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This work was concluded while visiting the ]Department of Physics, Zhejiang
Normal University, Jinhua 321004, Zhejiang, P. R. China.

# Suppressed Kondo screening in two-dimensional altermagnets

G. S. Diniz<EMAIL_ADDRESS>Curso de Física, ICET, Universidade Federal de
Jataí, Jataí, GO 75801-615, Brazil.  E. Vernek<EMAIL_ADDRESS>Instituto de
Física, Universidade Federal de Uberlândia, Uberlândia, MG 38400-902, Brazil
Department of Physics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P.
R. China.  [

(August 28, 2024)

###### Abstract

We have studied the Kondo effect of a spin-1/2 impurity coupled to a two-
dimensional altermagnet host material. To attain the low-temperature many-body
Kondo physics of the system, we have performed a numerical renormalization
group calculations that allows us to access the spectral properties of the
system at zero temperature. The impurity spectral function and the Kondo
temperature were calculated for different set of parameters, including Rashba
spin-orbit coupling (RSOC) and an external magnetic field. Interestingly, in
the RSOC and altermagnetic fields, the hybridization function is spin
independent, despite the characteristic broken time-reversal symmetry (TRS) of
the altermagnet. This is because the alternating sign of the spin splitting of
the bands in the momentum space renders equal contributions for both spin
components of the hybridization function. Our results demonstrate that,
although the hybridization function is TR symmetric, the Kondo temperature is
substantially suppressed by the altermagnet coupling. Moreover, we have
investigated the effect of an external magnetic field applied in the
altermagnet along different directions. Interestingly, we observe an important
restraining of the Kondo peak which depends strongly on the direction of the
field. This anisotropic effect is, however, masked if strong Zeeman splitting
takes place at the impurity, as it shatter the Kondo-singlet state.

##  I Introduction

Screening of localized magnetic moments by itinerant electrons in metallic
systems is paradigmatic in many-body physics in condensed matter. This
phenomenon, known as Kondo effect [1], is crucial for the understanding of
both electronic transport and magnetic properties of metallic systems doped
with magnetic atoms. The physical mechanism underlying this phenomenon was
first proposed by J. Kondo in 1963 [2], as spin-flip scatterings of conduction
electrons by magnetic impurities diluted in metallic systems, resulting in a
minimal resistivity observed at low temperatures [3]. After more than fifty
years of its discovery, Kondo effect has been studied in a variety of systems
resulting in an abundance of scientific studies [4, 5, 6, 7].

Since the Kondo effect results from an effective exchange interaction between
the localized moments and itinerant electrons, individual characteristics of
both may be equally important. Therefore, the conditions to which the
conduction electrons are subject to may modify the screening of localized
magnetic moments. For example, spin-polarized (ferromagnetic) bands are
detrimental to Kondo screening because time reversal symmetry (TRS) — which is
of key importance to spin-flip scattering process involved in the Kondo
mechanism — is broken [8, 9, 10]. Another example is the electron-phonon
interaction in the conduction band that results in a vanishing density of
states of the conduction band at the Fermi level and leading to rich Kondo
screening phases. More recently, Kondo screening of magnetic impurities by
spin-orbit coupled conduction electrons has been investigated [11, 12]. The
fundamental question in this context is how spin-orbit coupling (SOC) modifies
the Kondo temperature of the system [13].

Quite interestingly, although SOC produces important modification in the
energy bands of the host materials, numerical renormalization analysis have
shown that, at least in the high density regime, the effect is
straightforward. The increase of SOC produce a small widening of the
conduction band accompanied by a decreasing of the density of states at the
Fermi level. Nonetheless, the effective hybridization between the impurity and
the conduction electrons around the Fermi level decreases resulting in a
decreasing of $T_{K}$ [14]. It seems, therefore, that despite the fact that
SOC produces spin-splitting in the energy bands, because TRS is preserved, it
has a minor effect in the Kondo screening [15]. A natural follow up question
one may ask is whether this is the case for any system with spin-splitting
energy bands, for example, the recently discovered altermagnets [16, 17], with
peculiar TRS broken energy bands.

Altermagnets are a newly investigated class of collinear two- and three-
dimensional antiferromagnets that display a large broken spin degeneracy, even
in the absence of SOC [18, 19, 17]. Since the very first reports on these
materials, they are gaining considerable attention in the recent few years.
Most of these studies focus on the peculiar properties featured by this
emergent class of magnets materials, such as large anisotropic (non-
relativistic and) spin-splitting in momentum space with protected symmetry
points, accompanied by a zero (net) magnetization. This anisotropic spin-
splitting leads to the prediction of remarkable physical properties such as
gapless superconducting states with mirage gaps [20], “multi-piezo” effect in
${\rm V}_{2}{\rm Se}_{2}{\rm O}$ monolayer [21], spin-polarized Andreev levels
[22], Majorana zero modes with zero net magnetization [23], magnon bands with
an alternating chirality splitting [24], among many others exotic properties.

