Superconductivity in Empty Bands and Multiple Order Parameter Chirality

Recent experiments have shown rotation of the plane of polarization of light reflected from the surface of some superconductors. The photon energy exceeds the electronic bandwidth, so that completely filled or completely empty bands must play a role. We show that in strong-coupling theory a Coulomb interaction can produce an order parameter in the unoccupied band that explains the observations. Thus the phenomenology puts tight constraints on the form of the order parameter in different bands. We propose that the experiments have detected, for the first time, the existence of a superconducting order parameter in a band far from the Fermi energy. This is only possible because of the sensitivity to delicate symmetries: a positive Kerr effect indicates that time reversal and certain mirror symmetries are broken in the ordered phase. Furthermore, detailed analysis of the results implies that in UPt3 there exist bands that have different order parameter chiralities, opening up complex new possibilities for topological superconductivity.

The subject of unconventional superconductivity is now over 30 years old, and the prime driving force in the field has been the determination of the form of the order parameter. This is important in itself for possible applications in quantum computing, and also in order to determine the mechanism of superconductivity. This issue has become more urgent in the era of topological superconductivity. The determination has generally proved to be surprisingly difficult: unambiguous identifications remain remarkably few in number.
Experiments at optical frequencies are rarely useful in this connection. Superconductivity is fundamentally a low-frequency phenomenon, and modification of interband matrix elements by superconductivity are miniscule. The Kerr effect 1 , rotation of the plane of polarization of reflected light through an angle θ K , is an important possible exception to this venerable rule, since it tests time-reversal symmetry breaking.
The Kerr effect in superconductors was already an active area of research both in experiments 2 and theories 3,4 in the 1990s. However, positive experimental results are relatively recent. A nonzero signal that begins at the onset of superconductivity and grows as temperature decreases has been observed in Sr 2 RuO 4 5 , UPt 3 6 , and URu 2 Si 2 7 . The simplest theories of the pure system do not give a result large enough to explain the observed magnitude of θ K ≈ 10 −6 . In Sr 2 RuO 4 this has been interpreted in different ways. One is to invoke impurity scattering [8][9][10] . Another is to attribute the effect to interband transitions between bands that cross the Fermi energy 11,12 . This paper focuses on clean UPt 3 , though we comment on other compounds below. This is the simplest case for our purposes, since in UPt 3 the bandwidth is less than the energy (0.8 eV) of the light 13,14 (see Fig. 1 for a schematic plot). Thus the transitions induced by the light can only connect partially occupied with completely unoccupied bands and we have a clear-cut example of a material for which the above theories can be ruled out.
We propose that the Coulomb interaction induces a nonzero order parameter in the unoccupied bands that is responsible for the Kerr effect. This implies that the Kerr effect experiments probe, for the first time, a superconducting order parameter in an unoccupied band distant from the Fermi energy. A somewhat similar effect was proposed for semiconductors with small gaps 15 and may have been seen in a completely occupied band relatively near the Fermi energy in LiFeAs 16 . But this would be the first time that the order parameter in a distant band has been seen in experiments. Furthermore, since the present treatment is purely phenomenological the conclusions are independent of the microscopic mechanism of superconductivity.
Reflection of light of frequency ω incident on a sample along the z-axis is controlled by the components ε xx (ω), ε yy (ω), ε xy (ω), and ε yx (ω) of the dielectric tensor. UPt 3 is hexagonal and we can assume that tetragonal distortions that give rise to the inequality of ε xx (ω) and ε yy (ω) (linear birefringence) are absent so ε xx (ω) = ε yy (ω) = ε. In any case, the experiments are designed not to be sensitive to linear birefringence 1 . In a field → u that breaks time reversal symmetry, including an effective field from an order parameter, the Onsager relation is xy yx 17 and since the assumption of a linear relation between ε and → u is well-founded in the present case, we find ε xy (ω) = −ε yx (ω). Then the normal modes in the metal for a propagation direction → =k kz are circularly polarized. For each frequency there are two wavevectors k + and k − that correspond to the two helicities: and ε ± (ω) = ε xx (ω) ± i|ε xy (ω)|. The different dispersion and absorption for k + and k − gives the Kerr rotation θ ε ε = Re( / ) K x y xx 3/2 in a metal (for which |ε xx |  1). To have ε xy ≠ 0 or ε yx ≠ 0, we need breaking of both mirror symmetries x → −x and y → −y and time-reversal 18 . For a spatially uniform conventional singlet superconductor with order parameter Δ, time-reversal means Δ → Δ* and a change of gauge is Δ → Δe iφ , so any time-reversal transformation is equivalent to a gauge transformation and as a result there is no non-trivial notion of time-reversal. For an unconventional superconductor with a momentum-dependent order parameter, this is not the case: we might have Δ(p) = cΔ 0 p z (p x + ip y ). Then Δ*(p) = cΔ 0 p z (p x − ip y ) and no uniform phase factor relates Δ and Δ*.

