---
abstract: 'Electronic structure calculations combining the local-density approximation with an exact diagonalization of the Anderson impurity model show an intermediate 5$f^5$-5$f^6$-valence ground state and delocalization of the 5$f^5$ multiplet of the Pu atom 5$f$-shell in PuCoIn$_5$, PuCoGa$_5$, and $\delta$-Pu. The 5$f$-local magnetic moment is compensated by a moment formed in the surrounding cloud of conduction electrons. For PuCoGa$_5$ and $\delta$-Pu the compensation is complete and the Anderson impurity ground state is a singlet. For PuCoIn$_5$ the compensation is partial and the Pu ground state is magnetic. We suggest that the unconventional $d$-wave superconductivity is likely mediated by the 5$f$-states antiferromagnetic fluctuations in PuCoIn$_5$, and by valence fluctuations in PuCoGa$_5$.'
author:
- 'A. B. Shick'
- 'J. Kolorenc'
- 'J. Rusz'
- 'P. M. Oppeneer'
- 'A. I. Lichtenstein'
- 'M. I. Katsnelson'
- 'R. Caciuffo'
title: 'Unified character of correlation effects in unconventional Pu-based superconductors and $\delta$-Pu'
---

Providing a consistent description of correlation effects in the electronic structure of elemental actinides and their compounds is a complex problem due to the interplay between the localized and the itinerant nature of the 5f electrons. It is commonly accepted that 5$f$-electrons in light actinides form rather broad conduction bands whereas for the heavy actinides the 5$f$ states are atomic-like. Johansson [@Johansson1975] described this situation as a “Mott transition in the 5$f$-electron subsystem” taking place between Pu and Am when moving along the Periodic Table. Katsnelson [*et al.*]{} [@KatsnelsonJETPL92] linked the broadening of the 5$f$ band to the “atomic collapse” characterizing the transformation from the high-temperature expanded and the low-temperature compressed phases of Pu.

A quantitative description of the Mott transition in actinides [@Savrasov2001] was obtained by the dynamical mean-field theory (DMFT) [@Kotliar1996] more than 20 years after the concept was formulated. Further DMFT studies suggested an intermediate-valence nature of the Pu-atom 5$f$ shell [@Shim2007] and provided justification for the experimentally proved absence of magnetism in $\delta$-Pu [@Lashley2005].

The intermediate-valence and nonmagnetic character of the 5$f$ shell can play an important role in stabilizing the superconducting state exhibited by PuCoGa$_5$ below a critical temperature $T_c$ of 18.5 K. [@Sarrao2002; @Curro05; @Jutier08]. The unconventional character of superconductivity in this compound is now generally accepted but the microscopic mechanism responsible for electron pairing remains unknown. The $d$-wave symmetry of the superconducting gap in PuCoGa$_5$ has been proven by point-contact spectroscopy experiments [@daghero12] that also provided the first spectroscopic measurements of the gap amplitude and its temperature dependence.

Recently, superconductivity has been discovered also in PuCoIn$_5$ [@bauer12], with $T_c$ = 2.5 K. The experimental studies of this compound were immediately followed by conventional density functional theory (DFT) calculations in the local-density generalized-gradient approximation (LDA/GGA) [@zhu2012; @ronning12]. Keeping in mind a well known failure of DFT in the case of $\delta$-Pu [@Lashley2005], it can be expected that LDA/GGA does not provide an accurate description of the electronic structure for this strongly correlated material. A few static mean-field correlated band theory calculations were also performed [@zhu2012; @rusz12], making use of different flavors of the LDA/GGA plus Coulomb’s $U$ (LDA+$U$) method. While being an improvement over the conventional band theory, the LDA(GGA)+$U$ falls short in describing the itinerant-to-localized crossover of the 5$f$ manifold in $\delta$-Pu [@Shim2007] and PuCoGa$_5$ [@daghero12].

Here, we report electronic structure calculations of PuCoIn$_5$, PuCoGa$_5$ and $\delta$-Pu performed by combining LDA with the exact diagonalization (ED) [@J.Kolorenc2012] of a discretized single-impurity Anderson model [@Hewson]. In this approach, the band structure obtained by the relativistic version of the full-potential linearized augmented plane wave method (FP-LAPW) [@wimmer81] is consistently extended to account for the full structure of the 5$f$-orbital atomic multiplets and their hybridization with the conduction bands [@shick09].

