# Anderson Impurities In Edge States with Nonlinear Dispersion

Vinayak M. Kulkarni  Theoretical Sciences Unit, Jawaharlal Nehru Centre for
Advanced Scientific Research Jakkur, Bangalore - 560064,India

###### Abstract

A non-linear dispersion can significantly impact the Kondo problem, resulting
in anomalous effects on electronic transport. By analyzing a special bath with
a $\theta=\frac{\pi}{3}$ symmetry rotation in the Brillouin zone or 3-fold
symmetry in momentum, we derive an effective spin-spin interacting model.
Combining the anisotropic Dzyaloshinskii-Moriya (DM) interaction with non-
linear dispersion can lead to exceptional points $(E_{p})$ in a Hermitian
model. Our RG analysis reveals that the spin relaxation time has the signature
of coalescence in momentum-resolved couplings and an ideal logarithmic
divergence in resistivity over a range of nonlinearity ($\beta$). The
effective model at the impurity subspace has a Lie group structure of Dirac
matrices. We show nontrivial renormalization within a Poorman approximation
with the inclusion of potential scattering, and the invariant obtained will
not be altered by potential scattering. We expand the model to a two-impurity
Kondo model and investigate the Kondo destruction and anomalous spin transport
signature by calculating the spin-relaxation time ($\tau$).Analysis of RG
equations zeros and poles show a ”Sign Reversion”(SR) regime exists for a
Hermitian problem with a critical value of nonlinear coupling $J_{k^{3}}$. Our
results show the existence of an out-of-phase RKKY oscillation above and below
the critical value of the chemical potential.

††preprint: APS/123-QED

##  I Introduction

The occurrence of the Kondo peak and bound states in Topological Insulators[1,
2] (TIs) highly depends on various parameters such as topology and chemical
potential. The behavior of the resonance level is distinct from that of simple
metals and ordinary insulators. In the case of band inversion[3], the mixed-
valence regime is wider, and the coexistence of both the Kondo peak and in-gap
bound states is possible, unlike in ordinary insulators where only one exists.
If the impurity energy is far from the chemical potential, the in-gap bound
states merge into the bulk. Furthermore, a self-screening Kondo effect may
occur due to the interaction between the impurity and the bound-state spin.
The study[4] predicts the occurrence of quantized conductance in the presence
of strong disorder, even in parameters where the system is a metal in the
absence of disorder. Unlike previously studied topological insulators, the
Fermi energy is located within a mobility gap, and the existence of ballistic
edge states does not rely on band inversion. Further investigation is
necessary to determine the correlation between the presence of strong disorder
and the location of the topological Anderson insulator in the phase diagram.

The analysis presented in the study[5] investigates the impact of an Anderson
impurity on a 2D topological insulator. It demonstrates that the exchange
interaction between the impurity and an in-gap bound state can undergo dynamic
changes. The temperature dependence of the system exhibits crossover behavior,
which could provide experimental evidence for the theoretical analysis. In the
weak-coupling regime, both screened and underscreened Kondo effects display a
modification in their effective coupling constant as a function of
temperature. The interplay between topology and interactions[6] was analyzed;
analysis shows an interesting relationship between topology and interactions.
The critical strength for the interaction is smaller in situations where the
total charge of the system does not change during a transition. The critical
strength also does not change with a change in the system’s topology. The
rearrangement of the charge structure in the ordered phases can be studied
through spectroscopy or optical response measurements. Further studies of the
relationship between interactions and topology can provide new insights.

The effect of thermal bias on the two-impurity Kondo system was studied[7].
Findings show that Kondo correlations are destroyed for large thermal bias for
the dot connected to the hot reservoir. This leads to suppressed electrical
and heat flows as the dots get decoupled. Kondo correlations also influence
the interdot coupling. For non-negligible antiferromagnetic spin exchange
coupling, thermal bias affects the Kondo-to-AFM crossover. The critical value
for the crossover increases as thermal bias increases. These observations can
be experimentally tested due to advancements in thermoelectrical transport
through nanostructures. This work is also important for some engineered
nanostructures for thermoelectric systems.

The study of Two Impurity Models (TIK)[8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20] has previously focused on a variety of regimes, including over-
screened and unscreened phases, as well as impurity triplet and decoupled
single dot states. Unlike the single impurity case, TIK presents a complex
problem with numerous possible configurations and interactions between
impurities. These models have been studied extensively to understand the
interplay between magnetic interactions, Kondo physics, and electronic
transport properties in complex systems.

The article[21] examines the effects of Kondo screening and its interplay with
scalar disorder on transport properties in Weyl semimetals, leading to Weyl-
Kondo physics. At zero temperature, magnetic impurities generate a Kondo
resistivity that scales with the number of impurities. At finite temperatures,
the strength of the scalar disorder impacts the Kondo resistivity. In the case
of weak scalar disorder, the Kondo resistivity has a minimum, but it may not
be observed in strong disorder, even if part of the sample is Kondo screened.
We start with a generic model for the bath, which is well-considered for the
Topological Insulators(TI)[22, 6, 23].By projecting onto the singly-occupied
subspace of the dot, we obtain an effective Hamiltonian, which is an s-d
exchange model. We show its lie algebra and connection to conformal field
theory (CFT), and it’s topological properties. Later we also extend the
projected model to a two impurities case which will be a two-impurity kondo
model in a topological system. We performed two-loop RG calculations for one
impurity and one-loop for the TIK model within the Poorman scaling
approximations. We analytically and numerically solved the RG equations, found
invariants, and explored rich phase diagram of the problem. We highlight
transport signatures relevant for momentum-resolved measurements of the
impurity-bath systems.

##  II Formalism

We start with the proposed and studied model[22] in the context of topological
insulators. This was used to explicitly extract the bulk and surface
contributions from the model perspective. The model reads as the following,

$\begin{split}\hat{H}&=\alpha k^{2}_{\parallel}.\mathbb{I}+\beta
k^{3}_{\parallel}\cos(3\theta)\sigma_{z}+\hat{z}.(\vec{k}\times\vec{\sigma})\end{split}$
(1)

Where in the above we have $\vec{k}=k_{x}\hat{i}+k_{y}\hat{j}+k_{z}\hat{k}$
and $\vec{\sigma}=\sigma_{x}\hat{i}+\sigma_{y}\hat{j}+\sigma_{z}\hat{k}$ This
model in the eqn 1 is well studied in the context of topology with such a
nonlinear dispersion; the term spin coupling to momentum is studied in the
many body context by introducing spin-orbit coupling. We write the above in
the second quantized form for spinful fermions as the following,

$\begin{split}H&=\psi^{\dagger}\hat{H}\psi\end{split}$ (2)

In the above model, the basis vector
$\psi^{\dagger}=\begin{pmatrix}c^{\dagger}_{k\uparrow}&c^{\dagger}_{k\downarrow}\end{pmatrix}$.Now
we put an impurity with onsite interaction and hybridization with the edge
states Hamiltonian as the following,

$H_{siam}=\psi^{\dagger}\hat{H}\psi+H_{d}+\sum_{k\sigma}V_{k}(c^{\dagger}_{k\sigma}d_{\sigma}+hc)$
(3)

Figure 1: The band dispersion around the low k points from diagonalization of
the bath represents the emergent chiral bands.In this case we set $\beta=0.3$
and $\theta=\frac{\pi}{3}$ we observed more $\beta$ was flattening the bands.

Where in above single impurity model equation 3
$H_{d}=\sum_{\sigma}\epsilon_{d}d^{\dagger}_{\sigma}d_{\sigma}+Un_{d\uparrow}n_{d\downarrow}$
in the bath Hamiltonian we can replace the $k_{\parallel}$ with $k$ with a
parametrization $k_{x}\pm ik_{y}=k_{\parallel}e^{\pm i\theta}$.We do a unitary
operation which is k-dependent in bath operators preserving canonical
relations, which yields a nonlinear k dependent coefficients as the following,

$\begin{split}\mathcal{U}=\frac{1}{\sqrt{\mathcal{N}}}\begin{pmatrix}e^{i\frac{\theta}{2}}\alpha_{k1}&e^{-i\frac{\theta}{2}}\alpha_{k2}\\\
-ie^{i\frac{\theta}{2}}\alpha_{k2}&ie^{-i\frac{\theta}{2}}\alpha_{k1}\end{pmatrix}\end{split}$
(4)

In the above equation 4, we have $\alpha_{1}=\sqrt{\frac{\Delta}{2}+\beta
k^{3}\cos 3\theta}$ and $\alpha_{2}=\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}$, where
$\Delta=\sqrt{4\beta^{2}k^{6}\cos^{2}3\theta+4k^{2}}$, and the normalization
constant is given by $\mathcal{N}=\sqrt{|\alpha_{1}|^{2}+|\alpha_{2}|^{2}}$.

This unitary transformation will rotate the original bath operators as
$\tilde{\psi}_{k}=\mathcal{U}_{k}\psi_{k}$, and such a k-dependent operation
is possible as shown in the case of weyl multiplicity[24], which is also a
different form of nonlinear dispersion. An interesting work also discusses
weyl-like signatures in quasiparticle interference[25] in the different forms
of cubic dispersive $(k_{x}\pm ik_{y})^{3}$ baths. Also note that the
expression $\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}=-i\sqrt{\frac{\Delta}{2}-\beta k^{3}\cos 3\theta}$
is also a unitary operator. The reason for choosing this particular form is
that it leads to a simpler normalization constant for the unitary, which makes
subsequent analysis easier.

