# Altermagnetism in Heavy Fermion Systems

Miaomiao Zhao  Key Laboratory of Quantum Theory and Applications of MoE $\&$
School of Physical Science and Technology, Lanzhou University, Lanzhou 730000,
People Republic of China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, People Republic of China  Wei-Wei Yang  Beijing National
Laboratory for Condensed Matter Physics and Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  Xueming Guo  Key Laboratory of
Quantum Theory and Applications of MoE $\&$ School of Physical Science and
Technology, Lanzhou University, Lanzhou 730000, People Republic of China
Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou 730000, People Republic of
China  Hong-Gang Luo  Key Laboratory of Quantum Theory and Applications of MoE
$\&$ School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, People Republic of China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, People Republic of China  Yin Zhong<EMAIL_ADDRESS>Key
Laboratory of Quantum Theory and Applications of MoE $\&$ School of Physical
Science and Technology, Lanzhou University, Lanzhou 730000, People Republic of
China  Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, People Republic
of China

###### Abstract

Novel collinear magnet, the altermagnet (AM) with spin-splitting energy band
and zero net magnetization have attracted great interest due to its potential
spintronic applications. Here, we demonstrate AM-like phases in a microscopic
Kondo lattice model, widely used for heavy fermion compounds. With the
framework of fermionic parton mean-field theory, we find the $d$-wave AM state
can coexist with the intrinsic Kondo screening effect in such itinerant-local
electron system if an alternating next-nearest-neighbor-hopping (NNNH) is
included. Such alternating NNNH take nonmagnetic atoms, neglected in usual
antiferromagnetism study, into account when encountering real-life candidate
AM materials. The AM-like states are characterized by their spin-splitting
quasiparticle bands, Fermi surface, spin-resolved distribution function and
conductivity. It is suggested that the magnetic quantum oscillation and charge
transport measurement can detect those AM-like phases. We hope the present
work may be useful for exploring AM-like phases in $f$-electron compounds.

##  I Introduction

Recently, collinear magnetism beyond ferromagnetic and antiferromagnetic
states, i.e. the altermagnetism (AM) with spin-splitting energy band and zero
net magnetization have attracted great interest due to its potential
spintronic applications.Mazin2022 ; Smejkal2022 ; Smejkal2022b ; Bai2024
Current candidate materials of AM are mainly $d$-electron compounds such as
rutile metal RuO2, chalcogenide semiconductor MnTe and pnictide metal CrSb and
even the parent compound of high-$T_{c}$ cuprate superconductor
La2CuO4,Krempasky2024 ; Fedchenko2024 ; Bai2022 ; Feng2022 ; Gonzalez2023 ;
Bai2023 ; Hariki2024 ; Lee2024  wherein the smoking-gun feature of AM, namely
the spin-dependent band-splitting with $d$ or $g$-wave symmetry has been
unambiguously observed in RuO2, MnTe and CrSb by angle-resolved photoemission
spectroscopy (ARPES).Krempasky2024 ; Fedchenko2024 ; Lee2024 ; Osumi2024 ;
Yang2024

Although the density-functional-theory-based first-principle calculation has
successfully predicted these mentioned $d$-electron AM materials,Smejkal2020 ;
Mazin2021 ; Ahn2019 ; Yuan2020  more intuitive understanding, particularly the
many-body physics involving topological superconductivity and Kondo
effect,Chakraborty2023 ; Zhu2023 ; Diniz2024 ; Lee2023  can be extracted from
microscopic lattice models.Brekke2023 ; Mland2024 ; Bose2024 ; Maier2023 ;
Das2023 ; Leeb2024  In Ref. Das2023, , both metallic and insulating AM has
been found in Hartree-Fock treatment of single-band Hubbard model with an
alternating next-nearest-neighbor-hopping (NNNH), which could be realized in
optical lattice. Physically, such alternating NNNH takes (nonequivalent)
nonmagnetic atoms, neglected in usual antiferromagnetism study, into account
when encountering real-life AM materials.Smejkal2022  This alternating NNNH
combined with antiferromagnetic background from Hartree-Fock approximation
reproduces the desirable AM phase. On the other hand, Ref. Brekke2023, ;
Mland2024,  underline the importance of the $s-d$ exchange model where
itinerant electron interplays with local spin moment via Heisenberg/Hund-like
exchange interaction. It is indeed that the antiferromagnetic configuration of
local spin moment leads to the desirable AM state, in which fluctuations due
to magnon-exchange drive the unconventional $p$-wave pairing.

Actually, such itinerant-local exchange interaction is ubiquitous in heavy
fermion systems and the corresponding model turns to be the classic Kondo
lattice model (KLM).Tsunetsugu ; Coleman2015  Considering the
antiferromagnetic states have already occupied sensible parameter regime in
the phase diagram of KLM,Lacroix1979 ; Fazekas1991 ; Zhang2000 ; Li2015 ;
Watanabe2007 ; Asadzadeh2013 ; Martin2008 ; Lenz2008  it is promising to
explore AM in heavy fermion compounds if effects of nonmagnetic atoms are
suitably accounted for, e.g. by inserting the alternating NNNH into KLM.

In this work, we take the first step towards addressing this issue and study a
KLM with an alternating NNNH on square lattice. (Fig. 1(a)) With fermionic
large-N/parton mean-field theory, we uncover $d$-wave AM with spin-splitting
quasi-particle energy bands. Interestingly, although AM competes with Kondo
screening, we find there exists a coexisting regime involving both AM and
Kondo screening, which is reminiscent of the coexistence of antiferromagnetism
and paramagnetic Kondo screening states in original KLM without alternating
NNNH.Assaad1999 ; Zhang2000 ; Danu2021

In addition to the spin-splitting bands, spin-resolved density of state,
distribution function and conductivity are calculated and the latter two could
be used to identify AM-like phases in generic heavy fermion compounds. For
real-life heavy fermion materials, we suggest that the magnetic quantum
oscillation and charge transport measurement can detect those AM-like phases.
Going beyond the present mean-field theory tends to require sophisticated
numerical techniques such as the constrained path Monte Carlo.SZhang1995  It
will be interesting to perform such large-scale simulation in future work and
more importantly, we hope the present work is able to stimulate exploration of
AM candidates in realistic heavy fermion materials.

The remaining part of this article is organized as follows. In Sec. II, we
introduce the KLM on square lattice with an alternating NNNH. Formalism of
fermionic large-N mean-field theory is presented with details. Sec. III
provides the solution of mean-field theory and possible phases including the
Kondo screening state, AM state and their coexistent state are analyzed.
Physical observable like the spin-resolved density of state and conductivity
are considered. Sec. IV is devoted to some discussions, e.g. the implication
for experiment and realistic materials and alternative theoretical approaches.
Brief summary is given in Sec. V.

Figure 1:  (a) The Kondo lattice model (KLM) on square lattice with
alternating next-nearest-neighbor-hopping (NNNH) $t_{+},t_{-}$. The upper
layer denotes local spin moments, which interact with conduction electrons in
the lower layer via Kondo coupling $J$. (b) The schematic ground-state phase
diagram of KLM with alternating NNNH at half-filling. There exit three kinds
of states, i.e. altermagnet (AM), Kondo state (metallic or insulator) and the
coexistence of AM and Kondo state. (c) Mean-field order parameters
$m_{c},m_{f},V$ evolve with increasing Kondo coupling $J$ in the ground-state
of half-filled KLM. ($t=1,t_{+}=0.27,t_{-}=0.03,n_{c}=1,T=0$)

##  II Model and mean-field theory

Let us consider the following KLM on square lattice, (extension to other
lattice geometries is straightforward) which has been believed to be the
standard model for heavy fermion compoundsTsunetsugu ; Coleman2015

$\hat{H}=-\sum_{ij,\sigma}t_{ij}\hat{c}_{i\sigma}^{{\dagger}}\hat{c}_{j\sigma}+\frac{J}{2}\sum_{j,\sigma\sigma^{\prime}}\hat{c}_{j\sigma}^{{\dagger}}\vec{\sigma}_{\sigma\sigma^{\prime}}\hat{c}_{j\sigma^{\prime}}\cdot\hat{\vec{S}}_{j}$
(1)

Here, $\hat{c}_{j\sigma}^{{\dagger}}$ is the creation operator of conduction
electron ($c$-electron) with spin flavor $\sigma=\uparrow,\downarrow$ while
$\hat{\vec{S}}_{j}$ denotes local spin moment of $f$-electron at site $j$. The
$c$-electron obeys the usual anti-commutative rules, i.e.
$\\{\hat{c}_{i\sigma},\hat{c}_{j\sigma^{\prime}}^{{\dagger}}\\}=\delta_{ij}\delta_{\sigma\sigma^{\prime}}$,
$\\{\hat{c}_{i\sigma}^{{\dagger}},\hat{c}_{j\sigma^{\prime}}^{{\dagger}}\\}=\\{\hat{c}_{i\sigma},\hat{c}_{j\sigma^{\prime}}\\}=0$.
The local moment $\hat{\vec{S}}_{j}$ is encoded by the Pauli matrix
$\frac{1}{2}\hat{\vec{\sigma}}_{j}$ associated with
$[\hat{\sigma}_{m}^{\alpha},\hat{\sigma}_{n}^{\beta}]=2i\epsilon^{\alpha\beta\gamma}\hat{\sigma}^{\gamma}_{m}\delta_{mn}$.

