for field parallel to the crystal b axis, after low-field superconductivity had already disappeared. This is an orientation-specific re-entrant phenomenon. The original abstract suggests a connection to spin rotation; it does not uniquely establish a microscopic pairing kernel. R4

Pustogow and colleagues' 2019 Sr2RuO4 author manuscript reports a reduced oxygen-17 Knight shift and strong pulse-energy sensitivity. Its unstrained example uses an in-plane b-axis field of 0.7107 T; 20 mK is the base temperature, not a transition temperature. Selected methods explain why RF heating and different electronic/nuclear relaxation times can make a relaxation check insufficient. These observations challenge older unchanged-susceptibility interpretations; this entry does not infer a unique order parameter from them. R6

UTe2's discovery paper reported unusual anisotropic critical fields in a paramagnetic normal state. Its page 2 distinguishes a highest measured field of 20 T from an estimated zero-temperature b-axis value of 40 T. The estimate is not a measured endpoint. Historical triplet and ferromagnetic-pairing interpretations remain interpretations; the paper's schematic does not establish one continuous phase diagram shared by different uranium compounds. R7

Record field direction, pressure, composition, purity and temperature for each probe. Separate orbital/background contributions and pulse heating in Knight shifts; retain coupling, g-factor, multiband and orbital-limiting assumptions in critical-field comparisons. Combine symmetry-sensitive measurements with magnetic and superconducting volume evidence from the same sample. A bulk signal alone does not identify the mediating fluctuation.

Limits and disputes

In UTe2, Duan and colleagues reported a neutron resonance associated with antiferromagnetic wavevectors. Their extended-data comparison found no ferromagnetic signal above background in the stated measurement window. This is not absence of every possible small-momentum fluctuation at every field, energy and temperature; a resonance coupled to superconductivity also does not alone prove the direction of causation. R8

A study published on 29 April 2026 measured UTe2 magnetotropic response in pulsed fields up to 60 T at 4 K, where the studied superconducting phases were suppressed. It inferred enhanced transverse, near-zero-momentum susceptibility at high fields. Here transverse means perpendicular to the applied field, not an off-diagonal tensor component. These normal-state measurements differ from zero/low-field neutron conditions. The displayed 300 mK superconducting boundaries came from earlier work, and 4 K must not be recorded as this experiment's superconducting transition. R9

Fluctuations can enhance attraction while also renormalizing quasiparticles and affecting pair breaking. Fay–Appel's model gives a maximum followed by a decrease of calculated transition temperature toward the magnetic transition. Neither that model nor the elementary angular projection proves a universal optimum at a ferromagnetic quantum critical point. Longitudinal/transverse spectral weight, spin–orbit coupling and band structure must be specified. R1

Discovery implications

Treat small-momentum magnetic response as a candidate-screening observation. Test whether a proposed kernel simultaneously explains symmetry constraints, normal-state responses and field/pressure trends under matched conditions. Keep negative observations and alternative channels. The recent UTe2 comparison illustrates why distinct experiments should retain their conditions instead of being reduced to a single ferromagnetic or antiferromagnetic label. Selected source passages and model checks here leave full scientific acceptance open.

Related topics

Compare antiferromagnetic spin fluctuations, Eliashberg theory and the keyword catalogue for gap symmetry, critical fields and uranium material families.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.