Correlation-driven topological Kondo superconductors

Abstract

Searching for topological superconductors that host topological charge-neutral Majorana zero-modes at edges has become a central problem in condensed matter research due to their potential applications for quantum computations. Meanwhile, electron correlations in solid-state materials enhance quantum fluctuations and give rise to various quantum many-body phases. Whether these electron correlations alone would lead to topological superconductivity is a fundamentally important open problem. Here, we theoretically find the correlation-driven topological superconductivity in a class of Kondo lattice materials. Therein, the odd-parity Kondo hybridization mediates ferromagnetic spin-spin coupling and leads to spin-triplet pairing between local moments. Triplet $p\pm i{p}^{{\prime} }$-wave topological superconductivity with Majorana zero modes at edges is reached when Kondo hybridization co-exists with the triplet pairings. Our results offer a detailed understanding of the experimental observations on UTe₂, a ferromagnetic heavy-electron triplet superconductor. Our approach to topological superconductivity shows advantages over the heterostructure approach by proximity effect.

Introduction

Searching for topological superconductors has become one of the central problems in condensed matter physics due to the emergence of the long-sought fundamentally important self-dual charge-neutral quasi-particles, so-called Majorana zero-modes (MZM) as their excitations at edges, and their potential applications for quantum computations^1,2. To date, theoretical proposals and experimental realizations of topological superconductors are focused on heterostructures, combining strong spin-orbit coupled materials and conventional superconductors by proximity effect^3,4,5, where the topological edge states can be explained in terms of the single-particle band structure without considering many-body electron correlations. On the other hand, electron correlations in solid-state materials are known to give rise to various quantum many-body phases. Whether these electron correlations alone would lead to topological superconductivity without introducing the proximity effect is a fundamentally important open problem. The Kondo effect, which describes the screening of a local spin moment by conduction electrons, is a well-known example of strong electron correlation existing in heavy-electron compounds. The Kondo-mediated topological phases of matter have been studied in the context of topological Kondo insulators^6,7,8 and topological Kondo semi-metals^9,10, where the topological properties are driven either by the odd-parity Kondo hybridization or by the Kondo hybridization cooperating with strong spin-orbit coupling and crystalline symmetry.

Meanwhile, spin-triplet superconductors are known to be the prime candidates for topological superconductors. However, they are scarce in nature. While the existence of spin-triplet superconductivity in SrRu₂O₄^11,12,13 is still debatable, more convincing evidence has been observed in noncentrosymmetric superconductor BiPd by phase-sensitive measurement¹⁴. More recently, signatures of triplet chiral superconductivity were observed in heavy-electron compound UTe₂ at the edge of ferromagnetism, possibly marking the first example of a topological superconductor induced by the Kondo effect^{15,16,17,18,19}.

Motivated by these discoveries, in this paper, we theoretically find a distinct class of topological superconductivity driven purely by strong electron correlations without invoking proximity effect on heterostructures²⁰ in a distinct class of Kondo lattice materials with odd-parity Kondo hybridization. Beginning with the Anderson lattice model with odd-parity hybridization^6,7,8, we explicitly derive an effective low-energy Kondo term via the Schrieffer-Wolff transformation^21,22. Furthermore, we analytically obtain a short-distance ferromagnetic RKKY interaction among the local f-electrons mediated by this Kondo effect. This RKKY interaction results in the attraction of f-electrons and leads to the formation of pairings. We explore the mean-field phase diagram of this ferromagnetic Kondo-Heisenberg model. A time-reversal invariant topological superconducting phase is reached when the Kondo effect co-exists with the spin-triplet p-wave (S_z  =  ± 1) resonating valence bond (RVB) order parameter, see Fig. 1 below. Triplet superconducting pairing arising from Kondo correlation and triplet RVB has been proposed in the two-channel Hund’s-Kondo model in ref. ²³. However, this superconducting state via a simplified demonstration is topologically trivial; and it is not clear whether this approach would lead to robust topological superconductivity without substantially engineering the system. Here, we employ an alternative approach to realize a generic and robust Kondo-mediated topological superconductivity in a distinct class of Kondo-Heisenberg lattice model with an odd-parity Kondo hybridization. We rigorously demonstrate the topologically non-trivial properties in the superconducting phase by computing the Z₂ topological Chern number of the bulk band and the helical MZM emerging at the edges of a finite-sized ribbon. Since the topological states are pinned to the Fermi energy of the conduction band by the Kondo effect, they are easy to observe and do not require fine-tuning in experiments. Therefore, our approach offers a more feasible guiding principle to realize MZM and shows advantages^10,24 over the conventional heterostructure approach by proximity effect where experimental fine-tuning is often necessary (see Results). Our results offer a detailed understanding of the spin-triplet superconductivity recently observed in UTe₂, a promising candidate for topological Kondo superconductor (TKSc).

Fig. 1: Schematic illustration of the mechanism for TKSc in our Kondo lattice model.

a Schematic plot of the TKSc state, showing helical MZM (red and blue arrows). b shows the microscopic mechanism of TKSc: it involves both the Kondo hybridization (red wavy line) between the conduction electrons and local f fermions with opposite parity and anti-parallel spins and equal-spin RVB pairings between two local f fermions. The green eclipse enclosing the red wavy line connecting an even-parity ( ⊕ ) conduction electron and an odd-parity ( ⊖ ) local f fermion represents the Bose-condensed Kondo hybridization.

Results

From the odd-parity Anderson lattice model to the ferromagnetic Kondo-Heisenberg lattice model

Our perspective toward the problem of the topological superconductor with Kondo coherence is based on Doniach’s scenario for heavy-electron compounds²⁵—competition between the Kondo correlation and magnetism leads to phase transitions, whereas superconductivity can form when they collaborate. To capture the non-trivial topology in the superconducting state, we consider the odd-parity Kondo hybridization between conduction electrons and local f-electrons. A topological superconducting state emerges with the help of this type of Kondo hybridization.

To this end, it is natural to start with the odd-parity Anderson lattice model on a two-dimensional (2D) square lattice^6,7,8:

$${H}_{{{{\rm{ALM}}}}}={H}_{c}+{H}_{f}+{H}_{cf},$$

(1)

where the odd-parity Kondo hybridization was shown to give rise to topological insulators^6,7,8. This model involves the itinerant c and localized f electrons, corresponding to the physical electrons occupying the d and f orbital states. The dynamics of the c-electrons are described within the framework of a tightly binding model (H_c), showing the dispersion ${\varepsilon }_{{{{\boldsymbol{k}}}}}=-2t(\cos {k}_{x}+\cos {k}_{y})-\mu$ with the hopping strength t and the chemical potential (Fermi energy) μ. We set t  =  1 to be the energy unit in the following. The H_f term describes the f electrons with momentum-independent flat energy band ε_f < 0 relative to μ and a repulsive on-site Coulomb potential (the Hubbard-U term). The H_cf term,

$${H}_{cf}={\sum}_{\langle i,j\rangle }{\sum}_{\sigma ,{\sigma }^{{\prime} }=\uparrow \downarrow }{V}_{ij}^{\sigma {\sigma }^{{\prime} }}{c}_{i\sigma }^{{\dagger} }{f}_{j{\sigma }^{{\prime} }}+H.c.,$$

(2)

describes the hybridization of impurity and its neighboring conduction electrons with opposite spins, distinct from the conventional (onsite) and spin-conserving hybridization^6,7,8. The hybridization matrix is defined as ${V}_{ij}^{\sigma {\sigma }^{{\prime} }}\equiv {V}_{\hat{\alpha }}^{\sigma {\sigma }^{{\prime} }}=iV{\nu }_{\hat{\alpha }}{\sigma }_{\alpha }^{\sigma {\sigma }^{{\prime} }}$ with V being the hybridization strength. Here, ${\nu }_{\hat{\alpha }}$ satisfies ${\nu }_{\hat{\alpha }}={\nu }_{-\hat{\alpha }}$ with $\hat{\alpha }\equiv i-j\in \,\hat{x},\hat{y}\,(\alpha \in x,y)$ on a 2D square lattice and σ_α denotes the Pauli matrix of the α component. Due to the opposite parity of the f and c electrons as well as the odd parity of ${\nu }_{\hat{\alpha }}$, the parity symmetry of H_cf is preserved. The odd-parity hybridization between the f- and c-electrons in Eq. (2) implies an effective spin-orbit coupling. Candidate materials with odd-parity Kondo hybridization include SmB₆²⁶, SmS²⁷, YbB₁₂²⁸.