$t_{ij}$ are hopping integral between $i,j$ sites for $c$-electron. Inspired
by Ref. Das2023, , we consider nearest-neighbor-hopping (NNH) $t$ and NNNH
$t_{+},t_{-}$ so as to obtain AM in KLM. (The choice of the latter ones is
shown in Fig. 1(a)) The appearance of alternating NNNH breaks the
$\pi/2$-rotation ($C_{4z}$) symmetry of the original square lattice and this
feature is essential for producing AM phase. Physically, such alternating NNNH
takes (nonequivalent) nonmagnetic atoms, neglected in usual antiferromagnetism
study, into account when encountering real-life AM materials.Smejkal2022
Thus, AM materials may be explored in existing antiferromagnets with symmetry-
breaking such as the mentioned $\pi/2$-rotation.

$J$ denotes the strength of Kondo coupling between $c$-electron and local
spin. Furthermore, chemical potential $\mu$ can be added to fix electron’s
density $n_{c}$. It is noted that, when NNNH does not vanish, the KLM cannot
be solved by unbiased quantum Monte Carlo simulation even in half-filling case
(density of $c$-electron $n_{c}=1$).Assaad1999

For $t_{+}\neq t_{-}$, the lattice is divided into $A$ and $B$ sublattice,
thus we can rewrite the Hamiltonian as follows,

$\hat{H}=\hat{H}_{0}+\hat{H}_{K},$ (2)

where the non-interacting part reads

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle-t\sum_{i,\delta,\sigma}(\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta,B\sigma}+\hat{c}_{i+\delta,B\sigma}^{{\dagger}}\hat{c}_{iA\sigma})$
(3) $\displaystyle-$
$\displaystyle\sum_{i,\delta_{1}^{\prime},\sigma}(t_{+}\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta_{1}^{\prime},A\sigma}+t_{-}\hat{c}_{iB\sigma}^{{\dagger}}\hat{c}_{i+\delta_{1}^{\prime},B\sigma})$
$\displaystyle-$
$\displaystyle\sum_{i,\delta_{2}^{\prime},\sigma}(t_{-}\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta_{2}^{\prime},A\sigma}+t_{+}\hat{c}_{iB\sigma}^{{\dagger}}\hat{c}_{i+\delta_{2}^{\prime},B\sigma}).$

The $\delta=(\pm 1,0),(0,\pm 1)$ represents the vector of NNH. As for the
vector of NNNH, we have $\delta_{1}^{\prime}=(1,1),(-1,-1)$ and
$\delta_{2}^{\prime}=(1,-1),(-1,1)$. (Fig. 1(a))

$\displaystyle\hat{H}_{K}$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{j,\alpha=A,B,\sigma\sigma^{\prime}}\hat{c}_{j\alpha\sigma}^{{\dagger}}\vec{\sigma}_{\sigma\sigma^{\prime}}\hat{c}_{j\alpha\sigma^{\prime}}\cdot\hat{\vec{S}}_{j\alpha}$
(4) $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{j}(\hat{c}_{jA\uparrow}^{{\dagger}}\hat{c}_{jA\downarrow}\hat{S}_{jA}^{-}+\hat{c}_{jA\downarrow}^{{\dagger}}\hat{c}_{jA\uparrow}\hat{S}_{jA}^{+})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{j}(\hat{c}_{jB\uparrow}^{{\dagger}}\hat{c}_{jB\downarrow}\hat{S}_{jB}^{-}+\hat{c}_{jB\downarrow}^{{\dagger}}\hat{c}_{jB\uparrow}\hat{S}_{jB}^{+})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{j,\sigma}\sigma(\hat{c}_{jA\sigma}^{{\dagger}}\hat{c}_{jA\sigma}\hat{S}_{jA}^{z}+\hat{c}_{jB\sigma}^{{\dagger}}\hat{c}_{jB\sigma}\hat{S}_{jB}^{z}).$

Here, the first two lines correspond to the transverse Kondo coupling, which
gives rise to Kondo screening in the mean-field level. In contrast, the
longitudinal Kondo coupling in the last line is responsible for the appearance
of magnetic long-ranged order. Such magnetic orders compete with Kondo
screening and would lead to magnetic-paramagnetic transition.

Now, it is straightforward to have $\hat{H}_{0}$ in momentum space in terms of
Fourier transformation,

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{k\sigma}(\varepsilon_{k}\hat{c}_{kA\sigma}^{{\dagger}}\hat{c}_{kB\sigma}+h.c.+\varepsilon_{k}^{AA}\hat{c}_{kA\sigma}^{{\dagger}}\hat{c}_{kA\sigma}+\varepsilon_{k}^{BB}\hat{c}_{kB\sigma}^{{\dagger}}\hat{c}_{kB\sigma})$
(10) $\displaystyle=$
$\displaystyle\sum_{k\sigma}\left(\begin{array}[]{cc}\hat{c}_{kA\sigma}^{{\dagger}}&\hat{c}_{kB\sigma}^{{\dagger}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\varepsilon_{k}^{AA}&\varepsilon_{k}\\\
\varepsilon_{k}&\varepsilon_{k}^{BB}\\\
\end{array}\right)\left(\begin{array}[]{c}\hat{c}_{kA\sigma}\\\
\hat{c}_{kB\sigma}\\\ \end{array}\right)$

and

$\displaystyle\varepsilon_{k}=-2t(\cos k_{x}+\cos k_{y})$
$\displaystyle\varepsilon_{k}^{AA}=-2t_{+}\cos(k_{x}+k_{y})-2t_{-}\cos(k_{x}-k_{y})$
$\displaystyle\varepsilon_{k}^{BB}=-2t_{-}\cos(k_{x}+k_{y})-2t_{+}\cos(k_{x}-k_{y}).$

Here, $\varepsilon_{k},\varepsilon_{k}^{AA},\varepsilon_{k}^{BB}$ denote
intra-sublattice and inter-sublattice hopping energy, respectively. Since one
has chosen $A,B$ sublattice structure, the Brillouin zone (BZ) used for
summation over momentum $k$ must be the magnetic BZ or reduced BZ, which is
determined by $\cos k_{x}+\cos k_{y}\geqslant 0$. When $t_{+}=t_{-}=t_{1}$,
$\varepsilon_{k}^{AA}=\varepsilon_{k}^{BB}=-4t_{1}\cos k_{x}\cos k_{y}$, which
is just the standard result for an isotropic NNNH term on square lattice.

Although the Kondo interaction term $\hat{H}_{K}$ has simple structure in real
space (Hilbert space has $8$-dimension on each site), it cannot be
diagonalized via Fourier transformation, in contrast to the case of infinite-
ranged Hatsugai-Kohmoto-type interaction.Zhong2022 ; Wang2024  It is well-
known that treating $\hat{H}_{K}$ and $\hat{H}_{0}$ on the equal footing is
still challenging in terms of current analytical or numerical
techniques,Tsunetsugu ; Watanabe2007 ; Asadzadeh2013 ; Martin2008 ; Lenz2008
so we try the well-developed fermionic large-N mean-field theory.Coleman2015 ;
Zhang2000 ; Li2015 ; Hewson1993

In practise, the $S=1/2$ local spin moment is splitting into two flavor
fermions as

$\hat{\vec{S}}_{j}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\hat{f}^{{\dagger}}_{j\sigma}\vec{\sigma}_{\sigma\sigma^{\prime}}\hat{f}_{j\sigma^{\prime}}$
(11)

where $\hat{f}_{j\sigma}$ is the Abriksov/Schwinger fermion operator and
satisfies the anticommutative relation
$\\{\hat{f}_{i\sigma},\hat{f}_{j\sigma^{\prime}}^{{\dagger}}\\}=\delta_{ij}\delta_{\sigma\sigma^{\prime}}$.
The above process is an example of fractonalization because
$\hat{f}_{j\sigma}$ carries $S=1/2$ degree of freedom instead of magnon (the
$S=1$ excitation). Thus, this fermion is also called spinon in the literature
of quantum spin liquid and high-$T_{c}$ cuprate superconductor.Zhou2017 ;
Lee2006  Note that the local spin moment $\hat{\vec{S}}_{j}$ is invariant
under the $U(1)$ gauge transformation $\hat{f}_{j\sigma}\rightarrow
e^{i\theta_{j}}\hat{f}_{j\sigma}$. When Kondo screening is active, the $U(1)$
gauge symmetry is breaking and $\hat{f}_{j\sigma}$ acquires electric charge
via Higgs mechanism. In this situation, the Abriksov fermion can be seen as a
composite fermion
$\hat{f}_{j\sigma}\sim\hat{c}_{j\sigma^{\prime}}\vec{\sigma}_{\sigma^{\prime}\sigma}\cdot\hat{\vec{S}}_{j}$.Coleman2015
; Chen2024

One must be careful to apply above fermionic representation since those
fermions should satisfy the constraint
$\sum_{\sigma}\hat{f}^{{\dagger}}_{j\sigma}\hat{f}_{j\sigma}=1$ at each site
to exclude nonphysical Hilbert space. Violation of this constraint may lead to
nonphysical results even for the simplest case of Hubbard dimer.WWYang2019

After utilizing the Abriksov fermion representation, the Kondo coupling term
is rewritten as

