Unconventional full-gap superconductivity in Kondo lattice with semi-metallic conduction bands

TL;DR

This paper proposes a novel mechanism for superconductivity in Kondo lattices with semi-metallic bands, involving entangled states of localized spins and conduction electrons, leading to unconventional full-gap superconductivity.

Contribution

It introduces a new superconducting pairing mechanism based on composite three-body bound states in Kondo lattices with semi-metallic conduction bands.

Findings

The order parameter involves a three-body bound state of spins, electrons, and holes.

Superconductivity arises from induced pairing of conduction electrons, not directly from composite pairs.

Potential relevance to heavy-electron materials is discussed.

Abstract

A mechanism of superconductivity is proposed for the Kondo lattice which has semi-metallic conduction bands with electron and hole Fermi surfaces. At high temperatures, the $f$ electron's localized spins/pseudospins are fluctuating between electron and hole Fermi surfaces to seek for a partner to couple with. This system tries to resolve this frustration at low temperatures and chooses to construct a quantum mechanically entangled state composed of the Kondo singlet with electron surface and that with hole surface, to break the U(1) gauge symmetry. The corresponding order parameter is given by a composite pairing amplitude as a three-body bound states of localized spin/pseudospin, electron and hole. The electromagnetic response is considered, where composite pair itself does not contribute to the Meissner effect, but the induced pair between conduction electrons, which inevitably mixes due to e.g. a band cutoff effect at high energies, carries the superconducting current under the external field. Possible applications to real heavy-electron materials are also discussed.

Equations

$$\mathcal{H}$$

$$+\frac{1}{2}\sum _{\alpha \sigma {\sigma }^{\prime }}{J}_{\alpha }\int d\mathbf{r}\mathbf{S}(\mathbf{r})\cdot {\psi }_{\alpha \sigma }^{†}(\mathbf{r}){\mathbf{\sigma }}_{\sigma {\sigma }^{\prime }}{\psi }_{\alpha {\sigma }^{\prime }}(\mathbf{r}),$$

$${\mathcal{H}}_{\mathrm{int}}$$

$$V=\frac{3}{4}{J}_1⟨f_{\sigma }(\mathbf{r}){\psi }_{1\sigma }^{†}(\mathbf{r})⟩,$$

$$W{ϵ}_{\sigma {\sigma }^{\prime }}=\frac{3}{4}{J}_2⟨f_{\sigma }(\mathbf{r}){\psi }_{2{\sigma }^{\prime }}(\mathbf{r})⟩,$$

$$E_{\mathbf{k}\alpha \pm }$$

$${\Phi }_{\mu ,\sigma {\sigma }^{\prime }}^{\alpha {\alpha }^{\prime }}(\mathbf{R};\mathbf{r},{\mathbf{r}}^{\prime })=⟨S^{\mu }(\mathbf{R}){\psi }_{\alpha \sigma }(\mathbf{r}){\psi }_{{\alpha }^{\prime }{\sigma }^{\prime }}({\mathbf{r}}^{\prime })⟩$$

$$=\frac{1}{2}{\left({\hat{\sigma }}^{\mu }\widehat{ϵ}\right)}_{\sigma {\sigma }^{\prime }}{ϵ}_{\alpha {\alpha }^{\prime }}{V}^{\ast }W{F}_{\alpha }(\mathbf{r}-\mathbf{R})F_{{\alpha }^{\prime }}^{\ast }({\mathbf{r}}^{\prime }-\mathbf{R}),$$

$$F_{\alpha }(\mathbf{r}=0)={\rho }_{\alpha }(0)\log \left(\frac{D_{\alpha }|{\mu }_{\alpha }|}{{\left|V_{\alpha }\right|}^2}\right),$$

$$F_{\alpha }(|\mathbf{r}|\to \infty )=2{\mathrm{e}}^{i{\mathbf{K}}_{\alpha }\cdot \mathbf{r}}{\rho }_{\alpha }(0)\frac{\sin \left(k_{\mathrm{F}\alpha }r\right){\mathrm{e}}^{-r/{\xi }_{\alpha }}}{k_{\mathrm{F}\alpha }r}.$$

