Chiral pairing: time reversal, topology and boundary evidence

Scope

Chiral pairing describes a superconducting order with a handed orbital structure, such as a complex combination of two directional basis functions. Three questions must remain separate: whether time reversal is broken, whether the pairing is chiral, and whether a specified gapped Hamiltonian has a nonzero topological invariant. A Kerr signal, a complex coefficient or a named lattice does not answer all three. Kallin and Berlinsky's 2016 review is a historical guide; its candidate-material assignments are not current confirmations. R2

Broken time reversal need not mean orbital chirality. For example, frustrated interband pair scattering can favor complex relative phases among otherwise s-wave gaps. Stanev and Tešanović studied this possibility in a three-band model. An illustrative vector of gaps on three inequivalent labelled bands, $(1,e^{2\pi i/3},e^{-2\pi i/3})$, cannot be made equal to its conjugate by one common phase, although each component has no angular winding. This statement assumes only a common U(1) freedom; it is not a stability calculation or a material assignment. R8

Mathematical assumptions

Read and Green distinguish weak- and strong-pairing phases of a two-dimensional spinless complex p-wave model. Their continuum chemical potential is measured relative to the band bottom: its zero marks a gap-closing transition in that model. The broken symmetry alone does not distinguish the two phases. This continuum criterion must not be copied unchanged into a lattice or multiband calculation. R1

For an independent lattice illustration, set hopping and pairing scales to one and use $H(\mathbf{k})=\mathbf{h}(\mathbf{k})\cdot\boldsymbol{\tau}$, with $\mathbf{h}=(\sin k_x,\chi\sin k_y,-2\cos k_x-2\cos k_y-\mu)$ and $\chi=\pm1$. In this Nambu convention the off-diagonal gap is $\sin k_x-i\chi\sin k_y$. Define the occupied-band Berry connection as $\mathbf{A}=i\langle u_-|\nabla_{\mathbf{k}}u_-\rangle$. With the orientation $(k_x,k_y)$, the lower-band Chern number for $\chi=1$ is $0,-1,+1,0$ in the intervals separated by $\mu=-4,0,4$. Reversing $\chi$ reverses its sign. R1 R2

Sixteen calculations compare two mesh sizes, four chemical potentials and both handedness choices. Berry-link phases from explicit occupied eigenvectors agree with independently summed spherical solid angles. Four analytic gap-closing checks identify the excluded critical points; no gapped-band invariant is assigned there. The same complex gap can thus accompany a zero or nonzero Chern number. The calculation fixes the gap by hand and omits self-consistency, realistic bands and disorder. R1

A boundary calculation illustrates a different distinction. In Huang, Taylor and Kallin's ideal sharp-edge continuum treatment, occupied bound states and continuum states both contribute to the integrated current. For axial harmonic index $m=1$, their respective values are $1/2$ and $-1/4$ in units $\hbar k_F^2/(4\pi m^*)$ per spin component. For integer $m>1$, each integrated contribution vanishes in this limit despite the chiral edge branches. Independent angular quadrature reproduces the analytic sums for six values of $m$. R3

That cancellation is not a universal absence of current in higher-order chiral pairing. The same paper shows that making the density vary slowly over distances much longer than the coherence length restores a current; lattice effects also matter. A sharp specular boundary, a smooth trap and a disordered multiband surface are different problems. The calculation above does not predict a magnetic-field amplitude for a real sample. R3

Experimental criteria

Kerr rotation probes an optical response. In the clean Galilean-invariant limit, a spatially uniform electric field couples to center-of-mass motion and the charge Hall response vanishes even for complex pairing. At the single-particle level, the clean one-band current-vertex structure also gives a cancellation; multiband and impurity processes require separate treatment. Optical frequency, refractive response and geometry matter. Neither a null Kerr measurement nor one nonzero angle can be converted directly into a Chern number. R1 R2