$\displaystyle\hat{H}_{K}$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{j}(\hat{c}_{jA\uparrow}^{{\dagger}}\hat{c}_{jA\downarrow}\hat{f}_{jA\downarrow}^{{\dagger}}\hat{f}_{jA\uparrow}+\hat{c}_{jA\downarrow}^{{\dagger}}\hat{c}_{jA\uparrow}\hat{f}_{jA\uparrow}^{{\dagger}}\hat{f}_{jA\downarrow})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{j}(\hat{c}_{jB\uparrow}^{{\dagger}}\hat{c}_{jB\downarrow}\hat{f}_{jB\downarrow}^{{\dagger}}\hat{f}_{jB\uparrow}+\hat{c}_{jB\downarrow}^{{\dagger}}\hat{c}_{jB\uparrow}\hat{f}_{jB\uparrow}^{{\dagger}}\hat{f}_{jB\downarrow})$
$\displaystyle+$
$\displaystyle\frac{J}{4}\sum_{j,\sigma\sigma^{\prime}}\sigma\sigma^{\prime}(\hat{c}_{jA\sigma}^{{\dagger}}\hat{c}_{jA\sigma}\hat{f}_{jA\sigma^{\prime}}^{{\dagger}}\hat{f}_{jA\sigma^{\prime}}+\hat{c}_{jB\sigma}^{{\dagger}}\hat{c}_{jB\sigma}\hat{f}_{jB\sigma^{\prime}}^{{\dagger}}\hat{f}_{jB\sigma^{\prime}})$

Because, we will use mean-field approximation to treat $\hat{H}_{K}$, it is
helpful to introduce Kondo screening order parameters

$V=-\langle\hat{c}^{{\dagger}}_{j\alpha\sigma}\hat{f}_{j\alpha\sigma}\rangle=-\langle\hat{f}^{{\dagger}}_{j\alpha\sigma}\hat{c}_{j\alpha\sigma}\rangle,~{}\alpha=A,B$

and magnetic order parameters

$\displaystyle
m_{A}^{c}=\frac{1}{2}\sum_{\sigma}\sigma\langle\hat{c}_{jA\sigma}^{{\dagger}}\hat{c}_{jA\sigma}\rangle,m_{A}^{f}=\frac{1}{2}\sum_{\sigma}\sigma\langle\hat{f}_{jA\sigma}^{{\dagger}}\hat{f}_{jA\sigma}\rangle,$
$\displaystyle
m_{B}^{c}=\frac{1}{2}\sum_{\sigma}\sigma\langle\hat{c}_{jB\sigma}^{{\dagger}}\hat{c}_{jB\sigma}\rangle,m_{B}^{f}=\frac{1}{2}\sum_{\sigma}\sigma\langle\hat{f}_{jB\sigma}^{{\dagger}}\hat{f}_{jB\sigma}\rangle.$

As discussed in previous works on AM, one has to assume antiferromagnetic
magnetic structure to generate AM,Brekke2023 ; Mland2024 ; Bose2024 ;
Maier2023 ; Das2023 ; Leeb2024  which motivates us to consider

$m_{A}^{c}=-m_{B}^{c}=m_{c},~{}~{}m_{A}^{f}=-m_{B}^{f}=-m_{f}.$ (13)

This choice leads to the antiferromagnetic configuration shown in Fig. 1,
where $c$-electron and local spin on the same site have the opposite spin
direction, consisting with the antiferromagnetic Kondo coupling. (If this
coupling is ferromagnetic as in the Hund interaction, one may assume
$m_{A}^{c}=-m_{B}^{c}=m_{c},~{}~{}m_{A}^{f}=-m_{B}^{f}=m_{f}$.) Under this
antiferromagnetic configuration, the system is invariant under the combination
operation (time-reversal, $C_{4z}$ and translation), which is denoted by
$[C_{2}\|C_{4z}t]$ (the spin-group symmetry) and it gives rise to the so-
called $d$-wave AM.Smejkal2022  As comparison, the conventional
antiferromagnetism is invariant under the time-reversal combined with
translation (denoting $[C_{2}\|t]$) or just the single operation $C_{4z}$. In
this sense, AM may be considered as a secondary symmetry-breaking state of the
usual antiferromagnetic state.

Then, using the standard mean-field decoupling
$\hat{A}\hat{B}\simeq\langle\hat{A}\rangle\hat{B}+\hat{A}\langle\hat{B}\rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle$
and Fourier transformation, we obtain

$\displaystyle\hat{H}_{K}$ $\displaystyle\simeq$
$\displaystyle\frac{J}{2}V\sum_{k\sigma}(\hat{c}_{kA\sigma}^{{\dagger}}\hat{f}_{kA\sigma}+\hat{f}_{kA\sigma}^{{\dagger}}\hat{c}_{kA\sigma})$
(14) $\displaystyle+$
$\displaystyle\frac{J}{2}V\sum_{k\sigma}(\hat{c}_{kB\sigma}^{{\dagger}}\hat{f}_{kB\sigma}+\hat{f}_{kB\sigma}^{{\dagger}}\hat{c}_{kB\sigma})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{k\sigma}\sigma(m_{c}\hat{f}_{kA\sigma}^{{\dagger}}\hat{f}_{kA\sigma}-m_{f}\hat{c}_{kA\sigma}^{{\dagger}}\hat{c}_{kA\sigma})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{k\sigma}\sigma(-m_{c}\hat{f}_{kB\sigma}^{{\dagger}}\hat{f}_{kB\sigma}+m_{f}\hat{c}_{kB\sigma}^{{\dagger}}\hat{c}_{kB\sigma})$
$\displaystyle+$ $\displaystyle 2N_{s}J(V^{2}+m_{c}m_{f}).$

where $N_{s}$ is the number of unit-cell and the number of sites should be
$2N_{s}$.

Now, combining $\hat{H}_{0}$ with $\hat{H}_{K}$, we find the mean-field
Hamiltonian of KLM as

$\hat{H}=\sum_{k\sigma}\hat{\psi}_{k\sigma}^{{\dagger}}H_{\sigma}(k)\hat{\psi}_{k\sigma}+2N_{s}(JV^{2}+Jm_{c}m_{f}-\lambda)$
(15)

with the spinor
$\hat{\psi}_{k\sigma}^{{\dagger}}=(\hat{c}_{kA\sigma}^{{\dagger}},\hat{c}_{kB\sigma}^{{\dagger}},\hat{f}_{kA\sigma}^{{\dagger}},\hat{f}_{kB\sigma}^{{\dagger}})$
and $4\times 4$-matrix

$H_{\sigma}(k)=\left(\begin{array}[]{cc}H_{\sigma}^{cc}(k)&H_{\sigma}^{cf}(k)\\\
H_{\sigma}^{fc}(k)&H_{\sigma}^{ff}(k)\\\ \end{array}\right)$ (16)
$H_{\sigma}^{cc}(k)=\left(\begin{array}[]{cc}\varepsilon_{k}^{AA}-\mu-\frac{Jm_{f}\sigma}{2}&\varepsilon_{k}\\\
\varepsilon_{k}&\varepsilon_{k}^{BB}-\mu+\frac{Jm_{f}\sigma}{2}\\\
\end{array}\right)$ (17)
$H_{\sigma}^{ff}(k)=\left(\begin{array}[]{cc}\lambda+\frac{Jm_{c}\sigma}{2}&0\\\
0&\lambda-\frac{Jm_{c}\sigma}{2}\\\ \end{array}\right)$ (18)
$H_{\sigma}^{cf}(k)=H_{\sigma}^{fc}(k)=\left(\begin{array}[]{cc}\frac{JV}{2}&0\\\
0&\frac{JV}{2}\\\ \end{array}\right)$ (19)

Here, chemical potential $\mu$ and Lagrangian multiplier $\lambda$ are
included, whose effects are fixing conduction electron density $n_{c}$ and
enforcing the constraint on average,

$\frac{1}{2N_{s}}\sum_{k\sigma}(\langle\hat{c}_{kA\sigma}^{{\dagger}}\hat{c}_{kA\sigma}\rangle+\langle\hat{c}_{kB\sigma}^{{\dagger}}\hat{c}_{kB\sigma}\rangle)=n_{c}$
(20)

and

$\frac{1}{2N_{s}}\sum_{k\sigma}(\langle\hat{f}_{kA\sigma}^{{\dagger}}\hat{f}_{kA\sigma}\rangle+\langle\hat{f}_{kB\sigma}^{{\dagger}}\hat{f}_{kB\sigma}\rangle)=1.$
(21)

Unfortunately, the $4\times 4$-matrix $H_{\sigma}(k)$ have complicated
algebraic structures for its eigenstates and eigenvalues, thus it is not
possible to derive mean-field self-consistent equations required to give order
parameters $V,m_{c}$ and $m_{f}$. However, it is helpful to use the definition
of $V,m_{c},m_{f}$, which gives rise to

$V=-\frac{1}{N_{s}}\sum_{k}\langle\hat{c}^{{\dagger}}_{kA\sigma}\hat{f}_{kA\sigma}\rangle,$
(22)
$m_{c}=\frac{1}{2N_{s}}\sum_{k\sigma}\sigma\langle\hat{c}^{{\dagger}}_{kA\sigma}\hat{c}_{kA\sigma}\rangle,$
(23)
$m_{f}=-\frac{1}{2N_{s}}\sum_{k\sigma}\sigma\langle\hat{f}^{{\dagger}}_{kA\sigma}\hat{f}_{kA\sigma}\rangle.$
(24)

Thus, utilizing Eqs. 22,23,24 combining with Eqs. 20 and 21, we are able to
find all order parameters, chemical potential and Lagrangian multiplier via
the standard iterative algorithm supplemented with the bisection method.Li2015

##  III Results of mean-field theory

Guided by previous works on square lattice KLM without alternating NNNH at
half-filling,Assaad1999 ; Zhang2000 ; Li2015 ; Danu2021 ; Capponi2001  we
first explore the ground-state phase diagram of KLM with alternating NNNH at
half-filling ($n_{c}=1$). (We have used $t_{+}=0.57,t_{-}=0.03$ in the main
text and examples with other $t_{+},t_{-}$ have been discussed in Appendix.
A.)