Model Hamiltonian
An appropriate model Hamiltonian for the multiband superconductor UPt 3 is: N n p a n p a n p V n p n p a n p a n p a n p a n p where ξ → n p ( , ) are the single-particle energies measured relative to the chemical potential, σ is the pseudospin, n and n′ are band indices, → σ † a n p ( , ) creates an electron in the state σ → np , and → ′ → ′ V n p n p ( , ; , ) is a singlet pairing interaction. The sum runs over all bands within 0.8 eV of the Fermi energy. Since the bandwidths are of order B ≈ 0.2 to 0.3 eV, this includes completely full and completely empty bands as well as the usual partially occupied bands. Because of the narrow bandwidth, there is no pair of partially occupied bands that have energies as much as 0.8 eV apart. . The photon energy ħω = 0.8 eV is about the energy separation between occupied and empty bands G, while the band widths of the occupied and empty bands B 1 and B 2 ≈ 0.2 eV, which is smaller than ħω. Thus absorption of the light and the consequent polarization rotation is due to interband transitions.
We have restricted our model to give singlet pairing only for ease of presentation. The conclusions are essentially the same for triplet pairing. H is treated in the mean field approximation in a straightforward generalization of the usual BCS-Gor'kov procedure 19 . However, we include strong coupling in that we make no assumptions about frequency cutoffs for the function → ′ → ′ V n p n p ( , ; , ). This leads to a set of coupled gap equations The point group of the system is D 6h for UPt 3 . The case of interest is that of unconventional superconductivity. Let R be a group operation not the identity. We have that ∆ → ≠ ∆ → n Rp n p ( , ) ( , ) for all n. It is also true that → ′ → ′ = → ′ → ′ V n Rp n Rp V n p n p ( , ; , ) ( , ; , ) so V can be decomposed into channels corresponding to the irreducible representations of G. Regarded as a function of → p , we seek the highest eigenvalue of V, which then determines the representation actually realized. Calculations for UPt 3 using experimental data to estimate → ′ → ′ V n p n p ( , ; , ) were done years ago, but were not conclusive 20,21 and first principles calculations using the functional renormalization group have been done for other systems 22 , but not for UPt 3 . In any case, the conclusions of this paper are independent of the microscopic model.
The split transition in UPt 3 23,24 implies that this representation is multi-dimensional, which for singlet superconductivity means E 1g or E 2g . We choose the former for definiteness, but our conclusions apply equally to these two representations. It is important to note that in the linear regime, Eq. (2) determines the representation, but not which combination of basis functions is chosen by the system. This degeneracy is broken at higher order and there must be complex coefficients for a Kerr rotation to occur. Thus we have that ∆ → = ∆ → ± n p n p p p ip for all R.