The Kondo-Heisenberg model is the commonly-accepted minimum model for Doniach’s scenario²⁵ and the heavy-fermion superconductivity. Below, we apply the same method for which the conventional Kondo-Heisenberg lattice model is obtained to derive the corresponding one for the odd-parity Anderson lattice model^29,30: The effective Kondo term, denoted as H_K, can be derived by projecting out the valence fluctuations, the so-called Schrieffer-Wolff transformation^21,22,31; and the magnetic RKKY-like interaction, denoted as H_J, among the local f electrons is generated by perturbatively expanding the above Kondo term to the second order^31,32,33. The explicit forms of the H_K and H_J terms are provided in Eqs. (8) and (9) in the Methods section, with details of their derivations given in Supplementary Notes 1 and 2, respectively. In addition, the resulting H_J term can be expressed as a linear combination of spinon pairs with total spin S  =  0 (spin-singlet) and S  =  1 (spin-triplet), associated with opposite signs in the coupling [see Eq. (9) in Methods], reminiscent of the conventional spin-1/2 Heisenberg interaction. The RKKY-like coupling J(R) of the local f fermions exhibits an oscillatory behavior as a function of the distance $R\equiv \left\vert {{{{\boldsymbol{R}}}}}_{ij}\right\vert$ between two f fermions, accompanied by a decrease in its magnitude with increasing R:

$$J(R)= \frac{16J_{K}^{2}}{{{{\mathscr{N}}}}_{s}^{2}}{\sum}_{\varepsilon_{{{\boldsymbol{k}}}} < \mu}{\sum}_{\varepsilon_{{{\boldsymbol{k}}}^{\prime\prime}} > \mu} \frac{e^{i\left({{\boldsymbol{k}}}-{{\boldsymbol{k}}}^{\prime\prime}\right)\cdot{{\boldsymbol{R}}}_{ij}}}{\varepsilon_{{{\boldsymbol{k}}}}-\varepsilon_{{{\boldsymbol{k}}}^{\prime\prime}}} \\ \times \left({\sin}^{2}k_{x}+{\sin}^{2}k_{y}\right)\left({\sin}^{2}k_{x}^{\prime\prime}+{\sin}^{2}k_{y}^{\prime\prime}\right),$$

(3)

where J_K denotes the Kondo coupling (see Supplementary Note 1). Due to the rapid attenuation of J(R), we only consider the dominated nearest-neighbor interaction and assume J(R) to be a spatially homogeneous constant, i.e., J(R  =  a)  ≡  J_H. The dominating RKKY coupling J_H  <  0 is attractive (or of the ferromagnetic type, see Fig. 2), which energetically favors spin-triplet pairing. On the other hand, the effective RKKY coupling in the spin-singlet channel is repulsive and thus energetically disfavored. Finally, on a two-dimensional lattice, the triplet spin state of S_z  =  0 ($\left\vert \uparrow \downarrow \right\rangle +\left\vert \downarrow \uparrow \right\rangle$) with a p-wave form factor proportional to k_z does not exist here. Note that it is the odd-parity Kondo hybridization (or effective spin-orbit coupling) that stabilizes the ferromagnetic RKKY. When the odd-parity Kondo hybridization term is replaced by the conventional constant Kondo hybridization, the conventional RKKY coupling is recovered, which favors anti-ferromagnetic coupling between neighboring spins near half-filling.

Fig. 2: The effective RKKY coupling J(R).

The effective RKKY coupling J(R), normalized with ${J}_{K}^{2}$, for different (bare) chemical potentials μ is computed by Eq. (3) with R_ij∥(1, 1). The lattice constant a  =  1 is chosen. Inset: when R  = a (blue dashed line), we find J_H  <  0, indicating that two impurity spins (orange arrows) are parallel with each other, whereas the impurity spins show an anti-parallel alignment if J(R) >  0.

With the inclusions of H_K and H_J, the effective Kondo-Heisenberg lattice model reads H_FKH  =  H_c  +  H_K  +  H_J, subjected to the single-occupancy constraint of the f fermions on each site, ${n}_{i}^{\, f}={\sum }_{\sigma }{f}_{i\sigma }^{{{\dagger}} }{f}_{i\sigma }=1$. The Hamiltonian H_FKH offers a platform for discovering a distinct class of topological superconducting states induced by electron correlations via collaboration between the ferromagnetic RKKY coupling and the Kondo effect. To facilitate our numerical calculations of the mean-field phase diagram below, we treat J_K and J_H as independent couplings here since it is more convenient to explore the phase diagram by tuning the ratio of J_K/J_H^30,34. In experiments, by varying the non-thermal parameter (e.g., pressure, magnetic field, or chemical doping), one anticipates the system to follow a specific trajectory of J_K/J_H in the phase diagram.

Mean-field treatment

We now employ a mean-field analysis on the Kondo-Heisenberg Hamiltonian H_FKM. Performing the Hubbard-Stratonovich transformation, H_K and H_J can be factorized as

$${H}_{K} \to \sum\limits_{i,\alpha }\sum\limits_{\sigma {\sigma }^{{\prime} }}\left[{\chi }_{i}^{{{\dagger}} }\left(i{\nu }_{\hat{\alpha }}{\sigma }_{\alpha }^{\sigma {\sigma }^{{\prime} }}{f}_{i\sigma }^{{{\dagger}} }{c}_{i-\hat{\alpha },{\sigma }^{{\prime} }}\right)+H.c.\right]+\sum\limits_{i}\frac{| {\chi }_{i}{| }^{2}}{{J}_{K}},\\ {H}_{J} \to \sum\limits_{\left\langle i,j\right\rangle }\left[{\Delta }_{t}^{\uparrow }(i,j){f}_{i\uparrow }^{{{\dagger}} }{f}_{j\uparrow }^{{{\dagger}} }+{\Delta }_{t}^{\downarrow }(i,j){f}_{i\downarrow }^{{{\dagger}} }{f}_{j\downarrow }^{{{\dagger}} }+H.c.\right]\\ \quad +\sum\limits_{\langle i,j\rangle }\frac{{\left\vert {\Delta }_{t}^{\uparrow }(i,j)\right\vert }^{2}+{\left\vert {\Delta }_{t}^{\downarrow }(i,j)\right\vert }^{2}}{{J}_{H}},$$

(4)

where the mean-field values of the bosonic Hubbard-Stratonovich fields, χ_i and ${\Delta }_{t}^{\sigma = \uparrow /\downarrow }(i,j)$, represent the mean-field order parameters of the Kondo correlation and the S_z =  ±  1 t-RVB bonds, respectively.

To describe the Kondo-screened Fermi-liquid state, we allow χ_i field to acquire uniform Bose-condensation; hence, χ_i can be expressed as ${\chi }_{i}\to x+{\hat{\chi }}_{i}$ with $x=(-{J}_{K}/{{{{\mathscr{N}}}}}_{s}){\sum }_{i\sigma {\sigma }^{{\prime} }\alpha }\langle i{\nu }_{\hat{\alpha }}{\sigma }_{\alpha }^{\sigma {\sigma }^{{\prime} }}{f}_{i\sigma }^{{{\dagger}} }{c}_{i-\hat{\alpha },{\sigma }^{{\prime} }}\rangle$ being the Bose-condensed stiffness of χ_i while ${\hat{\chi }}_{i}$ represents its fluctuations. The mean-field order parameter of the t-RVB is given by ${\Delta }_{t}^{\sigma }=(-| {J}_{H}| /4{{{{\mathscr{N}}}}}_{s}){\sum }_{\langle i,j\rangle }\langle {f}_{j\sigma }{f}_{i\sigma }\rangle$. It has been known that the ferromagnetic spin-spin interaction (or ferromagnetism) is expected to favor spin-triplet pairing in unconventional superconductors³⁵ as well as in superfluid helium-3³⁶. Meanwhile, the odd-parity hybridization term H_cf in Eq. (2) leads to the momentum-dependent form factor $\sin {k}_{x}\pm i\sin {k}_{y}$ (see Eq. (5) below), reminiscent of the Rashba spin-orbit coupling which favors $p\pm i{p}^{{\prime} }$-wave spin-triplet pairing^37,38. While the form factor of the Kondo hybridization is odd under inversion transformation, the whole Kondo hybridization term in our model Hamiltonian maintains parity symmetry. Based on the above reasoning, in the following, we focus exclusively on the $p\pm i{p}^{{\prime} }$-wave spin-triplet pairing between the f electrons in our Kondo lattice model with odd-parity Kondo hybridization and ferromagnetic RKKY coupling, see Eqs. (13) and (14) below. The single-occupancy constraint of the local fermions can be equivalently enforced by adding $-{\sum }_{i}i{\lambda }_{i}\left[{\sum }_{\sigma }\left({f}_{i\sigma }^{{{\dagger}} }{f}_{i\sigma }\right)-1\right]$ in H_FKM. λ_i here is the Lagrange multiplier. In the mean-field approximation, we take iλ_i → λ at the mean-field level and neglect the fluctuations of λ_i, χ_i, and ${\Delta }_{t}^{\sigma }$, leading to the following mean-field Kondo-Heisenberg-like Hamiltonian of H_FKM:

$${H}_{MF} = \sum\limits_{{{{\boldsymbol{k}}}},\sigma }{\varepsilon }_{{{{\boldsymbol{k}}}}}{c}_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{c}_{{{{\boldsymbol{k}}}}\sigma }+\sum\limits_{{{{\boldsymbol{k}}}}\sigma }\lambda {f}_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{f}_{{{{\boldsymbol{k}}}}\sigma }+{{{\mathscr{C}}}}\\ +\sum\limits_{{{{\boldsymbol{k}}}}}\left[{V}_{1{{{\boldsymbol{k}}}}}\, {f}_{{{{\boldsymbol{k}}}}\uparrow }^{* }{c}_{{{{\boldsymbol{k}}}}\downarrow }+{V}_{2{{{\boldsymbol{k}}}}}{f}_{{{{\boldsymbol{k}}}}\downarrow }^{* }{c}_{{{{\boldsymbol{k}}}}\uparrow }+H.c.\right]\\ +\sum\limits_{{{{\boldsymbol{k}}}}}\left[{\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow }{f}_{{{{\boldsymbol{k}}}}\uparrow }^{{{\dagger}} }{f}_{-{{{\boldsymbol{k}}}}\uparrow }^{{{\dagger}} }+{\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow }{f}_{{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} }{f}_{-{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} }+H.c.\right],$$

(5)

where ${{{\mathscr{C}}}}=\frac{8{{{{\mathscr{N}}}}}_{s}{\Delta }_{t}^{2}}{| {J}_{H}| }+\frac{{{{{\mathscr{N}}}}}_{s}{x}^{2}}{{J}_{K}}-{{{{\mathscr{N}}}}}_{s}\lambda$ is a constant, ${V}_{1{{{\boldsymbol{k}}}}}=2x\left(\sin {k}_{x}-i\sin {k}_{y}\right)$ and ${V}_{2{{{\boldsymbol{k}}}}}=2x\left(\sin {k}_{x}+i\sin {k}_{y}\right)$. The Fourier transformation of a second-quantization operator is defined as ${\psi }_{i\sigma }=\frac{1}{\sqrt{{{{{\mathscr{N}}}}}_{s}}}{\sum }_{{{{\boldsymbol{k}}}}}{e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{r}}}}}_{i}}{\psi }_{{{{\boldsymbol{k}}}}\sigma }$. Note that the mean-field Kondo term of Eq. (5) is reminiscent of the topological Kondo insulator shown in ref. ³⁹. In Eq. (5), ${\Delta }_{{{{\boldsymbol{k}}}}}^{\sigma }$ are defined as ${\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow }={\Delta }_{t}\left(-\sin {k}_{y}-i\sin {k}_{x}\right)$ and $\quad {\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow }={\Delta }_{t}\left(\sin {k}_{y}-i\sin {k}_{x}\right)$ with Δ_t being the mean-field pairing potential of the t-RVB. Note that the momentum-dependent form factor of ${\Delta }_{{{{\boldsymbol{k}}}}}^{\sigma }$ is proportional to that of the odd-parity Kondo hybridization, ${\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow }\propto {V}_{2{{{\boldsymbol{k}}}}},{\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow }\propto {V}_{1{{{\boldsymbol{k}}}}}$, similar to the case of the p-wave triplet pairing favored by the Rashba spin-orbit coupling^37,38.

Our mean-field Kondo-Heisenberg Hamiltonian of Eq. (5) possesses time-reversal symmetry (TRS), whereas, due to the $p\pm i{p}^{{\prime} }$ RVB pairing, the parity symmetry is broken here. Owing to TRS, our model can be expressed as a summation of two decoupled matrices as follows ${H}_{MF}={\sum }_{I = A,B}{H}_{I}+{{{\mathscr{C}}}}={\sum }_{I,{{{\boldsymbol{k}}}}}{\phi }_{I}^{{{\dagger}} }({{{\boldsymbol{k}}}}){{{{\mathscr{H}}}}}_{I}({{{\boldsymbol{k}}}}){\phi }_{I}({{{\boldsymbol{k}}}})+{{{\mathscr{C}}}}$ with ${\phi }_{A}({{{\boldsymbol{k}}}})={\left({c}_{{{{\boldsymbol{k}}}}\uparrow },{c}_{-{{{\boldsymbol{k}}}}\uparrow }^{{{\dagger}} },{f}_{{{{\boldsymbol{k}}}}\downarrow },{f}_{-{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} }\right)}^{T}$ and ${\phi }_{B}({{{\boldsymbol{k}}}})={\left({c}_{{{{\boldsymbol{k}}}}\downarrow },{c}_{-{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} },{f}_{{{{\boldsymbol{k}}}}\uparrow },{f}_{-{{{\boldsymbol{k}}}}\uparrow }^{{{\dagger}} }\right)}^{T}$ being the Nambu spinors for H_A(B). One can show that H_A and H_B constitute the time-reversal partners. In this representation, the charge-conjugation (particle-hole) symmetry of our model can be readily seen. Our model Eq. (5) thus belongs to the DIII class of the ten-fold classification⁴⁰.

Mean-field phase diagram

The mean-field ground states are determined by minimizing the mean-field free energy per site ${{{{\mathscr{F}}}}}_{MF}=\frac{{{{\mathscr{C}}}}}{{{{{\mathscr{N}}}}}_{s}}-\frac{{k}_{B}T}{{{{{\mathscr{N}}}}}_{s}}{\sum }_{{{{\boldsymbol{k}}}},n}\ln \left[1+\exp \left(-\frac{{E}_{n{{{\boldsymbol{k}}}}}}{{k}_{B}T}\right)\right]$ with respect to the mean-field variables q ∈ (λ,  x,  Δ_t), i.e., $\partial {{{{\mathscr{F}}}}}_{MF}/\partial q=0$ (the explicit form for the dispersion is provided in Eq. (12) of Methods). Here, E_nk  <  0 is the n-th filled band of H_MF. The chemical potential μ is determined by the relation $\partial {{{{\mathscr{F}}}}}_{MF}/\partial \mu =-(1+\delta )$, with δ being the chemical doping of the c-electrons for which δ  ⋛  0 is for p/un/n− doped. By solving the above saddle-point equations self-consistently for both the bulk and the ribbon with N_y  =  44 chains, we obtain the ground-state phase diagrams of the model as shown in Fig. 3. Figure 3 contains three distinct mean-field phases: a pure Kondo phase is found at J_H/J_K  =  0 where x  ≠  0, Δ_t  =  0. At the opposite limit where the RKKY interaction dominates, the ground-state shows short-range magnetic correlation with p-wave t-RVB pairing (Δ_t  ≠  0,  x = 0). In the intermediate range of $0 < {J}_{H}/{J}_{K} < {({J}_{H}/{J}_{K})}_{c}$, we find a Kondo-t-RVB co-existing (superconducting) phase with x  ≠  0 and Δ_t  ≠  0, which can be explained via the mechanism of Kondo-stabilized spin liquid^30,41: The t-RVB state gains kinetic energy by coupling to the Kondo hybridization, leading to the stabilized co-existing superconducting state. The bulk band structure in the co-existing superconducting state, shown in Fig. 4, is fully-gapped. This superconducting gap is contributed by both the Kondo and t-RVB order parameters.

Fig. 3: The zero-temperature mean-field solutions of our Kondo lattice model.