According to the Kondo screening order parameter $V$, magnetic order
parameters $m_{c}$ and $m_{f}$, we expect four kinds of ground-states:

  1. 1.

Trivial state with $V=m_{c}=m_{f}=0$;

  2. 2.

Kondo screening state with $V\neq 0$ and $m_{c}=m_{f}=0$, it dominates at
large $J$;

  3. 3.

Magnetic state with $V=0$ and $m_{c},m_{f}\neq 0$, which is stable for small
$J$ regime;

  4. 4.

Coexistent state with all $V,m_{c},m_{f}$ being nonzero. It occupies the phase
diagram in the intermediate $J$ regime.

Except for the Kondo state and the decoupled trivial state, we will see below
that both the magnetic state and the coexistent state can be identified as AM
due to their spin-splitting quasiparticle energy bands. To emphasize the
coexistence with Kondo screening, we say there is a coexistent phase with AM
and Kondo screening. (Fig. 1(b) and (c)) This coexistent phase is not
contradicted Landau’s phase transition theory since orders in AM is the
antiferromagnetic order and Kondo screening state is a Higgs phase of the
underlying lattice gauge theory.Coleman2015

###  III.1 Trivial state

When all mean-field parameters $V,m_{c},m_{f}$ are zero, there is only a
trivial state, which is described by the Hamiltonian,

$\displaystyle\hat{H}=\sum_{k\sigma}\left(\begin{array}[]{cc}\hat{c}_{kA\sigma}^{{\dagger}}&\hat{c}_{kB\sigma}^{{\dagger}}\\\
\end{array}\right)\left(\begin{array}[]{cc}\varepsilon_{k}^{AA}-\mu&\varepsilon_{k}\\\
\varepsilon_{k}&\varepsilon_{k}^{BB}-\mu\\\
\end{array}\right)\left(\begin{array}[]{c}\hat{c}_{kA\sigma}\\\
\hat{c}_{kB\sigma}\\\ \end{array}\right).$ (30)

Its dispersion is

$E_{k\sigma\pm}=\frac{1}{2}(\varepsilon_{k}^{AA}+\varepsilon_{k}^{BB}\pm\sqrt{(\varepsilon_{k}^{AA}-\varepsilon_{k}^{BB})^{2}+4\varepsilon_{k}^{2}})-\mu,$

which tells us that the Kondo coupling does not work at the mean-field level.
Actually, fluctuations of the Kondo interaction term must exist and it may be
included in terms of perturbation theory or treated by equation of motion
method for the retarded Green’s function.Bernhard1999  At the present mean-
field level, this decoupled trivial state is unstable to other ordered states
and we will not consider it hereafter.

###  III.2 Kondo screening state

The Kondo screening state is rather simple to understand and it is the
dominating phase at strong coupling ($J$ is large). In this situation,
$m_{c}=m_{f}=0$, and it is safe to turn off $t_{+},t_{-}$ since no AM is
involved. Thus, the half-filling case has the particle-hole symmetry, which
dictates $\mu=\lambda=0$. Consequently, the mean-field Hamiltonian reads,

$\hat{H}=\sum_{k\sigma}\left(\varepsilon_{k}\hat{c}^{{\dagger}}_{k\sigma}\hat{c}_{k\sigma}+\frac{JV}{2}\hat{c}^{{\dagger}}_{k\sigma}\hat{f}_{k\sigma}+\frac{JV}{2}\hat{f}^{{\dagger}}_{k\sigma}\hat{c}_{k\sigma}\right).$
(31)

Careful reader may note that the sublattice index $A,B$ does not appear as a
result of vanishing $t_{+},t_{-}$. The key feature of the above Hamiltonian is
that it gives two spin-degenerated quasi-particle bands

$E_{k\sigma\pm}=\frac{1}{2}\left(\varepsilon_{k}\pm\sqrt{\varepsilon_{k}^{2}+J^{2}V^{2}}\right).$
(32)

At half-filling, the lower band $E_{k\sigma-}$ is fully occupied and the
system belongs to be an insulator, which is usually called Kondo insulator in
the language of heavy fermions.Coleman2015 ; Riseborough2000  This
interaction-driven insulator has direct gap $JV$ and the indirect gap is about
$J^{2}V^{2}/W$. ($W$ is the band-width of conduction electron) Alternatively,
the physics of Kondo insulator can be captured by the strong coupling
expansion, where the product of local spin singlet forms the basis and it
gives the same qualitative result with the present mean-field
theory.Coleman2015 ; Chen2024 ; Trebst2006 ; Huecker2023

When we turn on $t_{+},t_{-}$, the mean-field Hamiltonian reads

$\hat{H}=\sum_{k\sigma}\hat{\psi}_{k\sigma}^{{\dagger}}H_{\sigma}(k)\hat{\psi}_{k\sigma}+2N_{s}(JV^{2}-\lambda)$
(33)

and

$H_{\sigma}(k)=\left(\begin{array}[]{cccc}\varepsilon_{k}^{AA}-\mu&\varepsilon_{k}&\frac{JV}{2}&0\\\
\varepsilon_{k}&\varepsilon_{k}^{BB}-\mu&0&\frac{JV}{2}\\\
\frac{JV}{2}&0&\lambda&0\\\ 0&\frac{JV}{2}&0&\lambda\\\ \end{array}\right).$
(34)

It is clear that quantities in $H_{\sigma}(k)$ do not have dependence on
$\sigma$, thus $\hat{H}$ only has spin-degenerated bands
$E_{k\uparrow}=E_{k\downarrow}$, agreeing with the case of $t_{+}=t_{-}=0$.

Figure 2:  The spin-degenerated bands in the Kondo screening state of KLM.
This Kondo screening state corresponds to Kondo insulator with indirect gap
$J^{2}V^{2}/W$. ($t=1,t_{+}=0.57,t_{-}=0.03,J=2.5,n_{c}=1,T=0$)

The example in Fig. 2 confirms the above reasoning that the Kondo screening
state of KLM only has spin-degenerated bands. The indirect gap from Fig. 2 is
about $0.45$, which is consistent with expression $J^{2}V^{2}/W$ if we note
that $V=0.736,J=2.5,W\simeq 8$. Another noticeable feature is that the system
at half-filling is always a Kondo insulator in spite of nonzero alternating
NNNH. This fact seems to be valid for generic parameters if the Kondo
screening solution is stable.

###  III.3 AM state

In the pure AM state without Kondo screening, the conduction electron and the
local spin are decoupled approximately, so the mean-field Hamiltonian is
simplified into

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{k\sigma}\left(\begin{array}[]{cc}\hat{c}^{{\dagger}}_{kA\sigma}&\hat{c}^{{\dagger}}_{kB\sigma}\\\
\end{array}\right)H_{\sigma}^{cc}(k)\left(\begin{array}[]{c}\hat{c}_{kA\sigma}\\\
\hat{c}_{kB\sigma}\\\ \end{array}\right)$ (38) $\displaystyle+$
$\displaystyle\sum_{k\sigma}\left(\begin{array}[]{cc}\hat{f}^{{\dagger}}_{kA\sigma}&\hat{f}^{{\dagger}}_{kB\sigma}\\\
\end{array}\right)H_{\sigma}^{ff}(k)\left(\begin{array}[]{c}\hat{f}_{kA\sigma}\\\
\hat{f}_{kB\sigma}\\\ \end{array}\right).$ (42)

It is straightforward to find the following four quasiparticle bands,

$\displaystyle E_{k\sigma\pm}^{f}$ $\displaystyle=$
$\displaystyle\lambda\pm\frac{Jm_{c}\sigma}{2},$ $\displaystyle
E_{k\sigma\pm}^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\varepsilon^{AA}_{k}+\varepsilon^{BB}_{k}\pm\sqrt{(\varepsilon_{k}^{AA}-\varepsilon_{k}^{BB}-Jm_{f}\sigma)^{2}+4\varepsilon^{2}_{k}}\right)$
(43) $\displaystyle-$ $\displaystyle\mu.$

It is interesting to see that the Abriksov fermions have the flat band
$E_{k\sigma\pm}^{f}$ due to vanished Kondo hybridization $V$. In contrast, the
bands of conduction electrons $E_{k\sigma\pm}^{c}$ are dispersive and show the
characteristic spin-splitting of AM phase. (Fig. 3) In addition, under
$C_{4z}$ operation (rotation $\pi/2$ around $z$-axis,
$(k_{x},k_{y})\rightarrow(-k_{y},k_{x})$), $E_{k\sigma\pm}^{c}\rightarrow
E_{k-\sigma\pm}^{c}$, which implies the $d$-wave symmetry of AM found here.
Thus, the spin-resolved Fermi surface in Fig. 4(a) also exhibits the $C_{4z}$
symmetry.