$$1$$

$$k_{\mathrm{B}}{T}_{c,\alpha }=\frac{2{\mathrm{e}}^{\gamma }}{\pi }\sqrt{D_{\alpha }\left|{\mu }_{\alpha }\right|}\exp \left[-\frac{4}{3J_{\alpha }{\rho }_{\alpha }(0)}\right],$$

$$⟨{\psi }_{\mathbf{k}\alpha \sigma }{\psi }_{-\mathbf{k},{\alpha }^{\prime }{\sigma }^{\prime }}⟩=-\frac{V^{\ast }W}{E_c}{ϵ}_{\sigma {\sigma }^{\prime }}{\sigma }_{\alpha {\alpha }^{\prime }}^x\sum _{{\alpha }^{\prime \prime }}\frac{\theta \left(k_c-\left|\mathbf{k}-{\mathbf{K}}_{{\alpha }^{\prime \prime }}\right|\right)}{\sqrt{{\xi }_{\mathbf{k}{\alpha }^{\prime \prime }}^2+4{\left|V_{{\alpha }^{\prime \prime }}\right|}^2}}.$$

$$K_{\mu \nu }^p(\mathbf{q})$$

$$K_{\mu \nu }^d(\mathbf{q})$$

$${\hat{\mathbf{j}}}^p(\mathbf{q})=e\sum _{\alpha ,\sigma }\int d\mathbf{k}{\psi }_{\mathbf{k}-\mathbf{q}/2,\alpha \sigma }^{†}{\mathbf{v}}_{\mathbf{k}\alpha }{\psi }_{\mathbf{k}+\mathbf{q}/2,\alpha \sigma },$$

$$K_{\mu \nu }(\mathbf{q}\to 0)$$

Full text

Unconventional Full-Gap Superconductivity in Kondo Lattice with Semi-Metallic Conduction Bands

Shoma Iimura1

  

Motoaki Hirayama2

  

Shintaro Hoshino1

1Department of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan

2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

A mechanism of superconductivity is proposed for the Kondo lattice which has semi-metallic conduction bands with electron and hole Fermi surfaces. At high temperatures, the $f$ electron’s localized spins/pseudospins are fluctuating between electron and hole Fermi surfaces to seek for a partner to couple with. This system tries to resolve this frustration at low temperatures and chooses to construct a quantum mechanically entangled state composed of the Kondo singlet with electron surface and that with hole surface, to break the U(1) gauge symmetry. The corresponding order parameter is given by a composite pairing amplitude as a three-body bound states of localized spin/pseudospin, electron and hole. The electromagnetic response is considered, where composite pair itself does not contribute to the Meissner effect, but the induced pair between conduction electrons, which inevitably mixes due to e.g. a band cutoff effect at high energies, carries the superconducting current under the external field. Possible applications to real heavy-electron materials are also discussed.

The mechanism of unconventional superconductivities is an important issue in condensed matter physics Steglich et al. (1979); Jërome et al. (1980); Bednorz and Müller (1986) in designing a guiding principle to find new superconductors. The superconducting materials range over a broad class of correlated electron materials including cuprates, iron pnictides and organic compounds For a review, see M. R. Norman (2011). Among them, the heavy electron materials with nearly localized $f$ electrons are typical systems showing unconventional superconductivity, and the identification of the mechanisms has still remained open question since the discovery of CeCu2Si2 Steglich et al. (1979) and UBe13 Ott et al. (1983) in lanthanide- and actinide-based materials. Whereas the recent advanced experiments allow to determine the symmetry or structure of pairing gap functions, the underlying mechanisms have not yet been fully clarified.

Recently, specific heat measurements in a rotational magnetic field have revealed the full-gap nature of the superconducting states in CeCu2Si2 Kittaka et al. (2014) and UBe13 Shimizu et al. (2015), in addition to a well known cerium-based $s$-wave superconductor CeRu2 Hedo et al. (1998). This is in contrast with the conventional notion that the strongly correlated electrons favor a spatially non-local and anisotropic pairing. Hence, it is desirable to identify a new mechanism for full-gap superconductivity that is specific to heavy-electron materials. To this end, we here focus on the physics arising from a semi-metallic band structure having both electron and hole Fermi surfaces. One of the characteristics of the correlated semi-metal is an emergence of excitonic insulator Mott (1961); Halperin and Rice (1968); Kune$\check{\mathrm{s}}$ (2015), where the Coulomb interactions reconstruct the electronic states near the Fermi level and form a gap at the Fermi level. Another interesting problem, which is discussed in this paper, is to consider the situation where the semi-metal interacts quantum-mechanically with magnetic ions through the antiferromagnetic Kondo coupling as illustrated in Fig. 1.

