f-wave pairing: crystal representations and nodal evidence

Scope

In a rotationally invariant continuum, f-wave means orbital angular momentum $l=3$. In a crystal, specify the point group, spin or pseudospin structure and basis functions; nonsymmorphic materials also require the fractional translations of the space group. An odd cubic polynomial is not automatically a pure $l=3$ harmonic, and a node inferred from a response is not an irreducible-representation label. The discussion below assumes even-frequency pairing unless stated otherwise. Exchange symmetry must include momentum, spin, orbital indices and frequency before drawing parity conclusions in a multiorbital material. R5 R10

UPt3 is an important test case with competing descriptions of its superconducting phases. The historical E2u framework and a proposed E1u planar state organize different observations, but a candidate label is not a unique experimental determination. UTe2 has a different crystal symmetry and must not inherit UPt3's representation merely because both are uranium compounds. R2 R3 R5 R7

Mathematical assumptions

In the following local calculation, an illustrative hexagonal E2u orbital doublet is $[k_z(k_x^2-k_y^2),2k_zk_xk_y]$. With a fixed spin vector along z, its equal-amplitude chiral combination is proportional to $k_z(k_x+ik_y)^2$. It is odd under inversion. Under rotation by angle $\alpha$ about z, its orbital factor acquires $e^{2i\alpha}$: the continuum harmonic has $l=3$ but axial projection $m=2$, not $m=3$. Local transformation checks and finite-difference Laplacians verify these polynomial identities; they do not reproduce a cited material calculation. In a lattice, continuous angular momentum is only a comparison, not an exact conserved quantum number. R1 R3

On a unit sphere this chiral form has magnitude $|\cos\theta|\sin^2\theta$: an equatorial line zero and quadratic polar zeros. A planar E1u illustration uses two orthogonal spin vectors multiplying $k_x(5k_z^2-1)$ and $k_y(5k_z^2-1)$. Its magnitude is $\sin\theta|5\cos^2\theta-1|$, with polar point zeros and two horizontal zeros at $\cos^2\theta=1/5$. These are model angles, not measured node coordinates. Tsutsumi and colleagues use a related planar construction in their weak-spin-orbit scenario, with field-dependent spin orientation. R5

The constant one in the spherical E1u expressions represents $k_x^2+k_y^2+k_z^2$. Restoring it makes each component a homogeneous harmonic cubic. The lower-order pair $(k_x,k_y)$ belongs to the same orbital doublet. In a local illustration with fixed orthogonal spin vectors and a unit spherical Fermi surface, add a real coefficient $a$ times this pair. The gap magnitude becomes $\sin\theta|5\cos^2\theta-1+a|$. Two distinct horizontal nodal rings occur only for $-4<a<1$, at $k_z^2=(1-a)/5$. At $a=1$ they merge into a quadratic equatorial zero; at $a=-4$ they meet the poles, giving cubic polar zeros. For real $a$ outside this interval only the polar zeros remain. This admixture is our illustration, not a fitted state reported by Tsutsumi and colleagues. R5

For a complex coefficient, the same local model instead has squared magnitude $\sin^2\theta[(5\cos^2\theta-1+\operatorname{Re}a)^2+(\operatorname{Im}a)^2]$. Any nonzero imaginary part removes its horizontal zeros while retaining the poles. Odd parity and the orbital-doublet transformation rule survive, but time-reversal symmetry need not: a complex mixture cannot simply be called the same symmetry state as the real mixture. Symmetry-allowed basis mixing alone says neither which coefficients minimize a material's free energy nor where its measured nodes lie. R5

A separate local geometric calculation compares the polar zeros of the unmixed models. Restricting to small polar caps, count the fraction of a unit sphere with imposed gap below $\epsilon$. Doubling a sufficiently small cutoff approximately doubles this fraction for the quadratic E2u zero, but quadruples it for the linear E1u zero. Eight cap calculations verify these limits. This is a geometric illustration, not a density-of-states or thermal-conductivity calculation; real velocities, impurity scattering and other nodes still contribute to a measured power law. R1 R5

Our separate two-component free-energy illustration shows how split transitions can arise without determining a material representation. Set $u=|\eta_1|^2$, $v=|\eta_2|^2$ and relative phase $\delta$, with dimensionless free energy $F=(t-1)u+(t-0.8)v+(u^2+v^2)/2+(0.4+0.15\cos2\delta)uv$. When both amplitudes are present, $\delta=\pm\pi/2$ minimizes the phase term. The first onset is $t=1$; the second is $t=(0.8-0.25)/(1-0.25)\simeq0.7333$. Five independent positive-amplitude grid comparisons agree with the analytic minima. These coefficients are chosen, not fitted to UPt3 or UTe2. R1 R2

