Kondo Lattice model for field induced triplet pairing in CeRh$_2$As$_2$

Introduction

The possibility of superconductivity that survives strong magnetic fields has renewed interest in odd-parity and spin-triplet pairing, not least because some such states may possess nontrivial topological properties. In spin orbit coupled materials, however, the physical spin of a Cooper pair need not coincide with the pseudospin character of the gap projected onto the low energy quasiparticle bands. This distinction is especially important in multiband heavy fermion superconductors, where crystal symmetry and strong electronic correlations act together to determine the pairing state.

The heavy fermion compound CeRh$_2$As$_2$ exhibits a field-driven transition between two superconducting phases [1, 2]. For a magnetic field along the crystallographic $c$ axis, the low field SC1 phase gives way to a high field SC2 phase near 4 T, while superconductivity persists to an upper critical field of approximately 14 T, far above the conventional weak coupling Pauli scale for $T_{\mathrm{c}} \simeq0.3$ K. The pronounced contrast in field response motivates the interpretation of SC1 and SC2 as competing even and odd parity states, although the microscopic spin structure of the high field phase remains an active question [3].

A key feature of CeRh$_2$As$_2$ is that its crystal is globally inversion symmetric but the two Ce sites within a unit cell each lack a local inversion center. Inversion exchanges the Ce sublattices, allowing a staggered antisymmetric spin orbit coupling (ASOC) whose sign alternates between the layers. Symmetry based descriptions have shown how this structure can stabilize uniform even-parity and staggered odd parity superconducting states and allow a magnetic field to favor the latter [4, 5, 6]. The corresponding pairing patterns are illustrated in Figure 1.

Several theoretical approaches have clarified the competition between superconducting states in locally non-centrosymmetric systems. Möckli and Ramires [7] analyzed both even to odd and odd to odd field driven transitions within a two sublattice description, while Cavanagh et al. [4] emphasized the role of nonsymmorphic symmetry and the associated momentum dependent spin orbit structure. Nogaki and Yanase [8] demonstrated that electronic correlations can enhance the parity transition field. Extending the description to vortex degrees of freedom, Minamim and Yanase [9, 10] investigated the parity transition and possible unconventional vortex textures.

Recent microscopic studies provide complementary perspectives on the pairing mechanism. Lee, Agterberg, and Brydon [11] found competing superconducting instabilities of opposite parity and spin density wave tendencies associated with type-II van Hove singularities. Nogaki and Yanase [12], using a multiorbital Dirac–Anderson model, identified leading $B_{1u}$ and $B_{2g}$ superconducting instabilities influenced by quantum geometry and correlations. The appearance of a $B_{1u}$ channel provides a useful symmetry level comparison with the leading projected gap discussed below, although the microscopic origins and orbital structures of the states need not coincide. Work on the coexistence of magnetic order and superconductivity [13] further underscores the importance of the nearby ordered phases.

More recently, Ishizuka et al. [14] investigated the superconducting symmetry and possible topological properties of CeRh$_2$As$_2$ using realistic electronic structure calculations and crystalline symmetry analysis. In particular, their classification accommodates staggered spin singlet pairing in an odd parity representation, including a $d_{x^2-y^2}$-type form factor in the $B_u$ sector appropriate to a field along the $c$ axis. This provides a useful symmetry benchmark for a microscopic theory. The symmetry classification alone, however, does not determine how the staggered ASOC is generated in a heavy fermion Kondo lattice, or how the corresponding odd parity gap develops on the heavy quasiparticle Fermi surface.

These questions are sharpened by the strongly correlated normal state. The Curie–Weiss susceptibility, the onset of coherent transport on the Kondo scale, and the large superconducting specific heat anomaly indicate the importance of local moments and heavy quasiparticles [1, 2, 15]. Neutron measurements also reveal pronounced quasi-two-dimensional antiferromagnetic fluctuations [16], which naturally motivate predominantly intralayer $d$-wave spin singlet pairing. The challenge is to understand how such pairing can develop in a Kondo lattice with antiferromagnetic interactions, while also giving rise to an odd parity pseudospin triplet gap in the high field phase.

Here we construct a minimal Kondo lattice description in which local inversion symmetry breaking permits an additional onsite hybridization channel. Interference between this channel and the conventional intersite hybridization generates a staggered ASOC for the heavy quasiparticles. We then study the competition between uniform and staggered intralayer pairing states in a magnetic field, determining their phase boundary by comparing the self-consistent free energies. The staggered physical spin singlet order parameter acquires an effective odd parity pseudospin triplet structure upon projection onto the heavy bands. Its leading momentum dependence is compatible with the $B_u$ symmetry class discussed by Ishizuka et al., while our calculation identifies a microscopic route to that structure.

The paper is organized as follows. Section II introduces the crystal symmetries and the uniform and staggered pairing channels. Section III develops the local parity breaking hybridization and its symmetry allowed form factors. Section IV constructs the Kondo lattice model and mean-field formulation. Section V presents the emergent ASOC, the field dependent superconducting phase diagram, and the projected pseudospin gap. Section VI discusses the physical interpretation, limitations, and possible extensions to related heavy fermion materials. Technical details of the decoupling and charge conjugation conventions are given in the appendices.

Crystal Symmetry and Pairing

CeRh$_2$As$_2$ has a tetragonal $P4/nmm$ space group with two Ce atoms per unit cell. Although there is global inversion symmetry, locally, inversion symmetry is broken at the Ce sites due to the alternating RhAs and AsRh layers along the $c$ axis. The two Ce atoms in the unit cell are related by inversion, as shown in Figure 1. This leads to an additional sublattice degree of freedom and allows a staggered antisymmetric spin orbit coupling (ASOC) interaction of the form:

where $\alpha_i$ and $\sigma_i$ are Pauli matrices in sublattice and spin space respectively. The sublattice $\alpha_3$ reverses sign under

sublattice exchange and together with the momentum inversion, preserves global inversion.

In conventional superconductors, the allowed pairing symmetry is determined from the combined parity of spin and momentum which must be antisymmetric. The electron pair wavefunction with fermionic statistics must be antisymmetric upon particle exchange:

where $1 = \{\mathbf{R}_1, \alpha_1, \sigma_1\}$ and $2 = \{\mathbf{R}_2, \alpha_2, \sigma_2\}$ represents the position, sublattice, and spin of particle 1 and 2 respectively. Typically, in conventional superconductors with inversion symmetry, the allowed states are momentum even spin singlet or momentum odd spin triplet states. However, CeRh$_2$As$_2$ has an additional sublattice degree of freedom that enables a new combination of spin, momentum and sublattice. The quasi two dimensional nature of CeRh$_2$As$_2$ restricts to intralayer dominant pairing within the Ce layers. The symmetry allowed pairing states are given in Table I.