We show in this paper that the Kondo lattice with semi-metallic conduction bands exhibits a full-gap superconductivity as a consequence of resolution of a frustration associated with the multichannel Kondo effect. Underlying physical picture is the following: at high temperatures electron and hole bands are competing in screening the localized moments (see Fig. 1), and at low temperatures they decide to mix the two kinds of Kondo singlet states quantum-mechanically. As a result, the electrons and holes are superposed and superconductivity occurs since the U(1) gauge symmetry is broken. The resultant order parameter is the composite pair amplitude Emery and Kivelson (1992); Balatsky and Bon$\check{\mathrm{c}}$a (1993); Coleman et al. (1993, 1995), which has been proposed in the multi-channel Kondo/Anderson lattices Emery and Kivelson (1993); Coleman et al. (1999); Jarrell et al. (1997); Anders (2002a, b); Flint et al. (2008); Flint and Coleman (2010); Flint et al. (2011); Hoshino and Kuramoto (2014). A new perspective proposed here is that the stability of this composite pairing is closely related to the semi-metallic conduction bands. The mechanism is also related closely to the concept of frustration. It has been argued that the frustration generates intriguing quantum states out of classically degenerate states. The well-known example is the spin liquid caused by the geometrical frustration in the magnetically interacting systems For a recent review, see Y. Zhou, K. Kanoda, and T.-K. Ng (2017); For a recent review, see L. Savary and L. Balents (2017). In our setup, the frustration arises in the form of the competing screening channels in Kondo systems, which is resolved by utilizing the quantum states, and even realizes the gauge-symmetry broken state.

For theoretical analysis, we take the parabolic dispersion for conduction electrons, which makes the physical picture clear with a pure composite pairing state without conventional one-body pair amplitudes. The electromagnetic response is also considered based on the microscopic Hamiltonian, and the pure composite pair does not directly couple to electromagnetic field in linear response. Instead, with the consideration of lattice regularization (or band cutoff effect) that modifies the parabolic dispersions far away from Fermi surfaces, a usual one-body pair amplitude is secondarily induced and then show electromagnetic linear-response.

We use the following Hamiltonian for the Kondo lattice (KL) with semi-metallic conduction bands (SMCB):

[TABLE]

where $\psi_{\alpha\sigma}({\bm{r}})$ ($\psi_{\alpha\sigma}^{\dagger}({\bm{r}})$) is the annihilation (creation) operator of electrons at ${\bm{r}}$ in conduction band $\alpha$ with spin $\sigma$ and $\xi_{\bm{k}\alpha}$ is the one-body energy of the conducting electron, $\xi_{\bm{k}\alpha}=\hbar^{2}(\bm{k}-\bm{K}_{\alpha})^{2}/(2m_{\alpha})-\mu_{\alpha}$. $m_{\alpha}$ denotes an effective mass of the conduction electrons: $m_{1}>0$ (electron) and $m_{2}<0$ (hole). The chemical potential $\mu_{\alpha}$ is introduced for each bands. The condition $m_{1}\mu_{1}=m_{2}\mu_{2}$ is satisfied in compensated metals. $\bm{K}_{\alpha}$ denotes the centers of electron and hole pockets, and the wave vector summation is taken over the range $\left|\bm{k}-\bm{K}_{\alpha}\right|<k_{c}$ (denoted by $\mathcal{K}_{\alpha}$) where $k_{c}$ is a wavevector cutoff. $J_{\alpha}$ is the Kondo coupling. The localized spin operator is given by $\bm{S}(\bm{r})=\sum_{i}\bm{S}_{i}\delta(\bm{r}-\bm{R}_{i})$ where the $f$-electron spins $\bm{S}_{i}$ are localized at the lattice sites $\bm{R}_{i}$. ${\bm{\sigma}}$ is a Pauli matrix. Schematic illustration of our model is shown in Fig. 1.

