# Topological Kondo Superconductors

Yung-Yeh Chang  Physics Division, National Center for Theoretical Sciences,
Hsinchu 30013, Taiwan Republic of China  Department of Electrophysics,
National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan Republic of
China  Khoe Van Nguyen  Department of Electrophysics, National Yang Ming Chiao
Tung University, Hsinchu 30010, Taiwan Republic of China  Kuang-Lung Chen
Department of Electrophysics, National Yang Ming Chiao Tung University,
Hsinchu 30010, Taiwan Republic of China  Yen-Wen Lu  Department of Physics and
Astronomy, University of California, Riverside, California 92511, U.S.A.
Chung-Yu Mou  Department of Physics, National Tsing Hua University, Hsinchu
30043, Taiwan Republic of China  Chung-Hou Chung  Department of
Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010,
Taiwan Republic of China

###### Abstract

Spin-triplet $p$-wave superconductors are promising candidates for topological
superconductors. They have been proposed in various heterostructures where a
material with strong spin-orbit interaction is coupled to a conventional
$s$-wave superconductor by proximity effect. However, topological
superconductors existing in nature and driven purely by strong electron
correlations are yet to be studied. Here we propose a realization of such a
system in a class of Kondo lattice materials in the absence of spin-orbit
coupling and proximity effect. Therein, the odd-parity Kondo hybridization
mediates ferromagnetic spin-spin coupling and leads to spin-triplet resonant-
valence-bond ($t$-RVB) pairing between local moments. Spin-triplet $p\pm
ip^{\prime}$-wave topological superconductivity is reached when Kondo effect
co-exists with $t$-RVB. We identify the topological nature by the non-trivial
topological invariant and the Majorana fermions at edges. Our results offer a
comprehensive understanding of experimental observations on UTe2, a U-based
ferromagnetic heavy-electron superconductor.

##  I Introduction

Searching for topological superconductors (TSc) and the corresponding self-
dual charge neutral Majorana zero modes associated with their excitations at
edges has become one of the central problem in condensed matter physics [1,
2]. Theoretical proposals and experimental realizations of TSc are mostly
heterostructure combining strong spin-orbit coupled materials and conventional
superconductors by proximity effect [3, 4, 5]. The emergence of the
topological edge states in such systems can be explained in terms of the
single-particle band structure without considering many-body electron
correlations. Recently, the search for topological phases of matter has
focused on a more intriguing class of materials that exist in nature. Their
topological properties are driven by strong electron correlations instead of
the proximity effect. Kondo effect, describing the screening of a local spin
moment by conduction electrons, is a well-known strong correlation between
electrons 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], where the
topological properties are driven by either the odd-parity Kondo hybridization
or by the Kondo hybridization with strong spin-orbit coupling.

Spin-triplet $p$-wave superconductors are known to be the prime candidates for
TSc. However, they are scarce in nature. While it is still debatable for
SrRu2O4 [10, 11, 12], more convincing evidence for $p$-wave triplet
superconductivity was observed in noncentrosymmetric superconductor BiPd from
phase-sensitive measurement [13]. More recently, signatures of triplet chiral
$p$-wave superconductivity were observed in heavy-electron Kondo lattice
compound UTe2 at the edge of ferromagnetism, possibly marking the first
example of topological superconductor induced by the strongly correlated Kondo
effect [14, 15, 16, 17].

Motivated by these discoveries, in this paper, we propose a distinct class of
triplet $p$-wave superconductors in the absence of spin-orbit coupling or
proximity effect/heterostructure [18] in a two-dimensional Kondo lattice model
driven by odd-parity Kondo hybridization. We start from the Anderson lattice
model (ALM) with odd-parity hybridization, which occurs between $d$\- and
$f$-orbital electrons in various heavy-fermion compounds [6, 7, 8]. Via the
Schrieffer-Wolff transformation [19, 20], we derive an effective Kondo lattice
model with odd-parity hybridization. Furthermore, by integrating out the
conduction electron degrees of freedom, an effective ferromagnetic RKKY
interaction is generated. We explore the mean-field phase diagram of this
ferromagnetic Kondo-Heisenberg model. In the fermionic mean-field approach,
the ferromagnetic RKKY coupling describes the $p$-wave ($S_{z}=\pm 1$) $t$-RVB
spin-liquid state. A time-reversal invariant topological superconducting phase
is reached when the Kondo effect co-exists with the $p$-wave $t$-RVB order
parameter. The topological nature of this superconducting phase is manifested
by the non-trivial $Z_{2}$ topological Chern number of the bulk band and by
the existence of helical Majorana zero modes at the edges of a finite-sized
ribbon. Our results offer a qualitative and some quantitative understanding of
the spin-triplet superconductivity recently observed in UTe2 (see
Discussions).

##  II Model

###  II.1 Anderson lattice model with odd-parity hybridization

We start with the odd-parity Anderson lattice model (ALM) on a two-dimensional
(2D) square lattice, which has been shown to exhibit topologically non-trivial
states [6, 7, 8]:

$\displaystyle H_{PAM}=H_{c}+H_{f}+H_{cf},$ (1)

where
$H_{c}=\sum_{\mathbold{k},\sigma=\uparrow,\downarrow}\varepsilon_{\mathbold}{k}c^{\dagger}_{\mathbold{k}\sigma}c_{\mathbold{k}\sigma}$
describes the hopping of electrons in the $d$ orbits with orbital angular
momentum $l=2$ and dispersion $\varepsilon_{\mathbold}{k}=-2t(\cos k_{x}+\cos
k_{y})-\mu$. The Hamiltonian $H_{f}$ of the more localized electron in the $f$
orbits with orbital angular momentum $l=3$ is given by

$\displaystyle
H_{f}=\sum_{i,\sigma}\left[\varepsilon_{f}f^{\dagger}_{i\sigma}f_{i\sigma}+\frac{U}{2}n_{i\sigma}^{f}n_{i,-\sigma}^{f}\right],$
(2)

where $\varepsilon_{f}$ denote the energy level of the $f$-electron, and $U$
is the repulsive on-site Coulomb potential (the Hubbard-$U$ term).
Hybridization of the local and conduction electrons is described by

$\displaystyle H_{cf}=\sum_{\langle
i,j\rangle}\sum_{\sigma,\sigma^{\prime}=\uparrow\downarrow}V_{ij}^{\sigma\sigma^{\prime}}c^{\dagger}_{i\sigma}f_{j\sigma^{\prime}}+H.c..$
(3)

To conserve the parity symmetry of hybridization between electrons with their
angular momentum quantum numbers differing by one,
$V_{ij}^{\sigma\sigma^{\prime}}$ have to be odd under parity transformation.
This restriction results in the hybridization having to depend on sites and
spins [6, 7, 8]:

$\displaystyle V_{ij}^{\sigma\sigma^{\prime}}\equiv
V_{\hat{\alpha}}^{\sigma\sigma^{\prime}}=iV\nu_{\hat{\alpha}}\sigma_{{\alpha}}^{\sigma\sigma^{\prime}},$
(4)

distinct from the well-known onsite and spin-conserving Anderson
hybridization. In Eq. (4), $\nu_{ij}$ satisfies
$\nu_{ij}\equiv\nu_{\hat{\alpha}}=-\nu_{ji}$ with $\hat{\alpha}\equiv i-j\in\
\hat{x},\hat{y}\,(\alpha\in x,y)$ on a 2D square lattice, and
$\sigma_{\alpha}$ denotes the Pauli matrix of the $\alpha$ component.

###  II.2 The effective odd-parity ferromagnetic Kondo lattice model

Figure 1: The effective RKKY coupling $J_{H}$ (normalized with $J_{K}^{2}$) as
a function of $R/a$ for different chemical potentials $\mu$. $J_{H}$ is
computed by Eq. (7) with $\mathbold{R}_{ij}\parallel(1,1)$ and $a=1$ being
chosen.

In this paper, we focus on the competition of the Kondo and the magnetic
interaction among impurities–the Doniach scenario [21]. We, therefore, derive
the effective Kondo-Heisenberg lattice Hamiltonian from ALM in the Kondo limit
where the vacant and doubly-occupied states are projected out from the entire
Hilbert space, namely $1=\sum_{\sigma}f^{\dagger}_{i\sigma}f_{i\sigma}$. The
low-energy effective Kondo term from the odd-parity ALM of Eq. (1) can be
derived by applying the Schrieffer-Wolff transformation (SWT) [19, 20, 22],
yielding

$\displaystyle H_{K}=(-J_{K})$
$\displaystyle\sum_{i}\sum_{\sigma\sigma^{\prime}}\sum_{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\sum_{\alpha,\alpha^{\prime}}\left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}c_{i+\hat{\alpha},\sigma}^{\dagger}f_{i\sigma^{\prime}}\right)$
$\displaystyle\times\left(i\nu_{\hat{\alpha}^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}f_{i\sigma^{\prime\prime}}^{\dagger}c_{i-\hat{\alpha}^{\prime},\sigma^{\prime\prime\prime}}\right)$
(5)

with
$J_{K}=\frac{V^{2}}{U+\varepsilon_{f}-\varepsilon_{F}}+\frac{V^{2}}{\varepsilon_{F}-\varepsilon_{f}}>0$
(see Appendix A). The Kondo-like term of Eq. (5) describes the screening of an
impurity by its neighboring conduction electrons, distinct from the
conventional (on-site) Kondo term.