The two pairing solutions allowed by the sublattice is the uniform state which has the same pairing sign on all layers corresponding to an even parity state and the staggered state which has different pairing sign on alternating layers corresponding to an odd parity state. The two disTABLE I. The symmetry allowed pairing states in CeRh$_2$As$_2$. The total fermionic pair wavefunction must have odd parity upon particle exchange. The pairing wavefunction is given by $\Delta_s$, where $s = A, B$ which represent pairing within the layers which represent pairing between the same sublattice Ce atoms. The total parity is obtained by the product of the parities which include spin $P_\sigma$, momentum $P_l$, and sublattice $P_\alpha$ parity.

tinct solutions are shown in Figure 1. These two states are nearly degenerate in energy and exhibit a field induced transition in the presence of a field applied perpendicular to the plane.

Parity breaking onsite hybridization

A simple model of Ce based heavy fermion systems treats the high energy physics via an Anderson model, describing $f$ electrons hybridized with a conduction band formed from delocalized $d$ electrons. The low energy physics is then obtained by carrying out a Schrieffer-Wolff transformation that encodes the physics of the hybridization as a Kondo exchange interaction between the local moments and the conduction sea. If we also include the nearest neighbor RKKY interactions, the resulting Hamiltonian is called a Kondo Heisenberg lattice.

The hybridization between an $f$-state and a $d$-state is given by the overlap of the wavefunctions with the atomic potentials. The local hybridization is then

where $V(\mathbf{r})$ is the atomic potential at a Cerium site. In locally centrosymmetric systems, the $f$-state is odd parity $\psi_{f\sigma}(\mathbf{r}) = -\psi_{f\sigma}(-\mathbf{r})$ while the $d$-state is even parity $\psi_{d\sigma}(\mathbf{r}) = +\psi_{d\sigma}(-\mathbf{r})$, and since the electronic potential is inversion symmetric, $V(\mathbf{r}) = V(-\mathbf{r})$, $V_{fd} = 0$ vanishes. Thus in a centrosymmetric crystal, the $f-d$ hybridization takes place between neighboring Cerium sites and when written in momentum space, must be odd under inversion, yet even under time reversal. A minimal model for the non local odd parity hybridization within the planes of Ce atoms, has the form

where $V$ is the intersite hybridization strength and $\mathbf{k}_{\perp}$ is the in-plane wavevector. This model form is obtained in a Slater-Koster treatment of hybridizing $s$- and $p$-states and captures the essence of spin orbit coupled heavy fermion compounds allowing, for example, a simple description of topological Kondo insulators [17]. By contrast, CeRh$_2$As$_2$ has a locally noncentrosymmetric structure with inversion centers located in between the Cerium layers. In this case, the atomic potential contains a gradient along the $c$-axis that alternates between neighboring Cerium layers. This time, the atomic potential is not necessarily inversion symmetric:

where $\alpha= \pm1$ is a layer index. In this situation, the $f$- and $d$-states now acquire a $c$-axis dipole moment which alternates between layers, and they no longer a well defined parity. The resulting overlap between the $f$ and $d$ states at the same site given by (3) is then non-zero, leading to a local hybridization. A minimal model for the local parity breaking hybridization is given by:

where $\widetilde{W}$ is the local parity breaking hybridization amplitude. The appearance of the $\alpha_{3}$ term reflects the reversal of the $c$-axis electric field on neighboring layers. The hybridization is invariant under inversion, which inverts both the spin and sublattice index. The imaginary prefactor also guarantees that the hybridization is invariant under time-reversal. A model calculation of such a local hybridization mechanism is obtained by calculating the Slater-Koster overlap integral between a $j=1/2$ $s$ and $p$-state (see: Figure 2). A more realistic model of the hybridization, involving the $j=5/2$ overlap between a $d$ and $f$ state is expected to give rise to an even momentum dependence in the quantity $W$ which we neglect here. Combining the local (6) and intersite (4) terms, we can formulate a minimal model for the hybridization form factor

where $W=\widetilde{W}/V$ and $\mathbf{s}_{\mathbf{k}}=(\sin k_x,\sin k_y)$. The dimensionless parameter $W$ is the ratio of the local hybridization amplitude $\widetilde{W}$ to the intersite amplitude $V$. For real $W$ and $V$, the form factors obey the exact identity

The interference term is staggered in the layer index and has the antisymmetric spin-orbit structure. When an $f$ electron tunnels through the conduction band, the resulting tunneling amplitude will reflect an interference between the two terms in the hybridization. We can calculate this effect by integrating out the conduction electrons, to obtain an $f$ electron self-energy given by

where for simplicity we have approximated the conduction electrons by a single band of dispersion $\epsilon_{\mathbf{k}}$. The interference between the two channels creates a spin orbit cross term in this self-energy that has the form

where we denote $s_x=\sin k_x$, $s_y=\sin k_y$. This spin orbit term has the same momentum and spin dependence as the previous staggered ASOC interaction introduced from symmetry analysis. The Kondo lattice therefore demonstrates a mechanism of how the ASOC is generated through a local parity breaking hybridization induced by the locally broken inversion symmetry.

Model and Methods

Kondo lattice construction

We now formulate a minimal Kondo Heisenberg model for CeRh$_2$As$_2$, with the Hamiltonian $H=H_c+H_K+H_H$ where

where $c_{\mathbf{k},\alpha,\sigma}^{\dagger}$ creates a conduction electron with momentum $\mathbf{k}$, sublattice index $\alpha=\pm1$ and spin index $\sigma=\pm1$, while $\mathbf{S}_j=(S_j^x,S_j^y,S_j^z)$ is the three component spin-1/2 local moment operator for site $j$. We use Pauli matrix normalization for the local moments, i.e. $\operatorname{Tr}[S_j^\sigma S_j^{\sigma'}]=2\delta_{\sigma\sigma'}$. A summation convention is used for the sum over spin and sublattice indices $\alpha$. The first term $H_c$ describes the conduction electrons, with a dispersion obtained from a tight binding calculation on a tetragonal lattice

Here the dispersion elements

describe the intra, and inter-layer hopping processes. The normal state parameters are given by $t$ the intralayer hopping strength and $\mu$ the chemical potential and $t_s=t_u+t_d$ is the interlayer hopping strength, $t_u$ is hopping up and $t_d$ is hopping down. To maintain a connectivity with earlier work, we have followed the notation used in [1, 4].

