Superfluid response in heavy fermion superconductors

Abstract

Motivated by a recent London penetration depth measurement [H. Kim, et al., Phys. Rev. Lett. 114, 027003 (2015)] and novel composite pairing scenario [O. Erten, R. Flint, and P. Coleman, Phys. Rev. Lett. 114, 027002 (2015)] of the Yb-doped heavy fermion superconductor CeCoIn₅, we revisit the issue of superfluid response in the microscopic heavy fermion lattice model. However, from the literature, an explicit expression for the superfluid response function in heavy fermion superconductors is rare. In this paper, we investigate the superfluid density response function in the celebrated Kondo–Heisenberg model. To be specific, we derive the corresponding formalism from an effective fermionic large-N mean-field pairing Hamiltonian whose pairing interaction is assumed to originate from the effective local antiferromagnetic exchange interaction. Interestingly, we find that the physically correct, temperature-dependent superfluid density formula can only be obtained if the external electromagnetic field is directly coupled to the heavy fermion quasi-particle rather than the bare conduction electron or local moment. Such a unique feature emphasizes the key role of the Kondo-screening-renormalized heavy quasi-particle for low-temperature/energy thermodynamics and transport behaviors. As an important application, the theoretical result is compared to an experimental measurement in heavy fermion superconductors CeCoIn₅ and Yb-doped Ce_1-xYb_xCoIn₅ with fairly good agreement and the transition of the pairing symmetry in the latter material is explained as a simple doping effect. In addition, the requisite formalism for the commonly encountered nonmagnetic impurity and non-local electrodynamic effect are developed. Inspired by the success in explaining classic 115-series heavy fermion superconductors, we expect the present theory will be applied to understand other heavy fermion superconductors such as CeCu₂Si₂ and more generic multi-band superconductors.

Graphical abstract

Keywords

heavy fermion superconductor / Kondo lattice system / superfluid density

1 Introduction

Superfluid density, or London penetration depth measurement, is a fundamental experimental tool to detect low-temperature superconducting quasi-particle excitation in the highly entangled unconventional pairing ground-state [1-3]. The low-temperature behavior of such a quantity has provided invaluable information about the pairing symmetry for high-T_c cuprate, iron pnictide, and heavy fermion superconductors [4-13].

In the field of heavy fermion compounds, the power-law temperature dependence of superfluid density has been widely observed in the classic quasi-two-dimensional heavy fermion superconductor CeCoIn₅, with a remarkably high T_c=2.3 K [7-12, 14]; although its detailed power-law behavior is still controversial due to proximity to possible magnetic quantum critical points [7-11]. By combining thermodynamic and transport measurements, such as heat capacity [15, 16], thermal conductivity [15, 17], dc magnetization [18], NMR [19], tunneling spectrum [20], and the observation of a characteristic (\pi \mathrm{,}\pi \mathrm{,}\pi ) spin resonance mode and quasi-particle interference signal [21-24], CeCoIn₅ is now believed to be a nodal-d_{x^2-y^2}-wave superconductor, which has remarkable similarity to the well-known high-T_c cuprates.

Recently, London penetration depth measurements of Yb-doped CeCoIn₅, i.e., Ce_1-xYb_xCoIn₅, have been performed and, unexpectedly, the power-law behavior is gradually replaced with a fully gapped exponential behavior for increasing Yb-doping [13]. (However, a recent thermal conductivity measurement implies the nodal d-wave is robust and no pairing symmetry transition exists [25].) Such a surprising experimental finding is explained to be a change of pairing symmetry and even inspires an exotic local pairing scenario involving a composite molecular superfluid and Lifshitz transition of nodal Fermi surface [26].

However, when we try to understand the superfluid response in those heavy fermion superconductors from a theoretical standpoint, an explicit expression of the superfluid response function in the literature, particularly for the microscopic model Hamiltonian, is rare despite widespread and intensive experimental exploration. However, some phenomenological Fermi liquid-based arguments and superfluid density formulae of composite pairing do exist [27, 28]. In the present study, we investigate the superfluid density response function in the celebrated microscopic Kondo–Heisenberg model [29]. Specifically, we give a detailed derivation of the corresponding formalism from an effective fermionic large-N mean-field pairing Hamiltonian [30, 31], whose pairing interaction is assumed to originate from the effective local antiferromagnetic exchange interaction [32]. However, more conventional antiferromagnetic spin-fluctuation-exchange and resonance-valence-bond (RVB) theories give rise to the identical effective pairing Hamiltonian [29, 30, 33-36].

We find that the physically correct superfluid density formula can be obtained without any unsatisfactory breakdown only if the external electromagnetic field is assumed to directly couple to the heavy fermion quasi-particle rather than the bare conduction electron or local moment, see Fig. 1. Generally, this is the underlying feature when considering low-temperature/energy thermodynamics and transport; the physical fermionic excitation should be the (composite) heavy fermion quasi-particle [37], which results from nontrivial renormalization of collective Kondo screening effect. For superfluid density response considered in the present work to match the temperature/doping evolution of the superconducting heavy quasi-particle in the superfluid density formula, characteristic quantities, such as effective velocity and mass, have to describe the heavy quasi-particle, whose pairing drives the superconducting instability. These findings are presented below and contribute to the active field of heavy fermion systems.

As a crucial application of the derived superfluid response formula, we revisit undoped CeCoIn₅ and Yb-doped Ce_1-xYb_xCoIn₅. For the former, there is good agreement between theory and experiment, confirming the present theoretical formalism that underlies the dominating heavy fermion excitations. Furthermore, when applied to the latter, we successfully explain the doping evolution of pairing symmetry in Ce_1-xYb_xCoIn₅ as a simple doping effect by investigating its pairing strength. The calculated temperature-dependent superfluid density in both nodal d-wave and nodeless s-wave states is consistent with available measurements [13], which is a nontrivial confirmation of our theory. In addition, we have derived related formulas for general nonmagnetic impurities and non-local electrodynamic effects, which may be useful in realistic applications, e.g., La- and Nd-substituted CeCoIn₅ [13].

Having succeeded in explaining prototypical 115-series heavy fermion superconductors, we believe that the proposed formalism for superfluid response can be applied to understand other heavy fermion superconductors, such as the prototypical compound CeCu₂Si₂ and PrOs₄Sb₁₂ with possible time-reversal-symmetry breaking [38, 39], if proper modification and extensions are included. More generally, it is expected that our theory may also be useful for other multi-band superconductors, i.e., iron pnictide and LaOs₄Sb₁₂ [6, 40].

The remainder of this paper is organized as follows. In Section 2, the Kondo–Heisenberg model is introduced and its mean-field normal-state Hamiltonian is derived. In Section 3, the pairing interaction is discussed and the corresponding pairing Hamiltonian is derived. In Section 4, three different superfluid density formulas are derived and analyzed. We focus on the application of our theory to the heavy fermion superconductor CeCoIn₅ and its doped compound Ce_1-xYb_xCoIn₅ in Section 5. In Section 6, both the impurity effect and non-local effect are discussed. Finally, Section 7 is devoted to a brief discussion and conclusion.

2 The Kondo–Heisenberg model and its normal-state Hamiltonian

The standard definition of the Kondo–Heisenberg model reads [29, 41]

\begin{array}{}& H={\displaystyle \sum _{k\sigma }}{\epsilon }_k{c}_{k\sigma }^{†}{c}_{k\sigma }+J_K{\displaystyle \sum _i}{S}_i^c\cdot S_i^f+J_H{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}{S}_i^f\cdot S_j^f\end{array}

For simplicity, the system is located on a regular square lattice and the resulting conduction electron band is {\epsilon }_k=-2t(\cos k_x+\cos k_y)+4t^{\prime }\cos k_x\cos k_y-\mu with chemical potential, \mu. Extension to other lattices, such as triangular or honeycomb lattices, is straightforward [42, 43]. Next, we use fermionic representation to rewrite the degree of freedom of f-electron local spins as S_i^f=\frac{1}{2}{\displaystyle {\sum }_{\sigma {\sigma }^{\prime }}}{f}_{i\sigma }^{†}{\tau }_{\sigma {\sigma }^{\prime }}{f}_{i{\sigma }^{\prime }}, where \tau is the standard Pauli matrices. If magnetism and Kondo screening are required to treat on the same footing, the supersymmetric representation of local spins is helpful and it is of interest to see whether this novel method can be applied to heavy fermion superconductivity [44]. Moreover, to exclude nonphysical charge fluctuations of the auxiliary fermion, f_{\sigma }, it is crucial to enforce the local constraint {\displaystyle {\sum }_{\sigma }}{f}_{i\sigma }^{†}{f}_{i\sigma }=1 at each site.