In this paper, we study the Kondo screening of magnetic impurities in
altermagnets. More specifically, we aim at investigating how the unusual
magnetic properties of these materials modify the Kondo temperature of the
system. To this end, we propose a spin-1/2 magnetic impurity coupled to an
altermagnet host material described formally by a traditional single-impurity
Anderson model (SIAM). To accomplish this, we employ the well-known numerical
renormalization group (NRG) [25] that allows us to obtain the relevant low-
temperature physical quantities. For instance, within NRG calculations we can
obtain the local density of states LDOS at the impurity site, from which we
estimate the Kondo temperature of the system. Our results shows that
intriguing band structure of the altermagnets has interesting consequences on
the Kondo physics: (i) despite the spin-splitted band due to broken TRS, there
is no spin splitting in the impurity spectral function. This is because the
integration of the self-energy over the entire Brilouin zone renders a spin-
independent hybridization function. (ii) By comparing the effects of AM and
RSOC to the Kondo temperature, we find that $T_{\rm K}$ is up to two order of
magnitude smaller for AM coupling as compared to RSOC, for the _same
strength_. (iii) In the presence of external magnetic field, $\bm{B}$, on the
altermagnetic-magnetic impurity system, the suppression on the Kondo peak
depends strongly on the direction of the applied field. This anisotropy is
akin to what was obtained by one of us in quantum impurity coupled to SOC
quantum wires [14].

The rest of this paper is structured as follows. In Sec. II we present the
model Hamiltonian and the procedure to calculate the hybridization function.
Section. III the NRG results are presented. We summarize and conclude in Sec.
IV.

##  II Model and methods

Figure 1: (a) Schematic picture showing a spin-1/2 impurity coupled to an
altermagnet material, which is deposited on a substrate and under an external
magnetic field represented by the red arrow. (b) The $E_{\pm}(k_{x},k_{y})$
for _pristine_ altermagnet ($\lambda$ and ${\bm{B}}$ are both zero) the zone
boundary $-\pi/a\leq k_{x(y)}\leq\pi/a$. (c) $E_{\pm}(k_{x},k_{y})$ for the
case when $\lambda\neq$0 and ${\bm{B}}=0$. (d) $E_{\pm}(k_{x},k_{y})$ for the
case when $\lambda$=0 and $\vec{B}=B\bm{\hat{\textbf{x}}}$ (or
${\bm{B}}=B{\bm{\hat{\textbf{y}}}}$ by symmetry). (e) Same as in (d), but for
a magnetic field along the ${\bm{B}}=B\bm{\hat{\textbf{z}}}$.

###  II.1 Hamiltonian model

We consider a hybrid system composed by a spin-$1/2$ impurity coupled to a
d-wave AM with RSOC, as illustrated in Fig. 1 (a). The total Hamiltonian of
the system can be written in the form of a SIAM as,

$\displaystyle H=H_{\rm 0}+H_{\rm imp}+H_{\rm hyb},$ (1)

where $H_{\rm 0}$, $H_{\rm imp}$ and $H_{\rm hyb}$ represents the host
altermagnet material, the magnetic impurity and the coupling between electrons
of the altermagnet and the magnetic impurity, respectively. More precisely,
$H_{\rm 0}=H_{\rm AM}+H_{\rm RSOC}+H_{\rm Z}$, where $H_{\rm AM}$ describes,
for instance, a d-wave altermagnet on a substrate [19, 17, 26]. $H_{\rm RSOC}$
accounts for the contribution of the Rashba SOC (RSOC) induced by an inversion
asymmetry, which can naturally arise due to the substrate on which the AM is
deposited. Finally, $H_{\rm Z}$ represents an external magnetic field. More
explicitly, each of these terms are given by

$\displaystyle H_{\rm AM}=2t(\cos k_{x}\\!+\\!\cos
k_{y})\sigma_{0}\\!+\\!2t_{AM}(\cos k_{x}\\!-\\!\cos k_{y})\sigma_{z},$
$\displaystyle H_{\rm RSOC}=2\lambda(\sin k_{y}\sigma_{x}-\sin
k_{x}\sigma_{y})=\bm{B}_{\rm SOC}({\bm{k}})\cdot{\bm{\sigma}},$ (2)
$\displaystyle H_{\rm Z}=g\bm{B}\cdot\bm{S}.$

Here, $t$ the hopping amplitude parameter and $t_{AM}$ stands for the AM
exchange interaction. We have written $H_{\rm RSOC}$ as an effective
$k$-depended _magnetic_ field ${\bm{B}}_{\rm SOC}({\bm{k}})=2\lambda(\sin
k_{y},-\sin k_{x})$ is induced by the Rashba field, with $\lambda$ standing
for the strength of the interaction. The last term of Eq. (II.1), $H_{Z}$,
describes the Zeeman effect in the AM material due to an external magnetic
field ${\bm{B}}$, in which $g$ is the effective $g$-factor of the AM. In all
these terms, $\sigma_{i}$ represents the Pauli matrices, including the
identity $\sigma_{0}$ and $\mathbf{S}=\hbar\bm{\sigma}/2$.