Here, we go beyond the topological Kondo insulating phase by further deriving
the magnetic RKKY interaction among the local $f$-fermions. By perturbatively
expanding the Kondo term to second order [23, 24, 22], we obtain the effective
RKKY-like interaction between the local $f$ fermions $f_{i\sigma}$,

$\displaystyle H_{J}=$
$\displaystyle\sum_{i,j}\sum_{\sigma,\sigma^{\prime}}J_{ij}f_{i\sigma}^{\dagger}f_{j\sigma^{\prime}}^{\dagger}f_{j\sigma}f_{i\sigma^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\left\langle
i,j\right\rangle}J_{ij}\left(f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}f_{j\uparrow}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}f_{j\downarrow}f_{i\downarrow}\right)$
$\displaystyle+\sum_{\left\langle
i,j\right\rangle}\frac{J_{ij}}{2}\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}+f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{j\downarrow}f_{i\uparrow}+f_{j\uparrow}f_{i\downarrow}\right)$
$\displaystyle-\sum_{\left\langle
i,j\right\rangle}\frac{J_{ij}}{2}\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}-f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{j\downarrow}f_{i\uparrow}-f_{j\uparrow}f_{i\downarrow}\right),$
(6)

where

$\displaystyle J_{ij}\equiv J_{H}($ $\displaystyle
R)=\frac{16J_{K}^{2}}{\mathcal{N}_{s}^{2}}\sum_{\varepsilon_{\mathbold{k}}<\mu}\sum_{\varepsilon_{\mathbold{k}^{\prime\prime}}>\mu}\frac{e^{i\left(\mathbold{k}-\mathbold{k}^{\prime\prime}\right)\cdot\mathbold{R}_{ij}}}{\varepsilon_{\mathbold{k}}-\varepsilon_{\mathbold{k}^{\prime\prime}}}$
$\displaystyle\times\left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)\left(\sin^{2}k_{x}^{\prime\prime}+\sin^{2}k_{y}^{\prime\prime}\right)$
(7)

denotes the effective coupling of the spinons of sites $i$ and $j$ with
$R\equiv|\mathbold{R}_{ij}|\equiv|\mathbold{r}_{i}-\mathbold{r}_{j}|$. The
$H_{J}$ term of Eq. (6) can be re-expressed as a linear combination of a
spinon pair wave function with total spin $S=0$ (spin-singlet) and $S=1$
(spin-triplet). Note that the associated effective spinon coupling of the
spin-triplet channel is opposite to that of the spin-singlet. When $H_{J}$ is
expressed in terms of fermion pair with different spins, Eq. (6) is
reminiscent of the conventional Heisenberg interaction
$\mathbold{S}_{i}\cdot\mathbold{S}_{j}=-\frac{1}{2}\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}-f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{i\downarrow}f_{j\uparrow}-f_{i\uparrow}f_{j\downarrow}\right)+\frac{1}{4}n_{i}^{f}n_{j}^{f}$,
except for the difference in the constant coefficients of the pair operators.
As expected, the RKKY coupling $J_{ij}$ in Eq. (7) shows an oscillatory
behavior in $R$, accompanied by a decrease in its magnitude with increasing
$R$, similar to the behavior of the conventional RKKY coupling. Due to the
rapid attenuation of $J_{ij}$, we only consider the dominated nearest-neighbor
interaction and assume $J_{ij}$ to be spatially homogeneous, i.e. $J_{ij}\to
J(R=a)\equiv J_{H}$. Furthermore, when $R=a$, we find the effective RKKY
coupling is attractive (or of the ferromagnetic type), i.e., $J_{H}<0$ (see
Fig. 1), which energetically favors the spin-triplet pairing of spinons. On
the other hand, the effective RKKY coupling in the spin-singlet channel shows
repulsive interaction and can be neglected here since it is not energetically
favorable. Lastly, on a two-dimensional lattice, the triplet spin state
$|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle$ does not exist since
the corresponding structure factor is proportional to $k_{z}$, and $k_{z}=0$
is fixed here. Therefore, based on the above arguments, only the equal-spin
states, $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$,
survive, and the $H_{J}$ term is reduced to

$\displaystyle H_{J}\approx-\left|J_{H}\right|\sum_{\left\langle
i,j\right\rangle}\left(f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}f_{j\uparrow}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}f_{j\downarrow}f_{i\downarrow}\right).$
(8)

Combining $H_{K}$ and $H_{J}$ of Eqs. (5), (6) and (8), the effective Kondo-
Heisenberg lattice model with odd-parity Kondo hybridization reads
$H_{FKH}=H_{0}+H_{\lambda}+H_{K}+H_{J}$. Here,
$H_{\lambda}=-\sum_{i}i\lambda_{i}\left[\sum_{\sigma}(f^{\dagger}_{i\sigma}f_{i\sigma})-1\right]$
enforces the singly occupied local $f$-spinons with $\lambda_{i}$ being the
Lagrange multiplier. 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, 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 $J_{K}/J_{H}$ [25, 26]. In experiments, varying the non-
thermal parameter can be expected to follow a certain trajectory of
$J_{K}/J_{H}$ in the phase diagram.

##  III Mean-field treatment of the effective Kondo-Heisenberg-like model

Figure 2: The zero-temperature mean-field solutions of $t$-RVB order parameter
$\Delta_{t}$ (brown) and the Kondo correlation $x$ (black) as a function of
$J_{H}$. We fix $J_{K}=0.3$ and doping of the conduction band $\delta=-0.3$
($30$ percent hole doping). Without loss of generality, we set $t=1$. This
plot reveals a (co-existing) superconducting ground state with $x\neq
0,\,\Delta_{t}\neq 0$ for $0<J_{H}\lesssim 2.5$ and a pure $t$-RVB phase where
$x=0,\,\Delta_{t}\neq 0$ when $J_{H}\gtrsim 2.52$. A pure Kondo phase ($x\neq
0,\,\Delta_{t}=0$) only exists at $J_{H}=0$.

We now employ a mean-field analysis on the above effective Kondo-Heisenberg-
like Hamiltonian with an effective ferromagnetic RKKY interaction and odd-
parity Kondo hybridization.

Via performing Hubbard-Stratonovich transformation, $H_{K}$ and $H_{J}$ of
Eqs. (5) and (6) can be factorized as

$\displaystyle
H_{K}\to\sum_{i,\alpha}\sum_{\sigma\sigma^{\prime}}\left[\chi^{\dagger}_{i}\left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}}\right)+H.c.\right]$
$\displaystyle\qquad\qquad+\sum_{i}\frac{|\chi_{i}|^{2}}{J_{K}},$
$\displaystyle H_{J}\to\sum_{\left\langle
i,j\right\rangle}\left[\Delta^{\uparrow}_{t}(i,j)f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}+\Delta^{\downarrow}_{t}(i,j)f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}+H.c.\right]$
$\displaystyle\qquad\qquad+\sum_{\langle
i,j\rangle}\frac{\left|\Delta^{\uparrow}_{t}(i,j)\right|^{2}+\left|\Delta^{\downarrow}_{t}(i,j)\right|^{2}}{J_{H}}$
(9)

where the mean-field values of the bosonic Hubbard-Stratonovich fields,
$\chi_{i}$ and $\Delta^{\sigma}_{t}(i,j)$ ($\sigma=\uparrow,\downarrow$),
represent the order parameters of the Kondo correlation and the $S_{z}=\pm 1$
spin-triplet RVB bonds between two adjacent up/down spins, respectively.

To describe the Kondo-screened Fermi-liquid state, we allow the $\chi_{i}$
field to acquire uniformly Bose condensation over the real space; hence,
$\chi_{i}$ can be expressed as $\chi_{i}\to x+\hat{\chi}_{i}$ with
$x=(-J_{K}/\mathcal{N}_{s})\sum_{i\sigma\sigma^{\prime}\alpha}\langle
i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}}\rangle$
being the Bose-condensed stiffness of $\chi_{i}$ while $\hat{\chi}_{i}$
represents its fluctuations. The mean-field order parameter of the $t$RVB is
given by $\Delta_{t}^{\sigma}=(-J_{H}/4\mathcal{N}_{s})\sum_{\langle
i,j\rangle}\langle f_{j\sigma}f_{i\sigma}\rangle$. Since the ferromagnetic
coupling is expected to favor spin-triplet $p$-wave pairing similar to
superfluid helium-3 [27], we restrict ourselves to the $p$-wave pairing, i.e.,
$\Delta_{t}^{\sigma}(i,j)$ here is taken the $p$-wave form, see Eqs. (11) and
(12) below. We further fix the Lagrange multiplier at the mean-field level via
$i\lambda_{i}\to\lambda$ and neglect the fluctuations of $\lambda_{i}$,
$\chi_{i}$, and $\Delta_{t}^{\sigma}$, leading to the following mean-field
Kondo-Heisenberg-like Hamiltonian:

$\displaystyle H_{MF}$
$\displaystyle=\sum_{\mathbold{k},\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}\sigma}+\sum_{\mathbold{k}\sigma}\lambda
f_{\mathbold{k}\sigma}^{\dagger}f_{\mathbold{k}\sigma}$
$\displaystyle+\sum_{\mathbold{k}}\left[V_{1\mathbold{k}}f_{\mathbold{k}\uparrow}^{\ast}c_{\mathbold{k}\downarrow}+V_{2\mathbold{k}}f_{\mathbold{k}\downarrow}^{\ast}c_{\mathbold{k}\uparrow}+H.c.\right]$
$\displaystyle+\sum_{\mathbold{k}}\left[\Delta_{\mathbold{k}}^{\uparrow}f_{\mathbold{k}\uparrow}^{\dagger}f_{-\mathbold{k}\uparrow}^{\dagger}+\Delta_{\mathbold{k}}^{\downarrow}f_{\mathbold{k}\downarrow}^{\dagger}f_{-\mathbold{k}\downarrow}^{\dagger}+H.c.\right]$
$\displaystyle+\frac{8\mathcal{N}_{s}\Delta_{t}^{2}}{J_{H}}+\frac{\mathcal{N}_{s}x^{2}}{J_{K}}-\mathcal{N}_{s}\lambda,$
(10)

where $V_{1\mathbold{k}}=2x\left(\sin k_{x}-i\sin k_{y}\right)$ and
$V_{2\mathbold{k}}=2x\left(\sin k_{x}+i\sin k_{y}\right)$. The Fourier
transformation for the second-quantized operator is defined as
$\psi_{i\sigma}=\frac{1}{\sqrt{\mathcal{N}_{s}}}\sum_{\mathbold{k}}e^{-i\mathbold{k}\cdot\mathbold{r}_{i}}\psi_{\mathbold{k}\sigma}$.
Note that the mean-field Kondo term of Eq. (10) is reminiscent of the
topological Kondo insulator shown in Ref. [28]. In Eq. (10),
$\Delta_{t}^{\sigma}(\mathbold{k})$ represents the gap structure of the spin-
triplet $p$-wave RVB pairing in the momentum space for the spin-$\sigma$
sector, defined as $\Delta_{\mathbold{k}}^{\uparrow}=\Delta_{t}\left(-\sin
k_{y}-i\sin k_{x}\right)$ and
$\quad\Delta_{\mathbold{k}}^{\downarrow}=\Delta_{t}\left(\sin k_{y}-i\sin
k_{x}\right)$ with $\Delta_{t}$ being denoted the mean-field pairing potential
(see Appendix, Section II). This momentum-dependent gap structure for the up-
and down-spin sectors correspond to the following real-space patterns of
$\Delta^{\uparrow}_{t}(i,j)$ and $\Delta^{\downarrow}_{t}(i,j)$ of Eq. (9):

$\displaystyle\Delta_{t}^{\uparrow}(i,j)\to\Delta^{\uparrow}_{t}(i,i+\hat{x})=-\Delta^{\uparrow}_{t}(i,i-\hat{x})=-\Delta_{t},$
$\displaystyle\qquad\qquad\quad\Delta^{\uparrow}_{t}(i,i+\hat{y})=-\Delta^{\uparrow}_{t}(i,i-\hat{y})=i\Delta_{t},$
(11)

and

$\displaystyle\Delta_{t}^{\downarrow}(i,j)\to$
$\displaystyle\Delta^{\downarrow}_{t}(i,i+\hat{x})=-\Delta^{\downarrow}_{t}(i,i-\hat{x})=-\Delta_{t},$
$\displaystyle\Delta^{\downarrow}_{t}(i,i+\hat{y})=-\Delta^{\downarrow}_{t}(i,i-\hat{y})=-i\Delta_{t}.$
(12)

