Early after the discovery of HF superconductivity theoretical attempts were made to understand how a Cooper pair condensate can emerge out of the heavy Fermi liquid.
The theoretical studies included possible phononic origins of pairing in the Kondo lattice, potentially connected to the Kondo volume collapse [Raz84, Gre84], or pairing off electrons in the Anderson model with a small attractive interaction amongf electrons of unknown origin [Ohk84]. It turned out, that evens-wave superconductivity in the f-system would be anisotropic1: It can thus also lead to rather unconventional features such as a gap function featuring line nodes on the Fermi surface [Ohk84].
Since HF superconductivity was early associated with an unconventional pairing mechanism, it has in fact most often been assumed to appear among thef-electrons. Thus, the most promising model on this route is the periodic Anderson model, sometimes modified such that thef-electrons obtain a small dispersionεfk around the average f-level εf, i.e., the kinetic energy term for the f-electrons reads
Hkinf =X
kσ
εfk+εf
fkσ† fkσ .
The possibility of superconductivity in this Yoshimori-Kasai model [Yos83] has been studied early after the discovery of the first HF superconductors CeCu2Si2 and UBe13and gave first theoretical insight into the differences of HF superconductivity to conventional BCS theory [Tac84, Xu 87]. Some important issues of these studies should be noted here. It is assumed thatU,|Ef|, V are rather large such that the system constitutes a Kondo lattice but charge fluctuations are still allowed. Moreover, it is assumed that Cooper pairing takes place among f-electrons since the corresponding critical temperature seemed to be much higher than forc-electrons.2 While in BCS theory a frequency cutoff at the Debye frequency ωD is introduced, in HF systems the attractive interaction between the heavy quasiparticles is limited to a frequency range up to the Kondo temperature,ω ∼TK. Above TK, the Kondo effect leads to a repulsion of the electrons. The replacement ofωD byTK yields a critical temperature proportional to the Kondo temperature [Tac84]
Tc = 1.13 TKexp − 1 gc/fρ0(0)
!
,
where gc/f is the renormalized BCS coupling constant for either conduction band electrons or f electrons. The DOS of the conduction electrons at the Fermi energy is denoted by ρ0(0). One would thus already expect a rather lowTc since typically TK ≪ωD. Another
1 In two dimensions, an extended s-wave symmetry of the gap function is given by ∆sk =
∆s0(cos(kxa) + cos(kya)), whereadenotes the lattice constant.
2 Recent theoretical studies of HF superconductivity in fact focused onc-f-pairing, see e.g. [Mas13].
important difference to BCS theory is that the gap function is not equal to the order parameter [Xu 87].
5.1.1 Static Mean-Field Description of HF Superconductivity
Before the discussion of superconducting order emerging in the dynamical mean-field approximation for the KLM, it is instructive first consider the static MF treatment of superconductivity in the KLM. The MF description of the KLM in the paramagnetic phase has been introduced in chapter 3, cf. Eq. (3.5). The f degrees of freedom introduced there mainly contribute to the flat and weakly dispersive parts of the band. In case of a superconductor, these very parts of the band are of particular interest as will be shown in the following. As a first simple model, a mean-field BCS-type pairing term can be added to the Hamiltonian in Eq. Eq. (3.5) both in the conduction (c-) and f-band.
Another possibility is to add a hybrid c-f-pairing term ∆cf. The latter type of pairing was recently proposed to explains-wave superconductivity in HF systems [Mas13]. The general mean-field Hamiltonian for superconductivity in the Kondo-lattice can be written in a composite (c-f) ⊗(particle-hole) structure:
HKLMSC-MF =X
Although it should be noted that all three types of off-diagonal order lead to qualitatively similar quasiparticle bands and gap structures, here the focus is, however, on a c-f-pairing term. The corresponding bandstructure for a non-zero ∆cf is shown in (Fig. 5.1), illustrating several new features in comparison to the paramagnetic case in (Fig. 3.4). First of all, due to the introduction of hole degrees of freedom, the Hamiltonian matrix has four eigenvectors and thus the number of quasiparticle bands is doubled. On the one hand, a pair of narrow particle- and hole-like bands is present around the Fermi energy. They originate from the formerly non-dispersivef-levelεf. On the other hand, a corresponding pair of wide dispersive bands stemming from thec-electrons is present. The introduction of an off-diagonal superconducting order parameter (in this case∆cf) has two effects. First, it introduces a full superconducting gap∆0 around the Fermi energy. Secondly, it also gaps the narrow flat bands from the upper and lower dispersive band, respectively, atEk ≈0.1 in (Fig. 5.1). The corresponding “side-gap” will be referred to as∆>. Heavy-fermion pairs should appear as a pronounced weight in the flat particle and hole band close to the Fermi energy, with a full gap∆0. The connection of the gap∆> to Cooper pairing is, however, far from clear.
Actually, the gap ratio ∆0/∆> can be tuned by using different mean-field parameters
Figure 5.1: Typical renormalized quasiparticle bandstructure of the paramagnetic KLM on the two-dimensional square lattice in the mean-field picture. Left: Weight of c-states.
Right: Weight of f-states. The color intensity represents the amount of weight and gray corresponds to zero weight.
∆c, ∆f and ∆cf, even to an almost vanishing ∆> by increasing ∆f to high values together with a finite ∆cf. Due to quasiparticle scattering and finite lifetime effects, the gap ∆> is also likely to be filled. Below, these structures will be investigated in more detail by means of DMFT.
A similar effect in the bandstructure emerges in the AFM phase of the KLM (cf. Ch. 4), where the mean-field Hamiltonian can be written formally equivalent to (5.1). There the gapped structure of four bands in combination with finite lifetime effects eventually lead to the observation of the spin resonance in the DOS, cf. (Fig. 4.1). However, due to the non-zero imaginary part of the self-energy, ImΣ, the avoided crossing appears as a simple crossing and the spectral density in-between is either enhanced or reduced, depending on sign ofω and spin, which eventually produces the resonances. A similar observation can thus be expected for DMFT results in the SC phase.
Figure 5.2: Spectral functionA(εk,ω) for the KLM in the SC phase at a fillingn = 0.9 and with J/W = 0.2. A two-dimensional square-lattice DOS and dispersion is used for the plot.
For a comparison of the dispersion obtained in a static MF treatment, a brief “preview”
to DMFT results shall be discussed at this point already. In figure (Fig. 5.2) a typical spectral function Ak(ω) for the superconducting phase of the KLM at n= 0.9, J/W = 0.2 with a two-dimensional square-lattice DOS obtained via DMFT is shown.
One observes a hybridized bandstructure, fairly broadened by finite imaginary parts of the self-energy. Right at the Fermi energy ω = 0, a full gap in the flat band of heavy quasiparticles can be observed. Actually, the flat part of the lower band splits into particle and hole components as in the structures obtained in a static MF description, cf. (Fig. 5.1).
The flat structures are rather sharp and thus resemble quasiparticles of a well-defined heavy Fermi liquid. At frequenciesω ≈ ±0.15 the splitting between flat and strongly dispersive parts is visible. For positive frequencies the gap∆> is almost well developed, while for negative frequencies it is almost completely smeared out except for a slightly visible lack of spectral weight.
The mean-field description thus provides a good starting point to interpret the spectral structures arising in DMFT results of the KLM, not only in the paramagnetic, but also in the superconducting phase. Especially the splitting into a four-fold bandstructure observed in the static MF description helps to clearly identify the blurred quasiparticle bands in the DMFT-spectra.