Self-calibrating thermal interferometry of vortex parity in a two-dimensional chiral superconductor

Abstract: A chiral superconductor carries chiral Majorana modes along its boundary, and the integer that counts them fixes everything that follows, yet that integer has never been measured together with a local parity observable on one object. Proximitized one-dimensional wires read fermion parity rapidly but diagnose bulk topology through a separate protocol. Here we show that a reconfigurable domain wall between regions of opposite Chern number in an intrinsic two-dimensional chiral superconductor performs both functions. Opened to its contacts the wall is a ballistic channel whose quantized thermal conductance counts its Majorana modes; closed, the same wall is a Fabry--Pérot resonator whose spectrum shifts by half a level spacing when the parity of the enclosed vortices changes, giving a two-level heat conductance. We derive the exact transmission, the elastic heat full counting statistics, and a theorem showing that linear-response heat scattering of a fixed quadratic problem resolves vortex parity but not the fusion channel of well-separated cores. An outside vortex hybridized with the wall is an intrinsic false positive; temperature, geometry and a finite-bias mean--noise test separates it. Rhombohedral-graphene parameters place submicron loops in the resolved regime at millikelvin temperatures, where chiral-domain reconfiguration and noise thermometry are both established.

• The paper proposes a device utilizing a reconfigurable domain wall and is able both to calibrate the Chern number and to measure the parity of vortices, using quantized thermal conductance and a Fabry-Pérot resonator in one device.
• The calorimetry principle is the Fabry-Pérot interference of Majorana branches with the sector switch defined by vortex parity.
• System performances reach high specific contrasts, compounded by transparency and feasibility checks, investments both into plausible materials and proposal confirms the practical potential.

Summary of the proposal

This paper proposes a thermal-conductance measurement that reads out, on a single object, both the bulk topological invariant and the vortex parity of an intrinsic two-dimensional chiral superconductor. The central object is a reconfigurable domain wall between regions of opposite BdG Chern number $\nu=\pm C$ in symmetry class D. By bulk-boundary correspondence such a wall carries $N=2|C|$ co-propagating chiral Majorana channels; for $|C|=1$ this is two Majorana branches with total chiral central charge $|C|$. The device operates in two modes. With point contacts opened, the wall is a ballistic channel whose quantized thermal conductance fixes $|C|$ through the Majorana quantum $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$ W/K² per branch. With the contacts partially closed ($c_i=\cos\theta_i\approx0.9$), the same wall becomes a Fabry–Pérot resonator whose resonance comb is shifted by half a level spacing depending on whether the enclosed number of vortices $n_v$ is even or odd — the Neveu–Schwarz versus Ramond spin structure of the closed Majorana edge.

The key physical mechanism is the standard branch-cut argument: transporting a Majorana operator around a contour enclosing an odd number of vortices converts antiperiodic into periodic boundary conditions via the sign $\sigma=(-1)^{n_v+1}$. The single-particle spectrum then takes values $\varepsilon_n = (2\pi\hbar v/L)n$ (Ramond, containing a zero mode) versus $N=2|C|$0 (Neveu–Schwarz). Because the thermal weight $N=2|C|$1 vanishes quadratically at zero energy, the Ramond zero mode carries heat only through its finite width; the sectors are distinguished by where the first heat-carrying level sits. This yields a two-level thermal conductance that flips each time the enclosed vortex parity changes.

The paper derives the exact transmission by summing the multiple-traversal geometric series,

$N=2|C|$2

together with its elastic heat full counting statistics in Levitov–Lesovik form restricted to positive energies. Notably, only the total perimeter enters, not the individual arc lengths — a structural difference from two-arm Fu–Kane/Akhmerov–Nilsson–Beenakker interferometers, where parity moves the outgoing particle species rather than the spectrum and therefore produces no heat contrast.

A precise no-go: fusion blindness

A theorem delimits what this observable can and cannot measure. For well-separated vortex cores, every quantity entering linear-response heat transport is a functional of the real antisymmetric Majorana-basis Hamiltonian matrix $N=2|C|$3. The Ising fusion label $N=2|C|$4 (eigenvalue $N=2|C|$5 for vacuum $N=2|C|$6, $N=2|C|$7 for fermion $N=2|C|$8) labels the occupation of a degenerate many-body manifold without changing $N=2|C|$9, and hence cannot change the linear-response thermal conductance. Both even-vortex fusion outcomes carry Neveu–Schwarz boundary conditions and identical single-particle combs. The interferometer is therefore explicitly not a fusion-rule measurement; achieving fusion sensitivity requires non-Gaussian $|C|=1$0-tunneling or charging-energy constraints, as in existing proposals. This is stated plainly rather than left implicit, which strengthens the claim's credibility: the retained observable is vortex parity, weaker than fusion but more constrained than generic spectroscopy.