Physically, this model describes two competing tendencies. The first is Kondo screening, which leads to the formation of a collective spin-singlet state among local moments and conduction electrons. The second is the short-range antiferromagnetic fluctuations introduced by the Heisenberg interaction between local moments. The observed complicated phenomena in diverse heavy electron systems are believed to be captured by these two active factors. To obtain qualitatively correct information in the paramagnetic heavy fermion liquid state, the fermionic large-N or slave-boson mean-field theory is widely utilized [45, 46]. Here, we use the former to obtain an effective mean-field Hamiltonian.

2.1 Mean-field model in the paramagnetic state

After performing the standard large-N mean-field approximation [35, 41], we obtain the following Hamiltonian

\begin{array}{}\text{(1)}& \begin{array}{ll}H=\hfill & {\displaystyle \sum }_{k\sigma }[{\epsilon }_k{c}_{k\sigma }^{†}{c}_{k\sigma }+{\chi }_k{f}_{k\sigma }^{†}{f}_{k\sigma }\hfill \\ \hfill & +\frac{J_{K}V}{2}(c_{k\sigma }^{†}{f}_{k\sigma }+f_{k\sigma }^{†}{c}_{k\sigma })]+E_0.\hfill \end{array}\end{array}

Here, the Kondo screening effect is encoded with an effective hybridization between conduction electron and local spins via V=-{\displaystyle {\sum }_{\sigma }}\langle c_{i\sigma }^{†}{f}_{i\sigma }\rangle. Meanwhile, local spins acquire dissipation {\chi }_k=J_H\chi {\eta }_k+\lambda with {\eta }_k=\cos k_x+\cos k_y due to the formation of nearest-neighbor valence-bond order \chi ={\displaystyle {\sum }_{\sigma }}\langle f_{i\sigma }^{†}{f}_{j\sigma }\rangle. Intuitively, such valence-bond order reflects the quantum dynamics of disordered local spins, which competes with magnetic long-range order. In addition, the Lagrangian multiplier, λ, is introduced to impose the local constraint on average and a constant energy shift, E_0=N_s(J_K{V}^2/2+J_H{\chi }^2-\lambda ), where the number of lattice sites, N_s, is included.

Using the following quasi-particle transformation relation

\begin{array}{}\text{(2)}& \begin{array}{l}{c}_{k\sigma }={\alpha }_k{A}_{k\sigma }-{\beta }_k{B}_{k\sigma }\\ f_{k\sigma }={\beta }_k{A}_{k\sigma }+{\alpha }_k{B}_{k\sigma }\mathrm{,}\end{array}\end{array}

where {\alpha }_k^2=\frac{1}{2}\left(1+\frac{{\epsilon }_k-{\chi }_k}{E_{0k}}\right), {\beta }_k^2=\frac{1}{2}\left(1-\frac{{\epsilon }_k-{\chi }_k}{E_{0k}}\right), {\alpha }_k{\beta }_k=\frac{J_{K}V}{2E_{0k}}, and E_{0k}=\sqrt{{({\epsilon }_k-{\chi }_k)}^2+{(J_{K}V)}^2}. The original Hamiltonian, Eq. (1), is transformed into a diagonalized Hamiltonian

\begin{array}{}\text{(3)}& H={\displaystyle \sum _{k\sigma }}(E_k^{+}{A}_{k\sigma }^{†}{A}_{k\sigma }+E_k^{-}{B}_{k\sigma }^{†}{B}_{k\sigma })+E_0\mathrm{,}\end{array}

where E_k^{\pm }=\frac{1}{2}({\epsilon }_k+{\chi }_k\pm E_{0k}) is the quasi-particle energy for quasi-particles A_{k\sigma } and B_{k\sigma }, respectively.

3 Pairing interaction in the Kondo–Heisenberg model

We are mainly interested in the superfluid response in the corresponding superconducting pairing phase; therefore, a proper superconducting model is required to perform any realistic and sensible calculation.

It has long been proposed that the introduced Heisenberg interaction can mediate spin-singlet pairing between conduction electrons via the pairing of auxiliary fermions in the present Kondo–Heisenberg model [24, 29, 30, 47]. Microscopically, the underlying mechanism of auxiliary fermion pairing can be attributed to the celebrated RVB state or, alternatively, the spin-fluctuation-exchange theory [33-35, 48]. However, to date, there is still no consensus on the mechanism of such an unconventional pairing and we must reconsider this issue with a more phenomenological approach [37].

Theoretically, in both RVB and spin-fluctuation-exchange theory, the superconducting pairing is induced by a particular effective antiferromagnetic interaction. The main difference is that the former emphasizes local real-space correlations while the latter is focused on dynamic exchange of energy/momentum, i.e., the collective antiferromagnetic fluctuations. In practice, since the observed pairing structure in heavy fermion superconductors has a strong momentum-dependent behavior, one may expect that the driving force of pairing is the space-dependent part of the effective pairing interaction rather than its time/energy part. In other words, we may imagine a real-space pairing interaction and, if we take the magnetic (fluctuation) nature of interaction into account, we will arrive at the following phenomenological local magnetic exchange interaction [32]

\begin{array}{}\text{(3)}& H_{int}={\displaystyle \sum _{ij}}{J}_{ij}{S}_i^f\cdot S_j^f\mathrm{.}\end{array}

The spin density operator denotes the spin degree of freedom of the local f-electron, which should result from the existing RKKY interaction. Furthermore, if only nearest-neighbor coupling is involved, this term naturally evolves into the Heisenberg term in the Kondo–Heisenberg model. That is to say that the Kondo–Heisenberg model itself has already included the basic element of superconducting pairing.

Now, we can write the Heisenberg interaction term into its momentum-space version [37]

\begin{array}{}\text{(3)}& H_H=\frac{1}{2}{\displaystyle \sum _q}{J}_q{S}_q^f\cdot S_{-q}^f\mathrm{,}\end{array}

where J_q=2J_H(\cos q_x+\cos q_y) for a square lattice. Then, using the auxiliary fermionic representation of a spin operator, the Heisenberg term is rewritten as

\begin{array}{}\text{(3)}& \begin{array}{ll}{H}_H=\hfill & {\displaystyle \sum _{k\mathrm{,}{k}^{\prime }\mathrm{,}q}}{J}_q(\frac{-1}{2}{f}_{k+q\uparrow }^{†}{f}_{k^{\prime }-q\downarrow }^{†}{f}_{k^{\prime }\downarrow }{f}_{k\uparrow }\hfill \\ \hfill & +\frac{\sigma {\sigma }^{\prime }}{8}{f}_{k+q\sigma }^{†}{f}_{k^{\prime }-q{\sigma }^{\prime }}^{†}{f}_{k^{\prime }{\sigma }^{\prime }}{f}_{k\sigma })\mathrm{.}\hfill \end{array}\end{array}

In a heavy fermion system, it is well-known that the dominating magnetic correlation has an antiferromagnetic feature. Thus, the spin-triplet pairing channel is not favored and we will only consider the spin-singlet channel in our remaining derivation. Therefore, the effective interaction in the spin-singlet pairing channel is

\begin{array}{}\text{(4)}& H_H={\displaystyle \sum _{k\mathrm{,}{k}^{\prime }\mathrm{,}q}}\frac{-3}{4}{J}_q{f}_{k+q\uparrow }^{†}{f}_{k^{\prime }-q\downarrow }^{†}{f}_{k^{\prime }\downarrow }{f}_{k\uparrow }\mathrm{.}\end{array}

On physical grounds, such a term describes that a pair of fermions with opposite spins feel a pairing interaction, V_q=\frac{-3}{4}{J}_q.