The starting point of our approach is the multi-band Hubbard Hamiltonian [@A.I.Lichtenstein1998] $H = H^0 + H^{\rm int} $. $H^0 = \sum_{i,j,\gamma}  H^0_{i
\gamma_1, j \gamma_2}
                 c^{\dagger}_{i \gamma_1} c_{j \gamma_2}$, where $i,j$ label lattice sites and $\gamma = (l m \sigma)$ mark spinorbitals $\{ \phi_{\gamma}
\}$, is the one-particle Hamiltonian found from *ab initio* electronic structure calculations of a periodic crystal; $H^{\rm int}$ is the on-site Coulomb interaction [@A.I.Lichtenstein1998] describing the $f$-electron correlation. We assume that electron interactions in the $s$, $p$, and $d$ shells are well approximated in DFT.

The effects of the interaction Hamiltonian $H^{\rm int}$ on the electronic structure are described by a ${\bf k}$-independent one-particle selfenergy $\Sigma(z)$, where $z$ is a (complex) energy. The selfenergy is constructed with the aid of an auxiliary impurity model describing the complete seven-orbital 5$f$ shell. This multi-orbital impurity model includes the full spherically symmetric Coulomb interaction, the spin-orbit coupling (SOC), and the crystal field (CF). The corresponding Hamiltonian can be written as [@Hewson] $$\begin{aligned}
\label{eq:hamilt}
H_{\rm imp}  = & \sum_{\substack {k m m' \\ \sigma \sigma'}}
 [\epsilon^{k}]_{m m'}^{\sigma \; \; \sigma'} b^{\dagger}_{km\sigma}b_{km'\sigma'}
 +\sum_{m\sigma} \epsilon_f f^{\dagger}_{m \sigma}f_{m \sigma}
\nonumber \\
& + \sum_{mm'\sigma\sigma'} \bigl[\xi {\bf l}\cdot{\bf s}
  + \Delta_{\rm CF}\bigr]_{m m'}^{\sigma \; \; \sigma'}
  f_{m \sigma}^{\dagger}f_{m' \sigma'}
\nonumber \\
& +  \sum_{\substack {k m m' \\ \sigma \sigma'}}   \Bigl(
[V^{k}]_{m m'}^{\sigma \; \; \sigma'}
 f^{\dagger}_{m\sigma} b_{km' \sigma'} + \text{h.c.}
  \Bigr)
\\
& + \frac{1}{2} \sum_{\substack {m m' m''\\  m''' \sigma \sigma'}}
  U_{m m' m'' m'''} f^{\dagger}_{m\sigma} f^{\dagger}_{m' \sigma'}
  f_{m'''\sigma'} f_{m'' \sigma},
\nonumber\end{aligned}$$ where $f^{\dagger}_{m \sigma}$ creates an electron in the 5$f$ shell and $b^{\dagger}_{m\sigma}$ creates an electron in the “bath” that consists of those host-band states that hybridize with the impurity 5$f$ shell. The energy position $\epsilon_f$ of the impurity level, and the bath energies $\epsilon^{k}$ are measured from the chemical potential $\mu$. The parameter $\xi$ specifies the strength of the SOC and $\Delta_{\rm CF}$ is the crystal-field potential at the impurity. The parameter matrices $V^{k}$ describe the hybridization between the 5$f$ states and the bath orbitals at energy $\epsilon^{k}$.

The band Lanczos method [@J.Kolorenc2012] is employed to find the lowest-lying eigenstates of the many-body Hamiltonian $H_{\rm
imp}$ and to calculate the one-particle Green’s function $[G_{\rm
imp}(z)]_{m m'}^{\sigma \; \; \sigma'}$ in the subspace of the $f$ orbitals at low temperature ($k_{\rm B}T=1/500$ eV). The self-energy $[\Sigma (z)]_{m m'}^{\sigma \; \; \sigma'}$ is then obtained from the inverse of the Green’s-function matrix $[G_{\rm imp}]$.