$\begin{split}\tilde{\psi}&=\begin{pmatrix}c_{k+}\\\
c_{k-}\end{pmatrix}=\mathcal{U}\psi\\\ c_{k+}&=\frac{1}{\sqrt{\beta k^{3}\cos
3\theta}}\bigg{(}e^{i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos
3\theta}c_{k\uparrow}\\\ &+e^{-i\frac{\theta}{2}}\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}c_{k\downarrow}\bigg{)}\\\
c_{k-}&=\frac{1}{\sqrt{\beta k^{3}\cos
3\theta}}\bigg{(}-ie^{i\frac{\theta}{2}}\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}c_{k\uparrow}\\\
&+ie^{-i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos
3\theta}c_{k\downarrow}\bigg{)}\end{split}$ (5)

Inverting the above unitary operator to express the original spin basis to
these new chiral operators,

$\begin{split}c_{k\uparrow}&=\frac{1}{\sqrt{\beta k^{3}\cos
3\theta}}\bigg{(}e^{-i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos
3\theta}c_{k+}\\\ &+ie^{-i\frac{\theta}{2}}\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}c_{k-}\bigg{)}\\\
c_{k\downarrow}&=\frac{1}{\sqrt{\beta k^{3}\cos
3\theta}}\bigg{(}e^{i\frac{\theta}{2}}\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}c_{k+}\\\
&-ie^{-i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos
3\theta}c_{k-}\bigg{)}\end{split}$ (6)

This will effectively generate a chiral model with the eigenergies as
$\epsilon_{\pm}=\alpha k^{2}\pm\frac{1}{2}\Delta$ and the hybridization as the
following,

$\begin{split}\tilde{H}^{+}_{hyb}&=\sum_{k}\tilde{V}_{k}\bigg{(}e^{i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta
k^{3}\cos 3\theta}c^{\dagger}_{k+}d_{\uparrow}+hc)\\\
&+\sum_{k}\tilde{V}_{k}\bigg{(}e^{i\frac{\theta}{2}}\sqrt{\beta k^{3}\cos
3\theta-\frac{\Delta}{2}}c^{\dagger}_{k+}d_{\downarrow}+hc\bigg{)}\\\
\tilde{H}^{-}_{hyb}&=\sum_{k}\tilde{V}_{k}\bigg{(}-ie^{i\frac{\theta}{2}}\sqrt{\beta
k^{3}\cos 3\theta-\frac{\Delta}{2}}c^{\dagger}_{k-}d_{\uparrow}+hc\bigg{)}\\\
&+\sum_{k}\tilde{V}_{k}\bigg{(}ie^{i\frac{\theta}{2}}\sqrt{\frac{\Delta}{2}+\beta
k^{3}\cos 3\theta}c^{\dagger}_{k-}d_{\downarrow}+hc\bigg{)}\end{split}$ (7)

This will create the $\frac{3}{2}$ singularity of momentum in the
hybridization as $\tilde{V}_{k}=\frac{V_{k}}{\sqrt{\beta k^{3}\cos 3\theta}}$
distinguishing it from the square root singularity by the Rashba coupling
studies[26, 27, 28, 29].In order to simplify the symbolic cumbersome, let’s
introduce two k-dependent constants as
$\alpha_{k1}=\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos 3\theta}$ and
$\alpha_{k2}=\sqrt{\beta k^{3}\cos 3\theta-\frac{\Delta}{2}}$. If we look at
the $\theta=\frac{\pi}{3}$ 3-fold rotations might seem that the hybridization
has become complex, but it will introduce a phase factor, but overall
hybridization remains Hermitian. Hybridization in the new notation appears as
follows,

$\begin{split}\tilde{H}^{+}_{hyb}&=\sum_{k}\tilde{V}_{k}\alpha_{k1}\bigg{(}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\uparrow}+hc\bigg{)}\\\
&+\sum_{k}\tilde{V}_{k}\alpha_{k2}\bigg{(}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\downarrow}+hc\bigg{)}\\\
\tilde{H}^{-}_{hyb}&=\sum_{k}\tilde{V}_{k}\alpha_{k2}\bigg{(}-ie^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\uparrow}+hc\bigg{)}\\\
&+\sum_{k}\tilde{V}_{k}\alpha_{k1}\bigg{(}ie^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\downarrow}+hc\bigg{)}\end{split}$
(8)

We project this model to impurity subspace using the standard Hewson’s
projection operator method. The projection operators are
$P_{0}=(1-n_{\uparrow})(1-n_{\downarrow})$,
$P_{1}=n_{\uparrow}(1-n_{\downarrow})+n_{\downarrow}(1-n_{\uparrow})$ and
$P_{2}=n_{\uparrow}n_{\downarrow}$, respectively, for unoccupied, singly
occupied and doubly occupied states. We employ the projections and write the
following components of the effective model,

$\begin{split}\tilde{H}^{+}_{10}&=\sum_{k}\tilde{V}_{k}\alpha_{k1}e^{-i\frac{\theta}{2}}d^{\dagger}_{\uparrow}(1-n_{\downarrow})c_{k+}\\\
&+\sum_{k}e^{-i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k2}d^{\dagger}_{\downarrow}(1-n_{\uparrow})c_{k+}\\\
\tilde{H}^{-}_{10}&=\sum_{k}i\tilde{V}_{k}\alpha_{k2}e^{-i\frac{\theta}{2}}d^{\dagger}_{\uparrow}(1-n_{\downarrow})c_{k-}\\\
&+\sum_{k}-ie^{-i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k1}d^{\dagger}_{\downarrow}(1-n_{\uparrow})c_{k-}\end{split}$
(9)

Similarly, we have the projections connecting doubly and singly occupied
space. The $H_{02}$ and $H_{20}$ will vanish. $H_{12}$ can be written for this
form of hybridization,

$\begin{split}\tilde{H}^{+}_{12}&=\sum_{k}\tilde{V}_{k}\alpha_{k1}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}e^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k2}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
\tilde{H}^{-}_{12}&=-\sum_{k}i\tilde{V}_{k}\alpha_{k2}e^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+\sum_{k}ie^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k1}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}\end{split}$
(10)

Remaining components are
$H_{00}=\sum_{k\alpha}\epsilon_{k\alpha}c^{\dagger}_{k\alpha}c_{k\alpha}P_{0}+\sum_{\sigma}\epsilon_{d}n_{\sigma}P_{0}$
and
$H_{22}=\sum_{k\alpha}\epsilon_{k\alpha}c^{\dagger}_{k\alpha}c_{k\alpha}P_{2}+\sum_{\sigma}\epsilon_{d}n_{\sigma}P_{2}+Un_{\uparrow}n_{\downarrow}P_{2}$.
Now we have all the components to calculate the effective models in various
sub-spaces.

$\begin{split}H^{0}_{eff}&=H_{00}+\bigg{(}\sum_{\alpha}H^{\alpha}_{01}\bigg{)}\frac{1}{E-H_{11}}\bigg{(}\sum_{\alpha}H^{\alpha}_{10}\bigg{)}\\\
H^{1}_{eff}&=H_{11}+\bigg{(}\sum_{\alpha}H^{\alpha}_{10}\bigg{)}\frac{1}{E-H_{00}}\bigg{(}\sum_{\alpha}H^{\alpha}_{01}\bigg{)}\\\
&+\bigg{(}\sum_{\alpha}H^{\alpha}_{12}\bigg{)}\frac{1}{E-H_{22}}\bigg{(}\sum_{\alpha}H^{\alpha}_{21}\bigg{)}\\\
H^{2}_{eff}&=H_{22}+\bigg{(}\sum_{\alpha}H^{\alpha}_{21}\bigg{)}\frac{1}{E-H_{11}}\bigg{(}\sum_{\alpha}H^{\alpha}_{12}\bigg{)}\end{split}$
(11)

The singly occupied subspace is the low-energy effective model for the Kondo
regime. We want to explore the non-Kondo regime Hamiltonian to explore the
operator structures in the later sections. We detail the calculations in the
appendix for an effective sd exchange model.