The black, brown, and green curves represent the zero-temperature mean-field solutions of the Kondo correlation x, spin-triplet RVB (t-RVB) order parameter Δ_t, and the superconductor transition temperature T_c as a function of J_H/J_K. T_c shows a dome shape. This plot reveals three mean-field ground states: the (co-existing) superconducting state with x  ≠  0,  Δ_t  ≠  0 for 0 <  J_H/J_K  ≲  8.4 and a pure t-RVB phase where x  =  0,  Δ_t  ≠  0 when J_H/J_K  ≳ 8.4. A pure Kondo phase (x  ≠  0,  Δ_t  =  0) only exists at J_H/J_K =  0, as indicated by the yellow star. Here, we fix J_K  = 0.3 and doping of the conduction band δ  =  − 0.3 (30 percent hole doping).

Fig. 4: Bulk energy spectra of the co-existing phase.

Figures a (red curves) and b show the bulk energy spectrum of the co-existing superconducting state near the Fermi level μ. The Fermi level locates at E(k ) =  0. The coupling constants are J_K  = 0.3 and J_H  =  1.0. The inset of a displays the First Brillouin zone of a square lattice with indications of high-symmetry momenta, Γ,  X,  M.

The development of superconductivity in this co-existing phase can be understood in terms of higher-order processes involving both the Kondo and t-RVB terms as follows. When the Kondo hybridization field χ gets Bose-condensed (x  ≠  0), the local fermions delocalize into the conduction band and make the t-RVB pairs superconducting⁴². These processes can be described by the effective mean-field Hamiltonian ${H}_{sc}={\sum }_{{{{\boldsymbol{k}}}}}\left({\bar{\Delta }}_{{{{\boldsymbol{k}}}}}^{\downarrow * }{c}_{-{{{\boldsymbol{k}}}}\downarrow }{c}_{{{{\boldsymbol{k}}}}\downarrow }+{\bar{\Delta }}_{{{{\boldsymbol{k}}}}}^{\uparrow * }{c}_{-{{{\boldsymbol{k}}}}\uparrow }{c}_{{{{\boldsymbol{k}}}}\uparrow }+H.c.\right)$, where the effective gap functions take the form ${\bar{\Delta }}_{{{{\boldsymbol{k}}}}}^{\downarrow * }={V}_{1{{{\boldsymbol{k}}}}}{V}_{1,-{{{\boldsymbol{k}}}}}{\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow * } \sim {x}^{2}{\Delta }_{t}({\sin }^{2}{k}_{x}+{\sin }^{2}{k}_{y})(\sin {k}_{x}-i\sin {k}_{y})$ and ${\bar{\Delta }}_{{{{\boldsymbol{k}}}}}^{\uparrow * }={V}_{2{{{\boldsymbol{k}}}}}{V}_{2,-{{{\boldsymbol{k}}}}}{\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow * } \sim {x}^{2}{\Delta }_{t}({\sin }^{2}{k}_{x}+{\sin }^{2}{k}_{y})(\sin {k}_{x}+i\sin {k}_{y})$ with the size of the superconducting gap being proportional to x²Δ_t. The superconducting transition temperature T_c as a function of J_H/J_K shows a dome shape (Fig. 3). The superconducting gap function ${\overline{\Delta }}_{{{{\boldsymbol{k}}}}}^{\uparrow }$ we obtained here shows an f-wave-like pairing symmetry on a generic anisotropic 2D Fermi surface. Nevertheless, as we take the continuous limit of the conduction band here, ${\overline{\Delta }}_{{{{\boldsymbol{k}}}}}^{\uparrow }$ can be expressed as a product of s and $p\pm i{p}^{{\prime} }$ pairing orders, i.e., ${\bar{\Delta }}_{{{{\boldsymbol{k}}}}}^{\uparrow /\downarrow * } \sim {k}^{2}({k}_{x}\pm i{k}_{y})$ with ${k}^{2}\equiv {k}_{x}^{2}+{k}_{y}^{2}$ on a circular Fermi surface, only the $p\pm i{p}^{{\prime} }$ component plays a role here. The effective $p\pm i{p}^{{\prime} }$ pairing (${\overline{\Delta }}_{{{{\boldsymbol{k}}}}}^{\sigma }$) between the conduction electrons in H_sc gives rise to a fully superconducting gap on a 2D lattice. However, when considering a 3D conduction band, this effective pairing shows a point-node gap structure in the quasiparticle dispersion, see Applications for UTe₂ below and Supplementary Note 3. Note that we find the co-existing superconducting state persists for an arbitrarily small value of J_H/J_K → 0⁺. This behavior is likely due to overestimating the co-existing phase at the mean-field level. We expect a narrower co-existing superconducting phase after including fluctuations of the Kondo and t-RVB order parameters beyond the mean-field level. We also  notice that our mean-field solutions in Fig. 3 show a kink-like feature near J_H/J_K ≈ 6.5, possibly due to the specific mathematical form of the saddle-point equations. Nevertheless, they are continuous as J_K and J_H are varied, and no phase transition is found across the kink. A first-order transition is observed at the transition of the t-RVB and co-existing superconducting phases (see Fig. 3), similar to the results found in refs. ^30,43.

Topological properties of the co-existing phase

We now address the topological properties of the co-existing superconducting state. Given that our model preserves TRS, the bulk topological properties of the co-existing Kondo-t-RVB superconducting state can be characterized by the topological Z₂ index c_T (or time-reversal polarization)^44,45,46, given by

$${c}_{T}=\frac{{c}_{A}-{c}_{B}}{2}\,\,{\mbox{and}}\,\,{c}_{I}=\frac{1}{2\pi }{\int}_{ \!\!\!\! {{{\boldsymbol{k}}}}\in {\mbox{FBZ}}}d{{{{\boldsymbol{S}}}}}_{{{{\boldsymbol{k}}}}}\cdot \left({{{{\boldsymbol{\nabla }}}}}_{{{{\boldsymbol{k}}}}}\times {{{{\boldsymbol{{{{\mathscr{A}}}}}}}}}_{{{{\boldsymbol{k}}}}}^{I}\right),$$

(6)

with c_I (I ∈ A, B) being the Thouless-Kohmoto-Nightingal-den Nijs number⁴⁷ of H_I. The Berry’s connection ${{{{\boldsymbol{{{{\mathscr{A}}}}}}}}}_{{{{\boldsymbol{k}}}}}^{I}$ is given by ${{{{\boldsymbol{{{{\mathscr{A}}}}}}}}}_{{{{\boldsymbol{k}}}}}^{I}\equiv i{\sum }_{n\in I}\langle {u}_{n{{{\boldsymbol{k}}}}}^{I}| {{{{\boldsymbol{\nabla }}}}}_{{{{\boldsymbol{k}}}}}| {u}_{n{{{\boldsymbol{k}}}}}^{I}\rangle$ with $\left\vert {u}_{n{{{\boldsymbol{k}}}}}^{I}\right\rangle$ being the normalized Bloch state of the n-th filled band of H_I(k). We numerically find that c_A  =  − c_B  =  1 in the co-existing phase⁴⁸, indicating a topologically non-trivial Z₂ index c_T  =  1. By the bulk-edge correspondence, we expect this co-existing superconducting state to support a pair of counter-propagating MZM at the edges of a finite-sized ribbon. Further band structure calculations of our model on a ribbon in the following section confirm our expectations.

We now check whether our model would support helical Majorana edge states by examining the electronic band structures and edge-state wave functions of the co-existing state on a finite-sized strip near the Fermi level (details are provided in Supplementary Note 4). As shown in Figs. 5 and 6, the low-energy Bogoliubov quasiparticle excitations of both H_A and H_B exhibit linear-dispersed Dirac spectra around k_x  =  0 accompanied by an exponential decay in the wave functions, typical features of topological edge states. The excitations can be effectively described by the linear-dispersed Hamiltonian ${\tilde{H}}_{I}={\sum }_{{k}_{x}}{v}_{x}| {k}_{x}| \left({\gamma }_{I,{k}_{x}}^{R\,{{\dagger}} }{\gamma }_{I,{k}_{x}}^{R}-{\gamma }_{I,{k}_{x}}^{L\,{{\dagger}} }{\gamma }_{I,{k}_{x}}^{L}\right)$ with

$${\gamma }_{I,{k}_{x}}^{\Gamma } = {\sum}_{{y}_{i}}\left[{u}_{I,{k}_{x}}^{\Gamma }({y}_{i}){c}_{{k}_{x},{y}_{i},\uparrow }+{\bar{u}}_{I,{k}_{x}}^{\Gamma }({y}_{i}){c}_{-{k}_{x},{y}_{i},\uparrow }^{{{\dagger}} }\right.\\ \left.+{v}_{I,{k}_{x}}^{\Gamma }({y}_{i}){f}_{{k}_{x},{y}_{i},\downarrow }+{\bar{v}}_{I,{k}_{x}}^{\Gamma }({y}_{i}){f}_{-{k}_{x},{y}_{i},\downarrow }^{{{\dagger}} }\right]$$

(7)

with $u,\bar{u}$ and $v,\bar{v}$ being the coherent factors and v_x  ~  ∂E(k_x)/∂k_x being the velocity indicating the direction of propagation. In Eq. (7), Γ ∈ R, L and ${\gamma }_{I,{k}_{x}}^{R/L}$ represents the right/left-moving Bogoliubov quasiparticle of ${\tilde{H}}_{I}$. We find that the four edge states, shown in Fig. 6c, d, e, f, constitute two pairs of counter-propagating edge states located at opposite edges, revealing the nature of helical MZM (see Fig. 6g). The helical type of the MZM is the consequence of the TRS of our model. In addition to the Majorana fermions at zero energy, two pairs of counter-propagating edge states are observed at finite-energy, see Fig. 7. The two pairs of edge states correspond to the edge states of the topological Kondo insulator, where the t-RVB order parameter is absent (Δ_t  =  0)^6,7,8. Qualitative explanations for the emergence of the three pairs of edge states shown in Fig. 5 are provided in the next subsection and Fig. 8a.