$H_K$ is the local Kondo interaction between the conduction electrons and the local moments, where $J_K$ is the Kondo coupling strength and the site index $j\equiv(\mathbf{R}_j,\alpha)$ is a proxy for the position of the unit cell and the sublattice index of the Cerium site. Following the discussion in the introduction, the operator $\widetilde{c}_{j,\sigma}$ annihilates a conduction electron in a Wannier state localized at site $j$:

where $\mathcal{N}_s$ is the number of unit-cells in the crystal. The form factor matrix $\phi_{\mathbf{k}}=\phi(\mathbf{k})$ describes a Wannier state which carries the symmetry of the hybridization inherited from an underlying Anderson model. We take this to have the sublattice diagonal form introduced in (7),

where we use the short-hand $\mathbf{s}_{\mathbf{k}}=(\sin k_x,\sin k_y)$. The dominant source of spin orbit coupling in a Kondo lattice derives from the hybridization with the $f$ electrons. In our minimal model spin orbit coupling only enters through these terms in the Kondo coupling, and for simplicity, all spin orbit coupling terms in the conduction sea have been dropped.

Lastly, the Heisenberg interaction in Eq. (13) describes the AFM superexchange interaction between neighbouring spins where $J_H$ is the coupling strength.

We employ a large-$N$ path integral approach to formulate the model as a field theory and then develop a mean-field description. In the large-$N$ limit, all quantities that are extensive in $N$ behave as semi-classical variables, whose fluctuations are suppressed by powers of $1/N$, causing the path integral to focus into a mean field configuration. In the limit $N \to\infty$, the mean field solution becomes exact, allowing the basis for a controlled mean-field expansion.

An important aspect of our work is to use a large $N$ generalization of the SU(2) group to the symplectic group Sp($N$), where $N$ is even [18, 19, 20]. The first member of the Sp($N$) family is the SU(2) group, Sp(2) = SU(2), but at larger $N$, Sp($N$) $\in$ SU($N$) is a subgroup of SU($N$) that preserves time reversal symmetry. Sp($N$) loosely corresponds to an $N/2$-fold replication of the up and down spins, corresponding to $N/2$ “up” and $N/2$ “down” spin components:

where we identify the spin labels $\uparrow_\sigma\equiv\sigma$ and $\downarrow_\sigma\equiv-\sigma$, where $\sigma=(1,2,\ldots,N/2)$ label the replicas [20]. The spin operators that connect up and down spins in the form factor (17) are diagonal in replica space which allows us to replicate the spins at each site to controlling the mean field theory without interfering with the SU(2) spin algebra which rotates between up and down spins in the original model. Our final results are expressed in terms of the properties per spin pair component, equivalent to formally setting $N=2$ at the end of the calculation.

To formulate our model as a field theory, we use a fermionic representation

for the components of the Hermitian Sp($N$) spin operators, where $f^\dagger_\alpha$ and $f_\beta$ are fermionic creation and annihilation operators with spin component $\alpha\in(\pm1,\ldots\pm N/2)$. In passing to the large-$N$ theory we make the conventional rescaling $J_H\to J_H/N$ and $J_K\to J_K/N$, so that the interaction energy remains extensive in $N$ and the $N\to\infty$ limit is well defined. Using the corresponding Fierz decomposition (see appendix A for details), we rewrite the Kondo and Heisenberg interactions in the form:

where the

describe the matrix unit form of the spin operators. (These corresponds to an antisymmetrization of their SU($N$) spin counterpart $S^{\mathrm{SU}(N)}=f^\dagger_\alpha f_\beta-(n_f/N)\delta_{\alpha\beta}$ under time reversal). The parton decomposition of the local moments gives rise to a local SU(2) gauge invariance

which corresponds to rotations in particle-hole, or isospin space. We select amongst the various possible representations of the local moment by requiring that the $f$ parton state is an isospin singlet, requiring

corresponding to a half-filled $f$ state with local no $s$-wave pairing. These are the “Gutzwiller constraints” of the Kondo lattice, which inside a path integral, are naturally imposed by fluctuations over the SU(2) gauge fields.

Path integral formulation

The SU(2) path integral is given by the partition function as $Z=\operatorname{Tr}[e^{-\beta H}P_G]$ where $P_G$ imposes the Gutzwiller constraints (23). To develop the path integral, we adopt a Balian Werthammer notation,

where the spin labels are purely positive. We will combine these Nambu spinors into a row, as follows

In terms of these fields, the Gutzwiller constraints become $F_j^\dagger\vec{\tau}F_j=0$ where $\vec{\tau}=(\tau_1,\tau_2,\tau_3)$ are the isospin Pauli matrices, acting in particle-hole space. We can also form the conduction Wannier states as

where the invariance of the form factor (17) under charge conjugation allows us to extend its usage to the Nambu formulation of the Wannier states. (See appendix B for details). In terms of this representation, the Gutzwiller constraint is simply that the isospin vanishes $F_j^\dagger\vec{\tau} F_j = 0$, where the $\vec{\tau}$ are the Pauli matrices in the particle-hole space of the Nambu spinors. In terms of this representation, the path integral has the form:

where the Berry phase term

The Fierz identities of the Sp(N) representation allow the Heisenberg and Kondo interactions to be rewritten in the form (see Appendix A for details)

where the spin sums run over $\sigma,\sigma'\in(\pm1,\cdots\pm N/2)$.

These interactions are now decoupled using the Hubbard Stratonovich transformations. The Kondo interaction becomes

where the matrix

describes hybridization ($V_j$) and pairing ($\nu_j$) between the conduction and $f$ parton fields. Similarly, a Hubbard Stratonovich decoupling of the Heisenberg term gives

where

Here, the $t_{ij}^f$ and $\Delta_{ij}$ are the $f$ electron hopping and RVB pairing fields, respectively. Notice also that $U_{ji}=U_{ij}^\dagger$. The notation $\langle i,j\rangle$ denotes nearest neighbours within a given layer. Note that the nearest neighbour pairing $\Delta_{ij}\equiv\Delta_{R_i,\alpha;R_j,\alpha}$ can depend on the layers. With these transformations, the Lagrangian in the path integral now takes the form

Here $w_j=\vec{w}_j\cdot\vec{\tau}$, where $\vec{w}_j=(w_j^1,w_j^2,\lambda_j)$ are three SU(2) gauge potentials whose fluctuation along the imaginary axis impose the Gutzwiller constraints (23).