The second term of the Eq. (1) is given by,

$\displaystyle H_{\rm imp}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\varepsilon_{d}n_{\sigma}+Un_{\uparrow}n_{\downarrow}+g_{\rm
imp}\bm{B}\cdot\bm{S},$ (3)

where, $\varepsilon_{d}$ is the impurity on-site energy for spin $\sigma$
($\uparrow$ or $\downarrow$), $g_{\rm imp}$ is the impurity $g$-factor, $U$
represents the $e$-$e$ interaction at the impurity site, and
$n_{\sigma}=d^{\dagger}_{\sigma}d_{\sigma}$ is the electron number operator
with spin $\sigma$.

Finally, the third term of Eq. (1) has the form

$\displaystyle H_{\rm
hyb}\\!=\\!V\sum_{k\sigma}(d^{\dagger}_{\sigma}c_{k\sigma}+H.c.).$ (4)

where we have assumed for simplicity, that the coupling strength between the
impurity and the altermagnet is independent of momentum and spin component
[27].

###  II.2 Energy band properties

The main features of the electronic bands given by Eq. II.1 is shown in Fig.
1(b)-1(e), for different sets of parameter configurations. Fig. 1(b) shows the
_pristine_ case, which shows the alternating spin-splitting along the momentum
and with degenerate points at $\Gamma$ and also at the four corners of the
Brillouin zone. In Fig. 1 (c) we show the case where $\lambda\neq$ 0 (while
${\bm{B}}$=0), which fully breaks the spin degeneracy of the bands, except at
$\Gamma$ points and at the four corners. In Fig. 1(d)-1(e) a ${\bm{B}}$ is
considered along different directions (with $\lambda$=0), where now there is
no degeneracy at $\Gamma$ and at the boundaries of the Brillouin zone. Below,
we will return to the host band structure with a deeper analysis under
different interactions.

Figure 2: Energy bands along the $M$-$\Gamma$-$M^{\prime}$ path for different
parameters configurations: (a) ${\bm{B}}=0$, (b)
${\bm{B}}=0.05\bm{\hat{\textbf{x}}}$, (c) ${\bm{B}}=0.05\bm{\hat{\textbf{y}}}$
and (d) ${\bm{B}}=0.05\bm{\hat{\textbf{z}}}$. For all panels, $t$=0.5,
$t_{AM}$=0.1, $\lambda$=0.2, a Néel vector along the $z$-direction and the
color map is associated to the spin-projection, $\langle S_{i}\rangle$, for
each band. The insets correspond to each respective cases with $\lambda$=0.0.

Let us further analyze the host band structure in the presence of the external
magnetic field. For the sake of numerical convenience, throughout the paper we
will set, $g=k_{B}=1.0=\mu_{B}=\hbar=1.0$. Fig. 2(a)-2(d) show the electronic
band structure along the path $M$-$\Gamma$-$M^{\prime}$, which is a good
representative of the symmetries of the bands. In the absence of magnetic
field and RSOC, the system shows an intriguing broken TRS with a momentum
dependent alternating spin-splitting, as shown in Fig. 1(b). The band
degeneracy along the path $M$-$\Gamma$-$M^{\prime}$ is broken when RSOC is
considered. However, they are still degenerated at the $\Gamma$ and $M$
($M^{\prime}$) points, which is a consequence of the $C_{4z}\mathcal{T}$
symmetry that still holds [26]. This feature can be clearly seen in Fig. 2
(a), where we show the band structure for ${\bm{B}}$=0 and $\lambda=0.2$
($\lambda=0.0$ for the inset). Moreover, the spin projection $\langle
S_{z}\rangle$ is zero except at the spin-degenerate points, where the value is
exactly $\langle S_{z}\rangle$=0.5 (and -0.5). Indeed, the spin projection
would be non-zero (and equal, $\langle S_{x}\rangle$=$\langle S_{y}\rangle$)
only along the plane ($x-y$) (not shown in the panel). For an applied magnetic
field in the plane $x$-$y$, say, at the $x$-(or $y$-direction),
$C_{4z}\mathcal{T}$-symmetry is broken as shown in Fig. 2(b)-2(c). We observe
clear spin splittings of the bands at all points including $\Gamma$ and $M$
($M^{\prime}$) which acquires a finite gap of $2g|{\bm{B}}|$, even for the
$\lambda$=0 case (show in the insets). However, differently from the
$\lambda$=0, the presence of RSOC induces a spin mixing of the bands with a
broken inversion symmetry along $M$-$\Gamma$ and $\Gamma$-$M^{\prime}$ path.
Moreover, by alternating the direction of the in-plane ${\bm{B}}$ field, the
band structure around $\Gamma$ point is reversed, with significant changes in
the spin projection along the direction of the applied ${\bm{B}}$ for
$\lambda\neq 0$, with spin projections inverted at the $\Gamma$ an $M$
($M^{\prime}$) points. At last, in Fig. 2(d) we show the case when $\bm{B}$ is
applied along the $z$-direction. As previously observed, the system is TRS-
broken with a clear spin splitting. However, differently from the in-plane
case, the system still has a symmetry along the cut line $M$-$\Gamma$-$M$.
From these results, we can confirm that the interplay between RSOC and
magnetic field introduces important modifications to the altermagnet
electronic bands, with a relevant anisotropy that can be crucial for the Kondo
physics, as it will be further explored in the following sections.