$\begin{split}H^{1}_{eff}=H_{0}-i\alpha^{2}_{1}M_{kk^{\prime}}S_{-}s^{+}_{kk^{\prime}}+i\alpha^{2}_{2}M_{kk^{\prime}}s^{-}_{kk^{\prime}}S_{+}\\\
\alpha^{2}_{1}M_{kk^{\prime}}S_{z}s^{z}_{kk^{\prime}}+\alpha^{2}_{2}M_{kk^{\prime}}s^{z}_{kk^{\prime}}S_{z}\\\
\alpha_{1}\alpha_{2}M_{kk^{\prime}}S_{-}s^{z}_{kk^{\prime}}+\alpha_{1}\alpha_{2}M_{kk^{\prime}}s^{z}_{kk^{\prime}}S_{+}\\\
i\alpha_{1}\alpha_{2}M_{kk^{\prime}}S_{z}s^{-}_{kk^{\prime}}-i\alpha_{1}\alpha_{2}M_{kk^{\prime}}s^{+}_{kk^{\prime}}S_{z}\end{split}$
(12)

In above equation 12
$M_{kk^{\prime}}=\tilde{V}_{k}\tilde{V}_{k^{\prime}}\bigg{(}\frac{1}{\epsilon_{k^{\prime}}-\epsilon_{d}}+\frac{1}{\epsilon_{d}+U-\epsilon_{k}}\bigg{)}$
The above spin-spin interaction model has an algebraic structure similar to
the CFT paper[30]. We can notice some cross-product terms pop out in such a
nonlinear dispersion similar to the $\mathcal{DM}$ interaction term[26] for
the chiral channel but in the spin basis of the bath operators;

$\begin{split}H^{1}_{eff}&=H_{0}+\sum_{kk^{\prime}}J_{0}S.s_{kk^{\prime}}+i\sum_{kk^{\prime}}\vec{J}_{k^{3}}.(S\times
s_{kk^{\prime}})\\\ &+i\sum_{kk^{\prime}}\vec{J}_{k}.(S\times
s_{kk^{\prime}})+H_{pot}\end{split}$ (13)

where in the above $\vec{J}_{k^{3}}=\frac{M^{\theta}_{k}}{\beta k^{3}\cos
3\theta}\Delta\hat{z}$ and $\vec{J}_{k}=\frac{kM}{\beta k^{3}\cos
3\theta}\hat{x}+\frac{k\Delta}{\beta k^{3}\cos 3\theta}M^{\theta}\hat{y}$
Also, $H^{\dagger}=H$ since $\alpha_{1,2}$ are real constants and in the limit
$\Delta\to 0$, we have a standard hermitian problem with Anderson impurity in
a bath. Note that in this limit model no longer has
anisotropic$\mathcal{DM}$interaction. The band electrons’ left and right mover
can be defined and shown in the figure 2 for the above model. After Poorman
scaling is done for the magnitude of the vectors as defined in the equ 13, we
collect the contributions for RG equations in terms of these magnitudes.

Figure 2: A schematic representing two Anderson impurities in edge states. We
separate the left and right scatterers with green and blue
arrows.$k_{\parallel}\to k$ and $\theta\to\tan^{-1}\frac{k_{y}}{k_{x}}$ see
derivation. Scatterings for the $(k,\theta)$ to $(-k,\theta^{\prime})$ in
Poorman scaling are represented schematically in a 3D shell.

$\begin{split}\frac{dJ_{0}}{dl}&=J^{2}_{0}+J_{k^{3}}J_{k}+J^{2}_{k^{3}}+J^{2}_{k}\\\
\frac{dJ_{k^{3}}}{dl}&=J^{2}_{k}+J_{0}J_{k^{3}}\\\
\frac{dJ_{k}}{dl}&=J_{0}J_{k}\end{split}$ (14)

One solution of the above RG equations $J^{2}_{k}-J_{k}=mJ_{k^{3}}$ where m is
an invariant and remaining solution, we detail the analytic solution in the
appendix.

##  III Emergence Of Complex Solution

Notably, the emergence of complex solutions to RG equations due to nonlinear
dispersion has nontrivial renormalization. This is also seen in the one-loop
Poorman equations, which yield the emergent exceptional point scenario [30,
31] as RG reversion. Moreover, adding second-order self-energy to the original
Hamiltonian leads to non-Hermitian physics [32]. In open systems, adding
finite-order self-energies can also give rise to effective non-Hermitian
models [33, 34, 35]. In a theoretical study, it has been shown that complex
solutions emerge from various types of potentials and their symmetries [36].

Figure 3: The above plots show the emergence of coalescing point inside the
fermi-level. These points are solely due to the renormalization of the linear
and nonlinear couplings in the spin sector(1,1). Where m is RG invariant which
arises from the one loop poorman scaling. Figure 4: The above plots show the
spin sector(1,1) eigenvalues for large $J_{0}>1.0$. The eigenvalues become
flatter as the bare coupling increases, but coalescence remains, but it only
disappears for $J_{0}\to\infty$

Based on the plots in Figures 3 and 4, one might conclude that the Dirac cone
has disappeared, but in reality, it still exists in the bath. We are only
visualizing the eigenvalues of the sd model, which is more relevant to
impurity and dirac cone exist3 for resonant level case, which depend on the
flown couplings. The spectrum becomes gapped in impurity in large $J_{0}$
limit. By substituting the RG invariant
$J_{k}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4mJ_{k^{3}}}$ into the sd model
eigenvalues, we see that this is the source of the coalescing points.
Interestingly, these points disappear in the limit of $J_{0}\to\infty$. To
examine the relevance of potential scattering and its effect on particle-hole
symmetry, we diagonalize the flown $H^{1}_{eff}$ at the single occupancy
sector for only the sd part. In real materials, analogous exceptional points
have been observed, and in calculations, they have been tuned [37, 38]. Thus,
finite nonlinear dispersion can generate a complex solution and significantly
modify the RG flow of the Kondo problem, as shown by the Poorman solution.

##  IV Renormalization With Potential Scattering

To study the various momentum channel scattering shown in figure 2, which
interestingly yields rich physics. Largely we can write the effective model
derived13 with scattering terms as follows,

$\begin{split}(H^{\prime})^{1}_{eff}&=H^{1}_{eff}+\\\
&\begin{pmatrix}[k,\theta_{+2\frac{\pi}{3}}]\Leftrightarrow[k,\theta_{+\frac{\pi}{3}}]&[k,\theta_{+2\frac{\pi}{3}}]\Leftrightarrow[-k,\theta_{-\frac{\pi}{3}}]\\\
[-k,\theta_{-2\frac{\pi}{3}}]\Leftrightarrow[k,\theta_{+\frac{\pi}{3}}]&[-k,\theta_{-2\frac{\pi}{3}}]\Leftrightarrow[-k,\theta_{-\frac{\pi}{3}}]\end{pmatrix}\end{split}$
(15)

As discussed above, we can rewrite the effective Hamiltonian considering such
a potential scattering. We consider the basis of bath in the new scattering as
$\psi^{\dagger}=\begin{pmatrix}c^{\dagger}_{k_{\theta_{\frac{2\pi}{3}}}+},&c^{\dagger}_{k_{\theta_{\frac{\pi}{3}}}-},&c^{\dagger}_{-k_{\theta_{-\frac{2\pi}{3}}}+},&c^{\dagger}_{-k_{\theta_{-\frac{\pi}{3}}}-}\end{pmatrix}$
which makes the block structure of the kondo problem and represents the
couplings
$\vec{J}^{\theta,\theta^{\prime}}_{k^{3}}=M^{\theta,\theta^{\prime}}_{kk^{\prime}}(\Delta+\beta
k^{3}\cos(3\theta))\hat{z}$ and
$\vec{J}^{\theta,\theta^{\prime}}_{k}=\frac{k}{\beta k^{3}\cos
3\theta}M^{\theta,\theta^{\prime}}\hat{x}+\frac{k\Delta}{\beta k^{3}\cos
3\theta}M^{\theta,\theta^{\prime}}\hat{y}$

$\begin{split}H^{1}_{eff}&=H_{0}+\sum_{kk^{\prime}}J_{0}S.\psi^{\dagger}(\Sigma)\psi\\\
&+i\sum_{kk^{\prime}}\vec{J}^{(0,0)}_{k^{3}}.(S\times\psi^{\dagger}(\Sigma)\psi)\\\
&-\sum_{kk^{\prime}}\vec{J}^{\theta_{\pm\frac{2\pi}{3}},\theta_{\pm\frac{\pi}{3}}}_{k^{3}}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+\sum_{kk^{\prime}}\vec{J}^{\theta_{\frac{2\pi}{3}},\theta_{-\frac{\pi}{3}}}_{k^{3}}.(S\times\psi^{\dagger}(\Gamma)\psi)\\\
&+i\sum_{kk^{\prime}}\vec{J}^{(0,0)}_{k}.(S\times\psi^{\dagger}(\Sigma)\psi)\\\
&-\sum_{kk^{\prime}}\vec{J}^{\theta_{\pm\frac{2\pi}{3}},\theta_{\pm\frac{\pi}{3}}}_{k}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+i\sum_{kk^{\prime}}\vec{J}^{\theta_{\frac{2\pi}{3}},\theta_{-\frac{\pi}{3}}}_{k}.(S\times\psi^{\dagger}(\Gamma)\psi)\end{split}$
(16)

###  IV.1 Comparison With CFT scaling laws at two loops

We can see the algebraic consistency in the earlier works [30, 8, 39] using
the lie algebra of matrices, which is similar to the Poorman scaling for the
operator structures but CFT will be exact since it incorporates the bath
states more accurately. This idea can also be extended to the two impurity
problems. After introducing the potential scattering, we have a more general
model16 in the lie matrices, which are defined as follows,