Figure 3:  The spin-splitting bands in the AM state of KLM. The $C_{4z}$
symmetry of bands implies a $d$-wave AM.
($t=1,t_{+}=0.57,t_{-}=0.03,J=0.3,n_{c}=1,T=0$) Figure 4:  The spin-splitting
Fermi surface in (a) the AM with $J=0.3$ and (b) the coexistent states with
$J=1.5$ of KLM. The $C_{4z}$ symmetry of Fermi surface implies a $d$-wave AM.
($t=1,t_{+}=0.57,t_{-}=0.03,n_{c}=1,T=0$)

The advantage of the simplified Hamiltonian $\hat{H}$ is that one can derive
the mean-field equations for $m_{c},m_{f}$ via the following expression of
free energy (see Appendix. B for Green’s function and spin susceptibility in
the pure AM state)

$\displaystyle f$ $\displaystyle=$
$\displaystyle-T\sum_{k\sigma}(\ln(1+e^{-\beta
E_{k\sigma+}^{c}})+\ln(1+e^{-\beta E_{k\sigma-}^{c}}))$ (44) $\displaystyle-$
$\displaystyle T\sum_{k\sigma}(\ln(1+e^{-\beta
E_{k\sigma+}^{f}})+\ln(1+e^{-\beta E_{k\sigma-}^{f}}))$ $\displaystyle+$
$\displaystyle 2N_{s}(Jm_{c}m_{f}-\lambda).$

In this formula, $\beta=1/T$ is the inverse temperature. Now, minimizing free
energy $f$ ($\frac{\partial f}{\partial m_{c}}=\frac{\partial f}{\partial
m_{f}}=0$) leads to mean-field equations for $m_{c}$ and $m_{f}$:

$\displaystyle
m_{f}=\frac{1}{4N_{s}}\sum_{k\sigma}\sigma(f_{F}(E_{k\sigma-}^{f})-f_{F}(E_{k\sigma+}^{f}))$
$\displaystyle m_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{4N_{s}}\sum_{k\sigma}\sigma\frac{\varepsilon_{k}^{AA}-\varepsilon_{k}^{BB}-Jm_{f}\sigma}{\sqrt{(\varepsilon_{k}^{AA}-\varepsilon_{k}^{BB}-Jm_{f}\sigma)^{2}+4\varepsilon_{k}^{2}}}$
(45) $\displaystyle\times$
$\displaystyle(f_{F}(E_{k\sigma+}^{c})-f_{F}(E_{k\sigma-}^{c}))$

Note that $f_{F}(x)=1/(e^{x/T}+1)$ is the standard Fermi distribution
function.

The equations for $n_{c}$ and the constraint are also given,

$\displaystyle
n_{c}=\frac{1}{2N_{s}}\sum_{k\sigma}(f_{F}(E_{k\sigma+}^{c})+f_{F}(E_{k\sigma-}^{c}))$
$\displaystyle
1=\frac{1}{2N_{s}}\sum_{k\sigma}(f_{F}(E_{k\sigma-}^{f})+f_{F}(E_{k\sigma+}^{f})).$
(46)

For example, if we set $t=1,t_{+}=0.57,t_{-}=0.03,J=0.3,n_{c}=1$ at zero
temperature, by solving Eqs. 45 and 46, the resultant quasiparticle bands have
been plotted in Fig. 3. (the flat bands are not shown) It is amusing to see
that in different from the insulating nature of Kondo state at half-filling,
the AM state is metallic and satisfies the Luttinger theorem. (see Appendix. B
for a brief discussion on Luttinger theoremLuttinger1960 ; Oshikawa2000 ;
Dzyaloshinskii2003 ) Therefore, the transition from AM to Kondo insulator (if
it occurs) behaves as not only a magnetic-paramagnetic transition but also a
metal-insulator transition.

Naively, a finite-temperature phase transition out of AM may be obtained if
one solves Eqs. 45 and 46 for $T>0$. However, the Mermin-Hohenberg theorem
prohibits this putative thermodynamic transition since the KLM has the
continuous $SU(2)$ spin symmetry. Instead, we expect such finite-$T$ AM
transition appears in a three-dimensional KLM or a two-dimensional anisotropic
KLM (Ising-Kondo lattice) with alternating NNNH.WWYang2019b ; WWYang2020 ;
WWYang2021

###  III.4 Coexistent state

After learning the properties of Kondo and AM states, we turn to the case of
their coexistence. The coexistent state is expected to be stable in the
intermediate regime of $J$ and it is indeed found in our mean-field
calculation.

Fig. 1(b) shows all three kinds of ground-states and the corresponding order
parameters are exhibited in Fig. 1(c), particularly, the coexistent phase of
Kondo and AM occupies a finite parameter regime.

Since $V,m_{c},m_{f}$ are all nonzero in the coexistent phase, we are unable
to derive explicit formulas for quasiparticle bands and mean-field equations,
thus certain numerical iterative algorithm has to be used. A case study has
been shown in Fig. 5, where all of the eight quasiparticle bands
$E_{k\uparrow},E_{k\downarrow}$ are plotted. As expected, these bands are
spin-splitting as a result of non-vanishing contribution of AM order. At the
same time, they are all dispersive and have no flat bands. The fact is due to
the hybridization of Kondo screening in the coexistent state. The
hybridization mixes the original flat bands in AM with dispersive bands of
conduction electrons. Additionally, the $C_{4z}$ symmetry is preserved as in
the AM case, so the coexistent state also realizes a $d$-wave AM.

Figure 5:  The spin-splitting bands in the coexistent state of KLM. The
$C_{4z}$ symmetry still works as in AM case
($t=1,t_{+}=0.57,t_{-}=0.03,J=1.5,n_{c}=1,T=0$)

An interesting feature from Fig. 1(c) is that the order parameters have a jump
behavior around $J/t=1.76$. By examining the quasiparticle bands and the Fermi
surface, we find such jump is a result of Lifshitz transition, where the
topology of Fermi surface radically changes but without any symmetry-
breaking.Continentino  (Fig. 6) (Fermi surface centred around
$(\pm\pi/2,\pm\pi/2)$ transits into the one centred at $(0,0)$) This Lifshitz
transition is different from the short-ranged antiferromagnetic correlation-
driven counterpart in Kondo-Heisenberg model.GMZhang2011  Interestingly, the
spin-splitting Fermi surface (Fig. 6(b)) after the Lifshitz transition has the
similar structure of $d_{xy}$-wave ($\sim\sigma\sin k_{x}\sin k_{y}$), which
is relevant to the AM material RuO2.Smejkal2022

Figure 6:  The Fermi surface of quasiparticle in the coexistent state.
(a)$J/t=1.75$ versus (b) $J/t=1.8$. ($t=1,t_{+}=0.57,t_{-}=0.03,n_{c}=1,T=0$)

After all, we have uncovered $d$-wave AM-like phases in the half-filled KLM.
The defining feature of AM, the spin-splitting bands are unambiguously
observed in our mean-field calculation. We emphasize that these states are
driven by the antiferromagnetic magnetic order of local spin moments. If local
spin moments lose the antiferromagnetic order (due to magnetic/geometric
frustration), no AM-like states will survive. For example, when local spin
moments form valence-bond solids or quantum spin liquids,Zhou2017  the
antiferromagnetic long-ranged order is totally lost and the system may be in
the fractionalized Fermi liquid.Senthil2003

###  III.5 Phase transitions in phase diagram

We have clarified basic features of three candidate ground-states. From Fig.
1, there are transitions between AM and coexistent state, coexistent state and
Kondo state.

The phase transition between AM and coexistent state is characterized by the
appearance of Kondo screening ($V\neq 0$). The corresponding effective field
theory may be the one established in the study of Kondo-breakdown mechanism
although the effect of the magnetic background is not clear.Senthil2004 ;
Paul2007 ; Vojta2010 ; Zhong2012  (Frankly speaking, a small dispersion of
Abriksov fermions has to be added if one considers this seriously.) As
discussed in Appendix. C, the corresponding effective action is

$S_{eff}=\sum_{\Omega_{n},q}\left(J-(J/2)^{2}\chi_{0}^{\alpha}(q,\Omega_{n})\right)|V_{q\alpha}(\Omega_{n})|^{2}$
(47)

where $V_{q\alpha}(\Omega_{n})$ denotes the Kondo order field. In the sense of
Landau symmetry-breaking theory, we may treat Kondo order field as the order
parameter and its condensation leads to the coexistent state, where the Fermi
surface is enlarged by hybridizing with Abriksov fermions.Senthil2004 ;
Paul2007 ; Vojta2010 ; Zhong2012  In contrast, if $V$ does not condense, the
system is in the AM state and the Abriksov fermions have no contribution to
the Fermi surface. In light of above features, AM may be called a local moment
magnet while the coexistent state is an itinerant magnet. This fact underlies
that $f$-electron is localized in the former case and it hybridizes with
conduction electron in the latter one.

Finally, the coexistent state transits into the Kondo state when $J$ is large.
In this situation, magnetic order parameters $m_{c},m_{f}$ gradually decrease
and vanish at the second-order transition point. When magnetic fluctuations
are included, it is expected that this transition can be described by the
classic Hertz-Millis theory with the dynamic critical exponent $z=2$.Hertz ;
Millis  Here, $z=2$ is related to the critical Landau-damped paramagnetic
excitation around Fermi surface with characteristic wavevector $Q=(\pi,\pi)$.

Before ending this subsection, we note that the pure AM state, predicted by
the present mean-field theory, may disappear if one considers its counterpart
in KLM without NNNH on square lattice.Raczkowski2022  If such AM state indeed
exists, there will be a Kondo-breakdown transition between AM and the
coexistent state.Senthil2004 ; Paul2007 ; Vojta2010 ; Zhong2012  This non-
Landau symmetry-breaking phase transition is more possibly realized on
honeycomb lattice,Zhong2013 ; YLiu2017  where low-lying Dirac fermions are
susceptible to fluctuations neglected in the mean-field approximation.

###  III.6 Some observable

Except the spin-splitting bands and Fermi surface, let us inspect other
physical observable for the states involving AM orders.

####  III.6.1 Spin-resolved density of states

The spin-resolved density of states is defined by

$\displaystyle N_{\sigma}(\omega)=-\frac{1}{\pi
N_{s}}\sum_{k}\mathrm{Tr}\mathrm{Im}G_{\sigma}^{R}(k,\omega)$

and the retarded Green’s function is defined as

$\displaystyle G_{\sigma}^{R}(k,\omega)=(\omega+i0^{+}-H_{\sigma}(k))^{-1},$
(49)

which can be detected by the state-of-art spin-resolved scanning tunnel
microscope (STM).Wiesendanger2009 ; Bode2003  However, due to the intrinsic
$C_{4z}$ symmetry in $d$-wave AM-like states, one finds
$N_{\uparrow}(\omega)=N_{\downarrow}(\omega)$. Thus, it seems that the spin-
resolved density of states is not able to identify AM-like phases.