Once the self-energy is known, the local Green’s function $G(z)$ for the electrons in the solid, $$[G(z)]_{\gamma_1 \gamma_2} = \frac{1}{V_{\rm BZ}}
\int_{\rm BZ}{\rm d}^3 k \,\bigl[z+\mu-H_{\rm LDA}({\bf
k})-\Sigma(z)\bigr]^{-1}_{\gamma_1 \gamma_2}\,, \label{eq:gf}$$ is calculated in a single-site approximation as given in [@shick09]. Then, with the aid of the local Green’s function $G(z)$, we evaluate the occupation matrix $n_{\gamma_1 \gamma_2} = -\frac1{\pi}\,\mathop{\rm Im}
\int_{-\infty}^{E_{\rm{F}}} {\rm d} z \, [G(z)]_{\gamma_1 \gamma_2}$. The matrix $n_{\gamma_1 \gamma_2}$ is used to construct an effective LDA+$U$ potential ${V}_{U}$, which is inserted into Kohn–Sham-like equations: $$\begin{gathered}
[ -\nabla^{2} + V_{\rm LDA}(\mathbf{r}) + V_{U} + \xi ({\bf l} \cdot
{\bf s}) ]  \Phi_{\bf k}^b({\bf r}) = \epsilon_{\bf k}^b \Phi_{\bf
k}^b({\bf r}).
\label{eq:kohn_sham}\end{gathered}$$ These equations are iteratively solved until self-consistency over the charge density is reached. In each iteration, a new Green’s function ${G}_{\mathrm{LDA}}(z)$ \[which corresponds to $G(z)$ from Eq.(\[eq:gf\]) with the self-energy $\Sigma$ set to zero\] , and a new value of the 5$f$-shell occupation are obtained from the solution of Eq. (\[eq:kohn\_sham\]). Subsequently, a new self-energy $\Sigma(z)$ corresponding to the updated 5$f$-shell occupation is constructed. Finally, the next iteration is started by evaluating the new local Green’s function, Eq.(\[eq:gf\]).

![(Color online) The Pu atom LDA $j$ = 5/2, 7/2 projected DOS, and LDA hybridization function ${\Delta}(\epsilon) = -{1 \over
{\pi}} Im Tr [G^{-1}(\epsilon + i \delta)]$. The inset shows the PuCoIn$_5$ crystal structure.[]{data-label="hybridization"}](fig1s.eps){width="1.0\columnwidth"}

In order to determine the bath parameters $V^{k}$ and $\epsilon^{k}$, we assume that the LDA represents the non-interacting model. We then associate the LDA Green’s function ${G}_{\mathrm{LDA}}(z)$ with the Hamiltonian of Eq. (\[eq:hamilt\]) when the coefficients of the Coulomb interaction matrix are set to zero ($U_{mm'm''m'''}=0$). The hybridization function ${\Delta(\epsilon)}$ is then estimated as ${\Delta}(\epsilon) = -{\frac1{\pi}} \mathop{\rm Im}\mathop{\rm Tr}
[G^{-1}_{\rm LDA}(\epsilon + i \delta)]$. The curve obtained for ${\Delta}(\epsilon)$ is shown in Fig. \[hybridization\], together with the $j=5/2,7/2$-projected LDA densities of the $f$-states. The results also show that the hybridization matrix is, to a good approximation, diagonal in the $\{j,j_z\}$ representation. Thus, we assume the first and fourth terms in the impurity model, Eq. (\[eq:hamilt\]), to be diagonal in $\{j,j_z\}$, so that we only need to specify one bath state (six orbitals) with $\epsilon^{k=1}_{j=5/2}$ and $V^{k=1}_{j=5/2}$, and another bath state (eight orbitals) with $\epsilon^{k=1}_{j=7/2}$ and $V^{k=1}_{j=7/2}$. Assuming that the most important hybridization is the one occurring in the vicinity of $E_F$, the numerical values of the bath parameters $V^{k=1}_{5/2,7/2}$ are found from the relation [@Gunnarsson89] $\sum_{k} {|V_{k}^{j}|}^2
\delta(\epsilon_{k}^{j} - \epsilon) = - \Delta(\epsilon)/N_f$ integrated over the energy interval, $E_F - 0.5$ eV $\le \epsilon
\le E_F + 0.5$ eV, with $N_f=6$ for $j=5/2$ and $N_f=8$ for $j=7/2$. The bath-state energies $\epsilon^{k=1}_{5/2,7/2}$ shown in Table \[parameters\] are adjusted to approximately reproduce the LDA $5f$-state occupations $n_f^{5/2}$ and $n_f^{7/2}$.