###  II.3 Hybridization Function

The hybridization function of the impurity is of key importance in the NRG
approach to the Kondo impurity problem. To obtain it, we first calculate the
local impurity Green’s function ${\bm{G}}_{imp}(\omega)$, which is a $2\times
2$ matrix and can be written as [28, 14],

$\displaystyle{\bm{G}}_{\rm
imp}(\omega)\\!=\\!\left[(\varepsilon_{d}-\omega)\sigma_{0}-\bm{\Sigma}^{(0)}(\omega)-\bm{\Sigma}^{({\rm
int})}(\omega)\right]^{-1}.$ (5)

Here, $\bm{\Sigma}^{(0)}(\omega)=\sum_{k}\bm{V}\bm{G}_{\rm
host}({\bm{k}},\omega)\bm{V}^{\dagger}$ represents the non-interacting self-
energy [28], with $\hat{V}=V\sigma_{0}$ and $\bm{G}_{\rm
host}(k,\omega)=\left[\omega\sigma_{0}-H_{0}\right]^{-1}$, while
$\bm{\Sigma}^{(\rm int)}(\omega)$ is the interacting part (or proper) self-
energy. With the non-interacting self-energy, the hybridization function can
be written as,

$\displaystyle\bm{\Gamma}(\omega)\\!=\\!\frac{1}{2i}\\!\int\\!\\!\\!\int\left[\bm{\Sigma}^{(0)}({\bm{k}},\omega-i\eta)-\bm{\Sigma}^{(0)}(\bm{k}{},\omega+i\eta)\right]d^{2}k,$
(6)

where $\eta\rightarrow 0$. The integration defined in Eq. (6) is highly peaked
for small values of $\omega$, therefore a high precision numerical integration
is desirable to avoid numerical flaws. To achieve this purpose, we have used
cubature rules as implemented in the Cuba package [29]. Note that
$\bm{\Gamma}(\omega)$ is also a $2\times 2$ matrix and, in the presence of an
external in-plane magnetic field, there is a spin-mixing rendering non-
diagonal elements. To deal with the spin-mixing channels, a non-trivial
Wilsonian RG chain is required [14]. As the hybridization matrix has, in
general, non-zero off-diagonal complex elements, it is useful to decompose it
in terms of Pauli matrices as,
$\bm{\Gamma}(\omega)=\sum_{i}d_{i}(\omega)\sigma_{i}$ ($i$=0, x, y, z). For
convenience, we hereafter will rescale the bandwidth to unity ($D$=1), such
that the relevant energy range is defined as $-1<\omega<1$, a standard
procedure in impurity solvers. We also assume the impurity-altermagnet
coupling to be $V=$0.1.

Figure 3: Hybridization function $\hat{\Gamma}(\omega)$: (a) For
${\lambda}$=0.0 and $t_{AM}\neq$ 0; (b) $t_{AM}$=0.0 and $\lambda\neq$0\. For
both panels $\bm{B}$=0, and notice that we are showing the trace of
$d_{0}\sigma_{0}$, as the diagonal terms are equal and there is no off-
diagonal terms.