$\begin{split}\Sigma=\begin{pmatrix}\sigma&\sigma\\\
\sigma&\sigma\end{pmatrix},\Omega=\begin{pmatrix}\sigma&0\\\
0&-\sigma\end{pmatrix},\Gamma=\begin{pmatrix}\sigma&0\\\
0&\sigma\end{pmatrix}\end{split}$ (17)

where in the above equation17 $\sigma$ are the Pauli matrices

$\begin{split}[\Sigma^{a}\Sigma^{b},\Sigma^{c}]&=[\Gamma^{a}\Gamma^{b},\Gamma^{c}]=[\Omega^{a}\Omega^{b},\Omega^{c}]\\\
&=0\\\
[\Sigma^{\alpha}\Sigma^{\alpha^{\prime}},\Sigma^{\alpha}]&=-4i\delta_{\alpha,\alpha^{\prime}}\Sigma^{\alpha}\\\
[\Sigma^{b}\Sigma^{b},\Sigma^{c}]&=\Sigma^{b}[\Sigma^{b},\Sigma^{c}]+[\Sigma^{b},\Sigma^{c}]\Sigma^{b}\\\
&=+8i\epsilon^{abc}\Sigma^{c}\\\
[\Omega^{b}\Omega^{b},\Omega^{c}]&=\Omega^{b}[\Omega^{b},\Omega^{c}]+[\Omega^{b},\Omega^{c}]\Omega^{b}\\\
&=+4i\epsilon^{abc}\Omega^{c}\\\
[\Omega^{b}\Omega^{b},\Sigma^{c}]&=\Omega^{b}[\Omega^{b},\Sigma^{c}]+[\Omega^{b},\Sigma^{c}]\Omega^{b}\\\
&=+4i\epsilon^{abc}\Omega^{c}+4i\epsilon^{abc}\mathit{I}_{4X4}\\\
[\Sigma^{a}\Sigma^{b},\Upsilon]&=\Sigma^{a}[\Sigma^{b},\Upsilon]+[\Sigma^{a},\Upsilon]\Sigma^{b}\\\
&=0\end{split}$ (18)
$\begin{split}\frac{dJ_{0}}{dl}&=J^{2}_{0}+J^{2}_{k^{3}}+J^{2}_{k}+J_{k^{3}}J_{k}+J^{2}_{k}J_{0}+J^{2}_{k^{3}}J_{0}+J^{3}_{0}\\\
&+f(J_{k^{3}},g_{2k},g_{2k^{3}},J_{k})\\\
\frac{dJ_{k^{3}}}{dl}&=J^{2}_{k}+J_{0}J_{k^{3}}+J^{2}_{k}J_{k^{3}}+J^{2}_{0}J_{k^{3}}+J^{3}_{k^{3}}\\\
\frac{dJ_{k}}{dl}&=J_{0}J_{k}+J^{2}_{0}J_{k}+J_{k}J^{2}_{k^{3}}+J^{3}_{k}\\\
\frac{dg_{1k^{3}}}{dl}&=g^{3}_{1k^{3}}-g^{2}_{1k}g_{1k^{3}}\\\
\frac{dg_{1k}}{dl}&=g^{3}_{1k}-g^{2}_{1k^{2}}g_{1k}\\\
\frac{dg_{2k^{3}}}{dl}&=-g^{3}_{2k^{3}}+g^{2}_{1k^{3}}g_{2k^{3}}\\\
\frac{dg_{2k}}{dl}&=-g^{3}_{2k}+g^{2}_{1k}g_{2k}\end{split}$ (19)

We have given some details on calculations of the above rg equations in
Appendix B.

Figure 5: We set $g_{1k}=g_{1k^{3}}=1$ which is invariant derived in appendix
for all calculations $J_{1k}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4mJ_{1k^{3}}}$
is used and $m=-4$ we set for all above plots.We plot the grid as
$\begin{pmatrix}pot&pot1\\\ R^{+}R^{+}&R^{+}I^{+}\\\
I^{-}R^{-}&I^{-}I^{-}\end{pmatrix}$ first row is potential scattering, and the
second row is in both beta functions chosen real and imaginary part of the
positive root. We see at least two fixed points(FP), two spirals in (RR, RI)
row, and one spiral, and one is marginal. The dotted lines in the second plot
of the first row are a family of FP’s.

The spiral FP in non-hermitian systems are demonstrated earlier[40, 41]; also,
in three and higher dimensions, spiral invariants obtained[42]. We explored
the FP originating from the invariant by adding the potential scattering
terms. This made the problem richer and interestingly captured the one-loop
critical points discussed in analytic solutions to RG eq’s in the appendix and
didn’t alter the invariant of poorman. This indicates the additional terms are
very relevant for the problem, unlike in a homogeneous bath at particle-hole
symmetry, these potential scattering terms are irrelevant (note that potential
scattering terms are conventional $k\neq k^{\prime}$, not special like what is
considered here). In general, $\vec{S}\times\vec{s}$ terms are anti-hermitian
without the factor $\iota$; hence any finite order perturbation will bring
nonhermitian physics into play; in general, it does come into play as a
decohering interaction[43]. Various experiments relating to anisotropic
$\mathcal{DM}$ and topology of the spin texture[44] are discussed. We detail
the numerical solutions in Appendix G for confirming the ’sign reversion’. If
so, these points correspond to complex FP, independent of what potential
scattering terms we add.

##  V Generalization to Two Impurities

We generalize this model to two impurity kondo models with coupling between
the impurities. One can also start with the projection for two impurities
Anderson models to derive the two impurity kondo model from getting exact
matrix elements for coupling between the dots. Two dots with spin-orbit
coupling[45] is studied earlier, which shows that $\mathcal{DM}$ term exists
for only Y-component and linear dispersion in the bath. In this case, it
generates a more general form of $S_{1}\times S_{2}$ term in all components.

$\begin{split}H^{1}_{eff}&=H_{0}+\sum_{kk^{\prime}}J_{0}S_{\alpha}.s_{kk^{\prime}}+i\sum_{kk^{\prime}\alpha}\vec{J}_{k^{3}}.(S_{\alpha}\times
s_{kk^{\prime}})\\\ &+i\sum_{kk^{\prime}\alpha}\vec{J}_{k}.(S_{\alpha}\times
s_{kk^{\prime}})+H_{pot}\\\ &+J_{Y}S_{1}.S_{2}+iK.(S_{1}\times
S_{2})\end{split}$ (20)

With the new coupling, we see how it renormalizes the problem; immediately, we
can notice that the RKKY coupling $J_{Y}$ modifies the single impurity
invariant. Since the generalized problem has many coupling, we restrict our-
self to analyzing the one-loop RG equations.,

$\begin{split}\frac{dJ_{0}}{dl}&=J^{2}_{0}+J_{k^{3}}J_{k}+J^{2}_{k^{3}}+J^{2}_{k}+J_{Y}J_{0}+KJ_{0}\\\
\frac{dJ_{k^{3}}}{dl}&=J^{2}_{k}+J_{0}J_{k^{3}}+J_{Y}J_{k^{3}}+J_{Y}J_{k}+KJ_{k^{3}}\\\
\frac{dJ_{k}}{dl}&=J_{0}J_{k}+J_{Y}J_{k^{3}}+J_{Y}J_{k}+KJ_{k}\\\
\frac{dJ_{Y}}{dl}&=J^{2}_{Y}+K^{2}+J^{2}_{0}+J_{k^{3}}J_{k}+J^{2}_{k}+J^{2}_{k^{3}}\\\
\frac{dK}{dl}&=K^{2}+KJ_{Y}+J_{k^{3}}J_{k}\end{split}$ (21)

Solutions to the above equations are detailed in an appendix in various
limits. The beta function zeros for kondo destruction can be seen in the odd-
even couplings given in articles[46, 47]. Since we focus on the nonlinear
couplings $J_{k},J_{k^{3}}$, We look at the solutions around the anomalous
contributions to Spin-relaxation time and the FP’s in these couplings.

###  V.1 Kondo Scale in two Impurity

We integrate the RG equations to get the scales for the problem in various
limits so that how the nonlinear coupling affects this problem. First in limit
$J_{k^{3}}=J_{k}=K=0$ we have the solution $J_{0}=\frac{e^{-g}}{(1-g)^{2}}$ by
integrating the $J_{0},J_{Y}$ equations, where $g=\frac{J_{Y}}{J_{0}}$. We
solve when impurity $\mathcal{DM}$ interaction $K\neq 0$ in the appendix in
two limits which does not have a straightforward analytic solution by reducing
to bergurs equation . Figure 6 show the variation of $T_{K}$ with various
coupling limits. The two invariants $R_{1}$ modify the TIK problem
significantly, ”R” cause the destruction of the TIK bound state, and $R_{1}$
try to stabilize the bound state, which can be seen in the middle plot in
figure 6. A special bath can significantly affect the kondo problem, as shown
in NRG studies in single impurity[48]. Recent NRG studies[49, 50] on two
impurities anderson model yield a rich phase like kondo, local moment valence
fluctuation, RKKY, and spin liquid. Since we employed perturbative
renormalization, the flow will cuttoff and we need to go beyond; hence we
don’t have access to all phases, however we do have Kondo-RKKY phases in small
K limit inset fig 6 and qualitative physical picture of phases in homogeneous
bath cases.Now we solve the set of coupled equations21 numerically for various
IC’s found in flow diagrams and analytic solutions.

Figure 6: We plot the scales, as shown above, to show competition between RKKY
and impurity$\mathcal{DM}$ interactions.An Inset graph in K small label is an
analytic expression derived when impurity $\mathcal{DM}$ interaction is absent
and $g=\frac{J_{Y}}{J_{0}}$ in the x-axis. Discontinuity in the scale for K
Large limit is shown with the dotted line where it is ill-defined.  Figure 7:
RG flows for the two impurity problems when anisotropic $\mathcal{DM}$
interaction due to bath is absent. See the RG solutions in the appendix. There
are three FP’s from these solutions one is TIK point(blue), decoupled
dots(Green), and dot triplet state(blue in the right plot). The quadrant of
these points depends on the integral constants(either in third or first), and
we refer here to the $-ve$ sign for ferromagnetic couplings.

