Multiband strong-coupling superconductors with spontaneously broken time-reversal symmetry

Exceeding the Chandrasekhar-Clogston limit in flat-band superconductors: A multiband strong-coupling approach

“…The name "variational action" comes from the fact that inserting the stationary point conditions in the action yields an expression similar to the free energy derived in a variational approach [61]. From Z g = e −β = D[ φ, φ, ] exp(−S HS [ φ, φ, ]) ≈ e −S var [ φ,φ, ] under the assumption that the main contribution comes from the stationary point [62], the grand potential is ≈ S var [ φ, φ, ]/β [63,64,92]. We assume that the chemical potential is unchanged between the normal and superconducting states, as discussed in Refs.…”

“…(B14) λ −1 (q) defined as the FT of 1/λ(r i , τ ) satisfies Then, 1l = 1l , 2l = 2l . Using symmetries, the stationary point conditions can be rewritten as the Eliashberg equations given in Appendix A. λ −1 (q) is an oscillating function of frequency [64]. When T ω E , its magnitude becomes very large, meaning that any calculation involving λ −1 (q) and a sum over frequencies becomes a numerical sign problem.…”

“…[62] presented a way to derive the Eliashberg equations via functional integral methods. The derivation provides an expression for the free energy given in terms of solutions to the Eliashberg equations [63,64]. The latter approach is more easily generalizable to the case of multiple bands and the presence of a magnetic field and is therefore employed in this paper.…”

“…The latter approach is more easily generalizable to the case of multiple bands and the presence of a magnetic field and is therefore employed in this paper. Considerations of the free energy within Eliashberg theory are numerous [9,[57][58][59][60][61][62][63][64][65][66][67][68][69][70], but to the best of our knowledge no study of the CC limit within Eliashberg theory has been performed. One of the main results of this paper is an expression giving the free energy for a multiband system in a magnetic field in the spirit of Ref.…”

Proximity effects in a single- and two-band superconducting heterostructures: time-dependent Ginzburg-Landau approach.

“…Then, in this paper, we will study the vortex-state (Cooper pair density), magnetization as a proximity conditions function, via a new parameter λ ′ (see Fig. 1(b)) that consider the distance between these condensates, which depends on λ 0 , which is the penetration length, usually used in single-band condensates, which describes the entry of the magnetic field into the superconducting sample [1][2][3][4][5][6]15].…”