####  III.6.2 Spin-resolved conductivity

The spin-splitting bands are responsible for the spin-resolved conductivity in
AM-like phases, which is essential to spintronics, such as the spin splitting
torque effect.Smejkal2022 ; Bai2022 ; Feng2024  According to the linear-
response theory, the zero-temperature conductivity has the following
expression,

$\sigma_{xx}^{\sigma}=\frac{e^{2}}{\hbar}\frac{1}{N_{s}}\sum_{k}\left[\partial_{k_{x}}H_{\sigma}(k)A_{\sigma}(k)\partial_{k_{x}}H_{\sigma}(k)A_{\sigma}(k)\right].$
(50)

Here, the zero-frequency spectral function
$A_{\sigma}(k)=-\frac{1}{\pi}\mathrm{Im}G_{\sigma}^{R}(k,\omega=0)$.
$\sigma_{yy}^{\sigma}$ will have similar formula. In the AM state with
$J/t=0.3$, we find $\sigma_{xx}^{\uparrow}=1.5250\frac{e^{2}}{\hbar}$,
$\sigma_{xx}^{\downarrow}=1.5237\frac{e^{2}}{\hbar}$. The coexistent state
($J/t=1.5$) gives $\sigma_{xx}^{\uparrow}=1.9884\frac{e^{2}}{\hbar}$ and
$\sigma_{xx}^{\downarrow}=1.9856\frac{e^{2}}{\hbar}$. (a damping factor
$\Gamma=0.01t$ is used to obtain finite conductivity and the system size is
$300\times 300$) $\sigma_{xx}^{\uparrow}\neq\sigma_{xx}^{\downarrow}$ implies
a nonzero spin polarized current, thus heavy fermion compounds with AM-like
states have the potential for spintronics applications.

####  III.6.3 Distribution of conduction electron

The distribution function of conduction electron
$n_{k\sigma}^{c}=\langle\hat{c}_{kA\sigma}^{{\dagger}}\hat{c}_{kA\sigma}\rangle+\langle\hat{c}_{kB\sigma}^{{\dagger}}\hat{c}_{kB\sigma}\rangle$
is shown in Figs. 7 and 8. It is evident that $n_{k\sigma}^{c}$ is spin-
resolved in AM-like phases and gives the same information just as the spin-
splitting Fermi surface. The advantage of distribution function over the spin-
splitting Fermi surface/bands is that encoding the latter requires calculating
Green’s function of conduction electron, which is rather demanding for many-
body techniques due to its frequency-dependence. In contrast, calculating
$n_{k\sigma}^{c}$ is much easier.

Figure 7:  The distribution function of conduction electron (a)
$n_{k\uparrow}^{c}$ and (b) $n_{k\downarrow}^{c}$ in the AM
state.($t=1,t_{+}=0.57,t_{-}=0.03,J=0.3,n_{c}=1,T=0$) Figure 8:  The
distribution function of conduction electron (a) $n_{k\uparrow}^{c}$ and (b)
$n_{k\downarrow}^{c}$ in the coexistent
state.($t=1,t_{+}=0.57,t_{-}=0.03,J=1.5,n_{c}=1,T=0$)

##  IV Discussion

###  IV.1 Implication for experiment and realistic materials: how to detect AM
phases in heavy fermions

The key feature of AM-like phases in KLM or generic heavy fermion systems is
obviously the spin-splitting bands. These bands can be observed by ARPES as in
the case of $d$-electron compounds RuO2 and MnTe.Krempasky2024 ; Fedchenko2024
; Lee2024 ; Osumi2024

Because the AM-like states are driven by antiferromagnetic magnetic order of
local spin moments, exploring AM in heavy fermions can be considered as a task
to search antiferromagnetic heavy fermion compounds. However, the magnetic
transition temperature of heavy fermion materials is rather low (usually
several Kelvin),Kaczorowski2010 ; YLuo2011  which is a challenge for current
ARPES techniques.

In contrast, the low-temperature magnetic quantum oscillation widely used in
heavy fermion materials is able to probe the Landau quantization of
quasiparticles in applied magnetic field induced by the spin-splitting Fermi
surface, related to the spin-splitting bands.Shoenberg ; Tan2015 ;
Maksimovic2022  Thus, AM-like phases are more likely to be detected by the
quantum oscillation measurements, performed on thermodynamics and/or
transport.ZXLi2024

Additionally, the low-temperature transport measurement in heavy fermion
materials is standard, so AM-like phases could be detected by
charge/spin/thermal transport just like the spin-resolved conductivity
calculated in last section.

###  IV.2 Alternative theoretical approach

####  IV.2.1 Some mean-field theories

In the main text, we have used fermionic large-N mean-field theory to explore
AM-like phase in KLM on square lattice. Other mean-field theories like bond-
operator and Schwinger boson may also work.Jurecka2001 ; Eder2018 ;
Komijani2018 ; JWang2021 ; RHan2021  There seems to be no pure
antiferromagnetic phase in the bond-operator approach, so we expect AM to
coexist with Kondo screening in this situation.

The (dynamic) large-N theory of Schwinger boson has the same spirit with the
more conventional non-crossing approximation, which is widely used in the
study of single impurity Anderson/Kondo model.Hewson1993  It is not clear to
us whether it will work well for the coexistent state with both AM and Kondo
screening.

####  IV.2.2 Quantum Monte Carlo simulation

Theoretical tools beyond mean-field approximation are required if one desires
to have more exact information about possible AM and related phases. The
numerically exact determinant quantum Monte Carlo simulation works only for
half-filled KLM without NNNH.Assaad1999  When NNNH is nonzero, particle-hole
symmetry breaks and the corresponding fermion sign-problem prevents reliable
calculations. In this direction, we note that the constrained path Monte Carlo
(CPMC) method may be applicable if one starts with a well-tested trial
wavefunction.SZhang1995 ; MQin2020

A promising way to search AM-like phase is to study the following periodic
Anderson model with alternating NNNH,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle-t\sum_{i,\delta,\sigma}(\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta,B\sigma}+\hat{c}_{i+\delta,B\sigma}^{{\dagger}}\hat{c}_{iA\sigma})$
(51) $\displaystyle-$
$\displaystyle\sum_{i,\delta_{1}^{\prime},\sigma}(t_{+}\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta_{1}^{\prime},A\sigma}+t_{-}\hat{c}_{iB\sigma}^{{\dagger}}\hat{c}_{i+\delta_{1}^{\prime},B\sigma})$
$\displaystyle-$
$\displaystyle\sum_{i,\delta_{2}^{\prime},\sigma}(t_{-}\hat{c}_{iA\sigma}^{{\dagger}}\hat{c}_{i+\delta_{2}^{\prime},A\sigma}+t_{+}\hat{c}_{iB\sigma}^{{\dagger}}\hat{c}_{i+\delta_{2}^{\prime},B\sigma})$
$\displaystyle+$ $\displaystyle
E_{f}\sum_{i,\sigma}(\hat{f}_{iA\sigma}^{{\dagger}}\hat{f}_{iA\sigma}+\hat{f}_{iB\sigma}^{{\dagger}}\hat{f}_{iB\sigma})$
$\displaystyle+$ $\displaystyle
U\sum_{i}(\hat{f}_{iA\uparrow}^{{\dagger}}\hat{f}_{iA\uparrow}\hat{f}_{iA\downarrow}^{{\dagger}}\hat{f}_{iA\downarrow}+\hat{f}_{iB\uparrow}^{{\dagger}}\hat{f}_{iB\uparrow}\hat{f}_{iB\downarrow}^{{\dagger}}\hat{f}_{iB\downarrow})$
$\displaystyle+$ $\displaystyle
V\sum_{i,\sigma}(\hat{f}_{iA\sigma}^{{\dagger}}\hat{c}_{iA\sigma}+\hat{f}_{iB\sigma}^{{\dagger}}\hat{c}_{iB\sigma}+h.c.).$

Here, $\hat{f}_{iA\sigma},\hat{f}_{iB\sigma}$ represent the local $f$-electron
orbit. The $U$ term is the familiar Hubbard interaction for on-site
$f$-electrons and $E_{f}$ is the energy level of $f$-electron. A hybridization
$V$ between conduction electron and $f$-electron is crucial since it leads to
Kondo screening and other many-body effect. In the so-called Kondo limit
($E_{f}\ll E_{F}\ll E_{f}+U,V\ll U$ with $E_{F}$ being Fermi energy), the
above model can be mapped into KLM we have studied in the main text. We think
it is highly desirable to search the physics of AM in above model in terms of
the CPMC method.

##  V Conclusion and Future direction

In conclusion, we have uncovered AM-like phases in a prototypical model of
heavy fermion, the KLM on square lattice with an alternating NNNH. The
introduction of alternating NNNH takes nonmagnetic atoms (neglected in usual
antiferromagnetism study) into account and it underlies the $\pi/2$-rotation
in addition to the well-understood time-reversal and translation operation in
usual antiferromagnetism. The $d$-wave AM-like magnetic states are
characterized by their spin-splitting bands, Fermi surface, spin-resolved
distribution function and conductivity. Inspired by AM materials such as RuO2
with rotation-connected sublattices, we believe searching AM states in heavy
fermion compounds is promising when low-temperature transport and magnetic
quantum oscillation measurements are performed on antiferromagnetic heavy
fermion materials with rotation-connected sublattices.

