Condensation of fermion zero modes in the vortex

Abstract

The energy levels of the fermions bound to the vortex are considered for vortices in the superfluid/superconducting systems which contain the symmetry protected plane of zeroes in the gap function in bulk. The Caroli-de Gennes-Matricon branches with different angular momentum quantum number n approach zero energy level at pz → 0. Such condensation of the energy levels is the consequence of the bulk-vortex correspondence in topological superfluids/superconductors. In a given case this is the connection between the Dirac line of zeroes in the bulk spectrum and the level condensation in the vortex core. The density of states of the bound fermions diverges at zero energy giving rise to the √ Ω dependence of DoS in the polar phase of superfluid 3 He rotating with the angular velocity Ω and to the √ B dependence of DoS for superconductors in the (dxz + idyz)-wave pairing state. PACS numbers:The spectrum of the low-energy bound states in the core of the symmetric vortex with winding number m = ±1 in the isotropic model of s-wave superconductor was obtained in a microscopic (BCS) theory by Caroli, de Gennes and Matricon, 1 see Fig. 1 (left):Here p z is the momentum of the bound states along the vortex line, and n is related to the angular momentum quantum number L z . This spectrum is two-fold degenerate due to spin degrees of freedom. The level spacing -the so called minigap -is small compared to the energy gap of the quasiparticles outside the core, ω 0 ∼ ∆ 2 /E F ∆. For the chiral superfluid/superconductor with an odd winding number of the phase of the gap function in momentum space (i.e. ∆(p) ∝ (p x + ip y ) N with odd N ), the spectrum of fermions in the symmetric vortex is modified, 2,3 see Fig. 1 (right) for the most symmetric vortex in the Weyl superfluid 3 He-A:The spectrum contains the zero energy states at n = 0. In the two-dimensional case the n = 0 levels represent two Majorana modes. 4,5 The 2D half-quantum vortex, which is the vortex in one spin component, contains single Majorana mode. In the 3D case the Eq.(2) at n = 0 describes the flat band in the spectrum of the bound states:6 all the states in the interval −p 0 < p z < p 0 have zero energy, where p 0ẑ and −p 0ẑ mark the positions of two Weyl points in the bulk material.7 The connection between the positions of the Weyl points in bulk, and the boundary of the flat band in the vortex core is the consequence of the bulk-vortex correspondence in topological materials. For numerical simulation of the flat band in the core of the 3 He-A vortex see Ref.8 . The n = 0 level with the flat band appears also in spin-singlet superconductors with (d xz + id yz )-wave pairing. 9This is because the gap function ∆(p) ∝ p z (p x + ip y ) has an odd chirality number N = 1 and the Weyl points in bulk. Such pairing has been suggested in the heavy-fermion compound URu 2 Si 2 .10,11 The n = 0 level with the flat band does not appear in the core of superconductors with (d x 2 −y 2 + id xy )-wave pairing, 9 where the gap ∆(p) ∝ (p x + ip y ) 2 has N = 2, and the spectrum co...

Graphite on graphite

“…They lead to the singularity in the density of states of the fermions bound to the vortex. [30][31][32] .…”

“…They lead to the singularity in the density of states of the fermions bound to the vortex. [30][31][32] .Graphite is one of the realizations of the approximate nodal line semimetal (for the review, see 19 ). In the model, where some small hopping elements are neglected, the Bernal graphite has two vertical Dirac lines running between the H points in the 3D graphite band structure.…”

Preformed Cooper pairs in flat-band semimetals

“…It suffices to separate the spatial variables (r 1 + r 2 )/2 = R and r 1 − r 2 = r. After integrating out the fermions Ψ σ (r, τ ), Ψ σ (r, τ ) in (16), we arrive at the bosonic action for ∆(R, r). The solution of the linearized saddle point equation ( 4) of the bosonic action is given by (11) and Eq.…”

“…The study of flat-band materials is proposed to be of significant importance for understanding the mechanisms of high-temperature superconductivity [14]. One of the reasons for that is the singular density of states at the flat-band which might support elevated superconducting transition temperatures [15][16][17][18][19][20]. On the other hand, the flatness of the band dispersion guarantees the strong interaction limit of Cooper instability discussed above.…”

Dirac and Weyl fermions: from Gor’kov equations to Standard Model (in memory of Lev Petrovich Gor’kov), "Письма в Журнал экспериментальной и теоретической физики"

“…The Weyl fermions in bulk produce the dispersionless flat band of fermions with zero energy living in the vortex core 36,37 and the Fermi arc of the edge states 38,39 . The nodal lines in bulk superconductors lead to the condensation of the Caroli-de Gennes-Matricon levels in the vortex 40 and to the flat band of the edge states 41,42 .…”