Experimental criteria

The 2002 Joynt–Taillefer review weighs UPt3's phase diagram, nodal responses and magnetic coupling. Its abstract favors E1g or E2u on balance while acknowledging that neither explains all observations. That historical conclusion is not a current universal assignment. Machida and colleagues' later angle-dependent thermal-conductivity study instead interprets the B/C-phase data using E1u. The 2011 author abstract and 2012 journal abstract differ in the stated phase coverage; this entry uses the latter and does not claim a complete version comparison. R2 R4

Schemm and colleagues reported a UPt3 Kerr signal below about 0.46 K, separated from an upper susceptibility onset near 0.55 K. Their 1550 nm optical probe required an estimated local-heating allowance up to 50 mK; bulk heating inferred from mutual inductance was much smaller. Cooling in either sign of a 50 G field and then measuring with the field removed trained the signal's sign. These observations support additional time-reversal breaking in the lower-temperature phase within the experiment's controls; the optical signal alone does not select E2u. The Kerr onset is not the upper superconducting transition. R3

For interpretation, record field direction, heat-current direction, temperature and response model. Vorontsov and Vekhter find that minima can become maxima along a node in a quasi-two-dimensional d-wave vortex calculation as temperature or field changes. Their treatment includes vortex scattering and finite-energy thermal weighting; it neglects Zeeman splitting and has low-field limitations. This is a methodological warning, not a calculated f-wave response for UPt3. Comparing multiple axes and independent probes remains editorial guidance: the apparent success of one chosen response model does not uniquely select a material representation. R9

As editorial guidance for a null Kerr result, record illumination, surface, residual field, cooldown history and statistical sensitivity. Independent noise and domain signals add in variance, $\sigma_{\rm total}^2=\sigma_{\rm noise}^2+\sigma_{\rm signal}^2$. A local constructed example, not a reanalysis of the cited measurements, verifies this identity for uncorrelated sequences. The equation requires its independence assumption; a measured absence of increased scatter is a constraint at finite sensitivity, not proof that all possible microscopic time-reversal breaking is absent. R8

Limits and disputes

Hayes and colleagues reported UTe2 Kerr and calorimetric signatures supporting a multicomponent state. Their low-field warmups were compared with measurements following positive and negative training fields; the interpretation includes domains. Ajeesh and colleagues later found no spontaneous zero-field Kerr signal in four single-transition samples spanning two growth methods. Their heat-pulse procedure repeatedly crossed the transition, and trained signals varied with position and sample. The selected manuscript's optical and susceptibility sample sets are different: one must not attribute the separate double-transition susceptibility sample to the Kerr set. Retain this disagreement and its conditions rather than declaring all UTe2 samples chiral or all possible states excluded. R7 R8

Orthorhombic D2h has one-dimensional ordinary point-group representations, unlike hexagonal E2u. A complex combination of distinct components can still break time reversal if the energetics permit it; multidimensional point-group representations are not a universal prerequisite. Spin degeneracy, orbital structure and accidental near-degeneracy must be specified. UTe2's discovery evidence for unusual spin pairing does not independently choose an f-wave basis or a unique multicomponent state. R1 R6 R7

The local spherical polynomial examples omit the multiband and nonsymmorphic structure relevant to UPt3. Micklitz and Norman analyze its P63/mmc space group and explain how fractional-translation phases can alter odd-parity line-node constraints on a Brillouin-zone face intersected by the Fermi surface. The answer depends on spin-orbit coupling, time-reversal symmetry and band pairing; a point-group polynomial alone does not contain these ingredients. Their analysis supplies a missing theoretical context, not a unique experimental pairing assignment. Our chosen Ginzburg–Landau example does not reproduce the material phase diagram, and this entry supplies no calculated surface spectrum or bulk topological invariant. R10

Discovery implications

The following is editorial comparison guidance: specify the crystal symmetry and basis first. Then test whether one proposed state explains phase splitting, directional nodes, spin response and symmetry-breaking observations under compatible conditions. Preserve null results, competing interpretations and source-version differences alongside positive evidence. Full scientific acceptance, complete final-version comparisons and raw-data review remain open; the model calculations here constrain language and logic, not the pairing state of a new material. R1 R3 R5 R8

Related topics

Compare p-wave spin and chirality, dxy directional evidence, d-wave phase tests, ferromagnetic fluctuations, and topological superconductivity.

Material cases

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