Let q\to k-k^{\prime } and symmetrize with momentum; now we have the following pairing interaction [37]

\begin{array}{}\text{(4)}& \begin{array}{ll}{V}_{k\mathrm{,}{k}^{\prime }}\hfill & =\frac{1}{2}(V_{k-k^{\prime }}+V_{k+k^{\prime }})\hfill \\ \hfill & =\frac{-3J}{2}(\cos k_x\cos k_x^{\prime }+\cos k_y\cos k_y^{\prime })\mathrm{,}\hfill \end{array}\end{array}

which leads to the effective BCS pairing Hamiltonian

\begin{array}{}\text{(4)}& H_{pairing}={\displaystyle \sum _{k\mathrm{,}{k}^{\prime }}}{V}_{k\mathrm{,}{k}^{\prime }}{f}_{k\uparrow }^{†}{f}_{-k\downarrow }^{†}{f}_{-k^{\prime }\downarrow }{f}_{k^{\prime }\uparrow }\mathrm{.}\end{array}

Considering the C₄ rotation symmetry of a square lattice [49], the present pairing interaction can be split into extended s-wave ({\gamma }_k^s=\cos k_x+\cos k_y) and d_{x^2-y^2}-wave ({\gamma }_k^d=\cos k_x-\cos k_y) pairing channels

\begin{array}{}\text{(4)}& \begin{array}{l}{V}_{k\mathrm{,}{k}^{\prime }}=V_{k\mathrm{,}{k}^{\prime }}^s+V_{k\mathrm{,}{k}^{\prime }}^d\mathrm{,}\\ V_{k\mathrm{,}{k}^{\prime }}^s=-\frac{3}{4}{J}_H{\gamma }_k^s{\gamma }_{k^{\prime }}^s\mathrm{,}\\ V_{k\mathrm{,}{k}^{\prime }}^d=-\frac{3}{4}{J}_H{\gamma }_k^d{\gamma }_{k^{\prime }}^d\mathrm{.}\end{array}\end{array}

In other words, the Heisenberg-like local antiferromagnetic exchange interaction may mediate extended s- and d_{x^2-y^2}-wave pairing on a square lattice.

For each pairing symmetry, we obtain its pairing Hamiltonian as

\begin{array}{}\text{(4)}& H_{pairing}=-\frac{3}{4}{J}_H\left({\displaystyle \sum _k}{\gamma }_k{f}_{k\uparrow }^{†}{f}_{-k\downarrow }^{†}\right)\left({\displaystyle \sum _{k^{\prime }}}{\gamma }_{k^{\prime }}{f}_{-k^{\prime }\downarrow }{f}_{k^{\prime }\uparrow }\right)\mathrm{.}\end{array}

Following the spirit of classic BCS mean-field theory [2], we introduce the pairing order parameter

\begin{array}{}\text{(4)}& \text{Δ}=-\frac{3}{4}{\displaystyle \sum _k}{\gamma }_k\langle f_{k\uparrow }^{†}{f}_{-k\downarrow }^{†}\rangle =-\frac{3}{4}{\displaystyle \sum _k}{\gamma }_k\langle f_{-k\downarrow }{f}_{k\uparrow }\rangle \mathrm{,}\end{array}

and the corresponding mean-field Hamiltonian

\begin{array}{}\text{(5)}& \begin{array}{ll}{H}_{pairing}=\hfill & J_H\text{Δ}{\displaystyle \sum _k}{\gamma }_k(f_{k\uparrow }^{†}{f}_{-k\downarrow }^{†}+f_{-k\downarrow }{f}_{k\uparrow })\hfill \\ \hfill & +\frac{4J_H{\text{Δ}}^2}{3}\mathrm{.}\hfill \end{array}\end{array}

Note that such a pairing Hamiltonian is identical to those in Refs. [30, 31, 47], where the RVB-like approach is used [34], i.e., a spin-singlet pairing order of auxiliary fermions in real-space is assumed to be

\begin{array}{}\text{(5)}& {\text{Δ}}_{ij}=\langle f_{i\uparrow }^{†}{f}_{j\downarrow }^{†}-f_{i\downarrow }^{†}{f}_{j\uparrow }^{†}\rangle \mathrm{.}\end{array}

Although our method is similar to the conventional antiferromagnetic spin-fluctuation-exchange theory, it is equivalent to the RVB-pairing theory at least at the present mean-field level, which emphasizes the core role of local magnetic interactions [32]. In some sense, the mean-field pairing model misses the effect of the original pairing interaction; however, if the fluctuation of the pairing interaction is included, one may identify which microscopic interaction is responsible for pairing.

By adding this pairing Hamiltonian into the normal-state model Eq. (1), one obtains the desired model for heavy fermion superconductivity. In practice, the pairing between different heavy fermion quasi-particle bands is usually neglected due to the mismatch of the Fermi surface as done in Ref. [24]. Thus, the pairing Hamiltonian is simplified to

\begin{array}{}\text{(6)}& \begin{array}{lll}{H}_{pairing}\hfill & =\hfill & J_H\text{Δ}{\displaystyle \sum _k}{\gamma }_k({\beta }_k^2(A_{k\uparrow }^{†}{A}_{-k\downarrow }^{†})+{\alpha }_k^2(B_{k\uparrow }^{†}{B}_{-k\downarrow }^{†})\hfill \\ \hfill & \hfill & +{\alpha }_k{\beta }_k(A_{k\uparrow }^{†}{B}_{-k\downarrow }^{†}+B_{k\uparrow }^{†}{A}_{-k\downarrow }^{†})+\text{H}\mathrm{.}\text{c}\mathrm{.})+J_H\frac{4{\text{Δ}}^2}{3}\hfill \\ \hfill & \simeq \hfill & {\displaystyle \sum _k}[J_H{\text{Δ}}_k^A(A_{k\uparrow }^{†}{A}_{-k\downarrow }^{†}+A_{-k\downarrow }{A}_{k\uparrow })\hfill \\ \hfill & \hfill & +J_H{\text{Δ}}_k^B(B_{k\uparrow }^{†}{B}_{-k\downarrow }^{†}+B_{-k\downarrow }{B}_{k\uparrow })]+J_H\frac{4{\text{Δ}}^2}{3}\mathrm{.}\hfill \end{array}\end{array}

We have defined two gap functions, {\text{Δ}}_k^A=\text{Δ}{\gamma }_k{\beta }_k^2 and {\text{Δ}}_k^B=\text{Δ}{\gamma }_k{\alpha }_k^2, for quasi-particles A_{\sigma } and B_{\sigma }, respectively. Note that such a gap function is modified by the normal-state coherent factor, whose momentum dependence deviates from the pure pairing function, {\gamma }_k. However, the qualitative physics is unchanged by such modulation. The advantage of such a simplification is to decouple each heavy fermion quasi-particle band, which leads to two independent BCS-like mean-field Hamiltonians. Therefore, after introducing the Bogoliubov transformation for each quasi-particle band

\begin{array}{}\text{(7)}& \begin{array}{l}{A}_{k\sigma }={\mu }_k^A{\tilde{A}}_{k\sigma }-{\nu }_k^A{\tilde{A}}_{-k-\sigma }^{†}\mathrm{,}\\ B_{k\sigma }={\mu }_k^A{\tilde{B}}_{k\sigma }-{\nu }_k^A{\tilde{B}}_{-k-\sigma }^{†}\mathrm{,}\end{array}\end{array}

where

\begin{array}{}\text{(7)}& \begin{array}{l}{\mu }_k^A=1-{\nu }_k^A=\frac{1}{2}\left(1+\frac{E_k^{+}}{E_k^A}\right)\mathrm{,}\\ {\mu }_k^B=1-{\nu }_k^B=\frac{1}{2}\left(1+\frac{E_k^{-}}{E_k^B}\right)\mathrm{,}\end{array}\end{array}

the resultant Hamiltonian is

\begin{array}{}\text{(8)}& \begin{array}{ll}H=\hfill & {\displaystyle \sum _k}[E_k^A{\tilde{A}}_{k\sigma }^{†}{\tilde{A}}_{k\sigma }+E_k^B{\tilde{B}}_{k\sigma }^{†}{\tilde{B}}_{k\sigma }]\hfill \\ \hfill & +{\displaystyle \sum _k}(E_k^{+}+E_k^{-}-E_k^A-E_k^B)+N_s{E}_0^{\prime }\mathrm{,}\hfill \end{array}\end{array}

where the energy spectrum of the superconducting quasi-particles has the following form

\begin{array}{}\text{(8)}& \begin{array}{l}{E}_k^A=\sqrt{{(E_k^{+})}^2+{(J_H{\text{Δ}}_k^A)}^2}\mathrm{,}\\ E_k^B=\sqrt{{(E_k^{-})}^2+{(J_H{\text{Δ}}_k^B)}^2}\mathrm{.}\end{array}\end{array}

The corresponding free energy and mean-field self-consistent equations are given in Appendix B.