##  VI RG Equations Zeros And Poles Analysis

We realize that the solutions to couplings are divergent; therefore, it
becomes trickier to separate the two impurity kondo regime, single impurity
kondo regime, and the SR regime of couplings due to nonlinear coupling
$J_{k^{3}}$. We solve the RG ODE’s by allowing complex solutions and analyzing
solutions around the FP’s found in flow diagrams. If there is a phase
transition, then couplings will flow to a stable FP, and hence we should see
poles in functions and at unstable points only zero crossings. This serves as
the numerical diagnosis for the identification of various phases.

##  VII Impurity Transport Calculation

We have derived the anomalous contributions for the relaxation time in the
appendix due to the presence of the nonlinear dispersive bath following the
$\mathcal{T}_{kk^{\prime}}$ formalism[51] which is detailed in appendix F,

$\begin{split}\frac{1}{\tau(k)}\propto(1-2J\tilde{g}_{\alpha}(\epsilon_{k})-2J\tilde{g}_{\alpha}^{*}(\epsilon_{k})-2J_{k^{3}}\tilde{g}_{\alpha
k^{3}}(\epsilon_{k})\\\ -2J_{k^{3}}\tilde{g}_{\alpha
k^{3}}^{*}(\epsilon_{k})-2J_{k}\tilde{g}_{\alpha
k}(\epsilon_{k})-2J_{k}\tilde{g}_{\alpha k}^{*}(\epsilon_{k}))\end{split}$
(22)

We are addressing coalescing points from a hermitian model with any transport
signature. The appendix evaluates these functions as contour integrals for
polar components and definite integrals for radial fermi vector($k_{f}$). Spin
relaxation time will reflect the RG invariant as
$\frac{1}{\tau}\propto\tilde{g}_{\epsilon}=(1\pm\sqrt{1+4\epsilon})f(\epsilon,\epsilon^{\prime})$
implying the momentum resolved couplings have the signature of coalescing
points which can be seen in spin relaxation time.

Figure 8: We calculated the spin-relaxation time to see the coalescing point
signature with the nonlinear coupling $\beta$. as the $\beta$ decreases, ideal
log divergence reaches but dips persist in $\frac{1}{\tau}$. Left plot to
confirm if the anomalous contribution is from the coalescence, we divided each
curve by lowest $\beta=0.01$ curve and scaled for $\pm$ roots as
$\tau^{-1}_{\pm r}=\tau^{-1}*(1\pm\sqrt{4(\epsilon-0.25)+1})$ for the y-axis
and $D_{eff}*(1\pm\sqrt{4(\epsilon-0.25)+1})$ for the x-axis. Increasing
$\beta$, we see polynomial contributions dominate; hence the scaling will be
absent.

The matrix elements scale as
$M_{kk^{\prime}}\approx\frac{1}{\sqrt{k^{3}\cos(3\theta)}\sqrt{(k^{\prime})^{3}\cos{3\theta^{\prime}}}}$
and when $k=-k^{\prime}$ there is an imaginary coupling which creates a non-
Hermitian Kondo problem with a bath that has non-linear dispersion. Various
studies of non-Hermitian Kondo systems exist, considering different
perspectives, symmetry considerations, and complex spin exchange
interactions[31, 30, 52, 53]. In the presence of inversion symmetry, non-
Hermiticity can result from $\theta^{\prime}\to\frac{\pi}{3}-\theta$, where a
Dirac point exists in the Brillouin zone. In the appendix, it is discussed
that $\alpha_{1k}\alpha_{2k^{\prime}}$ beyond the poor man’s limit (i.e.,
$k\neq k^{\prime}$) results in a general non-Hermitian model at impurity due
to nonlinear parametric dispersion in the bath.

Figure 9: We perform The integrals for $J_{Y}$ given in the appendix and plot
by varying the bandwidth.RKKY is enhanced for $\beta<1.0$ and approaches to
ideal flatt band limit for $\beta>1.0$. As we saw in the relaxation time here,
we can see above and below a critical value of chemical potential
($\epsilon^{\prime}=\pm 0.25$), $J^{odd}_{Y}$ reverses sign and goes out of
phase in $J^{odd}_{Y}$.The other channel $J^{even}_{Y}$ has standard RKKY
oscillations and decay amplify around the EP. Figure 10: We plot the elliptic
functions which arose in calculating the RKKY integrals. For negative chemical
potential $\epsilon^{\prime}$, there is SR everywhere else we see the
exceptional point behaviors. The EP’s start shifting for increasing +ve
$\epsilon^{\prime}$, but there is no SR in the real part of these special
functions. This odd channel shows EP or SR behavior which is relevant in RG
when dominant odd channel scattering will emerge complex FP’s in such models.

##  VIII Discussion

This work investigates the possibility of coalescing points in a bulk system
having magnetic impurities through spin relaxation time calculations in
different momentum directions. Our results show that two distinct calculations
share a common invariant as renormalization group (RG) solutions, with or
without potential scattering. Increasing nonlinearity strength causes bands to
flatten in the bath spectrum while the impurity resistivity reaches the ideal
log divergence for a $\beta$ strength range. Furthermore, we analyze RG
calculations on single and two impurity problems in a homogeneous bath. The
competition between RKKY and Kondo interactions is only observed when the
impurity $\mathcal{DM}$-interaction is absent. We also show that to obtain
coalescing points in hermitian Kondo models, anisotropic $\mathcal{DM}$
interactions are necessary due to the $k^{3}$ and $k$ terms in the dispersion,
which naturally yields the roots as the RG invariants. These results are
significant as they provide a signature in transport, which we demonstrate
through numerical diagnosis for critical values of $J_{k^{3}}$ and $J_{k}$ in
the couplings, confirmed by flow diagrams. The SR regime in the couplings
corresponds to the spiral FP’s of the problem. Out-Of-Phase oscillations in
the RKKY in the odd channels are due to the presence of the special bath, as
we ruled out the other possibilities by analyzing the elliptic functions EP.
Future work could extend these findings to charge and thermal transport
calculations and investigate impurity effects in systems with anisotropic
$\mathcal{DM}$ interaction and in Weyl and topological systems with nonlinear
dispersion. One may expect a loss of unitarity in subsystem as we have seen
here in a single occupied subspace if we analyze the system in other subspaces
$H^{0}_{eff},H^{2}_{eff}$ maybe there will be global unitarity which is not
done in current work.

The results of our work demonstrate the potential for more advanced numerical
techniques, such as the Numerical Renormalization Group (NRG), to extract
valuable information about spin transport in complex systems. This opens up
new avenues for future research to investigate the behavior of impurities in
systems with anisotropic $\mathcal{DM}$ interactions and in Weyl and
topological systems with nonlinear dispersion.It is important to note that we
did not explore charge or thermal transport in our study. These are
interesting directions for future research, as they could provide a more
complete understanding of the transport properties of impurities in
topological systems. Nonhermitian problems can also originate from certain
defects[54, 55], and different boundary conditions can be used to tune[56] the
EP’s and realize them in quantum circuits, maybe if we map effective spin
models to Wilson chains, then we might look at these phases. Localization in
the Hatano-nelson type of models can have dramatic consequences of
impurities[57]. Our work contributes to a growing body of research to better
understand impurities’ behavior in topological systems. This can inform the
design of new materials and devices with enhanced transport properties and
deepen our fundamental understanding of phenomena associated with open and
closed condensed matter systems.

###### Acknowledgements.

The author extends gratitude to the JNCASR for the supportive research
environment provided. Acknowledgment is also given to Prof. N.S. Vidhyadhiraja
for his encouragement in publishing the research. The author is thankful to
Prof. Henrik Johannesson for his contribution to the conceptualization of the
TKSS model and to Prof. Ipsita Mandal and Dr. Ranjani Seshadri for introducing
the concept of Weyl and topological systems.

##  Appendix A Projection details for deriving effective sd model

$\begin{split}\tilde{H}^{+}_{12}&=\sum_{k}\tilde{V}_{k}\alpha_{k1}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}e^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k2}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
\tilde{H}^{-}_{12}&=-\sum_{k}i\tilde{V}_{k}\alpha_{k2}e^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+\sum_{k}ie^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k1}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}\end{split}$
(23)

