A quasiparticle mixes particle and hole amplitudes

For a translationally invariant spin-singlet mean-field state, group an electron amplitude at momentum k with the corresponding opposite-spin hole amplitude at −k. Relative to the chemical potential, their diagonal entries are ξ and −ξ. The off-diagonal pairing field mixes them. This particle–hole doubling is a useful representation of the fermionic problem, not two independent physical particles for each original electronic state. The negative eigenvalue accompanies the positive one through the redundancy of the BdG description.

The determinant gives energies whose squares equal ξ² plus the squared gap magnitude. Choose the positive branch as the quasiparticle excitation energy. Even at ξ=0 this energy is the nonzero gap magnitude in a fully gapped state. Thus the normal-state Fermi surface becomes the location of minimum excitation energy rather than a set of zero-energy electron states. A spatially varying gap, a vortex, or a boundary requires a differential or real-space matrix problem; the homogeneous block alone does not describe localized core or surface levels.

Solving the eigenvector equation and normalizing gives particle weight one half times one plus ξ/E, with the complementary hole weight. Far above the chemical potential the positive-energy excitation is mostly particle-like; far below it, mostly hole-like. At ξ=0 the weights are equal. The relative phase between u and v tracks the pairing phase under the declared matrix convention. Scattering and response matrix elements can depend on products of these amplitudes, so retaining only the energy spectrum can omit physically important interference factors.

The thermal Fermi factor f(E) specifies how often a quasiparticle mode is occupied. It is different from either coherence weight. At zero temperature, no positive-energy quasiparticles are occupied, yet the electron occupation is generally v squared. At finite temperature the occupation receives a contribution from both adding a quasiparticle and removing its paired partner. In the spin-balanced equilibrium model this gives v squared plus the particle-minus-hole weight times f. Setting electron occupation equal to f(E) would incorrectly erase the paired ground-state distribution.

In a clean isotropic fully gapped model, converting the normal-state energy density of states to quasiparticle energy produces an inverse-square-root gap-edge feature. Ideal singularities are rounded by lifetime effects, disorder, temperature, and instrumental resolution. A tunneling spectrum is also weighted by matrix elements. Observing a gap-like suppression does not by itself determine the sign of the pairing function, and the absence of an infinitely sharp peak does not invalidate particle–hole mixing. One needs to distinguish the underlying excitation structure from the measurement process.

The quantum lab illustrates normalized two-component weights without implementing fermionic quasiparticle dynamics. Reset and set both controls to θ=60 degrees and φ=0 degrees, predicting basis weights three quarters and one quarter. Reset and set θ=120 degrees and φ=0 degrees, predicting the reversed weights. Assign these geometrically to particle and hole components. They correspond to opposite ξ/E of magnitude one half if used as BdG eigenvector examples. Playback only precesses relative phase; the lab supplies neither a pairing field nor a thermal Fermi factor or electron-occupation readout.