The Fierz identities of the Sp(N) representation allow the Heisenberg and Kondo interactions to be rewritten in the form (see Appendix A for details)

Here $H=H_c+H_K+H_H$, while $H_K$ take the expanded forms shown in (31) and (33), respectively. The SU(2) gauge invariance (22) now includes the gauge fields,

where $g_j$ are members of the SU(2) group. We use this gauge degree of freedom to fix the gauge by writing $\mathcal{V}_j=g_j|V_j|\tau_3$, absorbing the $g_j$ by the above gauge transformation. In this “radial gauge”, $\mathcal{V}_j=|V_j|\tau_3$ describes a pure, real hybridization. The pure Kondo model, without the Heisenberg terms describes a Fermi liquid with an expanded Fermi surface volume which accommodates the hybridized $f$ electrons, according to Oshikawa’s theorem[21, 22]. In this phase, the difference between the internal SU(2) gauge field and the external electromagnetic field is Higgsed[23], in a fashion reminiscent of electroweak symmetry breaking. The finite mass of the difference field locks the internal gauge field to the external electromagnetic field, endowing the previously neutral $f$ partons with physical charge, thus converting them into heavy electrons. Once the Heisenberg coupling is restored, there are now further low temperature instabilities of the heavy Fermi liquid.

Mean field Theory

To develop a mean field theory we examine static, translationally invariant solutions. In the normal state, $\mathcal{V}_j=V$ and $\lambda_j=\lambda$, and $t_{ij}^f=-t_f$ are real, while the pairing order parameter variables $\Delta_{ij}=0$ are zero. We restrict our study of instabilities of the Fermi liquid to superconductivity, in which the nearest neighbour Heisenberg couplings develop an expectation value that can depend on the sublattice variable $\alpha$. Within a layer, energetics favours a $d$-wave pairing state, i.e. $\Delta_x=-\Delta_y=\Delta$. The $d$-wave order parameter within the layers guarantees there is no $s$-wave pairing at the $f$ sites, which allows us to set the pairing components of the constraint fields to zero, i.e. $\vec{w}=(w^1,w^2,\lambda)=(0,0,\lambda)$.

Now since the inversion centers lie between the layers, this allows for the possibility of both uniform and staggered pairing solutions in which the pairing develops a layer dependence. These two classes of solution are respectively, even and odd-parity under inversion.

where $\alpha=\pm1$ is the sublattice label. We will show that under the influence of the hybridization, these two states correspond to singlet and triplet pairing states, as originally reasoned by Yoshida, Sigrist and Yanase[24]. To accomodate a magnetic field, we introduce a Zeeman term into the Hamiltonian $\mathcal{H}\rightarrow\mathcal{H}-\sigma_zB$. We will show that while the uniform state has a higher $T_c$, the staggered odd parity triplet state is able to accomodate a Zeeman polarization, stabilizing it in a magnetic field.

Putting this all together, the resulting mean-field Hamiltonian takes the form

where the superconducting Hamiltonian is given by

and

combines the conduction and $f$ Nambu spinors on the two layers. Here $\alpha=\pm$ are the layer indices of $C_{\mathbf{k},\alpha}$ and $F_{\mathbf{k},\alpha}$. The off diagonal hybridization terms are given by $V_{\mathbf{k}}=V\phi_{\mathbf{k}}\tau_3$. The $f$ electron dispersion is

In a compact notation define $h_{\mathbf{k}}^f=t_f\gamma_{\mathbf{k}}+\lambda$, where $\gamma_{\mathbf{k}}=-2(\cos(k_x)+\cos(k_y))$. Note that the factor of $2NN_s$ in the mean-field term derives from the $N$ spin components and the $2N_s$ sites.

The mean field solutions are obtained by finding the stationary points of the free energy. Integrating out the fermionic fields leads to the free energy $F=-T\sum_{\mathbf{k},n}\ln(\det(-\mathcal{G}_{\mathbf{k}}^{-1}(i\omega_n)))$ where $\mathcal{G}_{\mathbf{k}}^{-1}(i\omega_n)=[i\omega_n-\mathcal{H}(\mathbf{k})]$ is the Greens function in imaginary time where $i\omega_n\beta=i\pi(2n+1)$ represents the fermionic Matsubara frequencies that runs over $n$ discrete frequencies. The mean field Hamiltonian is diagonalised by integrating out the $f$ electrons. With the energy eigenvalues $\xi_{\mathbf{k}n}$, using the identity $\ln(\det(A))=\operatorname{Tr}[\ln(A)]$ and computing the Matsubara sum leads to the mean field free energy

where $\pm\xi_{\mathbf{k}n}$ are the eigenvalues of the mean-field Hamiltonian, $\eta\in(1,8)$ is the eigenvalue index and $N_s$ the number of unit cells in the lattice[25]. The saddle point approximation is obtained by seeking the stationary points of the effective action $\partial S_{\mathrm{eff}}/\partial x=\partial F/\partial x=0$, where $x=V,t_f,\lambda,\Delta$. In practice, we impose these conditions on our numerical calculations, thereby imposing four self-consistency conditions:

where the normalizing prefactors derive from the $2N_s$ $f$ sites and the $N$ spin components. The last condition is the constraint of a half-filled $f$ band, $\langle F^\dagger\tau_3F\rangle=0$. In general, once a finite hybridization $V$ develops in the normal state, this gives rise to a finite hybridization between $f$ electrons at different sites, which by (44) induces a finite $t_f$.

Results

Normal state: Emergent Spin-Orbit Coupling

We begin with a discussion of the normal state. Here, the key point is that the absence of local inversion symmetry at the Cerium sites modifies the electrostatic environment of the atoms. This polarizes the atomic orbitals, giving them an ill defined parity which opens up previously forbidden mixing between even and odd parity states. We will see how this modifies the spin orbit coupling (SOC). To elucidate this effect, we examine the $f propagator obtained after integrating out the conduction electrons. This takes the form:

The $f$ electron self-energy $\Sigma_f(\mathbf{k},i\omega_n)$ is represented by the following diagram

where $\mathcal{V}_{\mathbf{k}}=V\phi(\mathbf{k})\tau_3$. Let us now examine the spin orbit coupling generated by this process. The conduction electron dispersion $h_{\mathbf{k}}$ can be divided into sublattice independent and dependent parts, $h_{\mathbf{k}}=h_{\mathbf{k}}^0+h_{\mathbf{k}}^1\alpha_1$. Now since $\tau_3$ and $\alpha_1$ are commuting operators, which can be simultaneously diagonalized, we can rewrite the conduction electron propagator in the form