  Material       $n_f^{5/2}$   $n_f^{7/2}$   $\epsilon_1^{5/2}$   $V_{1}^{5/2}$   $\epsilon_1^{7/2}$   $V_{1}^{7/2}$  
  ------------- ------------- ------------- -------------------- --------------- -------------------- --------------- --
  PuCoIn$_5$        4.78          0.39              0.36              0.21              -0.25              0.25       
  PuCoGa$_5$        4.38          0.76              0.25              0.29              -0.07              0.34       
  $\delta$-Pu       4.16          0.85              0.33              0.27              -0.01              0.36       

  : $5f$-states occupations $n_f^{5/2}$ and $n_f^{7/2}$, and bath state parameters $\epsilon^1_{5/2,7/2}$ (eV), $V^{1}_{5/2,7/2}$ (eV) for Pu-atom in PuCoIn$_5$, PuCoGa$_5$, and $\delta$-Pu from LDA calculations.[]{data-label="occup"}

\[parameters\]

In the calculations we used an in-house implementation [@shick99; @shick01] of the FP-LAPW method that includes both scalar-relativistic and spin-orbit coupling effects. The calculations were carried out assuming a paramagnetic state with crystal structure parameters for PuCoIn$_5$, PuCoGa$_5$, and $\delta$-Pu taken from Refs. [@bauer12; @Opahle2003; @ledbetter76], respectively. The Slater integrals were chosen as $F_0=4.0$ eV, and $F_2=7.76$ eV, $F_4=5.05$ eV, and $F_6=$ 3.07 eV [@KMoore2009]. They corresponds to commonly accepted values for Coulomb’s $U=4.0$ eV and exchange $J = 0.64$ eV. The SOC parameters $\xi=0.28$ eV for PuCoIn$_5$ and PuCoGa$_5$ and $0.29$ eV for $\delta$-Pu were determined from LDA calculations. CF effects were found to be negligible and $\Delta_{\rm CF}$ was set to zero. For the double-counting term entering the definition of the LDA+$U$ potential, $V_U$, we have adopted the fully-localized (or atomic-like) limit (FLL) $V_{dc} = U (n_f-1/2) - J(n_f-1)/2$. Furthermore, we set the radii of the atomic spheres to 3.1 a.u. (Pu), 2.3 a.u. (Co), 2.3 a.u. (Ga), and 2.5 a.u. (In). The parameter $R_{Pu} \times K_{\text{max}}=10.54$ determined the basis set size, and the Brillouin zone (BZ) sampling was performed with 1152 $k$ points. The self-consistent procedure defined by Eqs. – was repeated until the convergence of the 5$f$-manifold occupation $n_f$ was better than 0.01.

We are now ready to discuss the solution of Eq.(\[eq:hamilt\]). For PuCoIn$_5$, the ground state of the cluster formed by the 5$f$ shell and the bath is given by a superposition of a magnetic sextet (23%) and a non-magnetic singlet (77%), with occupation numbers $\langle n_f \rangle=5.40$ in the $f$ shell and $\langle
n_{bath} \rangle=8.40$ in the bath states. This ground state is not a singlet and carries a non-zero magnetic moment. For the 5$f$ shell alone, the expectation values of the spin $(S_f)$, orbital $(L_f)$ and total $(J_f)$ angular moments can be calculated as $\langle \hat
X_f^2 \rangle=X_f(X_f+1)$ ($X_f = S_f, L_f, J_f$), giving $S_f =
2.27$, $L_f = 3.90$, and $J_f = 2.09$. The individual components of the moments vanish, $\langle {\hat{S}^z}_f \rangle = \langle
{\hat{L}^z}_f \rangle=0$, unless the symmetry is broken by an external magnetic field.

In the case of PuCoGa$_5$, on the other hand, the hybridized ground state of the impurity is a non-magnetic singlet with all angular moments of the 5$f$-bath cluster equal to zero ($S = L = J =0$). It consists of $\langle n_f \rangle=5.30$ $f$ states and $\langle
n_{bath} \rangle=8.70$ bath states. In a pictorial way, we can imagine that the magnetic moment of the 5$f$ shell (for which we get $S_f = 2.18$, $L_f = 4.05$, $J_f = 2.43$) is completely compensated by the moment carried by the electrons in the conduction band. As the value of the 5$f$ magnetic moment fluctuates in time, because of the intermediate-valence electronic configuration, this compensation must be understood as dynamical in nature. The same situation is realized in $\delta$-Pu ($S_f = 2.11$, $L_f = 4.21$, $J_f= 2.62$), whose ground state is found to be a nonmagnetic singlet with $\langle n_f \rangle=5.21$ and $\langle n_{bath} \rangle=8.79$.