Choosing
$\Psi_{\mathbold{k}}=\left(\phi_{A\mathbold{k}},\phi_{B\mathbold{k}}\right)^{T}$
with the Nambu spinors defined by
$\phi_{A\mathbold{k}}=\left(c_{\mathbold{k}\uparrow},\,\,c_{-\mathbold{k}\uparrow}^{\dagger},\,\,f_{\mathbold{k}\downarrow},\,\,f_{-\mathbold{k}\downarrow}^{\dagger}\right)^{T}$
and
$\phi_{B\mathbold{k}}=\left(c_{\mathbold{k}\downarrow},\,\,c_{-\mathbold{k}\downarrow}^{\dagger},\,\,f_{\mathbold{k}\uparrow},\,\,f_{-\mathbold{k}\uparrow}^{\dagger}\right)^{T}$,
the mean-field Hamiltonian
$H_{MF}=\sum_{\mathbold{k}}\Psi_{\mathbold{k}}^{\dagger}\mathcal{H}_{\mathbold{k}}\Psi_{\mathbold{k}}+\mathcal{C}$
can be expressed as a summation of two decoupled $4\times 4$ matrices as
follows

$\displaystyle H_{MF}$ $\displaystyle=H_{A}+H_{B}+\mathcal{C},$ $\displaystyle
H_{A(B)}$
$\displaystyle=\sum_{\mathbold{k}}\phi_{A(B)\mathbold{k}}^{\dagger}\mathcal{H}_{\mathbold{k}}^{A(B)}\phi_{A(B)\mathbold{k}}$
(13)

with
$\mathcal{C}\equiv\sum_{\mathbold}{k}\varepsilon_{\mathbold}{k}+\frac{8\mathcal{N}_{s}\Delta_{t}^{2}}{J_{H}}+\frac{\mathcal{N}_{s}x^{2}}{J_{K}}$,
and

$\displaystyle\mathcal{H}_{\mathbold{k}}^{A}$
$\displaystyle=\begin{pmatrix}\frac{\varepsilon_{\mathbold{k}}}{2}&0&\frac{V_{2\mathbold{k}}^{\ast}}{2}&0\\\
0&-\frac{\varepsilon_{\mathbold{k}}}{2}&0&\frac{V_{2\mathbold{k}}}{2}\\\
\frac{V_{2\mathbold{k}}}{2}&0&\frac{\lambda}{2}&\Delta_{\mathbold{k}}^{\downarrow}\\\
0&\frac{V_{2\mathbold{k}}^{\ast}}{2}&\Delta_{\mathbold{k}}^{\downarrow\ast}&-\frac{\lambda}{2}\end{pmatrix},\,\,$
(14) $\displaystyle\mathcal{H}_{\mathbold{k}}^{B}$
$\displaystyle=\begin{pmatrix}\frac{\varepsilon_{\mathbold{k}}}{2}&0&\frac{V_{1\mathbold{k}}^{\ast}}{2}&0\\\
0&-\frac{\varepsilon_{\mathbold{k}}}{2}&0&\frac{V_{1\mathbold{k}}}{2}\\\
\frac{V_{1\mathbold{k}}}{2}&0&\frac{\lambda}{2}&\Delta_{\mathbold{k}}^{\uparrow}\\\
0&\frac{V_{1\mathbold{k}}^{\ast}}{2}&\Delta_{\mathbold{k}}^{\uparrow\ast}&-\frac{\lambda}{2}.\end{pmatrix}$
(15)

The Hamiltonian Eq. (13) possesses time-reversal symmetry: $\mathcal{H}_{A}$
and $\mathcal{H}_{B}$ constitute the time-reversal partner of each other, i.e.
$\Theta\mathcal{H}_{A(B)}\Theta^{-1}=\mathcal{H}_{B(A)}$ where the time-
reversal operator $\Theta=\rho^{0}\otimes(-i\sigma^{y})K$ with $\sigma^{y}$
being the $y$-component Pauli matrix on the spin subspace, $\rho^{0}$ being a
$2\times 2$ identity matrix on the orbital subspace while $K$ being the
complex-conjugate operator. Under time-reversal transformation, the spin and
quasi-momentum of conduction ($c$) and pseudofermion ($f$) operators are
flipped:
$\left(c_{\bm{k}\uparrow},\,c_{\bm{k}\downarrow},\,f_{\bm{k}\uparrow},\,f_{\bm{k}\downarrow}\right)\stackrel{{\scriptstyle\Theta}}{{\longrightarrow}}\left(c_{-\bm{k}\downarrow},\,-c_{-\bm{k}\uparrow},\,f_{-\bm{k}\downarrow},\,-f_{-\bm{k}\uparrow}\right)$.
Meanwhile, our Hamiltonian respects charge-conjugation (particle-hole)
symmetry:
$\mathcal{P}\mathcal{H}_{\bm{k}}\mathcal{P}^{-1}=-\mathcal{H}_{-\bm{k}}$ where
$\mathcal{P}\equiv\tau^{x}K$ is the particle-hole operator with $\tau_{x}$
being the $x$-component of the Pauli matrices on the particle-hole basis. Due
to the odd-parity $p\pm ip^{\prime}$ RVB pairing of our model, the parity
symmetry is broken here. Thus, our model Eq. (13) belongs to the DIII class of
topological symmetry [29].

##  IV Results

###  IV.1 Mean-field phase diagram

The mean-field ground states are determined by minimizing the mean-field free
energy per site
$\mathcal{F}_{MF}=\frac{\mathcal{C}}{\mathcal{N}_{s}}-\frac{k_{B}T}{\mathcal{N}_{s}}\sum_{n\bm{k}}\ln\left[1+\exp\left(-\frac{E_{n\bm{k}}}{k_{B}T}\right)\right]$
with respect to the mean-field variables $q=(\lambda,\,x,\,\Delta_{t})$, i.e.
$\partial\mathcal{F}_{MF}/\partial q_{i}=0$. Here, $E_{n\mathbold{k}}<0$ is
the $n$-th band of $\mathcal{H}_{\mathbold{k}}$. The chemical potential $\mu$
is determined by the relation
$\partial\mathcal{F}_{MF}/\partial\mu=-(1+\delta)$ with $\delta$ being the
chemical doping of the $c$-electrons for which $\delta\lesseqqgtr 0$ is for
$p/\text{un}/n-$ doped (half-filling corresponds to $\delta=0$). This leads to
the following saddle-point equations at zero temperature,

$\displaystyle\frac{1}{\mathcal{N}_{s}}\sum_{n\bm{k}}\frac{\partial
E_{n\bm{k}}}{\partial x}+\frac{2x}{J_{K}}=0,$
$\displaystyle\frac{1}{\mathcal{N}_{s}}\sum_{n\bm{k}}\frac{\partial
E_{n\bm{k}}}{\partial\Delta_{t}}+\frac{16\Delta_{t}}{J_{H}}=0,$
$\displaystyle\frac{1}{\mathcal{N}_{s}}\sum_{n\bm{k}}\frac{\partial
E_{n\bm{k}}}{\partial\lambda}=0,$
$\displaystyle\frac{1}{\mathcal{N}_{s}}\sum_{n\bm{k}}\frac{\partial
E_{n\bm{k}}}{\partial\mu}+\delta=0.$ (16)

Figure 3: Figures (a) (red curves) and (b) show the bulk energy spectrum of
the co-existing superconducting state near the Fermi level $\mu$. The Fermi
level locates at $E(\bm{k})=0$. The coupling constants are $J_{K}=0.3$ and
$J_{H}=1.0$. Inset of (a) displays the First Brillouin zone of a square
lattice with indications of high-symmetry points $\Gamma,\,X,\,M$.

The ground-state phase diagram (Fig. 2) of our model is obtained by solving
the saddle-point equations self-consistently. The phase diagram contains three
distinct mean-field phases: a pure Kondo phase is found at $J_{H}=0$ where
$x\neq 0$, $\Delta_{t}=0$. At the opposite limit where the RKKY interaction
dominates, the ground state shows short-range magnetic correlation with
$p$-wave spin-triplet RVB pairing ($\Delta_{t}\neq 0,\,x=0$). 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 $x\neq 0$ and
$\Delta_{t}\neq 0$, which can be explained via the mechanism of Kondo-
stabilized spin liquid [30, 26]. The development of superconductivity in this
co-existing phase requires higher-order processes involving both the Kondo and
$t$-RVB terms: the mean-field $t$-RVB pairings of the local $f$ fermions
provide preformed Cooper pairs. When the Kondo hybridization field $\chi$ gets
Bose-condensed ($x\neq 0$), the local fermions delocalize into the conduction
band and make the preformed $t$-RVB Cooper pairs superconduct [31]. These
processes can be described by the effective mean-field Hamiltonian
$H_{sc}=\sum_{\bm{k}}\left(\bar{\Delta}_{\bm{k}}^{\downarrow*}c_{-\bm{k}\downarrow}c_{\bm{k}\downarrow}+\bar{\Delta}_{\bm{k}}^{\uparrow*}c_{-\bm{k}\uparrow}c_{\bm{k}\uparrow}+H.c.\right)$,
where the effective gap functions take the form
$\bar{\Delta}_{\bm{k}}^{\downarrow*}=V_{1\bm{k}}V_{1,-\bm{k}}\Delta_{\bm{k}}^{\uparrow*}\sim
x^{2}\Delta_{t}(\sin^{2}k_{x}+\sin^{2}k_{y})(\sin k_{x}-i\sin k_{y})$ and
$\bar{\Delta}_{\bm{k}}^{\uparrow*}=V_{2\bm{k}}V_{2,-\bm{k}}\Delta_{\bm{k}}^{\downarrow*}\sim
x^{2}\Delta_{t}(\sin^{2}k_{x}+\sin^{2}k_{y})(\sin k_{x}+i\sin k_{y})$ with the
size of the superconducting gap being proportional to $x^{2}\Delta_{t}$. The
superconducting gap function $\bar{\Delta}_{\bm{k}}^{\uparrow}$ we obtained
here shows a $f$-wave-like pairing symmetry on a generic anisotropic (non-
circular) 2D Fermi surface. Nevertheless, as we are taking the continuous
limit of the conduction band here, $\bar{\Delta}_{\bm{k}}^{\uparrow}$ can be
expressed as a product of $s$ and $p\pm ip^{\prime}$ pairing orders, i.e.,
$\bar{\Delta}_{\bm{k}}^{\uparrow/\downarrow*}\sim k^{2}(k_{x}\pm ik_{y})$ with
$k^{2}\equiv k_{x}^{2}+k_{y}^{2}$ on a circular Fermi surface but only the
$p\pm ip^{\prime}$ component plays a role here. Note that we find the co-
existing superconducting state persists for an arbitrary small value of
$J_{H}/J_{K}\to 0^{+}$. This is likely due to the overestimation of the co-
existing phase at the mean-field level. Upon including fluctuations of the
Kondo and $t$-RVB order parameters beyond the mean-field level, we expect a
narrower co-existing superconducting phase. A first-order transition similar
to the results found in Refs. [26, 32] is observed at the transition of the
$t$-RVB and the co-existing superconducting phases (see Fig. 2). The bulk band
structure in the co-existing superconducting state is shown in Fig. 3.