When Heisenberg interaction between local moments is added, KLM is generalized
to the Kondo-Heisenberg model. Extension of our mean-field theory should
predict the existence of superconductivity due to the resonance-valence-bond
mechanism and its interplay with AM order will lead to topological
superconductivity.Zhu2023  This interesting issue will be explored in our
future work. Although AM phases found in the present work root on a specific
mean-field calculation, we hope they could be discovered in real-life
$f$-electron compounds.

## Acknowledgments

We thank Yu Li for bringing the author’s attention to the topic of
altermagnet. The work is partly supported by the National Key Research and
Development Program of China (Grant No. 2022YFA1402704) and the programs for
NSFC of China (Grant No. 11834005, Grant No. 12247101). This research was
supported in part by Supercomputing Center of Lanzhou University. We thank the
Supercomputing Center of Lanzhou University for allocation of CPU time.

##  Appendix A Examples with other $t_{+},t_{-}$

###  A.1 $t_{+}=t_{-}=0$

If the NNNH is turned off, ($t_{+}=t_{-}=0$) the model is just the symmetric
KLM on square lattice. The mean-field phase diagram and order parameters at
half-filling are shown in Fig. 9. Here, AFM denotes the conventional
antiferromagnetic state and it can coexist with the Kondo screening state. The
critical point between the coexistent state and the Kondo state is around
$J/t\simeq 2.1$, which is not too different from the exact value $J/t=1.45\pm
0.05$ extracted from the quantum Monte Carlo simulation.Assaad1999  (Note that
a different large-N mean-field theory used in Ref. Li2015,  gives a much
larger value $J/t\simeq 3.4$.) In this case, we have checked that all
quasiparticle bands in all three phases are not spin-splitting and no AM-like
phases appear as expected.

Figure 9:  Order parameters and phase diagram in half-filled KLM without NNNH.
($t=1,t_{+}=t_{-}=0,n_{c}=1,T=0$)

###  A.2 $t_{+}=0.4,t_{-}=0.2$

Then, let us consider NNNH is not zero and we set $t_{+}=0.4,t_{-}=0.2$. In
Fig. 10, we see that there exist three kinds of states just as the findings in
the main text. The AM and the coexistent states are also $d$-wave-like, as
what can be seen from Figs. 11 and 12. Additionally, a Lifshitz transition
appears around $J/t=1.51$, which is also found in the main text. Therefore, we
believe the results in the main text is robust for generic parameters.

Figure 10:  Order parameters and phase diagram in half-filled KLM without
NNNH. ($t=1,t_{+}=0.4,t_{-}=0.2,n_{c}=1,T=0$) Figure 11:  The spin-splitting
bands in the AM state of KLM. The $C_{4z}$ symmetry of bands implies a
$d$-wave AM. ($t=1,t_{+}=0.4,t_{-}=0.2,J=0.3,n_{c}=1,T=0$) Figure 12:  The
spin-splitting bands in the coexistent state of KLM. The $C_{4z}$ symmetry of
bands implies a $d$-wave AM. ($t=1,t_{+}=0.4,t_{-}=0.2,J=1.2,n_{c}=1,T=0$)

##  Appendix B Green’s function and spin susceptibility in the pure AM state

In the pure AM state without Kondo screening, the Green’s function of
conduction electron on the $A$-sublattice is easy to find as

$\displaystyle G_{A\sigma}^{c}(k,\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{i\omega_{n}-(\varepsilon_{k}^{BB}-\mu+J\sigma
S/2)}{(i\omega_{n}-E_{k\sigma+})(i\omega_{n}-E_{k\sigma-})}$ (52)
$\displaystyle=$
$\displaystyle\frac{\mu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma+}}+\frac{\nu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma-}},$

while the Green’s function on the $B$-sublattice is

$G_{B\sigma}^{c}(k,\omega_{n})=\frac{\nu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma+}}+\frac{\mu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma-}}.$
(53)

Here, the coherent factors are defined as

$\mu_{k\sigma}^{2}=\frac{E_{k\sigma+}-(\varepsilon_{k}^{BB}-\mu+J\sigma
S/2)}{E_{k\sigma+}-E_{k\sigma-}},$ (54)
$\nu_{k\sigma}^{2}=\frac{(\varepsilon_{k}^{BB}-\mu+J\sigma
S/2)-E_{k\sigma-}}{E_{k\sigma+}-E_{k\sigma-}},$ (55)

and they satisfy $\mu_{k\sigma}^{2}+\nu_{k\sigma}^{2}=1$. The poles of
$G_{A\sigma}^{c}(k,\omega_{n}),G_{b\sigma}^{c}(k,\omega_{n})$ determine the
quasiparticle bands. The zero-frequency part of these Green’s function
contributes to the Luttinger theoremLuttinger1960 ; Oshikawa2000 ;
Dzyaloshinskii2003

$n_{c}=\frac{1}{N_{s}}\sum_{k\sigma}\theta(G_{A\sigma}^{c}(k,0))=\frac{1}{N_{s}}\sum_{k\sigma}\theta(G_{B\sigma}^{c}(k,0)),$
(56)

where $\theta(x)$ is the unit-step function. ($\theta(x)=1$ for $x>0$, others
it gives zero) We have checked that the Luttinger theorem is satisfied in the
pure AM phase.

The (dynamic) spin susceptibility of conduction electron on $A$-sublattice is

$\chi_{A}^{-+}(q,\Omega_{n})=-\frac{1}{N_{s}}\sum_{k}\frac{1}{\beta}\sum_{\omega_{n}}G_{A\uparrow}^{c}(k,\omega_{n})G_{A\downarrow}^{c}(k+q,\omega_{n}+\omega_{n}),$
(57)

and its explicit expression is the following one (using
$G_{A\sigma}^{c}(k,\omega_{n})$ and summing over frequency $\omega_{n}$)

$\displaystyle\chi_{A}^{-+}(q,\Omega_{n})$ $\displaystyle=$
$\displaystyle-\frac{1}{N_{s}}\sum_{k}(\mu_{k\uparrow}^{2}\mu_{k+q\downarrow}^{2}\frac{f_{F}(E_{k\uparrow+})-f_{F}(E_{k+q\downarrow+})}{i\Omega_{n}-E_{k+q\downarrow+}+E_{k\uparrow+}}$
(58) $\displaystyle+$
$\displaystyle\nu_{k\uparrow}^{2}\nu_{k+q\downarrow}^{2}\frac{f_{F}(E_{k\uparrow-})-f_{F}(E_{k+q\downarrow-})}{i\Omega_{n}-E_{k+q\downarrow-}+E_{k\uparrow-}}$
$\displaystyle+$
$\displaystyle\mu_{k\uparrow}^{2}\nu_{k+q\downarrow}^{2}\frac{f_{F}(E_{k\uparrow+})-f_{F}(E_{k+q\downarrow-})}{i\Omega_{n}-E_{k+q\downarrow-}+E_{k\uparrow+}}$
$\displaystyle+$
$\displaystyle\nu_{k\uparrow}^{2}\mu_{k+q\downarrow}^{2}\frac{f_{F}(E_{k\uparrow-})-f_{F}(E_{k+q\downarrow+})}{i\Omega_{n}-E_{k+q\downarrow+}+E_{k\uparrow-}}).$

It is seen that all these terms correspond to inter-band transition.

##  Appendix C Field theory and effective action

Based on fermionic Hamiltonian Eqs. 3 and LABEL:eq5, one can obtain the
following imaginary-time action in terms of coherent states of fermions

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\sum_{j,\alpha=A,B,\sigma}\left(\bar{c}_{j\alpha\sigma}(\partial_{\tau}-\mu)c_{j\alpha\sigma}+\bar{f}_{j\alpha\sigma}(\partial_{\tau}+i\lambda_{j\alpha})f_{j\alpha\sigma}\right)-t\sum_{j,\delta,\sigma}(\bar{c}_{jA\sigma}c_{j+\delta,B\sigma}+\bar{c}_{j+\delta,B\sigma}c_{jA\sigma})$
(59) $\displaystyle-$
$\displaystyle\sum_{j,\delta_{1}^{\prime},\sigma}(t_{+}\bar{c}_{jA\sigma}c_{j+\delta_{1}^{\prime},A\sigma}+t_{-}\bar{c}_{jB\sigma}c_{j+\delta_{1}^{\prime},B\sigma})-\sum_{j,\delta_{2}^{\prime},\sigma}(t_{-}\bar{c}_{jA\sigma}c_{j+\delta_{2}^{\prime},A\sigma}+t_{+}\bar{c}_{jB\sigma}c_{j+\delta_{2}^{\prime},B\sigma})$
$\displaystyle+$
$\displaystyle\frac{J}{2}\sum_{j,\alpha}(\bar{c}_{j\alpha\uparrow}c_{j\alpha\downarrow}\bar{f}_{j\alpha\downarrow}f_{j\alpha\uparrow}+\bar{c}_{j\alpha\downarrow}c_{j\alpha\uparrow}\bar{f}_{j\alpha\uparrow}f_{j\alpha\downarrow})+\frac{J}{4}\sum_{j,\alpha,\sigma\sigma^{\prime}}\sigma\sigma^{\prime}\bar{c}_{j\alpha\sigma}c_{j\alpha\sigma}\bar{f}_{j\alpha\sigma^{\prime}}f_{j\alpha\sigma^{\prime}}-i\sum_{i,\alpha=A,B}\lambda_{i\alpha},$

where $\lambda_{j\alpha}$ is a dynamic field whose effect is to decode the
constraint $\hat{f}_{j\alpha\sigma}^{{\dagger}}\hat{f}_{j\alpha\sigma}=1$. The
partition function of the system is