![(Color online) $f$-electron density of states (DOS, $j$ = 5/2, 7/2 projected) for the Pu atom in PuCoIn$_5$ (a), PuCoGa$_5$ (b) and $\delta$-Pu (c).[]{data-label="fig:dos"}](fig2a.eps){width="0.975\columnwidth"}

![(Color online) $f$-electron density of states (DOS, $j$ = 5/2, 7/2 projected) for the Pu atom in PuCoIn$_5$ (a), PuCoGa$_5$ (b) and $\delta$-Pu (c).[]{data-label="fig:dos"}](fig2b.eps){width="0.975\columnwidth"}

![(Color online) $f$-electron density of states (DOS, $j$ = 5/2, 7/2 projected) for the Pu atom in PuCoIn$_5$ (a), PuCoGa$_5$ (b) and $\delta$-Pu (c).[]{data-label="fig:dos"}](fig2c.eps){width="0.975\columnwidth"}

The 5$f$-orbital density of states (DOS) obtained from Eq. (\[eq:gf\]) for the three investigated compounds is shown in Fig. \[fig:dos\]. Below the Fermi energy $E_F$ the DOS exhibits the three-peak structure typical for Pu and for a number of its compounds, and its shape is in good agreement with experimental photoemission spectra. It can be noticed that the multiplets for the atomic $f^6$ configuration ($f^6 \rightarrow f^5$ transition, lying closer to $E_F$) are better resolved than for the $f^5$ part of the spectrum ($f^5 \rightarrow f^4$ transition).

Comparison with previous LDA+Hubbard-I (HIA) calculation for $\delta$-Pu [@shick09], and PuCoGa$_5$ [@shick11] shows that the three-peak manifold lying above 2 eV binding energy has a slight upright shift towards $E_F$. At binding energies around 4 eV, the LDA+HIA peaks are substantially modified, and in the LDA+ED calculations they are spread over a $\sim$ 3 eV energy interval. These changes in the DOS are induced by the hybridization and suggest partial delocalization of the $f^5$ multiplet. This is a situation suggested first by Hanzawa [@hanzawa1998] in intermediate-valence rare-earth compounds such as SmS or SmB$_6$, where fluctuations occur between two atomic-like 4$f$ configurations. Here, the 5$f$ states remain localized for the $f^6$ configuration but become itinerant for the $f^5$ one.

As the many-body resonances lying closer to the Fermi energy are produced by $f^6 \rightarrow f^5$ multiplet transitions, they are in a way analogs to the [*Racah*]{} peaks, specific transitions between Racah multiplets [@Racah1949] of $f^n$ and $f^{n \pm 1}$. On the other hand, these structures determine the metallic character of the investigated materials that can therefore be considered as a realization of a [*Racah*]{} metal, situated between the two limiting cases represented by fully localized intermediate-valence rare-earth compounds and metallic systems (e.g., nickel) with a non-integer number of $d$ electrons.

Both PuCoGa$_5$ and $\delta$-Pu display a temperature-independent magnetic susceptibility at low temperatures [@Lashley2005; @hiess2008]. Analogous to the intermediate-valence rare-earth compounds [@Khomskii1979], the magnetic susceptibility is anticipated to behave as $\chi \sim 1/(T
+T_{fc})$, where the temperature $T_{fc}$ describes fluctuations between the 5$f$ and conduction band electron states. $T_{fc}$ corresponds indeed to the broadening of the quasiparticle resonance near $E_F$ due to valence fluctuations [@Varma1976]. As the ground state of the impurity is a singlet, we estimate $T_{fc}$ using a renormalized perturbation theory of the Anderson model [@Hewson], $T_{fc}=-\frac{\pi^2}{4}Z [\Delta(E_F)/N_f]$, where $[\Delta(E_F)/N_f]$ is the hybridization per orbital at $E_F$, and $Z$ is a quasiparticle weight, $ Z = ( \mathop{\rm
Tr}[N(E_F)(1-\frac{d
\Sigma(\epsilon)}{d\epsilon})|_{\epsilon=E_F}]/\mathop{\rm Tr}
[N(E_F)])^{-1}$. We get $T_{fc}= 72$ meV ($\sim
  850$ K) for PuCoGa$_5$ and $T_{fc}= 63$ meV ($\sim 750$ K) for $\delta$-Pu. Since $T_{fc}$ is high, $\chi$ remains constant for $T\ll T_{fc}$, as observed experimentally for PuCoGa$_5$ and $\delta$-Pu. The situation is different in the case of PuCoIn$_5$ where the ground state of the impurity is not a pure singlet due to weaker hybridization. Consequently, the temperature dependence of $\chi$ is expected to be more pronounced.