###  IV.2 Topological invariance

Figure 4: The left figure displays the electronic band structure of the
coexisting superconductor state for a strip with $N_{y}=81$ described by
$\mathcal{H}_{A}$ at $J_{K}/t=0.3$ and $J_{H}/t=1.0$. Three pairs of edge
states with Dirac spectra are observed near $k_{x}=0$ (the pink curves). The
edge states at zero energy correspond to the Majorana zero modes. Due to the
time-reversal symmetry of the model, the band structure for a strip for
$\mathcal{H}_{B}$ is identical to that of $\mathcal{H}_{A}$. The close-up band
structures near three pairs of edge states (pink curves) on the top, middle
and bottom bounded by the red squares are shown on the right figures.

We now address the topological properties of the coexisting superconducting
state. Since this system is invariant under time-reversal transformation, the
bulk topological properties of the coexisting Kondo-RVB superconducting state
with $p\pm ip^{\prime}$ spin-triplet RVB pairing can be thus characterized by
the $Z_{2}$ Chern number $c_{T}$ (or time-reversal polarization) [33, 34, 35],
given by

$\displaystyle c_{T}=\frac{c_{A}-c_{B}}{2}$ (17)

with $c_{I}$ $(I\in A,B)$ being the Thouless-Kohmoto-Nightingal-den Nijs
(TKNN) number [36] of $H_{I}$, defined as

$\displaystyle
c_{I}=\frac{1}{2\pi}\int_{\bm{k}\in\text{FBZ}}d\bm{S}_{\bm{k}}\cdot\left(\bm{\nabla}_{\bm{k}}\times\bm{\mathcal{A}}^{I}_{\bm{k}}\right).$
(18)

The Berry’s connection $\bm{\mathcal{A}}^{I}_{\bm{k}}$ for $H_{I}$ is given by
$\bm{\mathcal{A}}^{I}_{\bm{k}}\equiv i\sum_{n\in I}\langle
u_{n\bm{k}}^{I}|\bm{\nabla_{\bm{k}}}|u_{n\bm{k}}^{I}\rangle$ with
$|u_{n\bm{k}}^{I}\rangle$ being the normalized Bloch state of the $n$-th
filled band for $H^{I}_{\bm{k}}$. We numerically calculate the TKNN numbers
[37], $c_{A}$ and $c_{B}$, and find that $c_{A}=-c_{B}=1$ in the co-existing
phase, indicating a topologically non-trivial $Z_{2}$ Chern number $c_{T}=1$.
By the bulk-edge correspondence, we expect this co-existing superconducting
state to support a pair of counter-propagating Majorana zero modes at the
edges of a finite-sized strip. Further band structure calculations of our
model on a strip in the following subsection confirm our expectation.

###  IV.3 Edge states of the coexisting Kondo-RVB spin-triplet $p\pm
ip^{\prime}$-wave superconducting state

Figure 5: Figures (a) and (d) show the Bogoliubov excitation spectra of
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively, near the chemical
potential on a nano-strip with $N_{y}=81$ chains. Figures (b), (c) and (e),
(f) demonstrate the probability density of the Majorana edge state wave
functions of $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ as a function of atom
position $y_{i}$, $\left|\gamma^{\Gamma}_{I,k_{x}}(y_{i})\right|^{2}$ with
$I=A,B$ and $\Gamma=R,L$ (pink curves in (a) and (d)), at a fixed energy
$E\equiv E(k_{x}=\pm 0.03)$. The probability density is described by
$\left|\gamma^{\Gamma}_{I,k_{x}}(y_{i})\right|^{2}=\left(\left|u^{\Gamma}_{I,k_{x}}\right|^{2},\,\left|\bar{u}^{\Gamma}_{I,k_{x}}\right|^{2},\,\left|v^{\Gamma}_{I,k_{x}}\right|^{2},\,\left|\bar{v}^{\Gamma}_{I,k_{x}}\right|^{2}\right)(y_{i})$.
The parameters are $J_{K}/t=0.3$, $J_{H}/t=1.0$, and doping $\delta=-0.3$. The
edge states are of the helical type, as schematically represented in (g).
Figure 6: The finite-energy ($E(k_{x})>0$) Bogoliubov excitation spectra of
(a) $\mathcal{H}_{A}$ (shown on top right in Fig. 5) and (d)
$\mathcal{H}_{B}$. A pair of “helical” edge states is found to exist at finite
energy [pink curve in (a) and (d)], and their probability densities are shown
in (b) and (c), (e) and (f), respectively, at a fixed energy $E(k_{x}=\pm
0.22)$.

We now check whether our model would support helical Majorana zero modes at
the edge of a finite-sized system. We shall examine our model’s band
structures and edge-state wave functions on a finite-sized strip that extends
infinitely along the $x$ direction but contains a finite number of lattice
sites in $y$. The results are shown in Figs. 4 to 6. As shown in Fig. 4,
gapless Dirac spectra of the Bogoliubov excitations around $k_{x}=0$ near zero
energy are observed, exhibiting one of the typical features of topological
edge states. The excitations can be effectively described by the linear-
dispersed Hamiltonian
$\tilde{H}_{I}=\sum_{k_{x}}v_{x}|k_{x}|\left(\gamma^{R\,\dagger}_{I,k_{x}}\gamma^{R}_{I,k_{x}}-\gamma^{L\,\dagger}_{I,k_{x}}\gamma^{L}_{I,k_{x}}\right)$
with

$\displaystyle\gamma_{I,k_{x}}^{\Gamma}=$
$\displaystyle\sum_{y_{i}}\left[u^{\Gamma}_{I,k_{x}}(y_{i})c_{k_{x},y_{i},\uparrow}+\bar{u}^{\Gamma}_{I,k_{x}}(y_{i})c_{-k_{x},y_{i},\uparrow}^{\dagger}\right.$
$\displaystyle\left.+v^{\Gamma}_{I,k_{x}}(y_{i})f_{k_{x},y_{i},\downarrow}+\bar{v}^{\Gamma}_{I,k_{x}}(y_{i})f_{-k_{x},y_{i},\downarrow}^{\dagger}\right]$
(19)

with $u,\bar{u}$ and $v,\bar{v}$ being the coherent factors. In Eq. (19),
$I\in A,B$, $\Gamma\in R,L$, and $\gamma^{R/L}_{A/B,k_{x}}$ represents the
right/left-moving Bogoliubov quasiparticle of $\tilde{H}_{A/B}$. Here, $v_{x}$
in $\tilde{H}_{I}$ denotes the velocity. Due to time-reversal symmetry,
$H_{A}$ is the time-reversal partner of $H_{B}$, and thus their spectra are
identical. The low-energy eigenstates with Dirac spectra near $k_{x}=0$ for
both $H_{A}$ and $H_{B}$ exhibit the typical property of edge states, as their
probability densities accumulate mostly at the edges of strip, as shown in
Fig. 5. Combining the directions of propagation inferred from the velocity
$v_{x}\sim\partial E(k_{x})/\partial k_{x}$, we can classify these edge states
into two groups, each of them constitutes a pair of counter-propagating edge
states (see Fig. 5), revealing the nature of helical Majorana zero modes. The
helical type of the Majorana zero modes is the consequence of time-reversal
symmetry of our model, reminiscent of the well-known Kane-Mele model on a
single-layered graphene [38, 39]. Remarkably, in addition to the Majorana
fermions at zero energy, two pairs of counter-propagating edge-states are
observed at finite energy, see Fig. 6. The two pairs of edge states correspond
to the edge states of the topological Kondo insulator, where the spin-triplet
RVB order parameter is absent ($\Delta_{t}=0$) [8, 7, 6].

Figure 7: Plot of the temperature-dependent mean-field order parameters
$x(T)/t$ and $\Delta_{t}(T)/t$ with $k_{B}=1$, $J_{K}/t=0.3$ and $J_{H}/t=1.0$
fixed. Inset shows the enlarged plot of $\Delta_{t}(T)$. The single-impurity
Kondo temperature occurs at $T_{\text{onset}}/t\approx 0.16$ while the
transition of superconductivity takes places at temperature $T_{c}/t\approx
0.015$.

##  V Discussions and Conclusions

We now discuss the application of our results for heavy-electron
superconductors, particularly the Kondo lattice compound UTe2. Experimental
evidence indicates that this compound does not show long-range magnetic order
and is in the vicinity of the ferromagnetic quantum critical point, exhibiting
both strong ferromagnetic fluctuations, possibly due to magnetic frustrations
induced by sub-leading antiferromagnetic fluctuations [40, 41], and Kondo
screening [17, 14, 42]. 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 weak
$k_{z}$ dependence despite its 3D crystal structure [40]. Superconductivity is
reached at $T_{c}=1.6$K, while the resistivity maximum observed at
$T^{\star}\approx 15\sim 75$ K reveals signature of coherent Kondo scattering
[43, 14], indicating $T^{\star}/T_{c}\approx 10\sim 50$. The superconductivity
can, in general, co-exist and compete with the Kondo effect [17]. When a
magnetic field is applied along the hard-magnetic axis $b$ of UTe2 and before
entering the superconducting phase, a correlated paramagnetic phase is
observed below the temperature at which the magnetic susceptibility shows a
broad maximum [44]. Similar spin-liquid behavior has been observed in the
magnetic susceptibility of another heavy fermion compound CePdAl [45]. This
similarity suggests this correlated paramagnetic phase may feature short-range
magnetic order. Our theoretical framework based on competition and
collaboration between a Kondo-screened and a ferromagnetic $t$-RVB spin-liquid
states on a two-dimensional Kondo lattice is consistent with the above
observations in UTe2. It, therefore, constitutes a promising approach to
account for its exotic phenomena. On the other hand, the chiral in-gap state,
a signature of chiral topological superconductor, has been observed by
scanning tunneling spectroscopy in the superconducting phase of UTe2 [17].
Combining with the ferromagnetic fluctuations that are known to induce spin-
triplet pairing, people believe UTe2 is a promising candidate for the spin-
triplet chiral topological superconductor [17, 14]. Furthermore, the
superconducting phase co-existing with Kondo coherence in this material
strongly suggests the role played by the Kondo effect in this possible
topological superconductor. The topological Kondo superconducting 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) the $d$\- and $f$-orbitals electrons with their angular momentum
quantum number differing by $1$ in the uranium atoms of UTe2 likely give rise
to the odd-parity Kondo effect [8, 7, 6], (ii) the $t$-RVB state in our theory
may be considered as one possible realization of the short-ranged
ferromagnetic fluctuations 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 [14, 46] implies that the
superconducting state of UTe2 may have equal-spin Cooper pairs, and (v) the
effective pairing $\overline{\Delta}_{\bm{k}}^{\sigma}$ formed in the
conduction band mentioned in Section IV.1 shows characteristics of spin-
triplet point-node gap structure [47]. Various characteristic temperature
scales estimated from our mean-field calculations with $J_{H}/t=1.0$ and
$J_{K}/t=0.3$ at finite temperatures agree reasonably well with experimental
observations (see Fig. 7): The superconducting transition temperature $T_{c}$,
theoretically determined from our mean-field analysis
$T_{c}=\text{Min}[T(x=0),T(\Delta_{t}=0)]$, shows $T_{c}\approx 0.015t\approx
2.3\text{\,K}$ by taking estimated values of $t=150\,\text{K}$ and half-
bandwidth $D=1.25t$ [42]. The Kondo coherent scale can be obtained by
$T^{\star}=x^{2}(T=0)/D\approx 17.4\,\text{K}$ [48]. The ratio
$T^{\star}/T_{c}\approx 8$ is in reasonable agreement with experimental
observations. The onset temperature $T_{\text{onset}}$ of Kondo hybridization,
which occurs at $x(T=T_{\text{onset}})=0$, displays $T_{\text{onset}}\approx
0.16t\approx 24\,\text{K}$, within the theoretically estimated range
$10\text{K}<T_{\text{onset}}<100\text{K}$ by DMFT calculation [42]. Meanwhile,
there have been evidences of TRS breaking in UTe2 from the observed two
superconducting transitions and a finite polar Kerr effect at $T<T_{c}$ [49],
likely due to proximity to the ferromagnetic ordered phase. A number of
theoretical attempts were proposed based on these observations [50, 51].
However, the observed single superconducting transition near ambient pressure
and zero field [44, 52, 53] as well as the theoretically proposed unitary
triplet pairing [40] suggest TRS may be preserved in UTe2. Though our results
shown above are obtained in the presence of TRS, the chiral $p$-wave
superconducting state with chiral Majorana zero mode at edges is expected to
occur here once a time-reversal breaking magnetic field is applied [54]. Our
distinct predictions with and without fields serve as theoretical guidance for
future experiments to distinguish the time-reversal breaking from time-
reversal preserving triplet pairing states in UTe2. 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. Our approach and results are distinct from
the spin-triplet non-topological superconducting state recently proposed based
on the Hund’s-Kondo coupling and $S_{z}=0$ $t$-RVB state to account for UTe2
[51].

