Research

This work develops numerical methods for the quasiclassical theory of superconductivity and superfluid ³He. The specific target is confined geometry: films and slabs thin enough that surface effects reach the entire sample. In such systems the order parameter is strongly modified from its uniform bulk form, quasiparticle states appear that have no bulk analogue, and phases become stable that cannot exist in bulk at all.

The workhorse is the Eilenberger–Riccati formulation of quasiclassical theory. The current implementation is a finite-element solver, self-consistent on the Matsubara axis, executed on GPU.

Method

Quasiclassical theory reduces the Gor'kov equations to transport equations along classical trajectories on the Fermi surface. The Riccati parametrization rewrites those equations in terms of coherence amplitudes that satisfy the normalization condition identically, so it cannot be violated by discretization error. The cost is a nonlinear equation.

Seja and Löfwander (Phys. Rev. B 106, 144511) solve the scalar case with a discontinuous Galerkin finite-element method, self-consistently and for general device geometries. This work shares that formulation and extends it in three respects: the coherence amplitudes are matrix-valued rather than scalar, the self-consistency is carried on the Matsubara axis, and the solve is executed on a GPU.

The matrix generalization is not a change of bookkeeping. What remains on the coherence amplitude once the particle–hole index is absorbed is a 2×2 matrix in spin, carrying one singlet and three triplet components — all four required for the multi-component order parameters of superfluid ³He and of unconventional superconductors. In the numerics the components couple, the order parameter and the coherence amplitude no longer commute, and the Newton step acquires a Kronecker-structured Jacobian.

Verification

A converged solver has found a fixed point of its own iteration, which need not be the solution of the equations it was meant to discretize. This solver is therefore checked against references external to it: an exact analytic root, an independent Runge–Kutta integrator sharing no code with it, and a deliberately failing linearized control — because without such a control, two codes may agree merely by omitting the same term, and the comparison measures nothing.

Applied to a superconductor–normal-metal–superconductor junction, the solver reproduces the proximity effect: the order parameter is suppressed in the normal layer and recovers toward its bulk value, while the pair amplitude penetrates the normal metal. Superconducting correlations therefore exist where the order parameter does not.

Results to date lie in the singlet sector. No triplet state is constructed, and the triplet physics remains unexercised. They were produced by the continuous-Galerkin (SUPG) solver; a discontinuous formulation is under validation and its results are not reported here.

Figures and quantitative agreement are in the manuscript.

Direction

The scale that governs confinement is the coherence length ξ₀ = ħv_F/2πk_BT_c: the distance over which the order parameter can vary, and the scale on which the quasiclassical approximation is defined.

Vorontsov's review of Andreev bound states in films and slabs (Phil. Trans. R. Soc. A 376, 20150144) describes what occurs at D ∼ 10 ξ₀: transitions absent in bulk, phases breaking time-reversal or translational symmetry, and in superfluid ³He a competition between the B, planar and stripe phases decided by the structure of a multi-component order parameter rather than by its magnitude. Nanofluidic experiments have reached D/ξ₀ = 1 (Heikkinen et al., Phys. Rev. Lett. 134, 136001). At that thickness the sample has no bulk region.

Interpreting such experiments requires theory that is quasiclassical, matrix-valued, and free-energy-resolved. The first two are implemented. The third is not.

Background

B.S. Aerospace Engineering and Mechanics, University of Minnesota, followed by thirty years of independent consulting in computational modeling and systems analysis across thermal, fluid and acoustic systems. Graduate Research Associate in the Department of Electrical Engineering and Computer Science at Montana State University, 2011–2012, implementing CUDA parallel architectures for fluid-flow simulation and signal processing, alongside graduate study in electrical engineering.

The numerical machinery carries over more directly than the fields suggest: advection-dominated transport, characteristic sweeps, stabilization at gradient layers, and verification of a code against something that is not itself.

Graduate coursework in physics at Montana State University begins in fall 2026.

Manuscript in preparation. Correspondence welcome — elizabeth@elizabethburnim.com