Order Parameter Symmetry in Different Bands
We may separate the bands into partially filled bands, of which there are 5 in UPt 3 indexed by 1 ≤ n ≤ 5 and completely filled or empty bands, indexed by n > 5. There are 6 separate Fermi surfaces in UPt 3 . For n > 5, → F n p ( , ) is of order 1/B or less. We expect ∆ ′ → n p ( , ) for n′ > 5 to be induced by a coupling → ′ → ′ V n p n p ( , ; , ) that is off-diagonal in the band indices, connecting partially filled to completely filled or completely empty bands.
Then there are two questions that are crucial for the calculation of θ K . 1. How are the Ising-like variables ± in the equation ∆ → = ∆ → ± n p n p p p ip ( , ) ( , ) ( ) z x y 0 determined as n varies? 2. What is the order of magnitude of ∆ → n p ( , ) for n > 5?
1. The first question is fairly easy to answer in our model. → ′ → ′ V n p n p ( , ; , ) couples only bands with a (p x + ip y ) with other bands with a (p x + ip y ) gap and couples only bands with a (p x − ip y ) with other bands with a (p x − ip y ) gap, i.e., it is diagonal in the ± degree of freedom. However, this coupling can be of either sign. Of course there are no symmetries in the band index, so the couplings have no particular relation to each other. In a Ginzburg-Landau approach, we may define where the "internal" order parameter η η η → = ( , ) x y depends on the band index. The free energy in E 1g is then with a summation convention over the band indices m and n in effect. To break time-reversal symmetry we need the γ m to be positive, and we need some of the J mn to be positive for some pair (m, n) of bands that differ in energy by 0.8 eV. Then we have a problem of determining the ground state of an Ising magnet with more-or-less random couplings. We may expect both (p x + ip y ) and (p x − ip y ) to occur in the absence of physical considerations to the contrary.
2. The second question is more complicated. The size of ∆ → n p ( , ) for the partially occupied bands (n ≤ 5) is at least partially constrained by experiment. We expect at least one and perhaps more of the gaps to be of order 2k B T, i.e., about 10 −4 eV. The superconductivity for n > 5 is induced from the partially occupied bands. For estimation purposes, we choose 2 bands from Eq. (2), denoting them by g for partially occupied and e for empty. We consider the separable forms: . The assumption that ∆ e (0) is induced means that the corresponding component of → → ′ V e p e p ( , ; , ) is small and we set it to zero. Then Eq. (2) yields (0) is of order 1/ω c . The latter estimate also requires that the cutoff ω c is not too much less than the bandwidth, justified if the interaction comes from the Coulomb interaction. We find that . Since all the bands are f-like in UPt 3 , the Coulomb matrix elements at short distances are expected to be comparable, and this gives reason to suppose that |∆ ∆ | / e g (0) (0) is of order unity. Figure 2(a) gives a schematic plot showing that strong Coulomb interaction V c can generate superconducting order parameters in the empty bands.

Magnitude of θ K
In order to calculate θ K we need the diagonal complex dielectric function ε xx (ω = 0.8 eV). This has been determined by reflectivity measurements and a Kramers-Kronig analysis 25,26 . In this frequency range it is necessary to include several Lorentz oscillators to fit the data, showing that there are interband transitions at ω = 0.8 eV. This is in agreement with band calculations 13,14 . We extract the approximate values Reε xx (ω = 0.8 eV) ≈ 3 and Imε xx (ω = 0.8 eV) ≈ 25 from these results.
The key quantity is of course the off-diagonal dielectric function ε xy (ω = 0.8 eV). The result for a single pair of bands (m, n) is obtained from the anomalous part of the lowest-order bubble diagram. This is not the total dielectric function. To get that we must also sum over all pairs. Here → J p ( ) x y mn , ( ) are the interband matrix elements of the current operator between single-particle states in the partially occupied m-band and the empty n-band. An analogous expression would hold for transitions from a completely full band to a partially occupied band. This expression for ε xy is to be compared to that for the familiar normal-state dielectric function xx p x mn x mn n m 2 ( ) ( ) Although Eqs (5) and (6) contain quantities which are poorly known, only the ratio is involved in the Kerr angle θ K . It is only this that allows us to give an order of magnitude estimate for θ K . To make the estimate we consider two scenarios for the band structure. Scenario 1. We take the single-particle energies ξ to be random uncorrelated variables that are uniformly distributed over a bandwidth B, and the center of the m and n bands are separated by an energy G (see Fig. 1). In the model the averages over the current matrix elements are assumed to be the same for the two bands, and there are no correlations in momentum space between the gap functions ∆ → m p ( , ), ∆ → ⁎ n p ( , ) and . The computation of the ratio then reduces to a determination of the ratio of the density of states parts of Eqs (5) and (6). The result is: observation of this pairing field is novel: in fact we believe it is the first time that it has been observed in experiments in a band that is very distant in energy from the Fermi energy. The existence of this pairing field is also not enough to explain the phenomenon: there must be a subtle pattern of relative symmetry between the bands involved, which gives additional interest to the system. There must be superconductivity with opposite chiralities p z (p x + ip y ) and p z (p x − ip y ) coexisting in the same sample. This raises many interesting questions for the topology of the electronic states. We expect that both the theory of Majorana fermions in the vortex cores and that of the protected surface states must be generalized to accommodate this additional complexity.