Fig. 5: Energy spectra of the co-existing phase on a strip.

Figure a displays the electronic band structure of the co-existing superconductor state for ${{{{\mathscr{H}}}}}_{A}$ on a strip with 81 chains along the y direction (N_y  = 81). We choose J_K  = 0.3 and J_H  =  1.0. Three pairs of edge states are observed near k_x  =  0 (the pink curves). The edge states at zero energy correspond to the MZM. Figures b, c, d show the close-up band structures near three pairs of edge states (pink curves) bounded by the red squares in a. Due to the TRS of the model, the band structure of ${{{{\mathscr{H}}}}}_{A}$ is identical to that of ${{{{\mathscr{H}}}}}_{B}$.

Fig. 6: MZM of the co-existing phase.

Figures a and b show the Bogoliubov excitation spectra of ${{{{\mathscr{H}}}}}_{A}$ and ${{{{\mathscr{H}}}}}_{B}$ near the chemical potential on a nano-strip with N_y  =  81 chains. Figures c, d, and e, f demonstrate the probability density, ${\left\vert {\gamma }_{I,{k}_{x}}^{\Gamma }({y}_{i})\right\vert }^{2}$ with I  =  A, B and Γ  =  R, L, of the MZM wave functions of ${{{{\mathscr{H}}}}}_{A}$ and ${{{{\mathscr{H}}}}}_{B}$ as a function of atom position y_i (pink curves in a and b), at a fixed energy E  ≡  E(k_x  =  ±  0.03). The probability density is described by ${\left\vert {\gamma }_{I,{k}_{x}}^{\Gamma }({y}_{i})\right\vert }^{2}=({\left\vert {u}_{I,{k}_{x}}^{\Gamma }\right\vert }^{2},\,{\left\vert {\bar{u}}_{I,{k}_{x}}^{\Gamma }\right\vert }^{2},\,{\left\vert {v}_{I,{k}_{x}}^{\Gamma }\right\vert }^{2},\,{\left\vert {\bar{v}}_{I,{k}_{x}}^{\Gamma }\right\vert }^{2})({y}_{i})$. The parameters are J_K  =  0.3, J_H  =  1.0, and doping δ  =  − 0.3. The edge states are of the helical type, as schematically represented in g.

Fig. 7: The edge states at finite energy of the co-existing phase.

Figures a and b display the finite-energy (E(k_x)  >  0) Bogoliubov excitation spectra of ${{{{\mathscr{H}}}}}_{A}$ and ${{{{\mathscr{H}}}}}_{B}$, respectively. A pair of helical edge states [pink curve in a and b] is found to exist at a finite energy E(k_x  = ±  0.22), and their probability densities are shown in c and d, e and f, respectively as indicated by the corresponding numbers in each of the panels. Here, y_i labels the atom position along the y direction, see Fig. 6g.

Fig. 8: Schematic illustrations for TKSc and proximity-induced topological superconductor.

a In our topological Kondo system, when T_c  < T  < T_onset with T_onset being denoted as the onset temperature of the Kondo hybridization, the hole-doped conduction band (blue dashed curve) with chemical potential μ (orange thin dashed line) first Kondo-hybridizes with the flat f-band with effective energy level λ (blue dashed line) near μ. The Kondo hybridization leads to a shift in the system’s chemical potential to ${\mu }^{{\prime} }$ and the formation of a Kondo hybridization gap between the two hybridized bands. Inside the gap, a pair of edge states (green curves) emerge. b. As the temperature decreases below T_c, the RVB pairing sets in, leading to the emergence of energy gaps between the particle and hole branches of the hybridized bands above, at, and below ${\mu }^{{\prime} }$. One more pair of edge states, the MZM (pink curves), is generated inside the energy gap at ${\mu }^{{\prime} }$ (corresponding to the superconducting gap). c. In the topological superconductor by proximity effect in heterostructure, a s-wave superconductor with chemical potential ${\mu }^{{\prime} }$ is placed in proximity to a topological insulator whose topological edge states are pinned in general at a different chemical potential μ. Thus, a fine-tuning of the chemical potential of the conduction bands is required.

Physical insight to the search for topological superconductivity

The mechanism we propose here via co-existing Kondo hybridization and t-RVB offers a physical insight into the search for topological superconductivity. It shows advantages over the intensively studied heterostructure approach via the proximity effect for which the correlation effect plays a minor role. Nonetheless, we acknowledge the alternative approaches to correlation-driven topological superconductivity, including other possible proximity-induced approaches in heterostructures. As shown in Fig. 8a, for T_c  <  T  <  T_onset, with T_onset being the onset temperature of Kondo hybridization, the local f-electron band at an effective chemical potential λ first Kondo-hybridizes with the conduction band close to the Fermi energy μ. The non-trivial topological features of our model system are pinned near the Fermi energy of the conduction band when the Kondo effect emerges¹⁰ since these topological features are in-printed in the odd-parity Kondo hybridization. With further decreasing temperature below T_c, the formation of the p-wave t-RVB pairings opens up energy gaps between the particle and hole branches of the hybridized bands below, at, and above μ, as shown in Fig. 8b. The energy gap at μ corresponds to the superconducting gap, leading to the emergence of the topological superconducting state. Hence, the topologically non-trivial gapless MZM inside the superconducting gap is also pinned at μ. The pinning of MZM^10,24 by the Kondo hybridization in our system makes the topological superconductivity more feasible to form and to be observed compared to the heterostructure approach where the topological edge states may be far from the Fermi level of the s-wave superconductor (see Fig. 8b), and fine-tuning the chemical potential of either of the two sub-systems is required.