Reinstating $\alpha_1$ and $\tau_3$ as operators, the self-energy takes the form

where $\mathbf{s}_{\mathbf{k}}=(\sin k_x,\sin k_y)$ and $s_{\mathbf{k}}^2=\sin^2 k_x+\sin^2 k_y$. We see that the interference between the two terms in the hybridization generates staggered spin orbit coupling with a cross product structure between momentum and spin. At low energies,

where $W=\widetilde{W}/V$ demonstrating that the self-energy of the $f$ electron in (53) develops a frequency dependent form of the ASOC interaction deduced from symmetry analysis. Unlike previous models which added the ASOC interaction as a phenomenological term in the normal state, here the essential physics is produced from an interplay between the local parity breaking hybridization and the intersite $\mathbf{s}_{\mathbf{k}}\cdot\boldsymbol{\sigma}=\sin\mathbf{k}_{\perp}\cdot\boldsymbol{\sigma}$ hybridization that carries the strong SOC effects of the $f$-states. The new local parity breaking term $i\alpha_3W\sigma_3$ in the hybridization form factor $\phi_{\mathbf{k}}=(\mathbf{s}_{\mathbf{k}}\cdot\boldsymbol{\sigma}+iW\alpha_3\sigma_3)$ provides the spin flip necessary to swap the spin components of the intersite hybridization $\sin\mathbf{k}_{\perp}\cdot\boldsymbol{\sigma}$ generating an antisymmetric SOC. The frequency dependent prefactor encodes the virtual processes of the $f$ electrons fluctuating into the conduction sea and returning having acquired the necessary spin and momentum symmetry required to induce the ASOC interaction. In otherwords, Kondo screening in a noncentrosymmetric environment creates a heavy Fermi liquid with the necessary ingredients for a triplet superconducting state. The normal state properties can be seen in how the dispersion and Fermi surfaces change across the Brillouin zone.

The dispersion plot of the normal state is shown in Figure 3. The dispersion plotted along the path given by the high symmetry points $\Gamma=(0,0,0)$, $X=(\pi,0,0)$, $M=(\pi,\pi,0)$, $Z=(0,0,\pi)$, $R=(\pi,0,\pi)$, $A=(\pi,\pi,\pi)$. The 3D fermi surfaces of the normal state are shown in Figure 4.

Superconducting State

We now examine the superconductor, and the effect of a magnetic field in stabilizing an odd-parity condensate. In the absence of a magnetic field, the even parity solution with its lower kinetic energy, is expected to be more stable, exhibiting a higher $T_c$ than its odd parity counterpart. Interestingly, in the limit where the hybridization is a constant, $V_{\mathbf{k}}=V_0$, there is no staggered solution. The only solution is the uniform pairing which leads to the conventional $d$-wave singlet pairing within the layers commonly seen in the 115 heavy fermion superconductors [26, 27]. In CeRh$_2$As$_2$, neutron scattering does indeed detect a peak at the dynamic structure factor at $\mathbf{Q}=(\pi,\pi)$, a hallmark for AFM fluctuations [16].

To investigate the effect of the applied field on the superconducting state, we now switch on a Zeeman field as defined in (39). Although CeRh$_2$As$_2$ is a type II superconductor which develops a vortex lattice in a field, the effect of spatial modulation is assumed to be negligible for the low temperature and high field regime where Pauli limiting dominates, as described in (39).

Application of a Zeeman field affects the critical temperature $T_c$ which can be computed from the pair susceptibility $\chi_P$. Generally, in the normal state, the pair susceptibility enters into the Landau free energy functional via the quadratic term $F[\Delta]=|\Delta|^2(\chi_P(B_z)-2/J_H)+\beta|\Delta|^4$ calculated by taking the second derivative w.r.t the pairing field $\Delta$. At $T_c$, the $\partial^2F/\partial\Delta^2|_{\Delta=0}=0$ giving the well known criteria for $T_c$:

In our calculations, the pair susceptibility was obtained by evaluating the second derivative of the Free energy, $\chi_P(B_z,T_c)-2/J_H=-\partial^2F(B_z,T_c)/\partial\Delta^2$ for a range of magnetic field strengths for both uniform and staggered solutions. The corresponding phase boundaries for the uniform $\Delta_{\mathbf{k}}^{(0)}$ and staggered $\Delta_{\mathbf{k}}^{(3)}$ solutions are shown in Figure 5. The first order SC1 to SC2 boundary was determined by comparing the fully evaluated mean-field free energies of the self-consistent uniform and staggered superconducting solutions at the same $(B_z,T)$. The transition occurs where these two free energy branches cross; the pair susceptibility criterion in Eq. (54) determines the normal state instability lines, not the first order boundary. The uniform state (SC1) has a higher condensation energy than the staggered state since staggering the phase of the pairing field $\Delta$ requires a higher kinetic energy [24], and this results in a higher $T_c$ at zero field. However, the condensation energy of the singlet SC1 phase is more strongly susceptible to a Zeeman splitting, ultimately raising its energy above the staggered, triplet SC2 phase. Once a magnetic field is applied, there is a first order phase transition to the staggered state. There is a thus field induced parity switching between the uniform even parity state to the staggered odd parity state providing a concrete Kondo lattice demonstration of the field induced parity switching behavior observed in experiments.

For further investigation, a self consistent calculation was performed to analyze the uniform and staggered pairing states. The heavy Fermi liquid phase has already been developed and frozen with the Kondo hybridization parameters $V=3$ and $W=2$ which are the intersite and onsite hybridization respectively and the Kondo and Heisenberg coupling strengths are set to $J_K=0.3$ and $J_H=0.15$ respectively. The intralayer and interlayer hopping strengths are given by $t=5$ and $t_s=1.5$ respectively and the chemical potential was set to $\mu=-1$. The hopping scales are consistent with a quasi-2D structure. The $f$ electron hopping strength is set to $t_f=0.5$ with $t_f\ll t$ and the chemical potential for the $f$ electron was set to $\lambda=-1.7$ with a $q=1/3$ charge. The qualitative results do not depend on the microscopic details of the band structure as long as keeping the ratio of $t/t_s>1$ for the interlayer to intralayer coupling and $V W/t_s>1$ for the interlayer to hybridization strength. With these parameters, a Newton Raphson method was used to find the saddle point of the pairing field by minimizing the Free energy w.r.t the pairing field $\Delta$ for the uniform and staggered solutions. The resulting phase diagram of the pairing gap $\Delta$ vs temperature in the presence of magnetic field is given by Figure 6.

The zero field behaviour of the uniform and staggered states is shown in the top plot of Figure 6 which shows the pairing field and $T_c$ for the two states are nearly degenerate. The uniform state has a higher $T_c$ emphasising the dominance of the even parity singlet state at zero field. However, once a magnetic field of $B_z=0.8$ is applied, as shown in the lower plot Figure 6, the even parity state is suppressed by Pauli limiting effect. When the $T_c$ of the staggered odd parity state starts to overcome the uniform even parity state, a field induced transition takes place. The staggered odd parity state is more weakly susceptible to Pauli limiting effects. The upper critical field of the staggered state is $B_{c2}\approx3.2$ for which the uniform state completely vanishes. To understand how a spin singlet pairing in the layers staggered could realise a triplet odd parity state, the pairing gap will be projected onto the low energy pseudospin basis.