The mean-field superconducting Hamiltonian for the Kondo–Heisenberg model has now been obtained. When the temperature is above the critical temperature, T_c, there still exists the normal-state described by Eq. (3), where the heavy fermion quasi-particle moves freely in the renormalized heavy Fermi liquid. If we cool the system below T_c, the normal heavy Fermi liquid transitions into the heavy fermion superconducting state with pairing of heavy quasi-particles. To avoid the instability of a quantum spin liquid or fractionalized Fermi liquid states [41], the ratio J_H/J_K should not be larger than a particular critical value. These critical values are summarized in Fig. 2.

4 Superfluid response formula

In this section, we will study the superfluid response function in heavy fermion superconductors. The main point of this section is to determine the superfluid response in heavy fermion superconductors; is there coupling between A and conduction electrons, auxiliary fermions, or heavy fermion quasi-particles? As detailed in this section, we find that likely only the heavy fermion quasi-particle couples with A strongly, as depicted in Fig. 1.

4.1 Superfluid response without the auxiliary fermion

In the Kondo–Heisenberg model, by definition, A acts on a conduction electron by substitution t_{ij}\to t_{ij}{\text{e}}^{\text{i}e{A}_{ij}}, where t_{ij} is the hopping energy in real-space. Thus, the Hamiltonian is invariant under local U(1) charge gauge transformation via

\begin{array}{}\text{(8)}& \begin{array}{l}{c}_{\sigma }\to c_{\sigma }{\text{e}}^{\text{i}{\theta }_i}\mathrm{,}\\ A_{ij}\to A_{ij}+({\theta }_i-{\theta }_j)/e\mathrm{,}\end{array}\end{array}

while f_{\sigma } does not directly respond to A because it originates from the fermionic representation of spin and has no electric charge [37].

Therefore, we assume that only the conduction electron can be involved in the calculation of superfluid response.

To discuss the superfluid response function, we consider the conduction electron as being coupled to A_q via the following Hamiltonian [37]

\begin{array}{}\text{(8)}& \begin{array}{ll}{H}^{\prime }\hfill & ={\displaystyle \sum _{kq\sigma }}e{v}_k\cdot A_q{c}_{k+q\sigma }^{†}{c}_{k\sigma }\hfill \\ \hfill & ={\displaystyle \sum _{kq}}e{v}_k\cdot A_q(c_{k+q\uparrow }^{†}{c}_{k\uparrow }+c_{k+q\downarrow }^{†}{c}_{k\downarrow })\hfill \\ \hfill & ={\displaystyle \sum _{kq}}e{v}_k\cdot A_q(c_{k+q\uparrow }^{†}{c}_{k\uparrow }+c_{-k\downarrow }{c}_{-k-q\downarrow }^{†})\hfill \\ \hfill & ={\displaystyle \sum _{kq}}e{v}_k\cdot A_q{\psi }_{k+q}^{†}{\psi }_k\mathrm{.}\hfill \end{array}\end{array}

Here, we have defined conduction electron velocity, v_k=\frac{\partial {\epsilon }_k}{\partial k_i}, and the Nambu spinor, {\psi }_k^{†}=(c_{k\uparrow }^{†}\mathrm{,}{c}_{-k\downarrow }). Note that the bare vertex is simply ie{v}_k, which leads to paramagnetic current-current correlation function

\begin{array}{}\text{(8)}& \begin{array}{lll}\text{Π}{(q)}_{ij}^p\hfill & =\hfill & +e^2{\displaystyle \sum _k}{v}_k^i{v}_{k+q}^{j}Tr[G(k)G(k+q)]\hfill \\ \hfill & =\hfill & +e^2{\displaystyle \sum _k}{v}_k^i{v}_{k+q}^j[G_{11}(k)G_{11}(k+q)\hfill \\ \hfill & \hfill & +G_{12}(k)G_{21}(k+q)+(1\leftrightarrow 2)]\mathrm{,}\hfill \end{array}\end{array}

where G is a Nambu Green’s function 2\times 2-matrix for conduction electrons and its detailed formalism is given in Appendix A.

Inserting the explicit expression of the Green’s function and setting the zero-energy limit, i{\text{Ω}}_n=0, and zero-momentum limit, q=0, we obtain the contribution of paramagnetic current

\begin{array}{}\text{(8)}& Re\text{Π}{(q=0)}_{ij}^p=+2e^2{\displaystyle \sum _k}\text{}{v}_k^i{v}_k^j\left[{\alpha }_k^4\frac{\partial f_F(E_k^A)}{\partial E_k^A}+{\beta }_k^4\frac{\partial f_F(E_k^B)}{\partial E_k^B}\right]\mathrm{.}\end{array}

Alternatively, the diamagnetic current is coupled to A through

\begin{array}{}\text{(8)}& \begin{array}{ll}{H}^{\prime }\hfill & ={\displaystyle \sum _{kq\sigma }}\frac{e^2}{2m}{A}_{k^{\prime }-q}^{†}{A}_{k^{\prime }}{c}_{k+q\sigma }^{†}{c}_{k\sigma }\hfill \\ \hfill & ={\displaystyle \sum _{kq}}\frac{e^2}{2m}{A}_{k^{\prime }-q}^{†}\cdot A_{k^{\prime }}(c_{k+q\uparrow }^{†}{c}_{k\uparrow }-c_{-k-q\downarrow }{c}_{-k\downarrow }^{†})\hfill \\ \hfill & ={\displaystyle \sum _{kq}}\frac{e^2}{2m}{A}_{k^{\prime }-q}^{†}\cdot A_{k^{\prime }}{\psi }_{k+q}^{†}{\widehat{\tau }}_z{\psi }_k\mathrm{.}\hfill \end{array}\end{array}

The effective mass tensor for a conduction electron is defined as {\left(\frac{1}{m}\right)}_{ij}=\frac{{\partial }^2{\epsilon }_k}{\partial k_i\partial k_j}. Here, the interacting vertex is -{\left(\frac{e^2}{2m}\right)}_{ij}{\widehat{\tau }}_z and its corresponding zero-frequency diamagnetic current-current correlation is

\begin{array}{}\text{(8)}& \begin{array}{ll}\text{Π}{(q=0)}_{ij}^d\hfill & =+e^2{\displaystyle \sum _k}{\left(\frac{1}{m}\right)}_{ij}Tr[{\widehat{\tau }}_{z}G(k)]\hfill \\ \hfill & =e^2{\displaystyle \sum _k}{\left(\frac{1}{m}\right)}_{ij}(G_{11}-G_{22})\hfill \\ \hfill & =e^2{\displaystyle \sum _k}{\left(\frac{-1}{m}\right)}_{ij}\left({\alpha }_k^2\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}+{\beta }_k^2\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T}\right)\mathrm{.}\hfill \end{array}\end{array}