The electronic specific-heat coefficient can be estimated as $\gamma= \frac{\pi^2}{3}k_B^2 \mathop{\rm Tr}[N(E_F)(1 -\frac{d
\Sigma(\omega)}{d\omega})|_{\omega=0}]$. For $\delta$-Pu, we get $\approx$ 44 mJ K$^{-2}$ mol$^{-1}$, in very good agreement with experimental data. For PuCoGa$_5$, we get $\approx$ 43 mJ K$^{-2}$ mol$^{-1}$ which is smaller than the experimental value of 80–100 mJ K$^{-2}$ mol$^{-1}$. For PuCoIn$_5$, the estimated $\gamma$ value of $\approx$ 52 mJ K$^{-2}$ mol$^{-1}$ is even further away from the experimental value of $\approx$ 180 mJ K$^{-2}$ mol$^{-1}$. In this case, it is difficult to obtain an accurate value for $\gamma$ due to the sharp DOS peak in the vicinity of $E_F$ (see Fig. \[fig:dos\]). When taken right at the DOS peak position, the $\gamma$ value of 95 mJ K$^{-2}$ mol$^{-1}$ is obtained. Also, note that a possible enhancement of $\gamma$ due to the electron-phonon interaction is not taken into account.

![(Color online)(Top) The band structure with $f$-weight fatbands for PuCoIn$_5$, and (bottom) the Fermi surface of PuCoGa$_5$ and PuCoIn$_5$ obtained from LDA+ED calculations. The shade of colors encodes the size of the energy gradient. \[fig:ldma\_bands\]](fig3s.eps){width="1.05\columnwidth"}

Figure \[fig:ldma\_bands\] shows the band structure and the corresponding Fermi Surface (FS) for PuCoIn$_5$, calculated from the solutions of Eq. (\[eq:kohn\_sham\]), which represents an extended LDA+$U$ static-mean-field band structure with the 5$f$-states occupation matrix obtained from the local impurity Greens function Eq.(\[eq:gf\]). For comparison, Fig. \[fig:ldma\_bands\] shows also the FS for PuCoGa$_5$ (Fig. S2 of Ref. [@daghero12]). Close similarities in the band structure of the two compounds are immediately apparent. Both are compensated multiband metals, as the Fe-based superconductors, and for both materials the $f$ bands move away from the Fermi level when the Coulomb-$U$ is included, as can be seen by examining the $f$-weighted fatbands. The Fermi surfaces are composed by four sheets (1–4), one that is hole-like (FS-1) and three that are electron-like (FS-2,3,4). The Fermi velocities ratio $\langle v^2_{x,y} \rangle^{\frac{1}{2}}/\langle v^2_{z}
\rangle^{\frac{1}{2}}$ of 1.54 for PuCoIn$_5$, and 1.55 for PuCoGa$_5$ are calculated in reasonable agreement with the experimental anisotropy ratio of the critical field $H_{c2}$, $2-2.3$ for PuCoIn$_5$, and indicate a two-dimensional character of the electronic structure.

DFT electronic structure calculations for Pu-based 115 material have recently been reported by Ronning *et al.*  [@ronning12] and Zhu *et al.* [@zhu2012] Their analysis of the DFT band structure and FS (see, e.g., Figs. 3 and 4 of Ref. [@zhu2012]) indicated two possible superconducting gap symmetries, the so-called $s \pm$ and $d_{x^2-y^2}$, which correspond to a pairing potential peaked at the $(\pi,\pi,0)$ reciprocal lattice position. The conclusion was drawn that for Pu-based “115” superconductors, the $s \pm$ order parameter is more likely that the $d_{x^2-y^2}$ one. This is in contradiction with point-contact spectroscopy results [@daghero12] showing a zero-bias conductance anomaly that is not expected for $s \pm$ gap symmetry [@daghero11].

The presence of a 5$f$ local moment dynamically compensated by the surrounding conduction electrons together with the $f^5$-$f^6$ intermediate-valence ground state in PuCoGa$_5$ and PuCoIn$_5$ opens various possibilities for unconventional superconductivity. In PuCoIn$_5$ the Pu $f$-shell local moment is not fully compensated and superconductivity could be related to an antiferromagnetic quantum critical point [@bauer12; @das12]. On the other hand, in PuCoGa$_5$ the ground state is a singlet and it seems more plausible that superconductivity results from a valence instability, as in heavy-fermion superconductors [@miyake07].

We are grateful to D. Daghero and L. Havela for helpful comments and discussion. We acknowledge financial support from Czech Republic Grants No. GACR P204/10/0330 and No. GAAV IAA100100912 and from DFG Grant No. 436 TSE 113/53/0-1.

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