In the above, we have assumed the Kramers doublet for the $f$-electron degrees of freedom. On the other hand, for the non-Kramers doublet realized in $f^{2}$ configuration of U and Pr ions, which is not necessarily associated with time-reversal symmetry, the localized state is described by a pseudospin $\bm{T}$ Cox (1987); Cox and Zawadowski (1998). This pseudospin interacts with semi-metallic conduction bands through the nonmagnetic degrees of freedom $\alpha=1,2$. The simplest interaction takes the form

[TABLE]

In this case, the equivalence between $\sigma=\uparrow$ and $\downarrow$ for conduction electrons is protected by the time-reversal symmetry and the competition arises in forming a Kondo singlet state.Since the mean-field solution is same as the Kramers case [see Supplementary Materials (SM) A], in the following we mainly focus on the Kramers system described by Eq. (1) to simplify the discussion. Now we apply the mean-field theory which effectively describe the superconducting state resulting from the superposition of “electron” and “hole” through the multichannel Kondo effect. We first write the localized spin in terms of pseudofermion as ${\bm{S}}({\bm{r}})=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}f_{\sigma}^{\dagger}({\bm{r}}){\bm{\sigma}}_{\sigma\sigma^{\prime}}f_{\sigma^{\prime}}({\bm{r}}),$ with the local constraint $\sum_{\sigma}\langle f_{\sigma}^{\dagger}({\bm{r}})f_{\sigma}({\bm{r}})\rangle=n_{f}$, where $f_{\sigma}({\bm{r}})$ ($f_{\sigma}^{\dagger}({\bm{r}})$) is the annihilation (creation) operator of the $f$-electron and $n_{f}$ is an $f$-electron density. The mean-fields are introduced as

[TABLE]

where $\hat{\epsilon}=i\hat{\sigma}^{y}$ is the antisymmetric unit tensor. The mean-fields $V$ and $W$ respectively represent “hybridization” and “pair-condensation” between the pseudofermion and the conduction electrons Coleman et al. (1999); Flint et al. (2008); Hoshino (2014); Kusunose (2016). Although the mean-fields $V$ and $W$ are introduced asymmetrically with respect to the index $\alpha=1,2$, this situation is related to the fact that the way of introduction of pseudofermions is not unique Hoshino (2014). Hence, if we consider the physical quantities in terms of the original $\bm{S}$, the symmetries between $\alpha=1,2$ and between $\sigma=\uparrow,\downarrow$ are preserved. The nonzero self-consistent solution is obtained when the Kondo coupling is antiferromagnetic (see Eq.(9)). Our pseudofermion approach is justified in the low-temperature limit where the collective Kondo effects are fully activated as in the ordinary Kondo lattice Zhang and Yu (2000); Capponi and Assaad (2001). In this regime, the resonant $f$-electron level is generated near the Fermi surfaces.

We now show the energy dispersion relation in superconducting state, which is obtained as

[TABLE]

for each $\bm{k}\in\mathcal{K}_{\alpha}$. We have defined $V_{\alpha}$ by $V_{1}=V$, and $V_{2}=W$. The solid lines in Fig. 2 show the dispersion relation described by Eq. (5). The single-particle spectrum has the fully opened energy gap at the Fermi level. In addition, unlike the $s$-wave superconductor in the Bardeen-Cooper-Schrieffer (BCS) theory, the minimal energy gap is indirect with the amplitude $E_{\mathrm{gap},\alpha}\sim\left|V_{\alpha}\right|^{2}/|\mu_{\alpha}|+\left|V_{\alpha}\right|^{2}/D_{\alpha}$ where $D_{\alpha}$ ($\gtrsim|\mu_{\alpha}|$) denotes the energy range in which the conduction electrons are involved into the condensation. This energy gap is reflected in the thermodynamic properties such as a temperature dependence of the specific heat. On the other hand, the direct energy gap shows minimum at the Fermi momentum with its amplitude being $2\left|V_{\alpha}\right|$ ($\gg E_{\rm gap,\alpha}$), and is reflected in the optical conductivity.