In Fig. 3 (a) and (b), we show $\Gamma(\omega)$ for different values of
$t_{AM}$ and $\lambda$, respectively, in absence of $\bm{B}$. Notice that
regardless of the value of $\lambda$ or $t_{AM}$, the diagonal terms for both
panels are equal, as there is no spin-mixing terms in the hybridization
matrix. Therefore, we only showed the trace of $d_{0}(\omega)\sigma_{0}\propto
2d_{0}$, which is proportional to the total density of states. When $t_{AM}$
is considered (with $\lambda$=0), one can observe a strong suppression of the
peak around $\omega=0$, with the development of a splitting. This suppression
will be responsible to signatures in the Kondo temperature, as it will be
discussed in the following section. For the case when RSOC is non-zero (with
$t_{AM}$=0), shown in Fig. 3 (b), we can also notice a peak suppression
similar to the Fig. 3 (a), however, the suppression is less pronounced, and
there are peaks at the edge associated to the van Hove singularities. Since
they are far away from from the Fermi level, these van Hove singularities at
the edges near the edge of conduction band is unimportant for the Kondo
screening. The more pronounced suppression of the hybridization function near
the Fermi level produced by $t_{\rm AM}$ anticipates the stronger suppression
of Kondo screening caused by the altermagnet as compared to RSOC.

Figure 4: Decomposition of the matrix hybridization function
$\hat{\Gamma}(\omega)$ along different Pauli matrices $\sigma_{i}$ with
corresponding weights $d_{i}$. (a) For ${\bm{B}}$=0, (b) for a magnetic field
along $\bm{\hat{\textbf{x}}}$, (c) for a magnetic field along
$\bm{\hat{\textbf{y}}}$ and (c) for a magnetic field along
$\bm{\hat{\textbf{z}}}$. The solid line is for $\lambda$=0, while the dashed
line stands for $\lambda$=0.2. For panels (b)-(d), we have assumed $t_{\rm
AM}=0.1$ and $B=0.02$.

Now, we set the value of $t_{AM}$=0.1 and investigate how the presence of
$\bm{B}$ affects the hybridization matrix. In Fig. 4, we show
$\bm{\Gamma}(\omega)$ decomposed for different parameter configuration of
${\bm{B}}$ and $\lambda$. In the absence of a magnetic field and RSOC
($\lambda$=0), $d_{0}$ is the only nonzero value as shown by the black solid
line in Fig. 4 (a). However, notice that this result holds even for the case
of $\lambda$=0.2 (dashed line). This behavior is because (i) the impurity is
spin-independently coupled to the host, (ii) the AM Hamiltonian [$H_{0}$ in
Eq. (1)] preserves $C_{4z}\mathcal{T}$ symmetry and SOC is time-reversal
invariant, rendering thus a diagonal and spin-independent
$\bm{\Gamma}(\omega)$. Nonetheless, notice that because $\lambda\neq$0, we can
still see the emergent peaks close to the edges of the conduction band
associated to the appearance of van Hove singularities in the conduction band
around the $M$-point. For an in-plane magnetic field Fig. 4 (b)-(c), the off-
diagonal elements of $\hat{\Gamma}(\omega)$ will contribute to non-zero values
for the coefficients $d_{y}$ ($d_{x}$) for a magnetic field applied along $x$
($y$) conversely, while there is also a finite value for $d_{z}$ that changes
sign, i.e. $d_{z}\rightarrow-d_{z}$. Interestingly, for
$\bm{B}\parallel\bm{\hat{\textbf{z}}}$ shown in Fig. 4 (d), the only nonzero
components are along $d_{0}$ and $d_{z}$, even when $\lambda\neq$0 (which
intertwine the spin channels but does not break TRS). Furthermore, notice that
because of the interplay between the AM exchange interaction and the Zeeman
field, there is a enhancement of the $d_{z}$ component, and also a marked
_double_ splitting in the peaks of $d_{0}$, which will strongly reflect in the
impurity spectral function.

##  III Interacting regime and Kondo screening

Having obtained the hybridization function $\Gamma(\omega)$, we are ready to
use the NRG method to address the Kondo effect in the system. This numerical
technique is a powerful and standard impurity solver, which consists of a
logarithmic discretization of the continuum conduction band in energy scales
that decreases a $\Lambda^{-N/2}$, where $\Lambda$ is a discretization
parameter and $N$ is the number of iteration in the NRG procedure. The
discretized model is then mapped into a one-dimensional tight-binding-like
Hamiltonian (aka Wilson chain Hamiltonian) whose length increase iteratively
with $N$. The iterative diagonalization of a large-$N$ Wilson chain is
possible by proper truncation of Hilbert space at certain maximum dimension,
allowing to access the energy around the Fermi level in a controllable manner
[30]. Here, for practical purpose, we employ the NRG method to calculate the
impurity retarded Green’s function, from which we calculate the impurity
density of states in the interacting regime. Within the Zubarev’s notation
[31], our Green’s function can be written as