Combining the paramagnetic and diamagnetic contributions, we obtain

\begin{array}{}\text{(8)}& \begin{array}{ll}Re\text{Π}{(q=0)}_{ij}=\hfill & e^2{\displaystyle \sum _k}[\frac{-1}{m_{ij}}({\alpha }_k^2\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}\hfill \\ \hfill & +{\beta }_k^2\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T})+2v_k^i{v}_k^j({\alpha }_k^4\frac{\partial f_F(E_k^A)}{\partial E_k^A}\hfill \\ \hfill & +{\beta }_k^4\frac{\partial f_F(E_k^B)}{\partial E_k^B})]\mathrm{.}\hfill \end{array}\end{array}

Finally, by using of the definition of superfluid density, \frac{{\rho }_{ij}{e}^2}{m}=Re\text{Π}{(q=0)}_{ij}, we obtain the desirable superfluid density formula, {\rho }_s\equiv {\rho }_{ii},

\begin{array}{}\text{(9)}& \begin{array}{ll}\frac{{\rho }_s(T)}{m}=\hfill & {\displaystyle \sum _k}[\frac{{\partial }^2{\epsilon }_k}{\partial k_x^2}\left(-{\alpha }_k^2\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}-{\beta }_k^2\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T}\right)\hfill \\ \hfill & +2{\left(\frac{\partial {\epsilon }_k}{\partial k_x}\right)}^2\left({\alpha }_k^4\frac{\partial f_F(E_k^A)}{\partial E_k^A}+{\beta }_k^4\frac{\partial f_F(E_k^B)}{\partial E_k^B}\right)]\mathrm{.}\hfill \end{array}\end{array}

Unfortunately, as shown in Fig. 3, for both d_{x^2-y^2}- and extended s-wave pairing symmetry, the superfluid density does not vanish at T_c, thus the formula Eq. (9) is undoubtedly invalid. The parameters we used are t=1, t^{\prime }=0.3, J_K=2, J_H=0.6, and n_c=0.9, which are typical values in mean-field calculations using the Kondo–Heisenberg model [30]. We will explain why this formula is incorrect, although the derivation is rather straightforward.

A careful reader may note that in the normal-state mean-field Hamiltonian, Eq. (1), the conduction electron is hybridized with the auxiliary fermion via the formation of Kondo screening (V\ne 0 and is real). Thus repeating the argument of U(1) charge gauge transformation discussed in the beginning of this subsection, the auxiliary fermion has the transformation

\begin{array}{}\text{(9)}& f_{i\sigma }\to f_{i\sigma }{\text{e}}^{\text{i}{\theta }_i}\mathrm{,}\end{array}

which means that f_{\sigma } now effectively acquires electric charge due to the well-developed Kondo hybridization in the heavy fermion liquid state. This statement is more general and beyond the present mean-field level [37]. In Appendix C, we give a brief discussion on this issue using path integral formalism.

When considering the superfluid response to an external electromagnetic field, we also need to include the contribution of the auxiliary fermion and this will be analyzed in the following subsection.

4.2 Superfluid response including the auxiliary fermion

In this section, we assume that in addition to the conduction electron, the auxiliary fermion also electromagnetically couples to A. Thus, the total paramagnetic current is

\begin{array}{}\text{(9)}& \begin{array}{l}{j}_i^p(q\mathrm{,}\tau )=j_i^c(q\mathrm{,}\tau )+j_i^f(q\mathrm{,}\tau )\mathrm{,}\\ j_i^c(q\mathrm{,}\tau )=e{\displaystyle \sum _{k\sigma }}{v}_c^i{c}_{k+q\sigma }^{†}(\tau )c_{k\sigma }(\tau )\mathrm{,}\\ j_i^f(q\mathrm{,}\tau )=e{\displaystyle \sum _{k\sigma }}{v}_f^i{f}_{k+q\sigma }^{†}(\tau )f_{k\sigma }(\tau )\mathrm{.}\end{array}\end{array}

Here, v_c^i=\partial {\epsilon }_k/\partial k_i and v_f^i=\partial {\chi }_k/\partial k_i are the effective velocity of the conduction electron and auxiliary fermion, respectively.

Therefore, the paramagnetic response function is

\begin{array}{}\text{(9)}& \begin{array}{ll}{\text{Π}}_{ij}(q\mathrm{,}\tau )\hfill & =-\langle T_{\tau }{j}_i^p(q\mathrm{,}\tau )j_i^p(-q\mathrm{,0})\rangle \hfill \\ \hfill & ={\text{Π}}_{ij}^{cc}(q\mathrm{,}\tau )+{\text{Π}}_{ij}^{cf}(q\mathrm{,}\tau )+{\text{Π}}_{ij}^{fc}(q\mathrm{,}\tau )+{\text{Π}}_{ij}^{ff}(q\mathrm{,}\tau )\mathrm{.}\hfill \end{array}\end{array}

Note that there exist four terms, among which {\text{Π}}_{ij}^{cc} and {\text{Π}}_{ij}^{ff} denote the contribution of the conduction electron and auxiliary fermion, respectively. The remaining terms, {\text{Π}}_{ij}^{cc} and {\text{Π}}_{ij}^{ff}, are the cross-terms, which imply the quantum interference effect between different currents. In last subsection, we calculated {\text{Π}}_{ij}^{cc},

\begin{array}{}\text{(9)}& Re\text{Π}{(q=0)}_{ii}^{cc}=2e^2{\displaystyle \sum _k}{(v_c^i)}^2\left[{\alpha }_k^4\frac{\partial f_F(E_k^A)}{\partial E_k^A}+{\beta }_k^4\frac{\partial f_F(E_k^B)}{\partial E_k^B}\right]\mathrm{.}\end{array}

After a long, yet straightforward calculation, we obtain

\begin{array}{}\text{(9)}& Re\text{Π}{(q=0)}_{ii}^{ff}=2e^2{\displaystyle \sum _k}{(v_f^i)}^2\left[{\beta }_k^4\frac{\partial f_F(E_k^A)}{\partial E_k^A}+{\alpha }_k^4\frac{\partial f_F(E_k^B)}{\partial E_k^B}\right]\mathrm{.}\end{array}

Since Re\text{Π}{(q=0)}_{ii}^{cf}=Re\text{Π}{(q=0)}_{ii}^{fc}, the we calculate \text{Π}{(q=0)}_{ii}^{cf},

\begin{array}{}\text{(9)}& \begin{array}{ll}\text{Π}{(q)}_{ii}^{cf}=\hfill & e^2{\displaystyle \sum _k}{v}_c^i{v}_f^i[G_{13}(k)G_{31}(k+q)\hfill \\ \hfill & +G_{14}(k)G_{41}(k+q)+(1\leftrightarrow 2)]\mathrm{.}\hfill \end{array}\end{array}

After inserting the explicit form of G_{13} and G_{14}, the final result reads

\begin{array}{}\text{(9)}& Re\text{Π}{(q=0)}_{ii}^{cf}=2e^2{\displaystyle \sum _k}{v}_c^i{v}_f^i\left[{\alpha }_k^2{\beta }_k^2\frac{\partial f_F(E_k^A)}{\partial E_k^A}+{\alpha }_k^2{\beta }_k^2\frac{\partial f_F(E_k^B)}{\partial E_k^B}\right]\mathrm{.}\end{array}

Therefore, the paramagnetic current response is

\begin{array}{}\text{(9)}& \begin{array}{ll}Re\text{Π}{(q=0)}_{ii}^p=\hfill & 2e^2{\displaystyle \sum _k}[({\alpha }_k^4{(v_c^i)}^2+{\beta }_k^4{(v_f^i)}^2+2{\alpha }_k^2{\beta }_k^2{v}_c{v}_f)\hfill \\ \hfill & \times \frac{\partial f_F(E_k^A)}{\partial E_k^A}+({\beta }_k^4{(v_c^i)}^2+{\alpha }_k^4{(v_f^i)}^2\hfill \\ \hfill & +2{\alpha }_k^2{\beta }_k^2{v}_c{v}_f)\times \frac{\partial f_F(E_k^B)}{\partial E_k^B}]\hfill \end{array}\end{array}

and the diamagnetic current response is obtained

\begin{array}{}\text{(9)}& \begin{array}{l}Re\text{Π}{(q=0)}_{ii}^d\hfill \\ \begin{array}{ll}=\hfill & -e^2{\displaystyle \sum _k}\left[{\alpha }_k^2{\left(\frac{1}{m_c}\right)}_{ii}+{\beta }_k^2{\left(\frac{1}{m_f}\right)}_{ii}\right]\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}\hfill \\ \hfill & +\left[{\beta }_k^2{\left(\frac{1}{m_c}\right)}_{ii}+{\alpha }_k^2{\left(\frac{1}{m_f}\right)}_{ii}\right]\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T}\mathrm{.}\hfill \end{array}\hfill \end{array}\end{array}