Although $V$ and $W$ can be regarded as the order parameters in the effective model, the pseudofermions introduced to describe the low-energy excitations are not real but virtual physical degrees of freedom. Then we need to seek for the other pair amplitude that remains in the original model without considering pseudofermions. For a U(1) symmetry broken state, one naively expects a finite pair amplitude $\langle\psi_{\bm{k}\alpha\sigma}\psi_{-\bm{k},\alpha^{\prime}\sigma^{\prime}}\rangle$ for the conduction electrons. However, with the present effective low-energy Hamiltonian, any pair amplitude composed of conduction electrons becomes zero. This is due to the separation of the regions $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ in our model. Instead, the composite pair amplitude is the appropriate order parameter, which is given by

[TABLE]

where the function $F_{\alpha}$ behaves as

[TABLE]

See SM B for derivation. Here $\rho_{\alpha}(0)$ is the density of states of the conduction electron at the Fermi level. $k_{\mathrm{F}\alpha}=\sqrt{2m_{\alpha}\mu_{\alpha}}/\hbar$ stands for the Fermi momentum for each band. $\xi_{\alpha}$ represents the coherence length defined by $\xi_{\alpha}=\left|\mu_{\alpha}\right|/k_{\mathrm{F}\alpha}\left|V_{\alpha}\right|$, which is determined by the energy scale of $\left|V_{\alpha}\right|$. Since $E_{\mathrm{gap},\alpha}\ll\left|V_{\alpha}\right|$, the coherence length becomes much shorter than that expected by the amplitude of the energy gap. This is due to the nearly localized nature of the composite pairs involving the $f$-electron.

The composite pair amplitude $\Phi(\bm{R},\bm{r},\bm{r}^{\prime})$ depends on the relative coordinates $\bm{r}-\bm{R}$ and $\bm{r}^{\prime}-\bm{R}$ measured from the position of the localized spin, whereas it does not depend on that between conduction electrons, $\bm{r}-\bm{r}^{\prime}$. This nature reflects the superposition between the electron-Kondo singlet and the hole-Kondo singlet and not a simple pair-condensation of conduction electrons. We have thus demonstrated that the composite pair amplitude is generated in the SMCB-KL. We also comment on the relation between composite pair and odd-frequency pairing. The composite order parameter has been discussed in the context of the odd-frequency superconductor to date Emery and Kivelson (1992); Balatsky and Bon$\check{\mathrm{c}}$a (1993); Coleman et al. (1993). However, our result shows that, as long as the low-energy property is concerned, there is no pair amplitude between conduction electrons. In this sense, the composite order parameter does not directly mean the presence of odd-frequency pairing. This point can be clarified with our effective model in Eq. (1).

The values of mean-fields can be determined by solving the self-consistent equations given as follows:

[TABLE]

where $f(x)=1/(\mathrm{e}^{\beta x}+1)$ is the Fermi-Dirac distribution function and $\Omega$ is a volume of the system. The critical temperature $T_{c,\alpha}$ can be obtained for each mean fields from the above self-consistent equation as

[TABLE]

where $\gamma$ is the Euler constant and the prefactor is $2\mathrm{e}^{\gamma}/\pi\simeq 1.13$. There is also the relation $k_{\rm B}T_{c,\alpha}\propto E_{\mathrm{gap},\alpha}$, where the coefficient is order of unity but is still dependent on the parameters such as $D_{\alpha}$ and $\mu_{\alpha}$ due to the magnitude relation $D_{\alpha}\sim|\mu_{\alpha}|$. See SM C for more details. Since only the Kondo effect is involved in our theory, the appearance of the Kondo gap is natural as the characteristic energy scale, but it is much reduced from that in the usual heavy electrons because of a small density of states in semimetals. For a rough estimation, we use the expression $T_{K}=D\exp(-1/\lambda)$ where $D$ is a half bandwidth and $\lambda$ is a density of states at Fermi level multiplied with the Kondo coupling. If we take the bandwidth $2D=10^{4}$K and $\lambda=0.3$, we get $T_{K}\simeq 180$K. For SMCB, $\lambda$ is smaller and we get $T_{K}\simeq 0.2$K when we choose $\lambda=0.1$. We also examine the thermodynamic stability of the present ordered state in the low-temperature limit, and obtain the following free energy density $F(T)\simeq F_{0}(0)-2\sum_{\alpha}\rho_{\alpha}(0)\left|V_{\alpha}\right|^{2}$, where $F_{0}(T)$ denotes the component in the normal state. Therefore, the composite pair amplitude contributes to the energy-lowering in the low-temperature limit.