In conclusion, we propose a first realization of the topological
superconductivity in the Kondo lattice model, a distinct class of topological
superconductors due to purely strong electron correlations without employing
spin-orbit coupling or proximity effect. A topological Kondo superconductor
essentially constitutes of 1) itinerant $c$ and localized $f$ bands with
different orbital quantum numbers, 2) strong Hubbard interaction of the $f$
electrons, 3) odd-parity Kondo hybridization of the $c$ and $f$ bands, and 4)
the attractive exchange interaction of the $f$ electrons with spin-triplet
correlations. Starting from the odd-parity Anderson lattice model, we obtain
the unconventional type of Kondo hybridization and ferromagnetic RKKY-like
interaction via perturbation theory, leading to spin-triplet resonating-
valence-bond (RVB) pairing between $f$-electrons with time-reversal invariant
$p\pm ip^{\prime}$-wave gap symmetry. Via the mean-field approach, we find a
Kondo triplet-RVB coexisting phase in the intermediate range of the Kondo to
RKKY coupling ratio. This phase is shown as a time-reversal invariant
topological superconducting state with a spin-triplet $p\pm ip^{\prime}$-wave
RVB pairing gap. It exhibits non-trivial topology in the bulk band structure,
and supports helical Majorana zero modes at edges. Our prediction in the
presence of a time-reversal breaking field leads to chiral $p$-wave spin-
triplet topological Kondo superconductor. Our results on the superconducting
transition temperature, Kondo coherent scale, and onset temperature of Kondo
hybridization not only qualitatively but also quantitatively agree with the
observations for UTe2. The theoretical framework we propose here opens up the
search for topological superconductors induced by strongly electronic
correlations on the Kondo lattice compounds.

##  VI Acknowledgements

This work is supported by the Ministry of Science and Technology Grants
104-2112-M-009-004-MY3 and 107-2112-M-009-010-MY3, the National Center for
Theoretical Sciences of Taiwan, Republic of China (to C.-H. C.).

##  Appendix A The Schrieffer-Wolff transformation (SWT)

In this section, we provide derivations of the Kondo term via using the SWT.
We first perform the SWT on an odd-parity single-impurity Anderson model where
an impurity at an arbitrary site $i$ hybridizes with the conduction electrons
on the four nearest-neighbor sites of $i$. This result will be successively
generalized to the lattice version.

The single-impurity Anderson model takes the following form

$\displaystyle H=$
$\displaystyle\sum_{\mathbold{k}\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{{\mathbold{k}}\sigma}+\sum_{\sigma}\varepsilon_{f}f_{i\sigma}^{\dagger}f_{i\sigma}+Un_{i\uparrow}^{f}n_{i\downarrow}^{f}$
$\displaystyle+\sum_{\sigma\sigma^{\prime}}\sum_{\alpha=x,y}\left[iV\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}c_{i+\hat{\alpha},\sigma}^{\dagger}f_{i\sigma^{\prime}}+H.c.\right],$
(20)

where $\hat{\alpha}\equiv\pm\hat{x},\pm\hat{y}$ denotes the nearest-neighbor
vectors of a square lattice, and $\nu_{\hat{\alpha}}$ satisfies
$\nu_{\hat{\alpha}}=-\nu_{-\hat{\alpha}}$ and $\nu_{\hat{x}}=\nu_{\hat{y}}=1$.

The SWT aims at projecting out the empty and doubly occupied states to
generate the effective Hamiltonian $H_{eff}$ in the Kondo (singly-occupied)
limit. Following Ref. [20], we first use the states of impurity occupation as
the basis set, $\\{|f^{0}\rangle,\,\,|f^{1}\rangle,\,\,|f^{2}\rangle\\}$ with
the superscripts being denoted as the occupation of the localized electrons,
to expand the Hamiltonian of Eq. (20) in the following matrix form,

$\displaystyle H=\left[\begin{array}[]{ccc}H_{00}&H_{01}&H_{02}\\\
H_{10}&H_{11}&H_{12}\\\ H_{20}&H_{21}&H_{22}\end{array}\right].$ (24)

The matrix elements of Eq. (24), denoted as $H_{ij}\equiv\langle
f^{i}|H|f^{j}\rangle$ with $i,j=0,1,2$, are

$\displaystyle H_{10}$
$\displaystyle=\sum_{\sigma\sigma^{\prime}}\sum_{\alpha=\pm x,\pm
y}iV\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}}=H_{21},$
$\displaystyle H_{01}$
$\displaystyle=H_{10}^{\dagger}=\sum_{\sigma\sigma^{\prime}}\sum_{\alpha=\pm
x,\pm
y}iV\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}c_{i+\hat{\alpha},\sigma}^{\dagger}f_{i\sigma^{\prime}}=H_{12},$
$\displaystyle H_{11}$
$\displaystyle=\sum_{\mathbold{k}\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}\sigma}+\sum_{\sigma}\varepsilon_{f}f_{i\sigma}^{\dagger}f_{i\sigma},\,\,H_{00}=\sum_{\mathbold{k}\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}\sigma},$
$\displaystyle H_{22}$
$\displaystyle=\sum_{\mathbold{k}\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}\sigma}+\sum_{\sigma}\varepsilon_{f}f_{j\sigma}^{\dagger}f_{j\sigma}+Un_{i\uparrow}^{f}n_{i\downarrow}^{f}.$
(25)

We then project out $|f^{0}\rangle$ and $|f^{2}\rangle$ from the Hilbert space
to obtain the effective Hamiltonian $H_{eff}$ at the Kondo limit satisfying
$H_{eff}|f^{1}\rangle=E|f^{1}\rangle$ with $E$ being the eigenenergy. Via Eq.
(24), $H_{eff}$ can be expressed as $H_{eff}=H_{11}+H^{\prime}$ with

$\displaystyle H^{\prime}=$ $\displaystyle
H_{10}(E-H_{00})^{-1}H_{01}+H_{12}(E-H_{22})^{-1}H_{21}$ $\displaystyle=$
$\displaystyle\sum_{\alpha,\alpha^{\prime}=x,y}\sum_{\sigma\sigma^{\prime}}\sum_{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\left[\frac{V^{2}}{\varepsilon_{F}-\varepsilon_{f}-U}\left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}c_{i+\hat{\alpha},\sigma}^{\dagger}f_{i\sigma^{\prime}}\right)\left(i\nu_{\hat{\alpha}^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}f_{i\sigma^{\prime\prime}}^{\dagger}c_{i-\hat{\alpha}^{\prime},\sigma^{\prime\prime\prime}}\right)\right.$
(26)
$\displaystyle\left.\qquad\qquad+\frac{V^{2}}{\varepsilon_{f}-\varepsilon_{F}}\left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}}\right)\left(i\nu_{\hat{\alpha}^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}c_{i+\hat{\alpha}^{\prime},\sigma^{\prime\prime}}^{\dagger}f_{i\sigma^{\prime\prime\prime}}\right)\right]$
(27)

Here, we skip the derivations of $H_{10}(E-H_{00})^{-1}H_{01}$ and
$H_{12}(E-H_{22})^{-1}H_{21}$ in Eq. (27) as those are standard and can be
found in a number of references. See, for example, Refs. [19, 20].
$H^{\prime}$ can be further cast into the form similar to the conventional
single-impurity Kondo term, with the following antiferromagnetic Kondo
coupling

$\displaystyle
J_{K}=\frac{V^{2}}{U+\varepsilon_{f}-\varepsilon_{F}}+\frac{V^{2}}{\varepsilon_{F}-\varepsilon_{f}}>0,$
(28)

plus a potential scattering term. Eq. (27) can be generalized to the lattice
version by summing over all lattice sites, as described by Eq. (5).

##  Appendix B Derivation of the effective ferromagnetic RKKY-like interaction

In the section, we derive the RKKY-like interaction by perturbatively
expanding $H_{K}$ of Eq. (5) to second order.

The unperturbed state is described as

$\displaystyle\left|0,f\right\rangle=\left|k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right\rangle\left|f\right\rangle,$
(29)

where conduction electrons do not interact with the impurities. In Eq. (29),
$\left|k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right\rangle$ represents the
Fermi sea with all wave vectors lying below the Fermi wave vector, namely
$k_{i}<k_{F}$. After imposing perturbation, the unperturbed state acquires
correction and the corrected eigenenergy is expressed in powers of $J_{K}$,
$E=E_{0}+\Delta E^{(1)}+\Delta E^{(2)}+O(J_{K}^{3})$ with $E_{0}$ being the
eigenenergy of the unperturbed state.