Therefore, our current superfluid density formula is

\begin{array}{}\text{(10)}& \frac{{\rho }_s(T)}{m}=\frac{1}{e^2}(Re\text{Π}{(q=0)}_{ii}^d+Re\text{Π}{(q=0)}_{ii}^p)\mathrm{.}\end{array}

Figure 4 shows the result of the calculation using Eq. (10), which, again, exhibits an incorrect result (using the same parameters as those in the last subsection). The reason for this may be due to the mismatch of temperature dependence between bare particles, (v_c, v_f, m_c, and m_f), and the renormalized heavy fermion quasi-particle, E_k^A, E_k^B. In other words, the bare and renormalized particles have different temperature dependences; thus the paramagnetic contribution cannot cancel out the diamagnetic term when T>T_c. More generally, when calculating physical quantities at T=0, since no temperature effect is involved, the underlying breakdown of the corresponding formalism is hidden. However, at finite temperatures, one may encounter the aforementioned problem.

We indeed confront a serious problem on the calculation of superfluid density.

4.3 A useful and correct superfluid response formula

In the previous section, we showed that the standard process of calculating superfluid response breaks down. However, there exists a hint to resolve this question by noting that when collective Kondo screening develops, the quasi-particles of the system are the heavy fermion quasi-particles A_{k\sigma } and B_{k\sigma }. If all interesting physics occurs at energy-scales smaller than the width of the quasi-particle bands, E_k^{+} and E_k^{-}, the heavy fermion quasi-particle can be seen as the only fundamental particles in our problem. Thus we may expect the low-energy thermodynamics and transport to be dominated by those heavy quasi-particles alone. In heavy fermion superconductivity, it is commonly believed that the pairing results from heavy quasi-particles and not directly involving the original conduction electron [3, 14, 37, 38]. Therefore, by considering these two factors, we propose that the heavy fermion quasi-particle is directly responsible for superfluid electromagnetic response, thus the corresponding paramagnetic current is

\begin{array}{}\text{(10)}& j_i^p(q\mathrm{,}\tau )=e{\displaystyle \sum _{k\sigma }}(v_A^i{A}_{k+q\sigma }^{†}(\tau )A_{k\sigma }(\tau )+v_B^i{B}_{k+q\sigma }^{†}(\tau )B_{k\sigma }(\tau ))\mathrm{,}\end{array}

where the effective velocity for heavy fermion quasi-particles A_{\sigma } and B_{\sigma } are v_A^i=\frac{\partial E_k^{+}}{\partial k_i} and v_B^i=\frac{\partial E_k^{-}}{\partial k_i}, respectively. In addition, the diamagnetic current is found to be

\begin{array}{}\text{(10)}& \begin{array}{ll}{j}_i^d(q\mathrm{,}\tau )=\hfill & e^2{A}_j{\displaystyle \sum _{k\sigma }}[{\left(\frac{1}{m_A}\right)}_{ij}{A}_{k+q\sigma }^{†}(\tau )A_{k\sigma }(\tau )\hfill \\ \hfill & +{\left(\frac{1}{m_B}\right)}_{ij}{B}_{k+q\sigma }^{†}(\tau )B_{k\sigma }(\tau )]\mathrm{.}\hfill \end{array}\end{array}

With the same treatment as in last subsection, we use these two heavy quasi-particle currents to derive Eq. (11).

\begin{array}{}\text{(11)}& \begin{array}{ll}\frac{{\rho }_s(T)}{m}=\hfill & {\displaystyle \sum _k}[-\frac{{\partial }^2{E}_k^{+}}{\partial k_x^2}\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}\hfill \\ \hfill & +2{\left(\frac{\partial E_k^{+}}{\partial k_x}\right)}^2\frac{\partial f_F(E_k^A)}{\partial E_k^A}-\frac{{\partial }^2{E}_k^{-}}{\partial k_x^2}\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T}\hfill \\ \hfill & +2{\left(\frac{\partial E_k^{-}}{\partial k_x}\right)}^2\frac{\partial f_F(E_k^B)}{\partial E_k^B}]\mathrm{,}\hfill \end{array}\end{array}

which is a useful and physically correct superfluid density formula and is the most important finding of the present study. As shown in Fig. 5, the superfluid density correctly vanishes when T_c is approached from the heavy fermion superconducting phase. Meanwhile, the global behavior of the temperature dependence in {\rho }_s(T) is also consistent with general physical expectation, e.g., the extended s-wave gives rise to a flat T dependence while the nodal d_{x^2-y^2}-wave pairing usually leads to a power-law behavior in the low-temperature regime [2, 3].

We should emphasize that our formula, Eq. (11), is not only valid in the superconducting mean-field model, Eq. (8), but can be used in other heavy fermion superconducting models. As an example, we consider the effective model proposed in Ref. [24], which is obtained from fitting data in quasi-particle interference experiments on CeCoIn₅ [22, 23]. The details of their model are not given here, though we refer the reader to their original article; the calculation result is shown in Fig. 6.

When compared to the experimental data of undoped CeCoIn₅ in Ref. [7] (see also Fig. 6), we see that the calculated {\rho }_s(T) does not agree with the global behavior in the experimental measurement. This may be due to the fact that the model parameters in Ref. [24] are not temperature-dependent; thus certain physical quantities, such as superfluid density, cannot be reliably calculated. Therefore, it is important to include the thermal effect and we will revisit this issue in following section.

Before ending this section, we notice that since the heavy fermion superconductors have a much longer London penetration depth, {\lambda }_L, than ordinary superconductors, e.g., about 190 nm for CeCoIn₅ [7], the effective mass, (m^{\ast }=\frac{{\mu }_0{\lambda }_L^2}{{\rho }_s{e}^2}\propto {\lambda }_L^2), of heavy fermion superconductors is also strongly enhanced, which implies that the Cooper pairs in those superconductors should come from the pairing of heavy fermion quasi-particles.

5 Superfluid response in heavy fermion superconductors CeCoIn₅ and Ce_1-xYb_xCoIn₅

5.1 Superfluid response in heavy fermion superconductor CeCoIn₅

In this section, we investigate the superfluid response in a typical heavy fermion superconductor, CeCoIn₅. To meet with the realistic experimental measurement, we fix model parameters as t=-1, t^{\prime }=0.3, J_K=2, J_H=0.6, V=-0.2965, \chi =0.2222, and n_c=0.9 [36]. The resulting normal-state Fermi surface is shown in Fig. 8, which mimics the multi-Fermi surface structure observed in ARPES measurements [50, 51].

Performing the calculation with Eq. (11) and assuming d_{x^2-y^2}-wave pairing, we compare our theoretical result to the experimental data in Fig. 7. From Fig. 7 we see that our theory is consistent with the global behavior of the microwave surface impedance measurements of Ref. [7] and our previous work [36]. Particularly, the observed low-temperature linear/power-law behavior of superfluid density is reproduced by our theoretical model with only the assumption of d_{x^2-y^2}-wave symmetry [7-11].

5.2 Issues with Ce_1-xYb_xCoIn₅

Next, it is interesting to understand the possible pairing symmetry transition in Yb-doped CeCoIn₅, which is suggested by recent superfluid density measurements and inspires an exotic local electron pairing scenario [13, 26]. Effectively, Yb-doping adds an extra electron carrier into CeCoIn₅, thus the density of the charge carrier is modified as n_c(x)=0.9+0.8x, where x is the nominal doping [26]. Ref. [13] finds that when x>0.2, the low-temperature London penetration depth shows full gap behavior (\text{Δ}\lambda (T)\sim T^n, for n>2). Therefore, such anomalous behavior of the superfluid response strongly suggests that the superconducting pairing symmetry should change the structure from nodal to nodeless (see also Fig. 9).