The energy scales of the critical temperature should be considered separately for the Kramers case and the non-Kramers case. In general, the electron and hole bands can be asymmetric, and then $T_{c,1}$ is different from $T_{c,2}$ in the Kramers case. Since the self-consistent equations for $T_{c,1}$ and $T_{c,2}$ are independent in the low-energy effective model, the superconducting critical temperature is estimated by $T_{c}\sim\min(T_{c,1},T_{c,2})$. The larger value of $T_{c,\alpha}$ is regarded as a crossover scale for the single-channel Kondo effect. For a non-Kramers doublet system, on the other hand, the equivalence of the channels is protected by the time-reversal symmetry, thus only one energy scale appears for $T_{c}$.

So far, we have considered only the degrees of freedom near the Fermi level, and concluded the presence of composite pair amplitude in Eq. (6) and the absence of the conventional Cooper pairs. However, with this situation, the composite pairs are not carried by the external field which acts only on the conduction electrons. The Meissner effect is then absent. Hence we need to consider the secondarily induced pair amplitude among the conduction electrons for the electromagnetic response. The property of this induced pair amplitude depends on the detail of the high-energy region and is specific to the material details. As one of the origins, we consider the finite cut-off of the energy dispersion at high energy $E_{c}$ ($\sim D_{1},D_{2}$) (see Fig. 2). We note that the regions $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ now overlap with each other. This is regarded as a lattice-regularization for the energy dispersion. The contribution from the high-energy region can be included perturbatively by expanding the physical quantities as a series of the orders of $E_{c}^{-1}$. (See SM D for derivation.) Thus the pair amplitude composed of conduction electrons can be finite in general for a tight-binding lattice, which is obtained as

[TABLE]

The coherence length $\xi_{c}$ for this induced Cooper pair is $\xi_{c}=\max\{\xi_{1},\xi_{2}\}$ which is the same order of magnitude as that for the composite pairs in Eq. (6).

Electromagnetic response functions can be calculated based on the Kubo formula Kubo (1957). Within the linear response of the vector potential $\bm{A}(\bm{q})$, which is applied on the conduction electrons, we can write the total current as $j_{\text{tot}}^{\mu}=-\sum_{\nu}K_{\mu\nu}A_{\nu}$. The introduced kernel is separated into two terms as $K_{\mu\nu}=K_{\mu\nu}^{d}+K_{\mu\nu}^{p}$, which describes diamagnetic and paramagnetic contributions, respectively. Each term is given as

[TABLE]

The paramagnetic current operator $\hat{j}_{\mu}^{p}$ is defined as

[TABLE]

where $e<0$ denotes the charge of the electron and $\bm{v}_{\bm{k}\alpha}=\hbar^{-1}\partial_{\bm{k}}\varepsilon_{\bm{k}\alpha}$ is the velocity. We note that the energy $\varepsilon_{\bm{k}\alpha}$ represents the tight-binding band and includes the high-energy part shown in Fig. 2.

We neglect the vertex correction for the current correlation function as in the dynamical mean-field theory which is exact in infinite dimensions and is regarded as a good approximation for three dimensions Georges et al. (1996); Khurana (1990). Then the leading-order contribution to the response function is diagramatically shown in Fig. 3. At zero temperature, the electromagnetic kernel is evaluated as

[TABLE]

where $n_{\alpha}$ is the particle number of the electron $(\alpha=1)$ and the hole $(\alpha=2)$. $\tilde{m}_{\alpha}(\sim-m_{\alpha})$ denotes the effective mass $\varepsilon_{\bm{k}\bar{\alpha}}$ with $\bm{k}\in\mathcal{K}_{\alpha}$ (here $\bar{\alpha}$ is the complementary component of $\alpha$ such as $\bar{1}=2$). See SM E for the detailed derivation. The magnitude of the kernel is smaller than the usual BCS superconductors by the factor of $(|V_{\alpha}|/E_{c})^{2}$. Hence, the superconducting state has a large magnetic penetration depth reflecting the small Cooper pair amplitude $\langle\psi_{\bm{k}\alpha\sigma}\psi_{-\bm{k},\alpha^{\prime}\sigma^{\prime}}\rangle$. This is because both $K^{p}$ and $K^{d}$ have the finite and almost the same amplitudes at zero temperature. Therefore, the cancelation between them occurs, and the situation is similar to that for the normal Kondo insulator. In the presence of the induced pair amplitude, however, the paramagnetic (diamagnetic) term slightly decreases (increases). The remaining contribution, which is the order of $E_{c}^{-2}$, plays a role of the superconducting current (See SM E). In this way, although the induced pair amplitude does not contribute to the condensation energy as we mentioned in the previous section, it plays an essential role for the electromagnetic response expected in the superconducting state.