Applications for UTe₂

We will now discuss the application of our findings to a promising candidate for TKSc: ferromagnetic heavy-electron superconductor UTe₂. At ambient pressure, UTe₂ does not show long-ranged ferromagnetism, unlike its related uranium compounds UGe₂, URhGe, and UCoGe where superconductivity co-exists with long-range ferromagnetism. Nevertheless, UTe₂, located at the end of the above ferromagnetic superconducting Kondo lattice compounds, lives near the edge of ferromagnetism, suggesting short-range ferromagnetic fluctuations are still expected to play a role there. A characteristic feature of these ferromagnetic superconductors is that they exhibit very large upper critical fields (H_c2), far exceeding the ordinary Pauli paramagnetic limit (H_p/T_sc  =  1.86 T/K). Similar to UGe₂, URhGe, and UCoGe, the family of ferromagnetic superconductors, UTe₂ also shows very large H_c2^15,49. This strongly indicates the enhancement of ferromagnetic fluctuations there, similar to the other above-mentioned ferromagnetic superconductors. We will show that our results qualitatively agree with and, to some extent, quantitatively explain the experimental observations in UTe₂, despite our simplified two-dimensional square lattice system. Theoretical studies indicated competition or co-existence between ferromagnetic and anti-ferromagnetic fluctuations in UTe₂ when Hubbard U or Kondo hybridization is varied^19,50,51, possibly due to strong magnetic frustrations there. Indeed, both ferromagnetic^52,53 and anti-ferromagnetic⁵⁴ correlations in this compound were reported. In particular, it was predicted in ref. ⁵⁰ that the dominantly ferromagnetic correlations at ambient pressure change to anti-ferromagnetic correlations with increasing pressure, consistent with various experimental observations^49,52,53. Though magnetic correlations in UTe₂ are still an open issue, the ferromagnetic Kondo lattice model we propose here is consistent with theoretical expectations and various experimental observations mentioned above. The DFT+U calculations indicate that the dynamics of electron bands and the physical properties of UTe₂ are dominated by the electrons near the quasi-two-dimensional (cylindrical) Fermi surface with a weak k_z dependence despite its three-dimensional crystal structure⁵¹. The quasi-2D Fermi surface has been confirmed experimentally by magnetic quantum oscillation measurements^55,56, where the authors emphasized that they found no evidence for the presence of any 3D Fermi surface sections. This strong experimental evidence clearly justifies the relevance of our (simplified) 2D model for describing UTe₂. However, we also notice normal state electrical resistivity measurement⁵⁷ and thermal conductivity measurement⁵⁸ reveal the presence of 3D Fermi surfaces. While the Fermi surfaces of UTe₂ are quasi-2D or 3D-like is still under debate, at least, our theoretical approach is consistent with a number of experimental results that display 2D Fermi surfaces. Superconductivity is reached at T_c  =  1.6 K. The resistivity maximum observed at T^⋆  ≈  15 – 75 K^15,57 and the ARPES measurement⁵⁹ reveal the signature of Kondo coherence for T  <  T^⋆ with T^⋆/T_c  ≈  10– 50. This strongly suggests the co-existence between superconductivity and the Kondo correlation for T  <  T_c¹⁸. When a magnetic field is applied along the hard-magnetic axis b of UTe₂ and before the material enters the superconducting phase, a correlated paramagnetic phase is observed below the temperature at which the magnetic susceptibility shows a broad maximum⁶⁰. Similar spin-liquid behavior was also observed in the magnetic susceptibility of CePdAl⁶¹, suggesting the presence of short-range magnetic order in the correlated paramagnetic phase of UTe₂. The t-RVB state we proposed here can be naturally induced by the short-range ferromagnetic fluctuations and therefore relevant for UTe₂. Based on these experimental observations, our theoretical framework–competition and collaboration between Kondo-screened and ferromagnetic t-RVB spin-liquid states on a two-dimensional Kondo lattice–constitutes a promising approach to account for the above observations in UTe₂. Meanwhile, the chiral in-gap state, a signature of a chiral topological superconductor, has been observed by scanning tunneling spectroscopy in the superconducting phase of UTe₂¹⁸. Combined with the ferromagnetic fluctuations that are known to induce spin-triplet pairing, UTe₂ is considered a promising candidate for the spin-triplet chiral topological superconductor^15,18. Furthermore, the co-existence of superconductivity and Kondo coherence strongly suggests the significance of the Kondo effect in UTe₂, a promising candidate for TKSc.

The TKSc state with equal-spin spin-triplet p-wave pairings we proposed here bears striking similarities to and strong relevance for the experimental observations on UTe₂: (i) electrons in the d- and f-orbitals with their angular momentum quantum numbers differing by 1 in the uranium atoms of UTe₂ likely lead to the odd-parity Kondo effect^6,7,8, (ii) the t-RVB state in our theory may be considered as one possible consequence of the short-ranged ferromagnetic fluctuations observed in UTe₂, (iii) the Kondo-t-RVB co-existing superconducting state we find here qualitatively agrees with the co-existence between superconductivity and Kondo effect observed in UTe₂, (iv) the high upper critical field exceeding the Pauli limit^15,62 implies that the superconducting state of UTe₂ may have equal-spin Cooper pairs, and (v) recent measurements on London penetration depth strongly suggest the existence of point-nodes in the superconducting gap of UTe₂⁶³. We further checked that the point-node gap structure ${\overline{\Delta }}_{{{{\boldsymbol{k}}}}}^{\sigma }$ leads to a T³ dependence in the low-temperature specific heat in the 3D generalization of our model, in perfect agreement with experimental observation on UTe₂⁶⁴ (see Supplementary Note 3).

Moreover, various characteristic temperature scales estimated from our mean-field calculations with J_H  =  1.0 and J_K  =  0.3 at finite temperatures agree reasonably well with experimental observations (see Fig. 9): The superconducting transition temperature T_c, theoretically determined from our mean-field analysis T_c =  Min[T(x  =  0), T(Δ_t  =  0)], shows T_c  ≈  0.015t ≈ 2.3 K by taking estimated values of t  =  150 K and half-bandwidth D  =  1.25t⁶⁵. The Kondo coherent scale can be obtained by T^⋆  =  x²(T =  0)/D  ≈  17.4 K⁶⁶. The ratio T^⋆/T_c  ≈  8 is in reasonable agreement with experimental observations. The onset temperature T_onset of Kondo hybridization, which occurs at x(T  =  T_onset)  =  0, displays T_onset  ≈  0.15t  ≈  24 K, within the theoretically estimated range 10K  <  T_onset  <  100K by DMFT calculation⁶⁵. Meanwhile, there has been evidence of TRS breaking in UTe₂ from the observed two superconducting transitions and a finite polar Kerr effect at T  <  T_c⁶⁷, likely due to proximity to the ferromagnetic ordered phase. Based on the TRS breaking mentioned above, several theoretical attempts were proposed^23,68. However, the observed single superconducting transition near ambient pressure and zero field^60,69,70 and the theoretically proposed unitary triplet pairing⁵¹ suggest TRS may be preserved in UTe₂.

Fig. 9: Plot of the temperature-dependent mean-field order parameters x(T) and Δ_t(T).

The single-impurity Kondo temperature occurs at T_onset  ≈  0.15 while the transition of superconductivity takes place at a temperature T_c  ≈ 0.015. The inset shows the enlarged plot of Δ_t(T). Here, J_K  = 0.3 and J_H  =  1.0 are fixed, and we set t  = k_B =  1.

Discussion

Whether or not the superconducting state of UTe₂ preserves TRS requires more experimental evidence and deserves more discussions. Though our results shown above are obtained in the presence of TRS, the chiral p-wave superconducting state with chiral MZM at edges is expected to occur here once a time-reversal breaking magnetic field is applied³⁸. Our distinct predictions with and without applying fields serve as theoretical guidance for future experiments to distinguish the time-reversal breaking from time-reversal preserving triplet pairing states in UTe₂. However, a growing number of recent experimental evidence, particularly from cleaner UTe₂ samples, tend to support the existence of a time-reversal preserving superconducting state^71,72,73. These findings are in alignment with the predictions of our theoretical framework with TRS although the specific form factor of our superconducting order parameter may differ from the experimental predictions.

Below we make a few side remarks on our theoretical approach. Since the Kondo correlations stabilize the t-RVB spin liquid in the co-existing superconducting phase, it is expected to be robust against gauge-field fluctuations beyond the mean field. Meanwhile, we checked that the co-existing superconducting phase indeed has a lower energy than the ferromagnetic long-range ordered and t-RVB states for 0  <  J_H/J_K < 3.6 due to the mechanism of the Kondo-stabilized spin-liquid mentioned above. However, the superconducting phase is energetically unfavorable against the ferromagnetic long-range ordered state for J_H/J_K  >  3.6 (see Supplementary Note 5). In addition, when considering a 3D conduction band, the effective Cooper pairing manifests a point-node gap structure in the quasiparticle dispersion, resulting in the T³ behavior in specific heat. Nevertheless, recent thermal conductivity measurement on a high-quality UTe₂ crystal reveals a fully-gapped superconducting state⁵⁸, which is consistent with our findings based on a 2D model. Whether or not the Cooper pairing of UTe₂ is fully gapped needs further investigation. Our simplified model effectively captures many essential aspects of the topological Kondo superconductivity, despite our simplified treatment of the RKKY coupling.