$\displaystyle{\cal G}^{\sigma\sigma^{\prime}}_{\rm
imp}(\omega)=\langle\langle
d_{\sigma};d^{\dagger}_{\sigma^{\prime}}\rangle\rangle_{\omega},$ (7)

where $\langle\langle
d_{\sigma};d^{\dagger}_{\sigma^{\prime}}\rangle\rangle_{\omega}=-i\int\Theta(t)\langle\left[d_{\sigma}(t),d_{\sigma^{\prime}}^{\dagger}(0)\right]_{+}\rangle
e^{i\omega t}dt$. Here,
$\left[d_{\sigma}(t),d_{\sigma^{\prime}}^{\dagger}(0)\right]_{+}$ is the anti-
commutator between the annihilation operator at time $t$ with the creator
operator at $t=0$ at the impurity. The expectation value $\langle\cdot\rangle$
is taken in the ground state for zero temperature calculations.

The nontrivial energy dependence of the hybridization function requires an
improved discretization scheme to reproduce with high resolution the initial
function, thus reducing the numeric artifacts that can be present in the NRG
calculations. To this end, we have used the adaptive “z-averaging” scheme [32]
(with $N_{z}$=5), as implemented in the NRG Ljubljana open source code [33].
Furthermore, for the spectral functions calculations we have also used the
self-energy “trick”, as described in [34] to remove nonphysical oscillations
at the vicinity $\omega=0$. For all the NRG calculations, we have assumed
(unless stated otherwise) $\varepsilon_{d}=-U/2$ (particle-hole symmetry),
U=0.25, $\Lambda$=2 and number of kept states = 2000.

###  III.1 Kondo temperature

We start by investigating the effect of the AM exchange field in the Kondo
temperature of the system. The main quantity calculated here is the impurity
density of state,

$\displaystyle A_{\sigma}(\omega)=-\frac{1}{\pi}{\rm Im}[{\cal
G}^{\sigma\sigma}_{\rm imp}(\omega)].$ (8)

It is known that $A(\omega)=\sum_{\sigma}A_{\sigma}(\omega)$ exhibits a many-
body resonance at the Fermi level, known as Abrikosov-Shul resonance [35, 36]
(or simply Kondo peak), as a signature of the Kondo screening [1]. This
resonance not only signals the presence of Kondo screening in the system, but
also provides a useful way to estimate $T_{K}$. Indeed, the width of the Kondo
resonance can be directly associated to $T_{K}$ [37, 38, 39]. In Fig.
5(a)-5(d) we make an insightful analysis on the behavior of $T_{K}$, extracted
from the half-width at half-peak of the impurity spectral function, for
different configurations of $t_{AM}$ and $\lambda$, setting ${\bm{B}}=0$. Fig.
5(a) shows $A(\omega)$ for different values of $t_{AM}$. For $t_{AM}=0$ (black
line), we see that $A(\omega)$ exhibits a split peak around $\omega=0$. This
splitting is induced by the sharp peak in the hybridization function
$\Gamma(\omega)$ shown in the Fig. 3(a) for $t_{\rm AM}=0$ (black curve). This
splitting is analogous to the one observed in Ref. [40], resulting from an
Anderson impurity coupled to a structured host density of states provided by a
non-interacting energy level. Here, on the other hand, the sharp structure the
conduction band is naturally provided by the van Hove singularity of the two-
dimensional material. Now, as $t_{\rm AM}$ increases, we observe that Kondo
peak evolves rapidly to a single and sharper peak, as already observed for
$t_{\rm AM}=0.05$. This is because the sharp van Hove singularity is split for
finite $T_{\rm AM}$, leaving $\Gamma(\omega)$ relatively smooth near the Fermi
level, as seen in Fig. 3(a) for finite $t_{\rm AM}$. This prompt sharpening of
the Kondo resonance, as $t_{\rm AM}$ increases reveals a strong dependence of
$T_{\rm K}$ with $t_{\rm AM}$. Fig. 5(c) shows how $T_{\rm K}$ decreases as
$t_{\rm AM}$ increases as anticipated earlier in this paper.

Figure 5: (a) Impurity spectral function $A(\omega)$ for different values of
$t_{AM}$ with $\lambda$=0. (b) Impurity spectral function $A(\omega)$ for
different values of $\lambda$ with $t_{AM}$=0. (c) $T_{K}$ as function of
$t_{AM}$. (d) $T_{K}$ as function of $\lambda$. For all panels, $\bm{B}$ is
set to zero.