In this paper we have established the mechanism for the full-gap superconductivity characteristic for the semi-metals along with the localized spin/pseudospins. We here comment on the consideration of some competing orders. In the correlated semi-metals, excitonic insulator states Mott (1961); Halperin and Rice (1968); Kune$\check{\mathrm{s}}$ (2015) or metallic CDWs Mattheiss (1973) would compete with the present mechanism of superconductivity. While such instabilities arise only when the conduction bands have a nesting of the Fermi surface, the present mechanism works when there are the semi-metallic conduction bands regardless of the details of Fermi surfaces.

Finally, let us discuss the candidate materials of the present superconducting state. We propose the two relevant materials $\mathrm{U}\mathrm{Be}_{13}$ Ott et al. (1983) and $\mathrm{Pr}\mathrm{Ir}_{2}\mathrm{Zn}_{20}$ Onimaru et al. (2011). For $\mathrm{U}\mathrm{Be}_{13}$, the system undergoes a phase transition into unconventional superconductivity from a non-Fermi liquid state, and the multi-channel Kondo effect has been suspected as a cause Cox (1987); Cox and Zawadowski (1998). We speculate that $\rm{U}\rm{Be}_{13}$ can be a candidate material for our scenario based on the following two reasonable assumptions. First, as seen from its chemical composition, the number of $\mathrm{Be}$ inside the unit-cell is much larger than the magnetic uranium ion. Therefore, the presence of the uranium atom does not affect much the conduction band structure. Second, the electronic structure of $\mathrm{Be}$ in the atomic limit is partially filled L-shell, which is composed of fully filled $2s$-orbital and empty $2p$-orbital. In a crystal environment, the $s$-band and the $p$-band resulting from the inter-berylium hopping bury the atomic gap and the semi-metallic conduction bands are formed. Indeed, the electronic structures of the elemental substance of Be has a semi-metallic character Loucks and Cutler (1964); Inoue and Yamashita (1973). Hence, $\mathrm{U}\mathrm{Be}_{13}$ can be regarded as the embedded magnetic $\mathrm{U}$ atom in the sea of the electrons which have electron and hole conduction bands. Thus the SMCB-KL considered in this paper can be realized in this material.

As for Pr-based materials, $\mathrm{Pr}\mathrm{Ir}_{2}\mathrm{Zn}_{20}$, shows a peculiar superconductivity with the multi-channel Kondo behavior Onimaru et al. (2011); Yamane et al. (2018), can also be regarded as the promising candidate material for the SMCB-KL, since the $\mathrm{Zn}$ has the fully-filled $3d$ and $4s$ orbitals and an energy gap is needed to locate an additional electron in higher-energy $4p$ orbital. Indeed the elemental substance of $\mathrm{Zn}$ shows semi-metallic character Stark and Falicov (1967); Allen et al. (1968). In addition, the number of $\mathrm{Zn}$ is much larger than that of $\mathrm{Pr}$. Hence the situation is close to $\mathrm{U}\mathrm{Be}_{13}$, and the formation of the SMCB-KL is expected. Further studies such as realistic band calculations combined with electronic correlation effects are needed to establish the actual realization of our proposal in real materials, which are left as interesting future issues. In addition to these compounds, the idea of the SMCB-KL can be applied to a wider class of materials with the semi-metallic conduction bands plus localized moments, and gives a general guiding principle to find unconventional superconductors.

Acknowledgement: This work was supported by the Japan Society for Promotion of Science (JSPS) KAKENHI Grants No. 18K13490, No. 16H04021, and No. 18H01176.