Motivated by the above observation, authors in Ref. [26] propose that the change of pairing symmetry may result from the formation of composite pairing, i.e. electrons in two orthogonal orbits screening the same local f-electron spin to induce pairing. In this scenario, Yb-doping leads to a Lifshitz transition of a nodal Fermi surface into a fully gapped composite-pairing molecular superfluid. Thus, the nodeless behavior observed in superfluid density measurements is expected.

Here we would like to propose an alternative possibility, i.e., the change of pairing structure may simply be a doping effect and not related to the emergence of exotic electron pairing. To be specific, we utilize our normal-state model in Sec. 2 and calculate the doping dependence of the so-called pairing strength invented in Ref. [32],

\begin{array}{}\text{(11)}& {\lambda }^{\ast }\equiv {\displaystyle \sum _{k\in FS}}{\gamma }_k^2\mathrm{.}\end{array}

Here, the dimensionless {\lambda }^{\ast } measures the overlap of pairing function {\gamma }_k on the normal-state Fermi surface. Obviously, less (more) overlap between the nodal point/line (maximized gap zone) and Fermi surface is helpful to lower the ground-state energy and the corresponding pairing strength is larger. In other words, pairing symmetry with larger {\lambda }^{\ast } will ultimately become the true superconducting pairing instability. For the Fermi surface shown in Fig. 8, although the nodal line of d_{x^2-y^2}-wave has much overlap over Fermi surface, its high gap zone also has large overlap. Thus, the whole effect is to promote d_{x^2-y^2}-wave over the nodeless extended s-wave, which is consistent with the calculated value {\lambda }_d^{\ast }=0.957>{\lambda }_s^{\ast }=0.875 and agrees with the observed dominated nodal d-wave feature in CeCoIn₅. It should be noted that the nodal line of the extended s-wave exists; however, it is not in contact with the present Fermi surface, thus, it behaves like nodeless pairing.

When considering the doping case, we plot {\lambda }^{\ast } and the effective mass ratio, m^{\ast }/m, in Fig. 10. We use t=-1, t^{\prime }=0.14t, J_K=1.92, J_H=0.6, \chi =0.2222, and T=0.01. V, μ, and λ are self-consistently solved through mean-field equations.

We find that when , where n_c=1.06 corresponds to the observed critical doping x_c=0.2, d_{x^2-y^2}-wave pairing dominates over extended s-wave pairing. However, if n_c>1.06, the situation is inverted and the effectively nodeless extended s-wave is the dominating pairing symmetry. Therefore, this simple calculation indicates there may be a transition at n_c=1.06, for x_c=0.2, from nodal d-wave to nodeless s-wave symmetry. When checking the doping evolution of the effective mass, we find no clear signal of any singular behavior in either α (E_k^{+}) and β (E_k^{-}) heavy quasi-particle bands. Thus we expect the changes of electronic band or Lifshitz transition observed in ARPES and dHvA measurements may not be an active factor for the pairing symmetry transition, which is an issue in itself and needs further investigation [52, 53].

Furthermore, Fig. 11 shows the temperature-dependent superfluid density in both nodal d-wave and nodeless s-wave states for n_c=1.0, where , and n_c=1.10, where x=0.25>x_c=0.2, respectively.

Importantly, the behaviors of the calculated results are consistent with the experimental data in Ref. [13], where the low-temperature behavior of the London penetration depth is changed from power-law to exponential when doping surpasses x_c=0.2. In addition, it is noted that the pairing symmetry transition here is likely a first-order transition since the corresponding derivative of the ground-state energy is discontinuous at the putative transition point, x=0.2. Thus, no radical change of an observable, such as T_c, is expected around this transition, which is also consistent with the doping evolution of Yb in Ref. [13].

Before ending this section, it is interesting to note that for the heavy fermion superconductor CeRhIn₅ [54], where the coexistence of antiferromagnetism and superconductivity is firmly established [55], the present superfluid response formula may also be applicable if we use a four-quasi-particle energy-band model resulting from magnetic and pairing orders. Thus, we should sum the contributions from all four bands with corresponding quasi-particle effective mass and velocity. An appropriate test for our theory will be to fit to the future London penetration depth measurement of this compound.

6 Nonmagnetic impurity and non-local effects on the superfluid response

6.1 The nonmagnetic impurity effect

For realistic materials, nonmagnetic impurity scattering is ubiquitous and unavoidable. Particularly, it plays a key role in the low-temperature regime where linear-T dependence of nodal d-wave is replaced with the T² law [56]. Here, we give the corresponding formalism, which may be relevant to explaining realistic experimental data.

The main attribute of nonmagnetic impurity scattering is that it introduces effective damping rates, {\text{Γ}}_A and {\text{Γ}}_B, for the paired heavy quasi-particles. Then, the corresponding retarded heavy quasi-particle Green’s function in the superconducting state is

\begin{array}{}\text{(11)}& G_{AA}^R(k\mathrm{,}\omega )=\left(\begin{array}{cc}{G}_{11}^{AA}& G_{12}^{AA}\\ G_{21}^{AA}& G_{22}^{AA}\end{array}\right)\mathrm{,}\end{array}

where

\begin{array}{}\text{(11)}& \begin{array}{ll}{G}_{11}^{AA}\hfill & =\frac{{({\mu }_k^A)}^2}{\omega -E_k^A+\text{i}{\text{Γ}}_A}+\frac{{({\nu }_k^A)}^2}{\omega +E_k^A+\text{i}{\text{Γ}}_A}\hfill \\ G_{12}^{AA}\hfill & =G_{21}^{AA}\hfill \\ \hfill & ={\mu }_k^A{\nu }_k^A\left(\frac{1}{\omega -E_k^A+\text{i}{\text{Γ}}_A}-\frac{1}{\omega +E_k^A+\text{i}{\text{Γ}}_A}\right)\hfill \\ G_{22}^{AA}\hfill & =\frac{{({\nu }_k^A)}^2}{\omega -E_k^A+\text{i}{\text{Γ}}_A}+\frac{{({\mu }_k^A)}^2}{\omega +E_k^A+\text{i}{\text{Γ}}_A}\hfill \end{array}\end{array}

and the case for G_{BB}^R(k\mathrm{,}\omega ) is similar. Next, following the same treatment as above, we obtain the expression of superfluid density

\begin{array}{}\text{(11)}& \begin{array}{ll}\frac{{\rho }_s(T)}{m}=\hfill & {\displaystyle \sum _k}[\frac{{\partial }^2{E}_k^{+}}{\partial k_x^2}\frac{E_k^{+}}{E_k^A}{W}_k^A+2{\left(\frac{\partial E_k^{+}}{\partial k_x}\right)}^2{Z}_k^A\hfill \\ \hfill & +\frac{{\partial }^2{E}_k^{-}}{\partial k_x^2}\frac{E_k^{-}}{E_k^B}{W}_k^B+2{\left(\frac{\partial E_k^{-}}{\partial k_x}\right)}^2{Z}_k^B]\mathrm{.}\hfill \end{array}\end{array}

Where

\begin{array}{}\text{(11)}& \begin{array}{l}{W}_k^A={\displaystyle {\int }_{-\infty }^{\infty }}\text{d}\omega f_F(\omega )\left[\frac{{\text{Γ}}_A/\pi }{{(\omega -E_k^A)}^2+{\text{Γ}}_A^2}-\frac{{\text{Γ}}_A/\pi }{{(\omega +E_k^A)}^2+{\text{Γ}}_A^2}\right]\\ W_k^B={\displaystyle {\int }_{-\infty }^{\infty }}\text{d}\omega f_F(\omega )\left[\frac{{\text{Γ}}_B/\pi }{{(\omega -E_k^B)}^2+{\text{Γ}}_B^2}-\frac{{\text{Γ}}_B/\pi }{{(\omega +E_k^B)}^2+{\text{Γ}}_B^2}\right]\\ Z_k^A=-\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }}\text{d}\omega f_F(\omega )Im\frac{{(\omega +\text{i}{\text{Γ}}_A)}^2+{(E_k^A)}^2}{{[(\omega +\text{i}{\text{Γ}}_A{)}^2-{(E_k^A)}^2]}^2}\\ Z_k^B=-\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }}\text{d}\omega f_F(\omega )Im\frac{{(\omega +\text{i}{\text{Γ}}_B)}^2+{(E_k^B)}^2}{{[(\omega +\text{i}{\text{Γ}}_B{)}^2-{(E_k^B)}^2]}^2}\mathrm{.}\end{array}\end{array}

