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Spatial correlation of fluctuations in multi-component superconducting systems

We show that spatial variation and correlation of superconductivity fluctuations in a two-band model are scaled by two characteristic lengths. This results in substantially more complicated picture compared to one-band systems. In particular, short-range correlations are always present in a two-band scenario, even near the phase transition point.


Introduction
The multi-band superconductivity mechanisms with interband pairing channels have been became more and more actual [1,2,3]. The rich physics included here is reflected by the unusual properties of relevant compounds. In particular, the discovery of novel vortex structure in magnesium diboride [4], interpreted as type-1.5 superconductivity, points unambiguously to the nontrivial modification of coherency in multi-gap systems [5]. Recently the existence of type-1.5 superconductivity was suggested also in strontium ruthenate [6].

Correlation lengths in two-band superconductivity
The spatially inhomogeneous two-band superconductivity is described (see e. g. [11]) by the following gap equations in the vicinity of critical point Here v F α are the Fermi velocities and ρ α are the electron densities of states at the Fermi level in the corresponding bands, α = 1, 2. It is supposed that the superconducting ordering is created by the intraband effective electron-electron attractions with constants W 11,22 < 0 and by interband pair-transfer interaction (constant W 12 = W 21 ). These interaction channels are supposed to be operative in the energy layer ±hω D around the Fermi level intersecting both bands. The temperature of the bulk superconducting phase transition T c follows from the linearized equations (1) in the spatially homogeneous case.
We consider the small deviation from the bulk superconductivity representing ∆ α (r) = ∆ ∞ α +|∆ ∞ α | f α (r) where ∆ ∞ α are the bulk gaps. The system of linearized equations for the fluctuations f α (r) follows from Eq.
(1), namely with If interband interaction channel is absent, Ξ 12 = Ξ 21 = 0 and the system (5) splits into two independent equations. For this case ξ 1,2 are the coherence lengths which describe the spatial behaviour of superconductivity in the autonomous bands. For finite interband coupling the quantities ξ 1,2 cannot be treated as relevant coherence lengths. Instead, one obtains [11] on the basis of Eqs. (5) the length scales ξ ± for the spatial variation of f α (r), and The coherence lengths found cannot be attributed to the separate bands. Their temperature dependencies are shown in Fig. 1. As c 2 A1A2 = 1 at the phase transition temperature T c , one observes the critical (diverging) behaviour of ξ − (T ) near T c , while ξ + (T ) remains finite. The maximum of ξ − (T ) below T c is characteristic Fig. 1 The coherence lengths ξ − and ξ + vs temperature. Parameters: feature of a superconductor with tiny interband pairtransfer interaction. That non-monotonicity is correlated with the temperature where the band with weaker superconductivity becomes active. The second peak of ξ − (T ) disappears with increase of interband coupling.

Spatial behaviour of fluctuations
By considering the dependency on x-coordinate only, the general solution of the system (5) has the form where The temperature dependencies of the amplitudes p ± 1,2 are depicted in Fig. 2 for weak interband pairing. For T → T c the spatial behaviour in the band with stronger superconductivity is scaled predominantly by the critical coherence length (p − 1 ≫ p + 1 ). In this case the superconducting ordering is very close to the one-band situation where only one (critical) coherence length is of relevance. Larger discrepancy between intrinsic intraband critical points and the weakness of interband  interaction play in favour of that trend. With the lowering of the temperature (Fig. 2) or with increase of interband pairing (Fig. 3) the non-critical channel becomes important for the band considered. For the band with weaker superconductivity both critical and non-critical channels contribute near T c (p + 2 ∼ −p − 2 ). From Fig. 4 we clearly see that for shorter distances f 2 (x) scales by ξ + , while for longer distances by ξ − . Moreover, the interplay of critical and non-critical contributions can result in the non-monotonic dependence of f 2 (x).

Correlation functions of two-band superconductivity
Critical and non-critical coherence lengths (10) appear as scaling factors in the expressions for the correlation functions of superconductivity fluctuations (14) where η α (r) = δ α (r) − ∆ ∞ α and δ α (r) are the nonequilibrium gap order parameters. By using the statistics, determined by the non-equilibrium free energy, one finds in the Gaussian approximation [15] with As interband pairing vanishes, Γ αα transforms into the correlation function of independent band, scaled by the relevant coherence length, and Γ 12 → 0. If interband interaction is present, the picture becomes more complicated. For shorter distances |r − r ′ | ≪ ξ + < ξ − the functions Γ αα decrease in space as |r − r ′ | −1 , and this dependence may be caused by the different contributions in the bands stemming from critical or noncritical coherency channels. As the distance separating fluctuations increases, long range correlations, determined by ξ − , become dominating. Therefore, there exists a possibility for the interchange of the leading role between the critical and non-critical contributions in the spatial behaviour of Γ αα . At T ≈ T c the short- involved. In the one-band case only long range correlations ∼ |r − r ′ | −1 are present in this temperature region. More detailed analysis of the correlation functions (15) can be found in Ref. [15].

Conclusion
It was demonstrated that two-gap inhomogeneous superconductivity involves more complicated physics compared to one gap situation. Critical and non-critical coherence lengths, found in two-band scenario, scale the spatial behaviour of superconducting fluctuations as well as their correlation functions. The concrete picture depends substantially on the temperature and the strength of interband interaction. However, even in the vicinity of critical point both short-range and longrange correlations are essential.