To compare the effects of $t_{\rm AM}$ to $\lambda$ on $T_{\rm K}$, in Fig.
5(b) shows how $A(\omega)$ is affected the spin-orbit coupling $\lambda$. We
now set $t_{\rm AM}=0$. We note that $\lambda$ produces, qualitatively, a very
similar to the effect of $t_{\rm AM}$, although much less pronounced, as
$\lambda$ produces a very modest splitting of the $\Delta(\omega)$. Indeed,
the effect of $t_{\rm AM}$ on $T_{\rm K}$ is much more important as compared
to the effect of $\lambda$. By comparing Fig. 5(c) and Fig. 5(d) we observe
that $t_{\rm AM}=0.3$ produces a suppression of $t_{\rm K}$ almost three order
of magnitude, while for the same value of $\lambda$ the suppression is about
one order of magnitude.

Figure 6: Impurity spin-resolved local density of states $A_{\sigma}(\omega)$
projected along the direction of applied magnetic field $\mathbf{B}$. (a)-(d)
for $\lambda=0$ and
$\bm{B}=0,\,0.002\bm{\hat{\textbf{x}}},\,0.002\bm{\hat{\textbf{y}}},\,0.002\bm{\hat{\textbf{z}}}$,
respectively. (e)-(h) are the same results, but for $\lambda=0.2$. Solid lines
(dashed) are for $g_{imp}$=0.0 ($g/g_{imp}$=2.5). The insets are zooms for the
respective panel in the $\omega$ range [-0.005, 0.005].

Here, we should mention that the dependence of $T_{\rm K}$ with $\lambda$ has
been subject of interest over the past decades, and it is still source of
debate in the literature [15]. For instance, Zarea et al. reported that away
from the particle-hole symmetry in a 2DEG, the parity breaking Dzyaloshinsky-
Moriya term is able to renormalize the antiferromagnetic Kondo coupling with
an exponential enhancement of $T_{K}$ [13]. A similar result has also been
found by one of us for a quantum wire, even at the particle-hole symmetric
point [41]. Interestingly, depending on how drastic is the modification of
conduction band, we can also observe a suppression of $T_{K}$, as the RSOC is
increased, as reported for a quantum wire in the absence of $\bm{B}$ [14].

Within the NRG approach, its is quite clear that the change in the Kondo
temperature is directly associated to the modification around the Fermi level
in the density of states of the host material induced by properties of the
band structure. Therefore, for a given fixed coupling $V$ as in Eq. (4),
enhanced density of the host material produces an enhancement of the $T_{\rm
K}$. A splitting of the Kondo peak is produced if $\rho(\omega)$ exhibit a
peak at $\omega=0$ whose width is similar or smaller than $T_{K}$. See for
instance the discussion in Ref. [40].

###  III.2 Effect of magnetic field

In Sec. II, we have seen that the effect of magnetic field applied in the
altermagnet is anisotropic. Therefore, anisotropic suppression of the Kondo
should be observed. To see this, in Fig. 6 we show the spin-resolved local
density of states (LDOS) at the impurity, $A_{\sigma}(\omega)$, for the
magnetic field applied (${B}=0.02$) along different directions. We have set
the $g$-factor ($g=1$) for the host material (the altermagnet). The first and
second rows show, respectively, results for $\lambda=0.0$ and $\lambda=0.2$
while solid and dashed lines corresponds to $g_{\rm imp}=0$ and $g_{\rm
imp}=0.4g$, respectively.

Let us start looking the Fig. 6 (a) where there is no magnetic field applied,
therefore all the projected component of $A_{\sigma}$ are equal to each other.
Interestingly, even though TRS is broken by AM exchange, because of the
unusual spin splitting in momentum space, both spin-components of the
$\bm{G}_{\rm host}$ contributes equally to the integral of Eq. 6. As expected,
an inversion symmetry breaking induced by RSOC does not split
$A_{\sigma}(\omega)$, as shown in Fig. 6 (e). Moreover, the results are almost
unaffected when RSOC is included. Indeed, as observed in the insets of Fig.
6(a) and Fig. 6(b), the major splitting in the hybridization function is
produced by $t_{\rm AM}$.

Now, intriguing behavior occur when the magnetic field is considered. In Fig.
6(b)-(d) and Fig.6(f)-6(h) show LDOS projected along the different directions
of applied $\bm{B}$. As such, for instance, if the magnetic field is applied
along the $x$-direction, $\uparrow$ and $\downarrow$ refer to spin pointing
towards positive and negating $x$-direction 111To obtain the LDOS projected
along different spin directions we calculate the corresponding Green’s
function, which can be obtained by combining appropriately the components
given by Eq. (7). . Having this in mind, we observe, that regardless the
direction of the applied $\bm{B}$, a suppression of the Kondo peak accompanied
by splitting of the Kondo peak. However, for an in-plane magnetic field, its
interplay with the AM term and with $\bm{B}_{\rm SOC}$ brings a remarkable
anisotropic effect in the LDOS as shown in Fig. 6(b)-6(c). Indeed, we see a
stronger suppression of $A_{\sigma}(\omega)$ as compared to $\bm{B}$ along the
$z$-direction in Fig. 6(d) and (h). This anisotropic effect is observed only
if the direct Zeeman effect is negligible in the impurity ($g_{\rm imp}\approx
0$). If we consider $g_{\rm imp}=0.4$ as shown in dashed lines, the Kondo
resonance peak is almost fully suppressed, as at this magnetic field strength
both spins (band and impurity) are fully polarized, thus completely destroying
the Kondo singlet state.