The first and second order energy corrections take the form

$\displaystyle\Delta E^{(1)}=\left\langle
0,f\right|H_{K}\left|0,f\right\rangle,$ $\displaystyle\Delta
E^{(2)}=\sum_{(0,f)\neq(A,f^{\prime})}\frac{\left|\left\langle
0,f\right|H_{K}\left|A,f^{\prime}\right\rangle\right|^{2}}{E_{0}-E_{A}},$ (30)

where $|A,f^{\prime}\rangle$ denotes the excited state which can be expressed
as a direct product of the building blocks
$|k_{i}^{\prime\prime},m_{i}^{\prime\prime}\rangle$, with part of wave vectors
lying above the Fermi surface, i.e. $k_{i}^{\prime\prime}>k_{F}$.

Here, we first derive the effective interaction of the $f$ fermions for a
simpler two-impurity model and generalize the results to the lattice version.

$\Delta E^{(1)}$ can be evaluated by summing over the subspace of the
conduction electron, yielding

$\displaystyle\Delta E^{(1)}$ $\displaystyle=\left\langle
0,f\right|H_{K}\left|0,f\right\rangle$
$\displaystyle=\frac{4n_{f}J_{K}}{\mathcal{N}_{s}}\sum_{\bm{k}<k_{F}}\left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)+\mathcal{C},$
(31)

where $n_{f}=\sum_{i=1,2,\sigma}\left\langle
f\right|f_{i\sigma}^{\dagger}f_{i\sigma}\left|f\right\rangle$ and
$\mathcal{C}$ is a constant. $H_{K}$ in Eq. (31) denotes the two-impurity
Kondo term. It turns out that $\Delta E^{(1)}$ only introduces a constant
energy shift for the bare energy level of the $f$ fermions.

$\Delta E^{(2)}$ is given by

$\displaystyle\Delta E^{(2)}=$
$\displaystyle\frac{1}{\mathcal{N}_{s}^{4}}\sum_{f^{\prime\prime}}\sum_{k_{1}^{\prime\prime}}\sum_{m_{1}^{\prime\prime}}\cdots\sum_{k_{N}^{\prime\prime}}\sum_{m_{N}^{\prime\prime}}\left(\frac{J_{K}^{2}}{E_{0}-E_{A}}\right)\sum_{i=1}^{2}\sum_{\sigma\sigma^{\prime}}\sum_{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\sum_{\alpha,\alpha^{\prime}}\sum_{\mathbold{k},\mathbold{k}^{\prime}}\left(\sum_{j=1}^{2}\sum_{\tau\tau^{\prime}}\sum_{\tau^{\prime\prime}\tau^{\prime\prime\prime}}\sum_{\beta,\beta^{\prime}}\sum_{\mathbold{q},\mathbold{q}^{\prime}}\right)$
$\displaystyle\quad\times\left[e^{i\mathbold{k}\cdot(\mathbold{r}_{i}+\hat{\alpha})-i\mathbold{k}^{\prime}\cdot(\mathbold{r}_{i}-\hat{\alpha}^{\prime})}i\nu_{\hat{\alpha}}i\nu_{\hat{\alpha}^{\prime}}\right]\left[e^{i\mathbold{q}\cdot(\mathbold{r}_{j}+\hat{\beta})-i\mathbold{q}^{\prime}\cdot(\mathbold{r}_{j}-\hat{\beta}^{\prime})}i\nu_{\hat{\beta}}i\nu_{\hat{\beta}^{\prime}}\right]$
$\displaystyle\quad\times\sigma_{\alpha}^{\sigma\sigma^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\left\langle
f\right|f_{i\sigma^{\prime}}f_{i\sigma^{\prime\prime}}^{\dagger}\left|f^{\prime\prime}\right\rangle\left\langle
k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right|\left(c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}^{\prime}\sigma^{\prime\prime\prime}}\right)\left|k_{1}^{\prime\prime}m_{1}^{\prime\prime},\,k_{2}^{\prime\prime}m_{2}^{\prime\prime},\cdots,k_{N}^{\prime\prime}m_{N}^{\prime\prime}\right\rangle$
$\displaystyle\quad\times\sigma_{\beta}^{\tau\tau^{\prime}}\sigma_{\beta^{\prime}}^{\tau^{\prime\prime}\tau^{\prime\prime\prime}}\left\langle
f^{\prime\prime}\right|f_{j\tau^{\prime}}f_{j\tau^{\prime\prime}}^{\dagger}\left|f\right\rangle\left\langle
k_{1}^{\prime\prime}m_{1}^{\prime\prime},\,k_{2}^{\prime\prime}m_{2}^{\prime\prime},\cdots,k_{N}^{\prime\prime}m_{N}^{\prime\prime}\right|\left(c_{\mathbold{q}\tau}^{\dagger}c_{\mathbold{q}^{\prime}\tau^{\prime\prime\prime}}\right)\left|k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right\rangle.$
(32)

An annihilation operator acts on $\left|A,f^{\prime}\right\rangle$ can be
obtained as

$\displaystyle
c_{q\sigma^{\prime}}\left|k_{1}^{\prime\prime}m_{1}^{\prime\prime},\,k_{2}^{\prime\prime}m_{2}^{\prime\prime},\cdots,k_{N}^{\prime\prime}m_{N}^{\prime\prime}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}(-1)^{p_{\alpha}}\delta_{q,k_{\alpha}^{\prime\prime}}c_{\sigma^{\prime}}\left|\left(\prod_{l=1}^{\alpha-1}k_{l}^{\prime\prime}m_{l}^{\prime\prime}\right)m_{\alpha}^{\prime\prime}\left(\prod_{l=\alpha+1}^{N}k_{l}^{\prime\prime}m_{l}^{\prime\prime}\right)\right\rangle,$
(33)

we can thus obtain

$\displaystyle\left\langle
k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right|\left(c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}^{\prime}\sigma^{\prime\prime\prime}}\right)\left|k_{1}^{\prime\prime}m_{1}^{\prime\prime},\,k_{2}^{\prime\prime}m_{2}^{\prime\prime},\cdots,k_{N}^{\prime\prime}m_{N}^{\prime\prime}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}(-1)^{p_{\alpha}}(-1)^{p_{\beta}}\delta_{\mathbold{k},\mathbold{k}_{\alpha}}\delta_{\mathbold{k}^{\prime},\mathbold{k}_{\beta}^{\prime\prime}}\left\langle
m_{\alpha}\right|c_{\sigma}^{\dagger}c_{\sigma^{\prime\prime\prime}}\left|m_{\beta}^{\prime\prime}\right\rangle\left\langle\left(\prod_{l=1}^{\alpha-1}k_{l}m_{l}\right)\left(\prod_{l=\alpha+1}^{N}k_{l}m_{l}\right)\right|\left(\prod_{l=1}^{\beta-1}k_{l}^{\prime\prime}m_{l}^{\prime\prime}\right)\left(\prod_{l=\beta+1}^{N}k_{l}^{\prime\prime}m_{l}^{\prime\prime}\right)\Bigg{>}.$
(34)

The above matrix element is nonzero only if the momentum is restricted by
certain constraints and
$(k_{i},m_{i})=(k_{i}^{\prime\prime},m_{i}^{\prime\prime})$, signifying
$p_{\alpha}=p_{\beta}$:

$\displaystyle\left\langle
k_{1}m_{1},\cdots,k_{N}m_{N}\right|\left(c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}^{\prime}\sigma^{\prime\prime\prime}}\right)\left|k_{1}^{\prime\prime}m_{1}^{\prime\prime},\cdots,k_{N}^{\prime\prime}m_{N}^{\prime\prime}\right\rangle$
$\displaystyle=$
$\displaystyle\Theta(k_{F}-\left|\mathbold{k}\right|)\Theta\left(\left|\mathbold{k}^{\prime}\right|-k_{F}\right)$
$\displaystyle\times\sum_{\alpha=1}^{N}\left[\delta_{\mathbold{k},\mathbold{k}_{\alpha}}\delta_{\mathbold{k}^{\prime},\mathbold{k}_{\alpha}^{\prime\prime}}\left\langle
m_{\alpha}\right|c_{\sigma}^{\dagger}c_{\sigma^{\prime\prime\prime}}\left|m_{\alpha}^{\prime\prime}\right\rangle\prod_{l\neq\alpha}\delta_{\mathbold{k}_{l}\mathbold{k}_{l}^{\prime\prime}}\delta_{m_{l}m_{l}^{\prime\prime}}\right]$
(35)

Plugging this into $\Delta E^{(2)}$, we have

$\displaystyle\Delta E^{(2)}=$
$\displaystyle\frac{1}{\mathcal{N}_{s}^{4}}\sum_{f^{\prime\prime}}\sum_{a=1}^{N}\sum_{k_{a}^{\prime\prime}}\sum_{m_{a}^{\prime\prime}}\left(\frac{J_{K}^{2}}{E_{0}-E_{A}}\right)\sum_{i=1}^{2}\sum_{\sigma\sigma^{\prime}}\sum_{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\sum_{\alpha,\alpha^{\prime}}\left(\sum_{j=1}^{2}\sum_{\tau\tau^{\prime}}\sum_{\tau^{\prime\prime}\tau^{\prime\prime\prime}}\sum_{\beta,\beta^{\prime}}\sum_{\mathbold{q},\mathbold{q}^{\prime}}\right)$
$\displaystyle\qquad\times\Theta(k_{F}-\left|\mathbold{k}_{a}\right|)\Theta\left(\left|\mathbold{k}_{a}^{\prime\prime}\right|-k_{F}\right)\left[e^{i\mathbold{k}_{a}\cdot(\mathbold{r}_{i}+\hat{\alpha})-i\mathbold{k}_{a}^{\prime\prime}\cdot(\mathbold{r}_{i}-\hat{\alpha}^{\prime})}i\nu_{\hat{\alpha}}i\nu_{\hat{\alpha}^{\prime}}\right]\left[e^{i\mathbold{q}\cdot(\mathbold{r}_{j}+\hat{\beta})-i\mathbold{q}^{\prime}\cdot(\mathbold{r}_{j}-\hat{\beta}^{\prime})}i\nu_{\hat{\beta}}i\nu_{\hat{\beta}^{\prime}}\right]$
$\displaystyle\qquad\times\sigma_{\alpha}^{\sigma\sigma^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\sigma_{\beta}^{\tau\tau^{\prime}}\sigma_{\beta^{\prime}}^{\tau^{\prime\prime}\tau^{\prime\prime\prime}}\left\langle
f\right|f_{i\sigma^{\prime}}f_{i\sigma^{\prime\prime}}^{\dagger}\left|f^{\prime\prime}\right\rangle\left\langle
f^{\prime\prime}\right|f_{j\tau^{\prime}}f_{j\tau^{\prime\prime}}^{\dagger}\left|f\right\rangle\left\langle
m_{a}\right|c_{\sigma}^{\dagger}c_{\sigma^{\prime\prime\prime}}\left|m_{a}^{\prime\prime}\right\rangle$
$\displaystyle\qquad\times\left\langle\left(\prod_{l=1}^{a-1}k_{l}m_{l}\right)k_{a}^{\prime\prime}m_{a}^{\prime\prime}\left(\prod_{l=a+1}^{N}k_{l}m_{l}\right)\right|\left(c_{\mathbold{q}\tau}^{\dagger}c_{\mathbold{q}^{\prime}\tau^{\prime\prime\prime}}\right)\left|k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right\rangle$
(36)