We will derive the components of hamiltonian as done in the text[51] as
follows,

$\begin{split}H^{+}_{12}\frac{1}{E-H_{22}}H^{+}_{21}+H^{+}_{12}\frac{1}{E-H_{22}}H^{-}_{21}+H^{-}_{12}\frac{1}{E-H_{22}}H^{+}_{21}+H^{-}_{12}\frac{1}{E-H_{22}}H^{-}_{21}\\\
\bigg{(}\sum_{k}\tilde{V}_{k}\alpha_{k1}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}e^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k2}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\bigg{)}^{\dagger}\frac{1}{E-H_{22}}\bigg{(}\sum_{k^{\prime}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}1}e^{i\frac{\theta^{\prime}}{2}}c^{\dagger}_{k^{\prime}+}d_{\uparrow}n_{\downarrow}+\sum_{k^{\prime}}e^{i\frac{\theta^{\prime}}{2}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}2}c^{\dagger}_{k^{\prime}+}d_{\downarrow}n_{\uparrow}\bigg{)}\\\
\bigg{(}\sum_{k}\tilde{V}_{k}\alpha_{k1}e^{i\frac{\theta}{2}}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}e^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k2}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\bigg{)}^{\dagger}\frac{1}{E-H_{22}}\bigg{(}-\sum_{k^{\prime}}i\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}2}e^{i\frac{\theta^{\prime}}{2}}c^{\dagger}_{k^{\prime}-}d_{\uparrow}n_{\downarrow}+\sum_{k^{\prime}}ie^{i\frac{\theta}{2}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}1}c^{\dagger}_{k^{\prime}-}d_{\downarrow}n_{\uparrow}\bigg{)}\\\
\bigg{(}-\sum_{k}i\tilde{V}_{k}\alpha_{k2}e^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+\sum_{k}ie^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k1}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}\bigg{)}^{\dagger}\frac{1}{E-H_{22}}\bigg{(}\sum_{k^{\prime}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}1}e^{i\frac{\theta^{\prime}}{2}}c^{\dagger}_{k^{\prime}+}d_{\uparrow}n_{\downarrow}+\sum_{k^{\prime}}e^{i\frac{\theta^{\prime}}{2}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}2}c^{\dagger}_{k^{\prime}+}d_{\downarrow}n_{\uparrow}\bigg{)}\\\
\bigg{(}-\sum_{k}i\tilde{V}_{k}\alpha_{k2}e^{i\frac{\theta}{2}}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+\sum_{k}ie^{i\frac{\theta}{2}}\tilde{V}_{k}\alpha_{k1}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}\bigg{)}^{\dagger}\frac{1}{E-H_{22}}\bigg{(}-\sum_{k^{\prime}}i\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}2}e^{i\frac{\theta^{\prime}}{2}}c^{\dagger}_{k^{\prime}-}d_{\uparrow}n_{\downarrow}+\sum_{k^{\prime}}ie^{i\frac{\theta^{\prime}}{2}}\tilde{V}_{k^{\prime}}\alpha_{k^{\prime}1}c^{\dagger}_{k^{\prime}-}d_{\downarrow}n_{\uparrow}\bigg{)}\end{split}$
(24)

We calculate for the $\theta=\theta^{\prime}$ cases, and the remaining can be
included in potential scattering later.

$\begin{split}&\sum_{k}M^{22}_{k+}\alpha^{2}_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}M^{22}_{k+}\alpha_{k2}\alpha_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}\\\
&+\sum_{k}M^{22}_{k+}\alpha_{k1}\alpha_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k+}d_{\uparrow}n_{\uparrow}+\sum_{k}M^{22}_{k+}\alpha^{2}_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
&i\sum_{k}M^{22}_{k-}\alpha^{2}_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}-i\sum_{k}M^{22}_{k-}\alpha_{k2}\alpha_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\\\
&+i\sum_{k}M^{22}_{k-}\alpha_{k1}\alpha_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}-i\sum_{k}M^{22}_{k-}\alpha^{2}_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\\\
&-i\sum_{k}M^{22}_{k+}\alpha^{2}_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+i\sum_{k}M^{22}_{k+}\alpha_{k2}\alpha_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}\\\
&-i\sum_{k}M^{22}_{k+}\alpha_{k1}\alpha_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}+i\sum_{k}M^{22}_{k+}\alpha^{2}_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
&\sum_{k}M^{22}_{k-}\alpha^{2}_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}-\sum_{k}M^{22}_{k-}\alpha_{k2}\alpha_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\\\
&-\sum_{k}M^{22}_{k-}\alpha_{k1}\alpha_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}+\sum_{k}M^{22}_{k-}\alpha^{2}_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+(k\neq
k^{\prime}\text{terms})\end{split}$ (25)

Where in above
$M^{22}_{k\alpha}=\frac{\tilde{V}^{2}_{k}}{\epsilon_{d}+U-\epsilon_{k^{\prime}\alpha}}\approx
M^{22}_{k}(1-\alpha\Delta)$ with this substitution we can seperate the
original kondo model and kondo model from the edge states impurity
interaction.Collecting terms with
$\alpha^{2}_{k1},\alpha^{2}_{2},\alpha_{k1}\alpha_{k2}$ terms and with
simplifications using the commutation and some substitution at half filling
$n^{2}_{\sigma}=n_{\sigma}$, $n_{\uparrow}\to(\frac{1}{2}-n_{\downarrow})$,
$n_{\downarrow}\to(\frac{1}{2}-n_{\uparrow})$ and the dot operators
$n_{\sigma}d_{\sigma}=d_{\sigma}(1-n_{\sigma})$ can be written as follows,

$\begin{split}(\alpha^{2}_{k1})\longrightarrow\sum_{k}M^{22}_{k-}\alpha^{2}_{k1}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}-i\sum_{k}M^{22}_{k+}\alpha^{2}_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}\\\
i\sum_{k}M^{22}_{k-}\alpha^{2}_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}+\sum_{k}M^{22}_{k+}\alpha^{2}_{k1}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}\\\
=\sum_{k}M^{22}_{k}\alpha^{2}_{k1}S^{z}_{d}s^{z}_{kk^{\prime}}-i\sum_{k}M^{22}_{k}\alpha^{2}_{k1}S^{-}s^{+}_{kk^{\prime}}+i\sum_{k}M^{22}_{k}\alpha^{2}_{k1}s^{-}_{kk^{\prime}}S^{+}+(c^{\dagger}_{k\alpha}c_{k\bar{\alpha}}n_{\uparrow}n_{\downarrow}\text{terms})\\\
i\sum_{k}M^{22}_{k}\Delta\alpha^{2}_{k1}S^{-}s^{+}_{kk^{\prime}}-i\sum_{k}M^{22}_{k}\Delta\alpha^{2}_{k1}s^{-}_{kk^{\prime}}S^{+}\end{split}$
(26)

Note that The edge contribution is only appearing in the cross terms does not
contribute to $S^{z}_{d}s^{z}_{kk^{\prime}}$.Similarly for $\alpha^{2}_{k2}$
terms

$\begin{split}(\alpha^{2}_{k2})\longrightarrow\sum_{k}M^{22}_{k+}\alpha^{2}_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}-i\sum_{k}M^{22}_{k-}\alpha^{2}_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\\\
+i\sum_{k}M^{22}_{k+}\alpha^{2}_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}+\sum_{k}M^{22}_{k-}\alpha^{2}_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\\\
=\sum_{k}M^{22}_{k}\alpha^{2}_{k2}S^{z}_{d}s^{z}_{kk^{\prime}}-i\sum_{k}M^{22}_{k}\alpha^{2}_{k2}S^{-}s^{+}_{kk^{\prime}}+i\sum_{k}M^{22}_{k}\alpha^{2}_{k2}s^{-}_{kk^{\prime}}S^{+}+(c^{\dagger}_{k\alpha}c_{k\bar{\alpha}}n_{\uparrow}n_{\downarrow}\text{terms})\\\
i\sum_{k}M^{22}_{k}\Delta\alpha^{2}_{k2}S^{-}s^{+}_{kk^{\prime}}-i\sum_{k}M^{22}_{k}\Delta\alpha^{2}_{k2}s^{-}_{kk^{\prime}}S^{+}\end{split}$
(27)

Finally we can collect all terms with $\alpha_{1}\alpha_{2}$

$\begin{split}(\alpha_{k1}\alpha_{k2})\longrightarrow\sum_{k}M^{22}_{k+}\alpha_{k2}\alpha_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}+\sum_{k}M^{22}_{k+}\alpha_{k1}\alpha_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
-i\sum_{k}M^{22}_{k-}\alpha_{k2}\alpha_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k+}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}+i\sum_{k}M^{22}_{k-}\alpha_{k1}\alpha_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k+}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}\\\
+i\sum_{k}M^{22}_{k+}\alpha_{k2}\alpha_{k1}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k+}d_{\uparrow}n_{\downarrow}-i\sum_{k}M^{22}_{k+}\alpha_{k2}\alpha_{k1}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k+}d_{\downarrow}n_{\uparrow}\\\
-\sum_{k}M^{22}_{k-}\alpha_{k1}\alpha_{k2}n_{\downarrow}d^{\dagger}_{\uparrow}c_{k-}c^{\dagger}_{k-}d_{\downarrow}n_{\uparrow}-\sum_{k}M^{22}_{k-}\alpha_{k1}\alpha_{k2}n_{\uparrow}d^{\dagger}_{\downarrow}c_{k-}c^{\dagger}_{k-}d_{\uparrow}n_{\downarrow}\end{split}$
(28)

We simplify the above and use the Abrikosov representation[58] for spin for
impurity as $S=\psi^{\dagger}_{d}\sigma\psi_{d}$ and
$s_{kk^{\prime}}=\psi^{\dagger}_{k}\sigma\psi_{k^{\prime}}$ for bath
operators. Where in the representation $\sigma$ is pauli matrix and
$\psi^{\dagger}_{d}=\begin{pmatrix}d^{\dagger}_{\uparrow}&d^{\dagger}_{\downarrow}\end{pmatrix}$
$\psi^{\dagger}_{k}=\begin{pmatrix}c^{\dagger}_{k\uparrow}&c^{\dagger}_{k^{\prime}\downarrow}\end{pmatrix}$

$\begin{split}i\sum_{k}M^{22}_{k}\alpha_{k2}\alpha_{k1}(-s^{-}_{kk^{\prime}}S^{z}+S^{z}s^{+}_{kk^{\prime}})+\sum_{k}M^{22}_{k}\alpha_{k2}\alpha_{k1}(s^{z}_{kk^{\prime}}S^{+}-S^{-}s^{z}_{kk^{\prime}})\\\
+i\sum_{k}M^{22}_{k}\Delta\alpha_{k2}\alpha_{k1}(s^{-}_{kk^{\prime}}S^{z}-S^{z}s^{+}_{kk^{\prime}})-\sum_{k}M^{22}_{k}\Delta\alpha_{k2}\alpha_{k1}(s^{z}_{kk^{\prime}}S^{+}-S^{-}s^{z}_{kk^{\prime}})\end{split}$
(29)