Meanwhile, though we adapt the odd-parity Kondo hybridization used in previous works on topological Kondo insulators^6,7,8, we propose here a route to topological superconductivity via cooperation between Kondo and ferromagnetic RKKY interactions. This route to topological superconductivity has not yet been addressed. However, we acknowledge that the odd-parity Kondo hybridization we use here is not a necessity to reach the topological Kondo superconductivity. There exist alternative approaches to topological superconductivity in the heavy-fermion Kondo lattice systems without invoking odd-parity Kondo hybridization. For example, in the Kondo lattice model with Rashba spin-orbit coupled conduction band, conventional (on-site) Kondo hybridization and consequently ferromagnetic RKKY coupling, topological (p-wave triplet) superconductivity is expected to be induced by cooperation among Rashba spin-orbit, Kondo and ferromagnetic RKKY couplings despite the conventional Kondo hybridization. Meanwhile, having ferromagnetic RKKY coupling is not unique to the odd-parity Kondo hybridization. The band structure of the conduction band is usually the most important factor for the structure of RKKY coupling. In our case, we consider a simplified nearest-neighbor tight-binding conduction band near half-filling where nearly nesting of the Fermi surface may lead to competition between ferromagnetic and anti-ferromagnetic RKKY coupling through conventional (on-site) Kondo hybridization. Here, by introducing the odd-parity inter-site Kondo hybridization additional form factor is induced, which is essential to suppress anti-ferromagnetic RKKY coupling and favors ferromagnetic RKKY coupling. Nevertheless, one can alternatively start from different band structures that favor the ferromagnetic correlations. More generally, we acknowledge that there exist alternative approaches to topological superconductivity without invoking Kondo physics, such as a model with Ising-like ferromagnetic spin-spin interactions that would lead to the same pairing state. Though the heavy-fermion nature of the topological superconductivity is not a necessity, it shows great advantages in experimental realization as mentioned above.

Conclusion

In conclusion, we theoretically find a distinct type of topological superconductivity driven purely by electron correlations–the odd-parity Kondo hybridization–in a two-dimensional Kondo lattice without invoking the proximity effect on heterostructure. The proposed topological Kondo superconductor stems from the odd-parity Kondo hybridization between electrons in the itinerant c and localized f bands, with their orbital quantum numbers differing by 1. Starting from the odd-parity Anderson lattice model, we derive the odd-parity Kondo hybridization and ferromagnetic RKKY interaction mediated by the odd-parity Kondo coupling via perturbation theory, leading to spin-triplet resonating valence bond pairing between f-electrons with $p\pm i{p}^{{\prime} }$-wave gap symmetry. Via the mean-field approach, we find a Kondo triplet-resonating valence bond co-existing phase in the intermediate range of the Kondo to RKKY coupling ratio. We showed that this phase is a time-reversal invariant spin-triplet $p\pm i{p}^{{\prime} }$-wave topological superconducting state, exhibiting non-trivial topology in the bulk and supporting helical Majorana zero modes at edges. Our results offer a comprehensive understanding of the experimental observations on UTe₂, a promising candidate for topological Kondo superconductors. Our approach to topological superconductivity via Kondo physics shows advantages over the heterostructure approach by proximity effect and provides a paradigm to guide the search for topological superconductors induced by electron correlations.

Methods

The Kondo-Heisenberg-like lattice model

The explicit form of the Kondo and RKKY-like magnetic interaction terms, H_K and H_J, is specified here in more detail. The H_K term, by performing Schrieffer-Wolff transformation on Eq. (1), takes the form of

$${H}_{K} = \; (-{J}_{K})\sum\limits_{i}\sum\limits_{\sigma {\sigma }^{{\prime} }}\sum\limits_{{\sigma }^{\prime\prime}{\sigma }^{\prime\prime {\prime} }}\sum\limits_{\alpha ,{\alpha }^{{\prime} }}\left(i{\nu }_{\hat{\alpha }}{\sigma }_{\alpha }^{\sigma {\sigma }^{{\prime} }}{c}_{i+\hat{\alpha },\sigma }^{{{\dagger}} }{f}_{i{\sigma }^{{\prime} }}\right)\\ \times \left(i{\nu }_{{\hat{\alpha }}^{{\prime} }}{\sigma }_{{\alpha }^{{\prime} }}^{{\sigma }^{\prime\prime}{\sigma }^{\prime\prime {\prime} }}{f}_{i{\sigma }^{\prime\prime}}^{{{\dagger}} }{c}_{i-{\hat{\alpha }}^{{\prime} },{\sigma }^{\prime\prime {\prime} }}\right)\hfill$$

(8)

with ${J}_{K}=\frac{{V}^{2}}{U+{\varepsilon }_{f}-\mu }+\frac{{V}^{2}}{\mu -{\varepsilon }_{f}} > 0$. Different from the conventional on-site Kondo interaction, Eq. (8) describes the screening of impurity by its neighboring conduction electrons.

The RKKY-like magnetic interaction H_J is given by

$${H}_{J}= \sum\limits_{\left\langle i,j\right\rangle }{J}_{ij}\left({f}_{i\uparrow }^{{{\dagger}} }{f}_{j\uparrow }^{{{\dagger}} }{f}_{j\uparrow }{f}_{i\uparrow }+{f}_{i\downarrow }^{{{\dagger}} }{f}_{j\downarrow }^{{{\dagger}} }{f}_{j\downarrow }{f}_{i\downarrow }\right)\\ +\sum\limits_{\left\langle i,j\right\rangle }\frac{{J}_{ij}}{2}\left({f}_{i\uparrow }^{{{\dagger}} }{f}_{j\downarrow }^{{{\dagger}} }+{f}_{i\downarrow }^{{{\dagger}} }{f}_{j\uparrow }^{{{\dagger}} }\right)\left({f}_{j\downarrow }{f}_{i\uparrow }+{f}_{j\uparrow }{f}_{i\downarrow }\right)\\ -\sum\limits_{\left\langle i,j\right\rangle }\frac{{J}_{ij}}{2}\left({f}_{i\uparrow }^{{{\dagger}} }{f}_{j\downarrow }^{{{\dagger}} }-{f}_{i\downarrow }^{{{\dagger}} }{f}_{j\uparrow }^{{{\dagger}} }\right)\left({f}_{j\downarrow }{f}_{i\uparrow }-{f}_{j\uparrow }{f}_{i\downarrow }\right),$$

(9)

where ${J}_{ij}\equiv J\left(\left\vert {{{{\boldsymbol{R}}}}}_{ij}\right\vert \right)$ is shown in Eq. (3) with R_ij  ≡  r_i  −  r_j. Here, the first and second lines of Eq. (9) correspond to the spin-triplet (S  =  1) channel, while the last line corresponds to the single-singlet ((S =  0)) channel. Derivations of H_J above are provided in Supplementary Note 2. Since our model is on a two-dimensional lattice and the dominant contribution J_H  <  0, Eq. (9) can be simplified as

$${H}_{J}\approx -\left\vert {J}_{H}\right\vert {\sum}_{\left\langle i,j\right\rangle }\left({f}_{i\uparrow }^{{\dagger} }{f}_{j\uparrow }^{{\dagger} }{f}_{j\uparrow }{f}_{i\uparrow }+{f}_{i\downarrow }^{{\dagger} }{f}_{j\downarrow }^{{\dagger} }{f}_{j\downarrow }{f}_{i\downarrow }\right).$$

(10)

Eq. (10) is the origin of the equal-spin RVB pairing states, $\left\vert \uparrow \uparrow \right\rangle$ and $\left\vert \downarrow \downarrow \right\rangle$, we proposed in this paper.

Mean-field theory

With the choice of ϕ_A(k) and ϕ_B(k) shown in the main text, the mean-field Kondo-Heisenberg model H_MF can be expressed as a summation of two decoupled 4  ×  4 matrices: ${H}_{MF}={\sum }_{{{{\boldsymbol{k}}}},I}{\phi }_{I}^{{{\dagger}} }({{{\boldsymbol{k}}}}){{{{\mathscr{H}}}}}_{I}({{{\boldsymbol{k}}}}){\phi }_{I}({{{\boldsymbol{k}}}})+{{{\mathscr{C}}}}$ with

$$\begin{array}{rcl}{{{{\mathscr{H}}}}}_{A}({{{\boldsymbol{k}}}})&=&\left(\begin{array}{cccc}\frac{{\varepsilon }_{{{{\boldsymbol{k}}}}}}{2}&0&\frac{{V}_{2{{{\boldsymbol{k}}}}}^{* }}{2}&0\\ 0&-\frac{{\varepsilon }_{{{{\boldsymbol{k}}}}}}{2}&0&\frac{{V}_{2{{{\boldsymbol{k}}}}}}{2}\\ \frac{{V}_{2{{{\boldsymbol{k}}}}}}{2}&0&\frac{\lambda }{2}&{\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow }\\ 0&\frac{{V}_{2{{{\boldsymbol{k}}}}}^{* }}{2}&{\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow * }&-\frac{\lambda }{2}\end{array}\right),\\ {{{{\mathscr{H}}}}}_{B}({{{\boldsymbol{k}}}})&=&\left(\begin{array}{cccc}\frac{{\varepsilon }_{{{{\boldsymbol{k}}}}}}{2}&0&\frac{{V}_{1{{{\boldsymbol{k}}}}}^{* }}{2}&0\\ 0&-\frac{{\varepsilon }_{{{{\boldsymbol{k}}}}}}{2}&0&\frac{{V}_{1{{{\boldsymbol{k}}}}}}{2}\\ \frac{{V}_{1{{{\boldsymbol{k}}}}}}{2}&0&\frac{\lambda }{2}&{\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow }\\ 0&\frac{{V}_{1{{{\boldsymbol{k}}}}}^{* }}{2}&{\Delta }_{{{{\boldsymbol{k}}}}}^{\uparrow * }&-\frac{\lambda }{2}\end{array}\right).\end{array}$$