The matrix element of
$c^{\dagger}_{\mathbold{q}\tau}c_{\mathbold{q}^{\prime}\tau^{\prime\prime\prime}}$
in the fourth line of Eq. (36) can be evaluated as

$\displaystyle\left\langle\left(\prod_{l=1}^{a-1}k_{l}m_{l}\right)k_{a}^{\prime\prime}m_{a}^{\prime\prime}\left(\prod_{l=a+1}^{N}k_{l}m_{l}\right)\right|\left(c_{\mathbold{q}\tau}^{\dagger}c_{\mathbold{q}^{\prime}\tau^{\prime\prime\prime}}\right)\left|k_{1}m_{1},k_{2}m_{2},\cdots,k_{N}m_{N}\right\rangle=\delta_{\mathbold{q},\mathbold{k}_{a}^{\prime\prime}}\delta_{\mathbold{q}^{\prime},\mathbold{k}_{\alpha}}\langle
m_{a}^{\prime\prime}|c_{\tau}^{\dagger}c_{\tau^{\prime\prime\prime}}|m_{a}\rangle.$
(37)

Hence, the energy correction $\Delta E^{(2)}$ can be further simplified as
(sum over
$f^{\prime\prime},m_{a}^{\prime\prime},\mathbold{q},\mathbold{q}^{\prime}$ and
suppress the subscript $a$ below)

$\displaystyle\Delta E^{(2)}=$
$\displaystyle\frac{1}{\mathcal{N}_{s}^{4}}\sum_{i,j=1}^{2}\sum_{\alpha,\alpha^{\prime}}\sum_{\varepsilon_{\mathbold}{k}<\mu}\sum_{\varepsilon_{\mathbold{k}^{\prime\prime}}>\mu}\sum_{m,\tau=\pm}\sum_{\beta,\beta^{\prime}}\left(\frac{J_{K}^{2}}{\varepsilon_{\bm{k}}-\varepsilon_{\bm{k}^{\prime\prime}}}\right)\left(i\nu_{\hat{\alpha}}i\nu_{\hat{\alpha}^{\prime}}i\nu_{\hat{\beta}}i\nu_{\hat{\beta}^{\prime}}\right)$
$\displaystyle\quad\times
e^{i\mathbold{k}\cdot(\mathbold{r}_{i}+\hat{\alpha})-i\mathbold{k}^{\prime\prime}\cdot(\mathbold{r}_{i}-\hat{\alpha}^{\prime})}e^{i\mathbold{k}^{\prime\prime}\cdot(\mathbold{r}_{j}+\hat{\beta})-i\mathbold{k}\cdot(\mathbold{r}_{j}-\hat{\beta}^{\prime})}\sigma_{\alpha}^{m,-m}\sigma_{\alpha^{\prime}}^{-\tau,\tau}\sigma_{\beta}^{\tau,-\tau}\sigma_{\beta^{\prime}}^{-m,m}\left\langle
f\right|f_{i,-m}f_{i,-\tau}^{\dagger}f_{j,-\tau}f_{j,-m}^{\dagger}\left|f\right\rangle$
(38)

The effective interacting term among the $f$ fermions can be obtained by
removing the bracket $\langle f|\cdots|f\rangle$. This result can be simply
generalized to the lattice version by extending the summation of $i$ and $j$
over the entire lattice, as shown in Eqs. (6) and (7).

##  Appendix C The mean-field Kondo-Heisenberg Hamiltonian on a strip

In this section, we provide the details of the matrix elements of the Kondo-
Heisenberg Hamiltonian on a nano-strip with $N_{y}$ chains along $y$-axis. We
choose the basis of the Kondo-Heisenberg strip as

$\displaystyle\phi_{A,k}=\left(c_{k1\uparrow},c_{k2\uparrow},\cdots,c_{kN_{y}\uparrow},c_{-k1\uparrow}^{\dagger},c_{-k2\uparrow}^{\dagger},\cdots,c_{-kN_{y}\uparrow}^{\dagger},f_{k1\downarrow},f_{k2\downarrow},\cdots,f_{kN_{y}\downarrow},f_{-k1\downarrow}^{\dagger},f_{-k2\downarrow}^{\dagger},\cdots,f_{-kN_{y}\downarrow}^{\dagger}\right)^{T},$
$\displaystyle\phi_{B,k}=\left(c_{k1\downarrow},c_{k2\downarrow},\cdots,c_{kN_{y}\downarrow},c_{-k1\downarrow}^{\dagger},c_{-k2\downarrow}^{\dagger},\cdots,c_{-kN_{y}\downarrow}^{\dagger},f_{k1\uparrow},f_{k2\uparrow},\cdots,f_{kN_{y}\uparrow},f_{-k1\uparrow}^{\dagger},f_{-k2\uparrow}^{\dagger},\cdots,f_{-kN_{y}\uparrow}^{\dagger}\right)^{T},$
(39)

where we take $k_{x}\to k$. The total Hamiltonian $H$ is represented as a
summation of two decoupled Hamiltonians, $H_{A}$ and $H_{B}$, each of which is
$4N_{y}\times 4N_{y}$ in size, given by

$\displaystyle
H=\sum_{k}\phi_{A,k}^{\dagger}\mathcal{H}_{A}(k)\phi_{A,k}+\sum_{k}\phi_{B,k}^{\dagger}\mathcal{H}_{B}(k)\phi_{B,k}.$
(40)

Below, we provides the matrix elements of $\mathcal{H}_{A}(k)$ and
$\mathcal{H}_{B}$, respectively:

###  C.1 $\mathcal{H}_{A}$

The matrix elements of the hopping term for $\mathcal{H}_{A}$ are

$\displaystyle\mathcal{H}_{A}(y_{i},y_{i})=-t\cos k-\frac{\mu}{2},$
$\displaystyle\mathcal{H}_{A}(y_{i}+N_{y},y_{i}+N_{y})=t\cos k+\frac{\mu}{2}$
(41)

for $y_{i}=1,2,\cdots,N_{y}$ while

$\displaystyle\mathcal{H}_{A}(y_{i},y_{i}+1)=-\frac{t}{2},$
$\displaystyle\mathcal{H}_{A}(y_{i}+1,y_{i})=-\frac{t}{2},$
$\displaystyle\mathcal{H}_{A}(N_{y}+y_{i}+1,N_{y}+y_{i})=\frac{t}{2},$
$\displaystyle\mathcal{H}_{A}(N_{y}+y_{i},N_{y}+y_{i}+1)=\frac{t}{2}$ (42)

for $y_{i}=1,2,\cdots,N_{y}-1$.

For $H_{f}$, we have for $y_{i}=1,2,\cdots,N_{y}$

$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i},2N_{y}+y_{i})=\lambda/2,$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i},3N_{y}+y_{i})=-\lambda/2.$ (43)

The Kondo term $H_{K}$ for $\mathcal{H}_{A}$ describes the Kondo interaction
with the following matrix form: the Kondo hybridization of $c$ and $f$ with
the same $y$ chain are

$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i},y_{i})=x\sin k,$
$\displaystyle\mathcal{H}_{A}(N_{y}+y_{i},3N_{y}+y_{i})=x\sin k,$
$\displaystyle\mathcal{H}_{A}(y_{i},y_{i}+2N_{y})=x\sin k,$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i},N_{y}+y_{i})=x\sin k$ (44)

for $y_{i}=1,\cdots,N_{y}$. The matrix elements of the Kondo term for
$y_{i}=1,\cdots,N_{y}-1$ are

$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i}+1,y_{i})=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(N_{y}+y_{i},3N_{y}+y_{i}+1)=\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i},y_{i}+1)=\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(N_{y}+y_{i}+1,3N_{y}+y_{i})=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(y_{i},2N_{y}+y_{i}+1)=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i}+1,N_{y}+y_{i})=\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(y_{i}+1,2N_{y}+y_{i})=\frac{x}{2},$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i},N_{y}+y_{i}+1)=-\frac{x}{2},$ (45)

which corresponds to the hybridization of $c$ and $f$ with the nearest-
neighboring $y$ chains.

The RVB pairing term $H_{J}$ on a nano-strip is described by the following
matrix elements: for $y_{i}=1,\cdots,N_{y}$,

$\displaystyle\mathcal{H}_{\Delta}^{A}(2N_{y}+i,3N_{y}+i)=-i\Delta_{t}\sin k,$
$\displaystyle\mathcal{H}_{\Delta}^{A}(3N_{y}+i,2N_{y}+i)=i\Delta_{t}\sin k$
(46)

are the matrix elements for the pairing of spinons with the same $y_{i}$. For
$y_{i}=1,\cdots,N_{y}-1$, we have

$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i},3N_{y}+y_{i}+1)=-\frac{i}{2}\Delta_{t},$
$\displaystyle\mathcal{H}_{A}(2N_{y}+y_{i}+1,3N_{y}+y_{i})=\frac{i}{2}\Delta_{t}.$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i}+1,2N_{y}+y_{i})=\frac{i}{2}\Delta_{t},$
$\displaystyle\mathcal{H}_{A}(3N_{y}+y_{i},2N_{y}+y_{i}+1)=-\frac{i}{2}\Delta_{t}.$
(47)

###  C.2 $\mathcal{H}_{B}$

The matrix elements for the hopping term in $H_{B}$ are

$\displaystyle\mathcal{H}_{B}(y_{i},y_{i})=-t\cos k-\frac{\mu}{2},$
$\displaystyle\mathcal{H}_{B}(y_{i}+N_{y},y_{i}+N_{y})=t\cos k+\frac{\mu}{2}$
(48)

for $y_{i}=1,2,\cdots,N_{y}$. While, for for $y_{i}=1,2,\cdots,N_{y}-1$, we
obtain

$\displaystyle\mathcal{H}_{B}(y_{i},y_{i}+1)=-\frac{t}{2},$
$\displaystyle\mathcal{H}_{B}(y_{i}+1,y_{i})=-\frac{t}{2},$
$\displaystyle\mathcal{H}_{B}(N_{y}+y_{i}+1,N_{y}+y_{i})=\frac{t}{2},$
$\displaystyle\mathcal{H}_{B}(N_{y}+y_{i},N_{y}+y_{i}+1)=\frac{t}{2}.$ (49)

The matrix elements for $H_{f}$ are

$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i},2N_{y}+y_{i})=\lambda/2,$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i},3N_{y}+y_{i})=-\lambda/2$ (50)

with $y_{i}=1,2,\cdots,N_{y}$.