Similarly we can calculate
$H^{+}_{10}\frac{1}{E-H_{00}}H^{+}_{01}+H^{+}_{10}\frac{1}{E-H_{00}}H^{-}_{01}+H^{-}_{10}\frac{1}{E-H_{00}}H^{+}_{01}+H^{-}_{10}\frac{1}{E-H_{00}}H^{-}_{01}$
which yield same operator structure except we have $M^{00}_{k}$ Now
substituting back $\alpha_{1}=\sqrt{\frac{\Delta}{2}+\beta k^{3}\cos 3\theta}$
and $\alpha_{2}=\sqrt{\beta k^{3}\cos 3\theta-\frac{\Delta}{2}}$, where
$\Delta=\sqrt{4\beta^{2}k^{6}\cos^{2}3\theta+4k^{2}}$ and
$\tilde{V}_{k}=\frac{V_{k}}{\sqrt{k^{3}\cos{3\theta}}}$, collecting terms we
get various simplifications for example $\alpha^{2}_{k1}+\alpha^{2}_{k2}=\beta
k^{3}\cos(3\theta)$ and $\alpha_{k1}\alpha_{k2}=k$

$\begin{split}H^{1}_{eff}=H_{11}+\beta
k^{3}\cos{3\theta}\bigg{(}\sum_{k}M_{k}S^{z}_{d}s^{z}_{kk^{\prime}}-i\sum_{k}M_{k}S^{-}s^{+}_{kk^{\prime}}+i\sum_{k}M_{k}s^{-}_{kk^{\prime}}S^{+}\\\
i\sum_{k}M_{k}\Delta S^{-}s^{+}_{kk^{\prime}}-i\sum_{k}M_{k}\Delta
s^{-}_{kk^{\prime}}S^{+}\bigg{)}\\\
k\bigg{(}i\sum_{k}M_{k}(-s^{-}_{kk^{\prime}}S^{z}+S^{z}s^{+}_{kk^{\prime}})+\sum_{k}M_{k}(s^{z}_{kk^{\prime}}S^{+}-S^{-}s^{z}_{kk^{\prime}})\\\
+i\sum_{k}M_{k}\Delta(s^{-}_{kk^{\prime}}S^{z}-S^{z}s^{+}_{kk^{\prime}})-\sum_{k}M_{k}\Delta(s^{z}_{kk^{\prime}}S^{+}-S^{-}s^{z}_{kk^{\prime}})\bigg{)}\end{split}$
(30)

Collecting All terms from equations 26,27and 29 we can simplify and rewrite
the derived effective model as,

$\begin{split}H^{1}_{eff}&=H_{11}+\sum_{kk^{\prime}}J_{0}S.s_{kk^{\prime}}+i\sum_{kk^{\prime}}\Delta(S\times
s_{kk^{\prime}})_{z}+\sum_{kk^{\prime}}i\frac{k}{\beta k^{3}\cos
3\theta}M_{kk^{\prime}}.(S\times
s_{kk^{\prime}})_{y}+\sum_{kk^{\prime}}i\frac{k}{\beta k^{3}\cos
3\theta}\Delta M_{kk^{\prime}}.(S\times s_{kk^{\prime}})_{x}\\\ \therefore
H^{1}_{eff}&=H_{11}+\sum_{kk^{\prime}}J_{0}S.s_{kk^{\prime}}+i\sum_{kk^{\prime}}\vec{J}_{k^{3}}.(S\times
s_{kk^{\prime}})+i\sum_{kk^{\prime}}\vec{J}_{k}.(S\times
s_{kk^{\prime}})\end{split}$ (31)

##  Appendix B Including the potential scattering

The effective model indicates that the matrix elements are scaled as
$M^{\theta,\theta^{\prime}}_{kk^{\prime}}\propto\frac{V{k}}{\sqrt{k^{3}\cos
3\theta}}\frac{V_{k^{\prime}}}{\sqrt{k^{\prime 3}\cos 3\theta^{\prime}}}$.If
we go beyond poorman’s limit $k=k^{\prime}=k_{f}$ then this leads to four
possible potential scattering scenarios:
$\vec{J}^{\theta_{\pm\frac{2\pi}{3}},\theta_{\pm\frac{\pi}{3}}}_{k^{3},k}$,
$\vec{J}^{\theta{\frac{2\pi}{3}},\theta_{-\frac{\pi}{3}}}_{k^{3},k}$, and the
original couplings denoted as $\vec{J}^{0,0}_{k^{3},k}$,
$\vec{J}^{0,0}_{k^{3},k}$. It’s worth noting that the vector corresponding to
non-linear dispersion is only 1-dimensional and contains a z-component. In
contrast, the vector corresponding to linear dispersion is 2-dimensional and
contains x and y components. This introduces an anisotropic
$\mathcal{DM}$-interaction.

$\begin{split}H^{1}_{eff}&=H_{0}+\sum_{kk^{\prime}}J_{0}S.\psi^{\dagger}(\Sigma)\psi+i\sum_{kk^{\prime}}\vec{J}^{(0,0)}_{k^{3}}.(S\times\psi^{\dagger}(\Sigma)\psi)-\sum_{kk^{\prime}}\vec{J}^{\theta_{\pm\frac{2\pi}{3}},\theta_{\pm\frac{\pi}{3}}}_{k^{3}}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+\sum_{kk^{\prime}}\vec{J}^{\theta_{\frac{2\pi}{3}},\theta_{-\frac{\pi}{3}}}_{k^{3}}.(S\times\psi^{\dagger}(\Gamma)\psi)+i\sum_{kk^{\prime}}\vec{J}^{(0,0)}_{k}.(S\times\psi^{\dagger}(\Sigma)\psi)-\sum_{kk^{\prime}}\vec{J}^{\theta_{\pm\frac{2\pi}{3}},\theta_{\pm\frac{\pi}{3}}}_{k}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+i\sum_{kk^{\prime}}\vec{J}^{\theta_{\frac{2\pi}{3}},\theta_{-\frac{\pi}{3}}}_{k}.(S\times\psi^{\dagger}(\Gamma)\psi)\end{split}$
(32)

For notational simplicity we will rewite the coupling as following,

$\begin{split}H^{1}_{eff}&=H_{0}+\sum_{kk^{\prime}}J_{0}S.\psi^{\dagger}(\Sigma)\psi+i\sum_{kk^{\prime}}\vec{J}_{k^{3}}.(S\times\psi^{\dagger}(\Sigma)\psi)-\sum_{kk^{\prime}}\vec{g}_{1k^{3}}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+\sum_{kk^{\prime}}\vec{g}_{2k^{3}}.(S\times\psi^{\dagger}(\Gamma)\psi)+i\sum_{kk^{\prime}}\vec{J}_{k}.(S\times\psi^{\dagger}(\Sigma)\psi)-\sum_{kk^{\prime}}\vec{g}_{1k}.(S\times\psi^{\dagger}(\Omega)\psi)\\\
&+i\sum_{kk^{\prime}}\vec{g}_{2k}.(S\times\psi^{\dagger}(\Gamma)\psi)\\\
&=H_{0}+\sum_{kk^{\prime}}J_{0}\bigg{(}S_{x}\Sigma_{x}+S_{y}\Sigma_{y}+S_{z}\Sigma_{z}\bigg{)}+i\sum_{kk^{\prime}}|\vec{J}_{k^{3}}|(S_{x}\Sigma_{y}-\Sigma_{x}S_{y})-\sum_{kk^{\prime}}|\vec{g}_{1k^{3}}|(S_{x}\Omega_{y}-\Omega_{x}S_{y})\\\
&+\sum_{kk^{\prime}}|\vec{g}_{2k^{3}}|(S_{x}\Gamma_{y}-\Gamma_{x}S_{y})+i\sum_{kk^{\prime}}|\vec{J}_{k}|\bigg{(}(S_{y}\Sigma_{z}-\Sigma_{y}S_{z})+(S_{x}\Sigma_{z}-\Sigma_{x}S_{z})\bigg{)}\\\
&-\sum_{kk^{\prime}}|\vec{g}_{1k}|\bigg{(}(S_{y}\Omega_{z}-\Omega_{y}S_{z})+(S_{x}\Omega_{z}-\Omega_{x}S_{z})\bigg{)}\\\
&+i\sum_{kk^{\prime}}|\vec{g}_{2k}|\bigg{(}(S_{y}\Gamma_{z}-\Gamma_{y}S_{z})+(S_{x}\Gamma_{z}-\Gamma_{x}S_{z})\bigg{)}\end{split}$
(33)

We use the diagrams[59, 39, 60, 61, 62] with all permutations of vertices
using the following algebra,