$\mathcal{Z}=\int\mathcal{D}\bar{c}\mathcal{D}c\mathcal{D}\bar{f}\mathcal{D}f\mathcal{D}\lambda
e^{-S}.$ (60)

In order to match with mean-field theory in the main text, one uses the
Hubbard-Stratonovich transformation and the resulting action reads,Coleman2015

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\sum_{j,\alpha=A,B,\sigma}\left(\bar{c}_{j\alpha\sigma}(\partial_{\tau}-\mu)c_{j\alpha\sigma}+\bar{f}_{j\alpha\sigma}(\partial_{\tau}+i\lambda_{j\alpha})f_{j\alpha\sigma}\right)-t\sum_{j,\delta,\sigma}(\bar{c}_{jA\sigma}c_{j+\delta,B\sigma}+\bar{c}_{j+\delta,B\sigma}c_{jA\sigma})$
(61) $\displaystyle-$
$\displaystyle\sum_{j,\delta_{1}^{\prime},\sigma}(t_{+}\bar{c}_{jA\sigma}c_{j+\delta_{1}^{\prime},A\sigma}+t_{-}\bar{c}_{jB\sigma}c_{j+\delta_{1}^{\prime},B\sigma})-\sum_{j,\delta_{2}^{\prime},\sigma}(t_{-}\bar{c}_{jA\sigma}c_{j+\delta_{2}^{\prime},A\sigma}+t_{+}\bar{c}_{jB\sigma}c_{j+\delta_{2}^{\prime},B\sigma})$
$\displaystyle-$
$\displaystyle\frac{J}{2}\sum_{j,\alpha}(V_{j\alpha\uparrow}\bar{f}_{j\alpha\downarrow}c_{j\alpha\downarrow}+V_{j\alpha\downarrow}^{\ast}\bar{c}_{j\alpha\uparrow}f_{j\alpha\uparrow}-V_{j\alpha\downarrow}^{\ast}V_{j\alpha\uparrow})-\frac{J}{2}\sum_{j,\alpha}(V_{j\alpha\downarrow}\bar{f}_{j\alpha\uparrow}c_{j\alpha\uparrow}+V_{j\alpha\uparrow}^{\ast}\bar{c}_{j\alpha\downarrow}f_{j\alpha\downarrow}-V_{j\alpha\uparrow}^{\ast}V_{j\alpha\downarrow})$
$\displaystyle+$ $\displaystyle\frac{J}{4}\sum_{j,\alpha,\sigma}(\sigma
m_{j\alpha}^{c}\bar{f}_{j\alpha\sigma}f_{j\alpha\sigma}+\sigma
m_{j\alpha}^{f}\bar{c}_{j\alpha\sigma}c_{j\alpha\sigma}-m_{j\alpha}^{c}m_{j\alpha}^{f})-i\sum_{i,\alpha=A,B}\lambda_{i\alpha}.$

Note that this action is invariant under $U(1)$ gauge transformation
$f_{j\alpha\sigma}\rightarrow e^{i\theta_{j\alpha}}f_{j\alpha\sigma}$. Now, if
all dynamic fields are setting to be static,

$\displaystyle
V_{j\alpha\sigma}\rightarrow-V,~{}~{}i\lambda_{j\alpha}\rightarrow\lambda,~{}~{}m_{jA}^{c}\rightarrow
m^{c},~{}~{}m_{jB}^{c}\rightarrow-m^{c}$ $\displaystyle
m_{jA}^{f}\rightarrow-m^{f},~{}~{}m_{jB}^{f}\rightarrow m^{f},$ (62)

we recover the mean-field theory as expected.

When we discuss the phase transition between AM and the coexistent states, the
magnetic order is intact but the Kondo order is critical, thus the
corresponding action is

$S=S_{0}+S_{V}$ $\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\sum_{k\sigma}\bar{c}_{kA\sigma}(\partial_{\tau}-\mu+\varepsilon_{k}^{AA}-Jm_{f}\sigma/2)c_{kA\sigma}+\sum_{k\sigma}\bar{c}_{kB\sigma}(\partial_{\tau}-\mu+\varepsilon_{k}^{BB}+Jm_{f}\sigma/2)c_{kB\sigma}$
(63) $\displaystyle+$
$\displaystyle\sum_{k\sigma}\varepsilon_{k}(\bar{c}_{kA\sigma}c_{kB\sigma}+\bar{c}_{kB\sigma}c_{kA\sigma})+\sum_{k\sigma}\bar{f}_{kA\sigma}(\partial_{\tau}+\lambda+Jm_{c}\sigma/2)f_{kA\sigma}+\sum_{k\sigma}\bar{f}_{kB\sigma}(\partial_{\tau}+\lambda-
Jm_{c}\sigma/2)f_{kB\sigma}$

and

$\displaystyle S_{V}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\frac{J}{2}\sum_{j\alpha\sigma}\left(V_{j\alpha}\bar{f}_{j\alpha\sigma}c_{j\alpha\sigma}+V_{j\alpha}^{\ast}\bar{c}_{j\alpha\sigma}f_{j\alpha\sigma}+|V_{j\alpha}|^{2}\right)$
(64) $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\frac{J}{2}\sum_{kq\alpha\sigma}(V_{q\alpha}\bar{f}_{k+q,\alpha\sigma}c_{k\alpha\sigma}+V_{q\alpha}^{\ast}\bar{c}_{k-q,\alpha\sigma}f_{k\alpha\sigma}$
$\displaystyle+$ $\displaystyle\delta_{k=0}|V_{q\alpha}|^{2}).$

Here, the spin index in the Kondo order field $V,V^{\ast}$ is neglected as
guided by the mean-field theory. Generically, the interacting field theory
described by $S=S_{0}+S_{V}$ cannot be solved exactly and further
approximations has to be used.

In the sense of Landau symmetry-breaking theory, we may treat Kondo order
field as the order parameter and its condensation leads to the coexistent
state, where the Fermi surface is enlarged by hybridizing with Abriksov
fermions.Senthil2004 ; Paul2007 ; Vojta2010 ; Zhong2012  In contrast, if
$V,V^{\ast}$ do not condense, the system is in the AM state described by
$S_{0}$ and the Abriksov fermions have no contribution to the Fermi surface.

When, Kondo order field is nearly critical, one can integrate out all
fermionic degree of freedom and the resultant effective Gaussian action only
including Kondo order field,

$S_{eff}=\sum_{\Omega_{n},q}\left(J-(J/2)^{2}\chi_{0}^{\alpha}(q,\Omega_{n})\right)|V_{q\alpha}(\Omega_{n})|^{2}$
(65)

and the susceptibility $\chi_{0}^{\alpha}(q,\Omega_{n})$ is defined as

$\displaystyle\chi_{0}^{\alpha}(q,\Omega_{n})=-T\sum_{\omega_{n}}\sum_{k\sigma}G_{\alpha\sigma}^{c}(k,\omega_{n})G_{\alpha\sigma}^{f}(k+q,\omega_{n}+\Omega_{n}).$

The single-particle Green’s functions for conduction electron
$G_{\alpha\sigma}^{c}$ and Abriksov fermion $G_{\alpha\sigma}^{f}$ have the
following expressions (with the help of $S_{0}$)

$\displaystyle
G_{A\sigma}^{c}(k,\omega_{n})=\frac{i\omega_{n}-(\varepsilon_{k}^{BB}-\mu+Jm_{f}\sigma/2)}{(i\omega_{n}-E_{k\sigma+})(i\omega_{n}-E_{k\sigma-})}=\frac{\mu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma+}}+\frac{\nu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma-}}$
$\displaystyle
G_{B\sigma}^{c}(k,\omega_{n})=\frac{\nu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma+}}+\frac{\mu_{k\sigma}^{2}}{i\omega_{n}-E_{k\sigma-}}$
$\displaystyle G_{A\sigma}^{f}(k,\omega_{n})=\frac{1}{i\omega_{n}-\lambda-
Jm_{c}\sigma/2},~{}~{}G_{B\sigma}^{f}(k,\omega_{n})=\frac{1}{i\omega_{n}-\lambda+Jm_{c}\sigma/2}.$
(66)

The coherent factors are defined as

$\mu_{k\sigma}^{2}=1-\nu_{k\sigma}^{2}=\frac{E_{k\sigma+}-(\varepsilon_{k}^{BB}-\mu+Jm_{f}\sigma/2)}{E_{k\sigma+}-E_{k\sigma-}}.$
(67)

and the energy bands of quasiparticle in AM are

$\displaystyle E_{k\sigma\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\varepsilon^{AA}_{k}+\varepsilon^{BB}_{k}\pm\sqrt{(\varepsilon_{k}^{AA}-\varepsilon_{k}^{BB}-Jm_{f}\sigma)^{2}+4\varepsilon^{2}_{k}}\right)$
$\displaystyle-$ $\displaystyle\mu.$

Inserting Green’s function into the susceptibility
$\chi_{0}^{\alpha}(q,\Omega_{n})$ gives the explicit expression of effective
action of Kondo order field. If the Abriksov fermion has small dispersion
(induced by Heisenberg interaction between local spins), one finds that the
above action is similar to the one found in the study of Kondo-breakdown
mechanism for frustrated $f$-electron materials.Senthil2004 ; Paul2007 ;
Vojta2010 ; Zhong2012  However, it is still not clear that whether such
Gaussian action is able to capture the true feature of this transition, where
gapless fermions around Fermi surface couple to gapless bosons.Metlitski2010

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