(11)

Diagonalizing ${{{{\mathscr{H}}}}}_{A/B}$, we obtain eight quasiparticle energy bands,

$${E}_{A(B)}({{{\boldsymbol{k}}}})=\pm \sqrt{{{{{\mathscr{X}}}}}_{A(B)}({{{\boldsymbol{k}}}})\pm \sqrt{{{{{\mathscr{X}}}}}_{A(B)}^{2}({{{\boldsymbol{k}}}})-{{{{\mathscr{Y}}}}}_{A(B)}^{2}({{{\boldsymbol{k}}}})}},$$

(12)

where ${{{{\mathscr{X}}}}}_{A(B)}({{{\boldsymbol{k}}}})=({\varepsilon }_{{{{\boldsymbol{k}}}}}^{2}+{\lambda }^{2}+4{\vert {\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow (\uparrow )}\vert }^{2}+2{\vert {V}_{2(1){{{\boldsymbol{k}}}}}\vert }^{2})/8$ and ${{{{\mathscr{Y}}}}}_{A(B)}({{{\boldsymbol{k}}}})={({\varepsilon }_{{{{\boldsymbol{k}}}}}\lambda -{\vert {V}_{2(1){{{\boldsymbol{k}}}}}\vert }^{2})}^{2}/16+{\varepsilon }_{{{{\boldsymbol{k}}}}}^{2}{\vert {\Delta }_{{{{\boldsymbol{k}}}}}^{\downarrow (\uparrow )}\vert }^{2}/4$.

The mean-field Kondo-Heisenberg Hamiltonian possesses TRS: H_A and H_B constitute the time-reversal partner of each other, i.e. $\hat{\Theta }{H}_{A(B)}{\hat{\Theta }}^{-1}={H}_{B(A)}$ where the time-reversal operator $\hat{\Theta }={\tau }^{0}\otimes {\rho }^{0}\otimes (-i{\sigma }^{y})K$ with σ^y being the y-component Pauli matrix on the spin subspace, τ⁰(ρ⁰) being a 2  ×  2 identity matrix on the particle-hole (orbital) subspace while K being the complex-conjugate operator. The time-reversal operator flips the spin and quasi-momentum of the conduction (c) and local fermion (f) operators: $\left({c}_{{{{\boldsymbol{k}}}}\uparrow },\,{c}_{{{{\boldsymbol{k}}}}\downarrow },\,{f}_{{{{\boldsymbol{k}}}}\uparrow },\,{f}_{{{{\boldsymbol{k}}}}\downarrow }\right){\longrightarrow }^{\hat{\Theta }}\left({c}_{-{{{\boldsymbol{k}}}}\downarrow },\,-{c}_{-{{{\boldsymbol{k}}}}\uparrow },\,{f}_{-{{{\boldsymbol{k}}}}\downarrow },\,-{f}_{-{{{\boldsymbol{k}}}}\uparrow }\right)$. Meanwhile, H_MF respects charge-conjugation (particle-hole) symmetry: $\hat{{{{\mathscr{P}}}}}{{{{\mathscr{H}}}}}_{MF}({{{\boldsymbol{k}}}}){\hat{{{{\mathscr{P}}}}}}^{-1}=-{{{{\mathscr{H}}}}}_{MF}(-{{{\boldsymbol{k}}}})$ where $\hat{{{{\mathscr{P}}}}}\equiv {\tau }^{x}K$ is the particle-hole operator with τ_x being the x-component of the Pauli matrices. Due to the odd-parity $p\pm i{p}^{{\prime} }$ RVB pairing, the parity symmetry is broken in our model.

This $p\pm i{p}^{{\prime} }$ gap structure for the up- and down-spin sectors in Eq. (11) corresponds to the following real-space patterns, ${\Delta }_{t}^{\uparrow }(i,j)$ and ${\Delta }_{t}^{\downarrow }(i,j)$ as defined in Eq. (4):

$${\Delta }_{t}^{\uparrow }(i,j)\to {\Delta }_{t}^{\uparrow }(i,i+\hat{x}) = -{\Delta }_{t}^{\uparrow }(i,i-\hat{x})=-{\Delta }_{t},\\ {\Delta }_{t}^{\uparrow }(i,i+\hat{y}) = -{\Delta }_{t}^{\uparrow }(i,i-\hat{y})=i{\Delta }_{t},$$

(13)

and

$${\Delta }_{t}^{\downarrow }(i,j)\to {\Delta }_{t}^{\downarrow }(i,i+\hat{x})= -{\Delta }_{t}^{\downarrow }(i,i-\hat{x})=-{\Delta }_{t},\\ {\Delta }_{t}^{\downarrow }(i,i+\hat{y})= -{\Delta }_{t}^{\downarrow }(i,i-\hat{y})= -i{\Delta }_{t}.$$

(14)

The explicit form of the zero-temperature saddle-point equations via minimizing the mean-field free energy per site ${{{{\mathscr{F}}}}}_{MF}$ with respect to the mean-field variables q ∈ (λ,  x,  Δ_t, μ), i.e., $\partial {{{{\mathscr{F}}}}}_{MF}/\partial q=-(1+\delta ){\delta }_{q,\mu }$, is shown below:

$$ \frac{1}{{{{{\mathscr{N}}}}}_{s}}\sum\limits_{n{{{\boldsymbol{k}}}}}\frac{\partial {E}_{n{{{\boldsymbol{k}}}}}}{\partial x}+\frac{2x}{{J}_{K}}=0,\\ \frac{1}{{{{{\mathscr{N}}}}}_{s}}\sum\limits_{n{{{\boldsymbol{k}}}}}\frac{\partial {E}_{n{{{\boldsymbol{k}}}}}}{\partial {\Delta }_{t}}+\frac{16{\Delta }_{t}}{| {J}_{H}| }=0,\\ \frac{1}{{{{{\mathscr{N}}}}}_{s}}\sum\limits_{n{{{\boldsymbol{k}}}}}\frac{\partial {E}_{n{{{\boldsymbol{k}}}}}}{\partial \lambda }=0,\\ \frac{1}{{{{{\mathscr{N}}}}}_{s}}\sum\limits_{n{{{\boldsymbol{k}}}}}\frac{\partial {E}_{n{{{\boldsymbol{k}}}}}}{\partial \mu }+\delta =0.$$

(15)

Emergence of superconductivity from heavy-fermion bands

In this section, we would like to illustrate more clearly how the superconductivity develops out of the heavy-fermion bands by opening a superconducting gap at the Fermi energy through the BCS mechanism when the temperature gets below T_c (see Fig. 8 and Ref. ⁷⁴). To start with, we consider the temperature T < T_onset at which the hybridized bands are developed. We define the two hybridized bands as α_kσ and β_kσ, both of which are linear combinations of the original c and f fermions. We further assume that α_kσ (flatter) corresponds to the band close to the Fermi energy while β_kσ (more dispersive) is remote from the Fermi energy. As the temperature is decreased below the superconductivity transition temperature T_c, the Cooper pairs are formed out of the α- and β-bands. The superconductivity can be naturally described through the BCS process by including terms like ${\alpha }_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{\alpha }_{-{{{\boldsymbol{k}}}}{\sigma }^{{\prime} }}^{{{\dagger}} }$ and ${\beta }_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{\beta }_{-{{{\boldsymbol{k}}}}{\sigma }^{{\prime} }}^{{{\dagger}} }$ (neglecting the cross terms) in the model. If we consider low doping where the Fermi level intercepts with the flat α-band, the superconducting gap is expected to be opened on the Fermi level. Therefore, the pairing term on the remote β-band, ${\beta }_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{\beta }_{-{{{\boldsymbol{k}}}}{\sigma }^{{\prime} }}^{{{\dagger}} }$, can be neglected.