The matrix elements of the Kondo term for $c$ and $f$ lying on the same chain
$y_{i}$ are

$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i},y_{i})=x\sin k,$
$\displaystyle\mathcal{H}_{B}(N_{y}+y_{i},3N_{y}+y_{i})=x\sin k,$
$\displaystyle\mathcal{H}_{B}(y_{i},2N_{y}+y_{i})=x\sin k,$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i},N_{y}+y_{i})=x\sin k,$ (51)

where $y_{i}=1,\cdots,N_{y}$. For Kondo term where the hybridization is
happening between nearest-neighboring chains, we have

$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i}+1,y_{i})=\frac{x}{2},$
$\displaystyle\mathcal{H}_{B}(N_{y}+y_{i},3N_{y}+y_{i}+1)=-\frac{x}{2}$
$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i},y_{i}+1)=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{B}(N_{y}+y_{i}+1,3N_{y}+y_{i})=\frac{x}{2},$
$\displaystyle\mathcal{H}_{B}(y_{i},2N_{y}+y_{i}+1)=\frac{x}{2}$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i}+1,N_{y}+y_{i})=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{B}(y_{i}+1,2N_{y}+y_{i})=-\frac{x}{2},$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i},N_{y}+y_{i}+1)=\frac{x}{2}$ (52)

for $y_{i}=1,\cdots,N_{y}-1$.

The matrix elements for the RVB spinon-pairing term are

$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i},3N_{y}+y_{i})=-i\Delta_{t}\sin k,$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i},2N_{y}+y_{i})=i\Delta_{t}\sin k$
(53)

for $y_{i}=1,\cdots,N_{y}$, and

$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i},3N_{y}+y_{i}+1)=\frac{i}{2}\Delta_{t},$
$\displaystyle\mathcal{H}_{B}(2N_{y}+y_{i}+1,3N_{y}+y_{i})=-\frac{i}{2}\Delta_{t},$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i}+1,2N_{y}+y_{i})=-\frac{i}{2}\Delta_{t},$
$\displaystyle\mathcal{H}_{B}(3N_{y}+y_{i},2N_{y}+y_{i}+1)=\frac{i}{2}\Delta_{t}$
(54)

for $y_{i}=1,\cdots,N_{y}-1$.

## References

  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Alicea [2012] J. Alicea, Reports on Progress in Physics 75, 076501 (2012).
  * Lutchyn  _et al._ [2010] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010).
  * Oreg  _et al._ [2010] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010).
  * Gaidamauskas  _et al._ [2014] E. Gaidamauskas, J. Paaske, and K. Flensberg, Phys. Rev. Lett. 112, 126402 (2014).
  * Dzero  _et al._ [2016] M. Dzero, J. Xia, V. Galitski, and P. Coleman, Annual Review of Condensed Matter Physics 7, 249 (2016).
  * Dzero  _et al._ [2012] M. Dzero, K. Sun, P. Coleman, and V. Galitski, Phys. Rev. B 85, 045130 (2012).
  * Dzero  _et al._ [2010] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev. Lett. 104, 106408 (2010).
  * Lai  _et al._ [2018] H.-H. Lai, S. E. Grefe, S. Paschen, and Q. Si, Proceedings of the National Academy of Sciences 115, 93 (2018).
  * Mackenzie and Maeno [2003] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).
  * Maeno  _et al._ [2012] Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, Journal of the Physical Society of Japan 81, 011009 (2012), https://doi.org/10.1143/JPSJ.81.011009 .
  * Kallin and Berlinsky [2009] C. Kallin and A. J. Berlinsky, Journal of Physics: Condensed Matter 21, 164210 (2009).
  * Xu  _et al._ [2020] X. Xu, Y. Li, and C. L. Chien, Phys. Rev. Lett. 124, 167001 (2020).
  * Ran  _et al._ [2019a] S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Science 365, 684 (2019a), https://www.science.org/doi/pdf/10.1126/science.aav8645 .
  * Ran  _et al._ [2019b] S. Ran, I.-L. Liu, Y. S. Eo, D. J. Campbell, P. M. Neves, W. T. Fuhrman, S. R. Saha, C. Eckberg, H. Kim, D. Graf, F. Balakirev, J. Singleton, J. Paglione, and N. P. Butch, Nature Physics 15, 1250 (2019b).
  * Aoki  _et al._ [2019] D. Aoki, A. Nakamura, F. Honda, D. Li, Y. Homma, Y. Shimizu, Y. J. Sato, G. Knebel, J.-P. Brison, A. Pourret, D. Braithwaite, G. Lapertot, Q. Niu, M. Vališka, H. Harima, and J. Flouquet, Journal of the Physical Society of Japan 88, 043702 (2019).
  * Jiao  _et al._ [2020] L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez, M. Sigrist, Z. Wang, N. P. Butch, and V. Madhavan, Nature 579, 523 (2020).
  * Choi  _et al._ [2018] W. Choi, P. W. Klein, A. Rosch, and Y. B. Kim, Phys. Rev. B 98, 155123 (2018).
  * Schrieffer and Wolff [1966] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).
  * Hewson [1997] A. C. Hewson,  _The Kondo problem to heavy fermions_ , Vol. 2 (Cambridge university press, 1997).
  * Doniach [1977] S. Doniach, Physica B+C 91, 231 (1977).
  * Legner [2016] M. Legner,  _Topological Kondo insulators: materials at the interface of topology and strong correlations_ , Doctoral thesis, ETH Zurich, Zürich (2016).
  * Ruderman and Kittel [1954] M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).
  * Van Vleck [1962] J. H. Van Vleck, Rev. Mod. Phys. 34, 681 (1962).
  * Kirchner  _et al._ [2020] S. Kirchner, S. Paschen, Q. Chen, S. Wirth, D. Feng, J. D. Thompson, and Q. Si, Rev. Mod. Phys. 92, 011002 (2020).
  * Wang  _et al._ [2022] J. Wang, Y.-Y. Chang, and C.-H. Chung, Proceedings of the National Academy of Sciences 119, e2116980119 (2022).
  * Mineev  _et al._ [1999] V. Mineev, K. Samokhin, L. Landau, and L. Landau,  _Introduction to Unconventional Superconductivity_ (Taylor & Francis, 1999).
  * Coleman [2015] P. Coleman,  _Introduction to Many-Body Physics_ (Cambridge University Press, 2015).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
  * Coleman and Andrei [1989] P. Coleman and N. Andrei, Journal of Physics: Condensed Matter 1, 4057 (1989).
  * Coleman and Nevidomskyy [2010] P. Coleman and A. H. Nevidomskyy, Journal of Low Temperature Physics 161, 182 (2010).
  * Senthil  _et al._ [2003] T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003).
  * Fu and Kane [2006] L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006).
  * Fu  _et al._ [2007] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
  * Sheng  _et al._ [2006] D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, Phys. Rev. Lett. 97, 036808 (2006).
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Fukui  _et al._ [2005] T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Physical Society of Japan 74, 1674 (2005).
  * Kane and Mele [2005a] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005a).
  * Kane and Mele [2005b] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005b).
  * Xu  _et al._ [2019] Y. Xu, Y. Sheng, and Y.-f. Yang, Phys. Rev. Lett. 123, 217002 (2019).
  * Duan  _et al._ [2021] C. Duan, R. E. Baumbach, A. Podlesnyak, Y. Deng, C. Moir, A. J. Breindel, M. B. Maple, E. M. Nica, Q. Si, and P. Dai, Nature 600, 636 (2021).
  * Miao  _et al._ [2020] L. Miao, S. Liu, Y. Xu, E. C. Kotta, C.-J. Kang, S. Ran, J. Paglione, G. Kotliar, N. P. Butch, J. D. Denlinger, and L. A. Wray, Phys. Rev. Lett. 124, 076401 (2020).
  * Eo  _et al._ [2022] Y. S. Eo, S. Liu, S. R. Saha, H. Kim, S. Ran, J. A. Horn, H. Hodovanets, J. Collini, T. Metz, W. T. Fuhrman, A. H. Nevidomskyy, J. D. Denlinger, N. P. Butch, M. S. Fuhrer, L. A. Wray, and J. Paglione, Phys. Rev. B 106, L060505 (2022).
  * Braithwaite  _et al._ [2019] D. Braithwaite, M. Vališka, G. Knebel, G. Lapertot, J. P. Brison, A. Pourret, M. E. Zhitomirsky, J. Flouquet, F. Honda, and D. Aoki, Communications Physics 2, 147 (2019).
  * Zhao  _et al._ [2019] H. Zhao, J. Zhang, M. Lyu, S. Bachus, Y. Tokiwa, P. Gegenwart, S. Zhang, J. Cheng, Y.-f. Yang, G. Chen, Y. Isikawa, Q. Si, F. Steglich, and P. Sun, Nat. Phys. 15, 1261 (2019).
  * [46] D. Aoki, A. Nakamura, F. Honda, D. Li, Y. Homma, Y. Shimizu, Y. J. Sato, G. Knebel, J.-P. Brison, A. Pourret, D. Braithwaite, G. Lapertot, Q. Niu, M. Vališka, H. Harima, and J. Flouquet, Spin-Triplet Superconductivity in UTe2 and Ferromagnetic Superconductors, in  _Proceedings of the International Conference on Strongly Correlated Electron Systems (SCES2019)_, https://journals.jps.jp/doi/pdf/10.7566/JPSCP.30.011065 .
  * Metz  _et al._ [2019] T. Metz, S. Bae, S. Ran, I.-L. Liu, Y. S. Eo, W. T. Fuhrman, D. F. Agterberg, S. M. Anlage, N. P. Butch, and J. Paglione, Phys. Rev. B 100, 220504 (2019).
  * Burdin  _et al._ [2000] S. Burdin, A. Georges, and D. R. Grempel, Phys. Rev. Lett. 85, 1048 (2000).
  * Hayes  _et al._ [2021] I. M. Hayes, D. S. Wei, T. Metz, J. Zhang, Y. S. Eo, S. Ran, S. R. Saha, J. Collini, N. P. Butch, D. F. Agterberg, A. Kapitulnik, and J. Paglione, Science 373, 797 (2021), https://www.science.org/doi/pdf/10.1126/science.abb0272 .
  * Shishidou  _et al._ [2021] T. Shishidou, H. G. Suh, P. M. R. Brydon, M. Weinert, and D. F. Agterberg, Phys. Rev. B 103, 104504 (2021).
  * Hazra and Coleman [2022] T. Hazra and P. Coleman, Triplet pairing mechanisms from Hund’s-Kondo models: applications to UTe2 and CeRh2As2 (2022), arXiv:2205.13529 [cond-mat.supr-con] .
  * Rosa  _et al._ [2022] P. F. S. Rosa, A. Weiland, S. S. Fender, B. L. Scott, F. Ronning, J. D. Thompson, E. D. Bauer, and S. M. Thomas, Communications Materials 3, 33 (2022).
  * Rosuel  _et al._ [2022] A. Rosuel, C. Marcenat, G. Knebel, T. Klein, A. Pourret, N. Marquardt, Q. Niu, S. Rousseau, A. Demuer, G. Seyfarth, G. Lapertot, D. Aoki, D. Braithwaite, J. Flouquet, and J.-P. Brison, Field-induced tuning of the pairing state in a superconductor (2022).
  * Sato and Fujimoto [2009] M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).