For UPt3, retain Schemm and colleagues' distinction between the Kerr onset near 0.46 K and the upper susceptibility onset near 0.55 K, including optical heating and field-training controls. For Sr2RuO4, Pustogow and colleagues' oxygen NMR response constrains a simple fixed-d-vector interpretation. The later 2026 muon Knight-shift study also has field-dependent extraction and low-temperature magnetization limits. These are complementary constraints, not a license to restore a historically favored chiral p-wave label as settled fact. R4 R5 R9

Mielke and colleagues studied KV3Sb5, not the whole AV3Sb5 family interchangeably. The additional zero-field exponential relaxation rate $\Gamma$ rises below charge order near 80 K and stays approximately constant across the superconducting transition near 1.1 K. This is not a statement about every muon signal: field-cooled transverse-field measurements at 5 and 10 mT show increased Gaussian damping below that superconducting transition, attributed to the flux-line lattice. Zero-field data alone do not establish time-reversal breaking: nuclear and electric-field-gradient effects can contribute. The authors combine zero-field and high-field evidence for their charge-order interpretation without resolving the orbital-current pattern. Residual-field compensation better than 30 mG and longitudinal decoupling at 50 G belong to the zero-field controls, not the transverse-field vortex measurement. The 80 K feature is not a superconducting chirality onset. R6

Chen and colleagues studied CsV3Sb5 with tunneling and Josephson spectroscopy. Their selected manuscript distinguishes bulk signatures near 2.8 K from a surface transition near 2.3 K, and uses a calibrated 300 mK electron temperature. Spatial gap and conductance modulations motivate their roton-PDW interpretation. The authors explicitly state that roton excitations were not directly observed. A proposed complex vortex–antivortex pattern is therefore an interpretation to test, not a measured uniform chiral d-wave order parameter. R7

Limits and disputes

Local magnitude alone does not resolve handedness: $|\Delta|^2=|\Delta^*|^2$. Four complex spatial examples verify this identity without making a tunneling-response model. A Josephson signal can support superconducting coherence, while determining orbital phase structure requires suitable reference geometry and controls. Fourier peaks, gap-depth maps and zero-bias peaks must retain their individual evidential roles. R7

Do not require every time-reversal-breaking state to belong to a two-dimensional crystal representation. Complex combinations of distinct representations or interband phases can behave differently. Likewise, half-quantum vortices can occur in equal-spin-pairing states without establishing orbital chirality. A strain response or vortex signature must be interpreted in the actual symmetry and spin-orbit setting. R2 R8

The preceding Chern numbers apply only to gapped two-dimensional Hamiltonians. A local three-dimensional extension illustrates the restriction: add $-2\cos k_z$ to $h_z$ and fix $\mu=-4$. Each fixed-$k_z$ slice has $\mu_{\rm eff}=-4+2\cos k_z$. In the same occupied-band convention, $C(k_z)=-\chi$ for $|k_z|<\pi/2$ and zero for $\pi/2<|k_z|\leq\pi$; it is undefined on the nodal slices $k_z=\pm\pi/2$. Solving all three components of $\mathbf{h}=0$ gives only $(0,0,\pm\pi/2)$. Sixteen numerical comparisons and this analytic zero classification check the chosen model, not a material or a calculation reproduced from the references. Thus a single gapped two-dimensional integer cannot be assigned unchanged to this gapless three-dimensional example. Edge spectra, thermal transport, optical response and screened equilibrium charge currents still require their own calculations. Complete scientific acceptance, final-version comparisons, raw-data review and material-specific modelling remain open. R2

Discovery implications

First identify the transition and its probe. Then specify the order-parameter basis, domain controls, bulk gap and Hamiltonian used for any topology claim. Compare positive and null evidence under compatible conditions, retaining charge-order and superconducting temperature scales separately. A useful discovery target is a state that satisfies these linked tests, not a material selected merely for a complex gap notation or a kagome lattice. R1 R4 R6 R7

Related topics

Continue with p-wave spin structure, f-wave candidate states, d-wave phase tests, topological superconductivity, and charge and density-wave pairing.

Material cases

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