Figure 7: Impurity local density of states
$A(\omega)=\sum_{\sigma}A_{\sigma}(\omega)$ projected along the direction of
applied magnetic field $\mathbf{B}$. (a) $A(\omega)$ along
$\bm{\hat{\textbf{x}}}$ (or $\bm{\hat{\textbf{y}}}$, by symmetry) for $g_{\rm
imp}=0$ and $B=0,\,0.004,\,0.008,\,0.012,\,0.016$ and $0.020$, respectively.
(b) Same as in (a), but for $A(\omega)$ along $\bm{\hat{\textbf{z}}}$. (c)-(d)
are for $g/g_{\rm imp}=2.5$. Solid lines (dashed) are for $\lambda$=0.0
($\lambda=0.2$).

Before concluding, in Fig. 7 we show how the anisotropic magnetic response
manifests in the total density of states, $A(\omega)$. To this end, we show
$A(\omega)$ for different strengths and direction of the applied magnetic
field. Upper and lower panels correspond to $g_{\rm imp}=0$ and $g_{\rm
imp}=0.4g$. For $B=0$ the Kondo peak can be clearly seen as shown in black
line in Fig 7(a). Now, as $\bm{B}$ increases the Kondo peak is progressively
suppressed for magnetic field along either $\bm{\hat{\textbf{x}}}$,
$\bm{\hat{\textbf{y}}}$ and $\bm{\hat{\textbf{z}}}$ direction. As shown in
Fig. 7(a) and 7(b) for $g_{\rm imp}$=0. However, for field along $z$-direction
the peak is less affected, as compared to the other directions. Again, this a
signature of the anisotropic magnetic response of the system induced by the
interplay between $\bm{B}$, AM and RSOC. Finally, for $g_{\rm imp}=0.4g$, the
Kondo peak is strongly suppressed with an enhanced splitting, as shown in Fig.
7 (c)-(d), providing support that the system no longer sustain the Kondo
single state.

##  IV Conclusions

In summary, we have studied the Kondo screening effect in a spin-1/2 impurity
coupled to a two-dimensional altermagnet in the presence of external magnetic
field. By describing the system within a single-impurity Anderson Hamiltonian,
we determined the hybridization function by exact calculation of the local
Green’s function, which allows us to garner the complete influence of the host
material onto the impurity. We employ a numerical renormalization group
approach to calculate the spin-resolved local density of states in the
interacting regime, which allowed us to attain the low-temperature Kondo
physics in the system. Our results shows a drastic decrease of the Kondo
temperature $T_{\rm K}$ as the altermagnet coupling $t_{\rm AM}$ increases. By
comparing this suppression with the one produced by the RSOC $\lambda$, we
showed that the $t_{\rm AM}$ has a dominant effect in the suppression of
$T_{\rm K}$. Furthermore, despite the time-reversal symmetry broken by $t_{\rm
AM}$, it does not produce spin-spliting in the hybridization function. This is
a direct consequence of the alternating momentum-dependent band spin-splitting
that renders zero net magnetization to the impurity upon integration of the
self-energy over the entire Brillouin zone. Moreover, we have investigated the
effect of an applied magnetic field along different directions. The results
showed an important anisotropic response of the system to the applied field.
Indeed, a in-plane field has strong effect in Kondo screening marked by a
sizable suppression of Kondo peak. In contrast, this suppression is less
pronounced for a magnetic field along the $z$-direction. When the Zeeman
coupling takes place directly in the impurity, the Kondo peak is
overwhelmingly suppressed by the field and the anisotropic effect blurred. Our
work shed light on the Kondo correlation in altermagnets and paves the way for
future theoretical as well as experimental investigations of correlated
phenomena in these materials.

###### Acknowledgements.

The authors acknowledge financial support from CAPES, FAPEMIG and CNPq. EV
thanks FAPEMIG (Process PPM- 00631-17) and CNPq (Process 311366/2021-0). This
work used resources of the “Centro Nacional de Processamento de Alto
Desempenho em São Paulo (CENAPAD-SP).”

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  * Note [1] To obtain the LDOS projected along different spin directions we calculate the corresponding Green’s function, which can be obtained by combining appropriately the components given by Eq. (7).