Projection of the Gap onto pseudospin basis

To understand the nature of the gap structure, the gap must be projected onto the Fermi surface. The procedure requires the gap operator originally in the high energy sublattice and spin subspace to be evaluated within the Kramers pair of energy eigenstates in a given band [4]

Here $\xi_{\mathbf{k}n}^{(\sigma)}$, $(\sigma\in(\uparrow,\downarrow))$ denotes the Kramers doublet of eigenstates of the superconducting mean field Hamiltonian at momentum $\mathbf{k}$ in the $n$-th band and $P_f$ projects onto the $f$ sector. The sublattice matrices can take either $\alpha_m=\alpha_0$ or $\alpha_3$ which denotes the sublattice for the uniform, and staggered gap functions respectively.

To consistently calculate this quantity, we need to consistently fix the phase relationship between the pseudospin up and down eigenstates. We calculate the up eigenstate by applying a small field to split the Kramers degeneracy, selecting it as the “up” state as the lowest energy member of the doublet. We then apply time-reversal and parity inversion to the up state to generate its down-spin partner, $\xi_{\mathbf{k}n}^{\downarrow}=i\alpha_1\sigma_2(\xi_{\mathbf{k}n}^{\uparrow})^*$.

The Fermi surface projection of the uniform gap $\Delta_{\mathbf{k}\alpha_0}$ is shown in Figure 7. As expected from (55), the gap has zeros along the $k_x=k_y$ derived from the in-plane $d$-wave gap structure. The colors show the sign of the gap $\operatorname{sgn}(\Delta)$ where the $+$ and $-$ signs are denoted by pink and blue respectively. Thus the projected gap structure of the uniform gap on the Fermi surface is an even parity $d$-wave spin singlet $\Delta^{(0)}_{\mathbf{k}n}$ where $n$ represents the band index

Here $\psi_0(\mathbf{k})$ is an even, generally signed gap function. Throughout this subsection, $\Delta_{\mathbf{k}}$ denotes the reduced gap matrix, while the full anomalous gap matrix is $\hat{\Delta}_{\mathbf{k}}=\Delta_{\mathbf{k}}(i\sigma_2)$ in the pseudospin basis.

The Symmetry of the singlet and pseudo-spin triplet states

Symmetry of the gap

We now discuss the symmetry and physical origin of the spin-orbit generated pseudospin triplet state. The projection of the staggered gap $\Delta^{(t)}_{\mathbf{k}}$ is shown in Figure 8. The key result is that even though the in plane pairing is singlet, the staggered pair potential on alternating layers transforms it into an odd parity triplet state on

the Fermi surface. The projected odd-parity pseudospin-triplet gap, with the universal factor $i\sigma_2$ suppressed, has the structure

where $d_{\pm}(\mathbf{k})=d_x(\mathbf{k})\pm id_y(\mathbf{k})$, If we rewrite the pairing Hamiltonian in terms of the quasiparticle creation operators, $a^\dagger_{\mathbf{k},\sigma}$,

Thus the off-diagonal structure of the reduced gap matrix corresponds, after multiplication by $i\sigma_{2}$, to diagonal equal pseudospin pairing amplitudes $\hat{\Delta}_{\uparrow\uparrow}=-d_{-}$ and $\hat{\Delta}_{\downarrow\downarrow}=d_{+}$. These are pseudospin labels, not necessarily the physical electron spin quantum numbers. Fig. 8 shows the variation of $\vec{d}_{\mathbf{k}}$ around the Fermi surface. To a good approximation $\Delta_{\downarrow\downarrow}(\mathbf{k})=d_{x}(\mathbf{k})+id_{y}(\mathbf{k})\propto i\cos(2\theta)e^{-i\theta}$ is a product of a $d$-wave and $p$-wave gap function. We can decompose this into $\Delta_{\downarrow\downarrow}(\mathbf{k})\propto(e^{i\theta}+e^{-3i\theta})$, corresponding to angular harmonics of winding numbers $+1$ and $-3$ (often described as $p$- and $f$-wave components); the crystalline representation is determined by the full tetragonal symmetry. In the tetragonal point group approximation, the uniform intralayer gap $\cos k_{x}-\cos k_{y}$ belongs to $B_{1g}$ of $D_{4h}$. For the projected staggered state, the approximate form $\vec{d}_{\mathbf{k}}\propto(\cos k_{x}-\cos k_{y})(\sin k_{y},-\sin k_{x},0)$ is odd under inversion, changes sign under a fourfold rotation, and is even under a vertical mirror when the axial vector transformation of $\vec{d}$ is included. It therefore transforms as $B_{1u}$ in this pseudospin convention. This identification is for the displayed leading harmonic; a complete space group classification must also account for the nonsymmorphic sublattice representation and the momentum dependent pseudospin gauge. The interesting feature of the $\vec{d}_{\mathbf{k}}$ vector is that it lies in the basal plane, and this makes the resulting triplet state robust against fields along the $z$-axis. This projected symmetry is consistent with the analysis of Ishizuka et al. [14], who classified candidate superconducting gaps in CeRh$_2$As$_2$ using electronic structure calculations and crystalline symmetry. For a magnetic field along the $c$ axis, the relevant point group classification is reduced to $C_{4h}$; the leading $B_{1u}$ gap identified above belongs to its $B_{u}$ sector. In particular, their staggered spin singlet $B_{u}$ basis includes $k_{x}^{2}-k_{y}^{2}$, the long wavelength limit of our $\cos k_{x}-\cos k_{y}$ form factor. Our projected gap $\vec{d}_{\mathbf{k}}\propto(\cos k_{x}-\cos k_{y})(\sin k_{y},-\sin k_{x},0)$ provides a higher harmonic realization of the same $B_{u}$ symmetry class. The diagonal zeros of this leading form factor need not be enforced for every gap in that representation. The approaches are complementary: their work establishes crystalline symmetry constraints and possible topological properties, whereas our Kondo lattice calculation provides a microscopic mechanism for the emergence of the projected pseudospin triplet structure. This agreement at the point group level does not replace a full nonsymmorphic analysis of our heavy band gap.

Physical origin of the triplet gap

The origin of this pseudospin triplet structure can be understood physically by considering the bonding and antibonding combinations of $f$ states on neighboring Ce layers. In the absence of staggered spin orbit coupling, the heavy quasiparticle bands are approximately eigenstates of the interlayer hopping matrix $\alpha_{1}$, with eigenvalues $\pm1$. Since $\{\alpha_{1},\alpha_{3}\}=0$, the staggered pairing and staggered spin-orbit coupling, both proportional to $\alpha_{3}$, are off-diagonal in this approximate band basis.