By using the above equations, one can inspect the explicit behavior of superfluid density in an all-temperature regime. Here, we investigate the low-temperature behavior by expanding Z_k^{\alpha =A\mathrm{,}B} [2] (W_k^{\alpha } is approximated to be constant in this case)

\begin{array}{}\text{(11)}& Z_k^{\alpha }=-\frac{1}{\pi }\left[\frac{{\text{Γ}}_{\alpha }}{{\text{Γ}}_{\alpha }^2+{(E_k^{\alpha })}^2}-\frac{{\text{Γ}}_{\alpha }({\text{Γ}}_{\alpha }^2-3(E_k^{\alpha }{)}^2)T^2}{3({\text{Γ}}_{\alpha }^2+{(E_k^{\alpha })}^2{)}^3}+O(T^4)\right]\text{}\mathrm{,}\end{array}

where the first term denotes a correction to zero-temperature superfluid density and the second term introduces the impurity-dominated T² dependence since this term vanishes as {\text{Γ}}_{\alpha }\to 0.

Then, for nodal superconductors, such as d_{x^2-y^2}-wave in undoped CeCoIn₅, it is found that the low-temperature superfluid density behaves as

\begin{array}{}\text{(11)}& \begin{array}{ll}{\rho }_s(T)\simeq \hfill & {\rho }_A(0)\left(1-\frac{2{\text{Γ}}_A}{\pi {\text{Δ}}_A}\ln \frac{2{\text{Δ}}_A}{{\text{Γ}}_A}-\frac{T^2}{3\pi {\text{Γ}}_A{\text{Δ}}_A}\right)\mathrm{,}\hfill \\ \hfill & +{\rho }_B(0)\left(1-\frac{2{\text{Γ}}_B}{\pi {\text{Δ}}_B}\ln \frac{2{\text{Δ}}_B}{{\text{Γ}}_B}-\frac{T^2}{3\pi {\text{Γ}}_B{\text{Δ}}_B}\right)\mathrm{,}\hfill \end{array}\end{array}

which is in contrast to the linear-T behavior of the pure sample. Here, {\rho }_{\alpha }(0) and {\text{Δ}}_{\alpha } are the zero-temperature superfluid density and effective gap for each quasi-particle band. We should emphasize that both the pure and impure nodal d-waves are at least qualitatively consistent with the observed power-law behaviors in CeCoIn₅. For La- and Nd-substituted CeCoIn₅ [13], this impurity induces a T² dirty d-wave behavior as observed by using a tunnel-diode resonator technique. Furthermore, for the possible nodeless pairing state in Yb-doped Ce_1-xYb_xCoIn₅, no qualitative behaviors are changed by nonmagnetic impurity scattering since all low-energy excitations are fully gapped.

6.2 The non-local effect

In nodal pairing states, such as the studied d_{x^2-y^2}-wave, when approaching gap nodal points, its coherence length, \xi \sim v_F/\text{Δ}, is larger than its {\lambda }_L. Thus, the so-called non-local effects are important [57], as they may be responsible for the observed power-law behavior of London penetration depth in CeCoIn₅ [7, 9, 10].

Following Ref. [57], we find that, for a specular boundary, the temperature-dependent London penetration depth is

\begin{array}{}\text{(11)}& \lambda (T)=\frac{2}{\pi }{\displaystyle {\int }_0^{\infty }}\frac{\text{d}q}{q^2+Re\text{Π}(q\mathrm{,}\omega =0)}\mathrm{,}\end{array}

where the momentum-dependent superfluid response function is

\begin{array}{}\text{(11)}& \begin{array}{l}Re\text{Π}(q\mathrm{,}\omega =0)\hfill \\ \begin{array}{ll}=\hfill & -e^2{\displaystyle \sum _k}[\frac{{\partial }^2{E}_k^{+}}{\partial k_x^2}\frac{E_k^{+}}{E_k^A}\tanh \frac{E_k^A}{2T}\hfill \\ \hfill & +{({\mu }_k^A{\mu }_{k+q}^A+{\nu }_k^A{\nu }_{k+q}^A)}^2\frac{\tanh \frac{\beta E_k^A}{2}-\tanh \frac{\beta E_{k+q}^A}{2}}{E_k^A-E_{k+q}^A}\hfill \\ \hfill & +{({\mu }_k^A{\nu }_{k+q}^A-{\nu }_k^A{\mu }_{k+q}^A)}^2\frac{\tanh \frac{\beta E_k^A}{2}+\tanh \frac{\beta E_{k+q}^A}{2}}{E_k^A+E_{k+q}^A}\hfill \\ \hfill & +\frac{{\partial }^2{E}_k^{-}}{\partial k_x^2}\frac{E_k^{-}}{E_k^B}\tanh \frac{E_k^B}{2T}\hfill \\ \hfill & +{({\mu }_k^B{\mu }_{k+q}^B+{\nu }_k^B{\nu }_{k+q}^B)}^2\frac{\tanh \frac{\beta E_k^B}{2}-\tanh \frac{\beta E_{k+q}^B}{2}}{E_k^B-E_{k+q}^B}\hfill \\ \hfill & +{({\mu }_k^B{\nu }_{k+q}^B-{\nu }_k^B{\mu }_{k+q}^B)}^2\frac{\tanh \frac{\beta E_k^B}{2}+\tanh \frac{\beta E_{k+q}^B}{2}}{E_k^B+E_{k+q}^B}]\mathrm{.}\hfill \end{array}\hfill \end{array}\end{array}

At low temperatures (), for nodal superconducting states, the above equation gives \text{Δ}\lambda (T)\equiv \lambda (T)-\lambda (0)\sim T^2, which is similar to the nonmagnetic impurity effect and modifies the linear-T behavior. We note that Ref. [9] argued that the penetration depth of CeCoIn₅ is governed by this non-local electrodynamic effect rather than the nonmagnetic impurity effect.

7 Discussion and conclusion

In this work, we have systematically studied the superfluid density response formula in a pure sample and one with a nonmagnetic impurity. These formulas have been successfully applied to undoped heavy fermion superconductor, CeCoIn₅ and its Yb-doped counterpart, Ce_1-xYb_xCoIn₅. Theory and experiment of those typical heavy fermion superconductors are in good agreement, which confirms our basic formalism and emphasizes the core role of heavy fermion quasi-particles. The present work may provide a useful formalism to understand and explain existing and future superfluid density experiments in heavy fermion superconductivity.

We have noted that the measurement of thermal conductivity suggests there is no pairing symmetry transition in Ce_1-xYb_xCoIn₅, which is in contrast to the result of London penetration depth experiments. Frankly, we are still unable to judge which experiment gives the true physical behavior; however, if we use the model in Ref. [24] with our superfluid density formula, we find only a nodal d-wave state is possible.

It is intriguing to note the possibility that if the critical quasi-particle picture is still valid in heavy fermion superconducting states in proximity to the quantum critical point [58]; the paramagnetic current should contribute an extra T^{\mathrm{1/2}} due to the critical enhancement of the effective velocity. Based on our superfluid response formula, the nodal d-wave states may have T^{\mathrm{3/2}} behavior, which agrees well with existing data [7, 9, 10].

In addition, as suggested in Refs. [26, 28], since both interstitial and onsite two-channel effects are likely important for heavy fermion superconductivity, the composite pairing has to be considered seriously, which is an interesting and crucial issue for future study.

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