Numerical contrast and material feasibility

At the dimensionless finite-size ratio $|C|=1$1 and contact return amplitudes $|C|=1$2, the Neveu–Schwarz sector conducts $|C|=1$3 times more heat than Ramond; the signed contrast exceeds $|C|=1$4 in magnitude in the sharp-resonance corner. A sign reversal of the contrast occurs near $|C|=1$5, where the ordering of sector conductances flips as the NS level exits the thermal window. Representative rhombohedral-graphene parameters ($|C|=1$6 mK from Han et al., $|C|=1$7 nm$|C|=1$8 from octalayer Landau fans, gap ratios $|C|=1$9–$|C|$0) give edge velocities $|C|$1–$|C|$2 m/s, three orders below the graphene Fermi velocity. At $|C|$3 mK, submicron loops reach $|C|$4 of order unity, placing them in the resolved regime accessible to existing graphene Johnson-noise thermometry (~5% quantized-thermal-conductance accuracy). Against Sr$|C|$5RuO$|C|$6, UTe$|C|$7 and Fe(Te,Se), RHG offers the largest resolving perimeter relative to the bulk coherence length at fixed $|C|$8, owing to its small Fermi wavevector.

Two caveats temper these numbers. The gap ratio is an illustrative benchmark, not a spectroscopic determination, so direct measurement of $|C|$9 would reduce the dominant scale uncertainty. And class D does not protect branch independence: intervalley mixing can shift or even null the parity contrast at isolated holonomies (e.g., $|C|$0 maps the two sectors' combs onto one another), although the open-contact plateau remains universal.

Systematics: false positives and coherence

The paper identifies an intrinsic false positive: a vortex hybridized with the wall from outside the contour contributes a low-energy $|C|$1 phase shift spectrally indistinguishable from an enclosed vortex, with spurious contrast growing to 89.6% of the true value at $|C|$2. Two discriminators are provided. Temperature dependence separates them because the true switch depends only on $|C|$3 (crossing zero at $|C|$4 for $|C|$5), while the artefact requires an additional parameter and does not collapse onto the same curve. More powerfully, the fixed-field geometric switch — rewriting the contour so the same pinned vortex passes inside or outside — toggles the true signal while leaving the exponentially distance-controlled hybridization approximately fixed; this control exists only because the wall is rewritable.

Heat full counting statistics supplies a third test: at a deliberately constructed mean-degenerate point (a clean NS loop and a hybridized R loop tuned to identical average current), the normalized second cumulants differ by $|C|$6, falsifying the mimic without any braiding operation. Partial coherence is treated exactly within a Büttiker-type damping model: the contrast halves at $|C|$7 for representative parameters. Since $|C|$8 has not been measured for an RHG chiral Majorana wall, coherent propagation around the loop remains an explicit experimental requirement rather than a derived property — though intrinsic inelastic scattering is expected weak because charge conjugation forbids ordinary electron–phonon coupling and the lowest local self-interaction carries six derivatives.

Experimental protocol

Three thermal tests are specified. First, the open-contact plateau calibrates $|C|$9 against a known quantum on the same wall before resonant operation. Second, field-driven sweeps give a two-level conductance switch correlated with independently identified vortex-entry events (using nanoSQUID derivative analysis to separate Meissner background from discrete entries); entry fields need not be equally spaced, and the signature is the switch itself, not periodicity. Third, and most cleanly, the fixed-field geometric switch rewrites the contour around a pinned vortex at fixed field, eliminating field-sweep ambiguity entirely. Falsification criteria are explicit: a sinusoidal response persisting at fixed vortex configuration indicates trivial continuous-flux interference; a nonquantized plateau invalidates the gapped-Chern interpretation (a Bogoliubov Fermi surface being one possible origin); and the clean null control is tuning $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$0, since even-$\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$1 phases are not strict nulls.

Limitations and open questions

The construction rests on several assumptions stated candidly. Applying the $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$2 wall picture to imaged RHG domains assumes opposite chirality implies opposite BdG Chern number — present imaging establishes chirality reversal but not this topological identification, so the gappedness criterion $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$3 must hold along the relevant gate path. Arbitrary micron-scale closed-contour writing around an individual Abrikosov vortex exceeds what deterministic domain reconfiguration has demonstrated; chip-integrated vortex manipulation in NbSe$\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$4 shows single-vortex positioning is feasible, but its combination with rewritable chiral walls in RHG does not exist yet. The optional charge-based cross-check (Fu–Kane conductance $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$5 or $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$6) would require local coexistence of a charged QAH edge with the superconductor, also undemonstrated. Finally, whether the predicted signatures can be observed at all reduces to two open experimental quantities: the actual coherence length $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$7 of an RHG domain-wall Majorana mode, and a spectroscopic determination of $\pi^2 k_B^2 T/6h \simeq 4.732\times10^{-13}$8.

Conclusion

The paper establishes a self-calibrating scheme in which one rewritable domain wall serves simultaneously as a Chern-number thermometer (quantized plateau) and a vortex-parity interferometer (half-level spectral shift read as a two-level heat conductance). The information content is sharply delimited — parity yes, fusion channel no — and the dominant systematics are identified and given concrete discriminators. Whether the required contour pinning, bulk-gap topology identification, and loop-scale phase coherence can be realized in rhombohedral graphene are precisely formulated questions the theory leaves to experiment.