Virtual transitions between the low energy Fermi surface and higher energy quasiparticle bands then generate an effective intraband pairing interaction. To leading order, this contribution has the symmetry of the product of the staggered $d$-wave pairing and the staggered Rashba spin orbit coupling:

where $\delta E_{\mathbf{k}}$ denotes the characteristic energy separation between the bands. The two staggered factors combine through $\alpha_{3}^{2}=1$, producing a sublattice even, momentum odd pseudospin triplet gap.

This mechanism illustrates how an odd-parity pseudospin triplet superconducting state can emerge from staggered spin singlet pairing in the presence of locally broken inversion symmetry. More generally, analogous processes may occur in multi sublattice superconductors where staggered pairing interacts with staggered spin orbit coupling.

Conclusion

In this work we have developed a Kondo lattice framework for the field induced parity transition for the heavy fermion superconductor CeRh$_2$As$_2$. Our central result combines the Kondo hybridization and local inversion symmetry breaking which provides a microscopic mechanism by which the two sublattice structure of CeRh$_2$As$_2$ can generate nearly competing superconducting states of opposite parity. Local inversion symmetry breaking at the Ce sites generates a new onsite spin flip hybridization which combines with the non local odd parity hybridization to generate a staggered antisymmetric spin orbit coupling (ASOC). For pairing driven by antiferromagnetic exchange interaction within the Ce layers, the uniform and staggered solutions are then determined by how they transform under inversion, which are even and odd respectively. In this model, the uniform intralayer $d$-wave pairing produces the low field, even parity state, while the staggered intralayer $d$-wave pairing with opposite signs of the pairing gap between the two Ce sublattices produces the high field, odd-parity pseudospin-triplet state after projection onto the heavy quasiparticle bands. A magnetic field along the $c$ axis suppresses the uniform phase much more strongly than the staggered phase leading to the field induced phase transition between the two superconducting phases.

Projection onto the normal state describing the heavy Fermi surface further shows an important consequence of the two pairing states. Although the underlying interaction was decoupled through an intralayer spin singlet $d$-wave pairing, the staggered state projected onto the Fermi surface reveals odd parity pseudospin triplet gap structure with an in-plane $\vec{d}_k$ vector. The resulting odd parity state is not affected by the $c$-axis magnetic field as the $\vec{d}_k \perp B_z$. More generally, this illustrates that in a locally noncentrosymmetric Kondo lattice system, the spin, sublattice and parity structure of the microscopic pairing mechanism cannot be thought of independently. The interplay between the Kondo hybridization and ASOC can modify the structure of superconducting gap function when it is projected onto the normal state heavy quasiparticle basis. The key physics here is not to identify the particular $d$- or $p$-wave pairing interaction within the layers, but rather a mechanism by which the locally broken inversion symmetry with the two sublattice structure leads to a closely competing superconducting states with opposite global parity.

This distinction is particularly important because the microscopic pairing symmetry of CeRh$_2$As$_2$ has not yet been established experimentally. We have assumed that antiferromagnetic exchange favours an in-plane $d$-wave singlet pairing, but an intrinsic spin triplet pairing interaction is also possible. Indeed, a Hund’s–Kondo mechanism has been proposed in which valence fluctuations into high spin configurations generate triplet correlations, with the two sublattice structural motif allowing these correlations to acquire an odd parity form factor on the heavy Fermi surface [28]. Starting from an intralayer triplet pairing rather than a singlet state would alter the correspondence between the uniform/staggered sublattice structure and the parity of the superconducting state, while retaining the central importance of the locally noncentrosymmetric environment. It is therefore an important open problem to repeat the present microscopic analysis allowing singlet and triplet pairing channels to compete on an equal footing. Measurements capable of determining the spin susceptibility and gap anisotropy in the two superconducting phases would be especially valuable in distinguishing the different possibilities.

A second direction is suggested by the recently discovered van der Waals heavy fermion material CeSiI. CeSiI provides a striking two dimensional realization of a Kondo lattice. Within a single van der Waals layer, two triangular Ce layers surround a conducting silicene layer, and heavy fermion coherence and magnetic order persist into the ultrathin limit [29, 30]. More recently, pressure has been shown to suppress the antiferromagnetic state and produce a superconducting dome, establishing CeSiI as a two dimensional heavy fermion superconductor [31]. Its two Cerium layer structure that naturally leads to the question whether a sublattice dependent Kondo hybridization can generate an analogous relation between local symmetry, ASOC and superconducting parity. Unlike CeRh$_2$As$_2$, however, the Ce ions form triangular layers surrounding a conducting silicene layer, so the microscopic hybridization form factors and crystal symmetry are distinct. Extending the current Kondo lattice framework to the actual symmetry and electronic structure of CeSiI would therefore test which aspects of the parity switching mechanism are generic to a two sublattice Kondo lattice system and which are specific to a noncentrosymmetric CeRh$_2$As$_2$. The ability to exfoliate, gate and assemble CeSiI into van der Waals heterostructure additionally raises the prospect of controlling the relative Kondo coupling of the two Ce layers and hence tuning the balance between competing superconducting states.

An equally revealing comparison is provided by $\beta$-CeNi$_2$As$_2$. Recent work has established superconductivity in the CaBe$_2$Ge$_2$ isomorph of CeNi$_2$As$_2$, which is a structural analogue of CeRh$_2$As$_2$ and possesses the same essential locally noncentrosymmetric two sublattice structural motif [32]. Yet $\beta$-CeNi$_2$As$_2$ does not display the characteristic field induced high field phase observed in CeRh$_2$As$_2$. This comparison to CeRh$_2$As$_2$ is important. It shows that the locally noncentrosymmetric crystal structure by itself is not sufficient to a field induced parity transition. Within the framework developed here, the occurrence of such a transition also depends on quantitative microscopic ingredients. These are the strength and momentum dependence of the Kondo hybridization, the resulting staggered ASOC, the interlayer coupling, the structure of the heavy Fermi surface, the pairing interaction, and the degree of nearly degenerate uniform and staggered superconducting states. A microscopic comparison of CeRh$_2$As$_2$ and $\beta$-CeNi$_2$As$_2$ could provide a stringent test of the mechanism proposed here. Rather than asking only why CeRh$_2$As$_2$ has the high field phase, one can ask which microscopic ingredient present in CeRh$_2$As$_2$, but absent or weaker in $\beta$-CeNi$_2$As$_2$, is responsible for the high field phase.

A further challenge is to establish the superconducting symmetry classification at the full nonsymmorphic space group level. The assignments discussed here are based on the tetragonal point-group transformation properties of the leading gap form factors and their projected pseudospin structure. A complete analysis for $P4/nmm$ must additionally track the fractional translations, the sublattice dependent Bloch phases, and the momentum dependent symmetry sewing matrices of the heavy quasiparticle bands, especially near Brillouin zone boundaries. Such a treatment would test the robustness of the proposed irreducible representation assignments and clarify any symmetry enforced constraints on the gap nodes.

There remain further unresolved questions within CeRh$_2$As$_2$ itself. The superconductivity develops in close proximity to an additional ordered phase, Phase I whose microscopic origin remains uncertain, and antiferromagnetic order has also been reported at temperatures below $T_c$ [3, 15, 33, 34]. Understanding how these orders couple to the two superconducting phases may be important for a quantitative theory of the phase diagram. Similarly, the anomalous normal state transport and indications of a non Fermi liquid behavior suggests that the superconductivity develops from a strongly correlated normal state close to quantum criticality [35]. These effects lie beyond the static mean field treatment used here and motivate extensions in which the Kondo coherence, magnetic fluctuations and superconducting instability are treated on an equal footing.

Our results suggest a broader perspective for locally noncentrosymmetric heavy fermion superconductors. The essential ingredient identified here is the interplay between Kondo hybridization with the crystal structure where the inversion exchanges two locally noncentrosymmetric magnetic atoms. This structure allows local parity breaking to become encapsulated as a staggered ASOC of the heavy quasiparticles and provides a natural route to competing superconducting states of opposite global parity. CeSiI and $\beta$-CeNi$_2$As$_2$ now provide complementary tests of this idea. CeSiI extends the problem toward a tunable two dimensional Kondo lattice, while the $\beta$-CeNi$_2$As$_2$ enables an isostructural superconducting control system in which the high field phase is absent. Together with a systematic treatment of intrinsic triplet as well as singlet pairing, these materials should help determine whether the mechanism presented here is peculiar to CeRh$_2$As$_2$ or forms part of a more general organizing principle for unconventional superconductivity in two sublattice Kondo lattices.

Acknowledgments

We gratefully acknowledge discussions David Vanderbilt, Kevin Lucht, Andreas Gleis, Po-Jui Chen, Liam Lau, Elena Hassinger, Joerg Schmalian, Daniel Schultz and particularly, Philip Brydon. The early part of the work in this paper was completed under the support from the Division of Materials Research, US National Science Foundation grant DMR-1830707(PC). We acknowledge support from the US National Science Foundation (NSF) Grant Number 2201516 under the Accelnet program of Office of International Science and Engineering (OISE)

Symplectic decoupling of the Heisenberg interaction

Here we derive the dot product of the spins in the Heisenberg interaction. More specifically, the fermionic form of the Heisenberg interaction used in Eq. (29) will be obtained. The main framework for the symplectic $N$ construction comes from the work [20]. For an even $N$, the spin can have $N$ number of components:

and the antisymmetric tensor which time-reverses the spins is

There are $N(N+1)/2$ components of Hermitian spin generators $S^a$ in the $\operatorname{Sp}(N)$ subgroup. These symplectic spin generators are odd under time reversal operation:

The normalization condition of the symplectic spin generators are defined as:

The spin in terms of the fermionic operator and symplectic spin generator is given by:

In this convention, there is no additional factor of 1/2. For completeness, we now derive the Fierz identity used for decoupling the spin. An arbitrary $N\times N$ matrix $A$ can be decomposed into parts that are odd and even under time reversal as $A=A_S+A_A$, where

The projector onto the symplectic subspace which has the property that inverts under time reversal is therefore

Here $P_S$ represents the projector onto the symplectic subgroup. Writing this relation in component form,

then it follows from (A7) that

The same projector $P_G$ can be expanded in terms of the symplectic part consisting of the complete set of Symplectic $N$ generators,

Using the normalisation $\operatorname{Tr}(S^aS^b)=2\delta_{ab}$ leads to

where the second equality comes from the fact that the symplectic and antisymplectic subgroups are orthogonal to each other. Then using the definition of the prefactors the projection onto the symplectic subgroup in terms of components:

Then the projector $P_G$ in terms of the spin generator representation has the form:

Combining both the definition of the projector in Eq. (68) and Eq. (73) leads to the symplectic completeness relation:

This is the Fierz identity for the SP($N$) generators. The second term imposes the inversion of spins under the time reversal operation.

In the main text which uses the large-$N$ treatment, the Heisenberg interaction with the rescaled interaction has the explicit form:

Using the completeness relation in Eq. (74), and normal ordering fermions on the two different sites leads to:

where $\sigma,\sigma'=\pm1,\ldots,\pm N/2$ which are the spin components. This is the decoupled Heisenberg interaction in Eq. (29) of the main text. Physically, the two terms in Eq. (76) are the particle hole and particle particle channels respectively. It is useful to introduce the bond orientation operators:

Since this is Hermitian construction, their Hermitian conjugates are given by:

Rewriting the Heisenberg interaction in terms of the bond operators leads to the simple form:

Equation (A20) gives the explicit form of the SU(2) isospin structure of the symplectic-$N$ representation. The antiferromagnetic exchange interaction generates degenerate particle hole and particle particle singlet decoupling channels. Using a Hubbard Stratonovich decoupling of the two channels leads to the $f$ electron hopping and pairing fields used in the main text.

Charge conjugation transformation

We now show explicitly develop the Balian-Werthammer (BW) isospin structure used in the main text. The definitions of the fermionic operators used in the path integral formulation in the main text are:

with $\sigma>0$. The Pauli matrices $\tau_i$ act in this particle hole also known as the isospin space. Combining the sublattice and the conduction and $f$ electron sectors leads to the spinor used in the mean field Hamiltonian,

To transform normal state bilinears of the form $a_k^\dagger A(k)a_k$ into the Balian Werthammer form, we combine it with the corresponding hole component, obtained by charge conjugation. The Balian Werthammer representation of this operator then becomes:

Now in general, operators are either even, or odd under time-reversal, so that

then the corresponding operators in the Balian-Werthammer representation take the form

For example, in the normal state the Hamiltonian contains a dispersion and a Zeeman term $\mathcal{H}(k)=H_N(k)-\sigma_zB_z$. Since the first term is even under time-reversal, while the second is odd under time reversal, in the Balian Werthammer notation,

Thus, in the BW formulation, the normal-state Hamiltonian takes the form

which accounts for the form used in (39) of the main text.

References