$\begin{split}[\Sigma_{a}\Sigma_{b},\Sigma_{c}]=\Sigma_{a}[\Sigma_{b},\Sigma_{c}]+[\Sigma_{a},\Sigma_{c}]\Sigma_{b}=0\\\
[\Sigma_{a}\Sigma_{b},\Sigma_{a}]=\Sigma_{a}[\Sigma_{b},\Sigma_{c}]+[\Sigma_{a},\Sigma_{a}]\Sigma_{b}=8i\Sigma_{b}\\\
[\Gamma_{a}\Gamma_{b},\Gamma_{a}]=\Gamma_{a}[\Gamma_{b},\Gamma_{c}]+[\Gamma_{a},\Gamma_{a}]\Gamma_{b}=8i\Gamma_{b}\\\
[\Omega_{a}\Omega_{b},\Omega_{a}]=\Omega_{a}[\Omega_{b},\Omega_{c}]+[\Omega_{a},\Omega_{a}]\Omega_{b}=8i\Omega_{b}\\\
[\Omega_{a}\Sigma_{b},\Sigma_{a}]=\Omega_{a}[\Sigma_{b},\Sigma_{c}]+[\Omega_{a},\Sigma_{a}]\Sigma_{b}=8i\Sigma_{b}\\\
[\Sigma_{a}\Omega_{b},\Gamma_{a}]=\Sigma_{a}[\Omega_{b},\Gamma_{c}]+[\Sigma_{a},\Gamma_{a}]\Omega_{b}=8i\Upsilon\end{split}$
(34)

With the above algebra, we find the added potential scattering renormalizes
only at the second loop, and we derive these equations as follows,

$\begin{split}\frac{dJ_{0}}{dl}&=J^{2}_{0}+J^{2}_{k^{3}}+J^{2}_{k}+J_{k^{3}}J_{k}+J^{2}_{k}J_{0}+J^{2}_{k^{3}}J_{0}+J^{3}_{0}-J_{k^{3}}g_{2k}-J_{k^{3}}g_{2k^{3}}-J_{k}g_{2k}+g^{2}_{2k}J_{k^{3}}+g^{2}_{2k^{3}}J_{k}-J^{2}_{k}g_{2k^{3}}\\\
\frac{dJ_{k^{3}}}{dl}&=J^{2}_{k}+J_{0}J_{k^{3}}+J^{2}_{k}J_{k^{3}}+J^{2}_{0}J_{k^{3}}+J^{3}_{k^{3}}\\\
\frac{dJ_{k}}{dl}&=J_{0}J_{k}+J^{2}_{0}J_{k}+J_{k}J^{2}_{k^{3}}+J^{3}_{k}\\\
\frac{dg_{1k^{3}}}{dl}&=g^{3}_{1k^{3}}-g^{2}_{1k}g_{1k^{3}}\\\
\frac{dg_{1k}}{dl}&=g^{3}_{1k}-g^{2}_{1k^{2}}g_{1k}\\\
\frac{dg_{2k^{3}}}{dl}&=-g^{3}_{2k^{3}}+g^{2}_{1k^{3}}g_{2k^{3}}\\\
\frac{dg_{2k}}{dl}&=-g^{3}_{2k}+g^{2}_{1k}g_{2k}\end{split}$ (35)

From above, we identify various RG invariants as
$\frac{g_{1k^{3}}}{g_{1k}}=m_{1}$, and we can notice that after adding the
potential scattering terms, we still have the invariant
$J^{2}_{k}+J_{k}=mJ_{k^{3}}$ and
$\frac{g^{2}_{2k}(g_{2k}+\sqrt{m_{2}})}{(g_{2k}-\sqrt{m_{2}})}=\frac{g^{2}_{2k^{3}}(g_{2k^{3}}+\sqrt{m_{2}/m_{1}})}{(g_{2k^{3}}-\sqrt{m_{2}/m_{1}})}$

##  Appendix C Rg equations For Two impurity

To study the renormalization of RKKY and Kondo couplings, we show here to
introduce a 4-operator vertex in a dot, specifically
$J_{Y}:d^{\dagger}_{\alpha\sigma}d^{\dagger}_{\alpha^{\prime}\sigma^{\prime}}d_{\alpha\sigma}d_{\alpha^{\prime}\sigma^{\prime}}:$,
among all the possible vertices in two dots. By using spin algebra, we can
write
$[S^{+}{\alpha},S^{-}{\alpha^{\prime}}]=\delta_{\alpha,\alpha^{\prime}}S^{z}_{\alpha}-d^{\dagger}_{\alpha\uparrow}d^{\dagger}_{\alpha\downarrow}d_{\alpha\downarrow}d_{\alpha^{\prime}\uparrow}+d^{\dagger}_{\alpha\uparrow}d^{\dagger}_{\alpha\downarrow}d_{\alpha\downarrow}d_{\alpha^{\prime}\uparrow}$
in normal ordering, where fermionic algebra is used except for $S^{z}$, and
all terms except for $S^{z}$ vanish. Summing all one flip vertices, we get
$[S^{z}_{\alpha},S^{\pm}{\alpha^{\prime}}]=2S^{\pm}\delta{\alpha\alpha^{\prime}}-d^{\dagger}_{\alpha\uparrow}d^{\dagger}_{\alpha\sigma}d_{\alpha\downarrow}d_{\alpha^{\prime}\sigma}+d^{\dagger}_{\alpha\uparrow}d^{\dagger}_{\alpha\sigma}d_{\alpha\downarrow}d_{\alpha^{\prime}\sigma}$.
We have written the full operator terms to demonstrate that they obey spin
algebra, and the commutator yields one spin vertex with no scattering terms in
the dot. However, this is not the case in the bath.

Figure 11: To study the renormalization of RKKY and usual Kondo couplings, we
can extend the Poorman diagrams to include RKKY vertices represented by
crossed circles and the usual J vertices with black circles. Three spin-flip
processes are absent in the single impurity case at the second-order process.
To distinguish between different types of interactions, we can use a wavy line
to denote RKKY interactions and a dotted line to denote impurity-$mathcal{DM}$
interactions.

The above discussion highlights the difficulty in computing all diagrams at
third order in the presence of both RKKY and $\mathcal{DM}$ interactions. With
${}^{7}C_{2}$ diagrams possible at second order and ${}^{7}C_{2}+{}^{7}C_{3}$
diagrams possible at third order, the calculations become extremely tedious.
However, one can use algebraic techniques to simplify the computation in
scenarios with potential scattering. Lie matrices can be used to compute the
contributions in such cases.

Figure 12: The above panels in the first row correspond to the positive root
and $m=-4$ and in the real, imaginary plane, similarly below for the negative
root and $m=-4$, showing the spiral FP in RG flow.

##  Appendix D RKKY Interaction

We can write the RKKY interaction in edge states by expanding in the Fourier
series for the spins,

$\begin{split}J_{Y}(r_{i}-r_{j})&\propto\sum_{kk^{\prime}hh^{\prime}}\frac{V_{ij}}{\epsilon_{kh}-\epsilon_{k^{\prime}h^{\prime}}}e^{ik(r_{i}-r_{j})}V^{*}_{ij}e^{-ik^{\prime}(r_{i}-r_{j})}\\\
&=\int^{2\pi}_{0}\int^{\pi}_{0}\int^{k_{f}}_{0}\frac{k^{2}sin\theta
e^{ikR}dkd\theta d\phi}{\sqrt{k^{6}\beta^{2}\cos^{2}3\theta+k^{2}}}+(k\neq
k^{\prime},h=h^{\prime})\hskip 5.69054pt\text{terms}\end{split}$ (36)

We solve the above integral the same way as we evaluate in impurity transport
sections and show that the leading contribution for $J_{Y}$ is the sum of
special functions as $E_{1}(\epsilon)+E_{2}(\epsilon)$ with substitution
$\cos\theta=t$ we get following also we don’t set
$\epsilon_{k}=\epsilon_{k^{\prime}}$ as in the Poorman limit instead we
consider a chemical potential such that integral does not diverge in this
limit,

$\begin{split}J_{Y}&\propto\pi\int_{k}\frac{e^{ikR}}{k\beta}\oint\frac{qdt}{q^{2}-(4t^{3}-3t)^{2}}+\sum_{kk^{\prime}}\frac{|V_{ij}|}{k^{2}-(k^{\prime})^{2}}e^{ikr}e^{-ik^{\prime}r}\\\
&\text{where}\hskip 5.69054ptq=\frac{\sqrt{(k^{2}-\epsilon)^{2}-k^{2}}}{\beta
k^{3}}\\\ \therefore J_{Y}&\propto\int
e^{i\sqrt{\epsilon}r}\frac{\sqrt{(\epsilon-\epsilon^{\prime})^{2}-\epsilon}}{\beta\epsilon^{\frac{3}{2}}}\sqrt{\epsilon}d\epsilon+\frac{\sin(\sqrt{\epsilon}r)}{\epsilon^{\frac{3}{2}}}+\frac{\sin(\sqrt{\epsilon}r)}{\sqrt{\epsilon}}..\end{split}$
(37)

We can immediately see the sum of the roots will vanish for such cubic
equations since there is no $t^{2}$ coefficient, but the quantity in the root
is not the same for all residues; hence it does contribute. Here we show the
first part of the integral has a root structure and can be scaled with the
root and can be identified why there is critical $\epsilon^{\prime}=-0.25$ for
rkky although there are even and odd terms in series. The second term is
standard considered in various studies shown explicitly in pointlike contact
potential, it can be expressed as bessel function[63] or elliptic
function[64](can be expressed in both special functions).

$\begin{split}J_{Y}&\propto\sum^{\infty}_{n=0}i^{n}\frac{2\sqrt{\epsilon^{\prime
2}-2\epsilon^{\prime}\epsilon+(\epsilon-1)\epsilon}r^{n}\epsilon^{n/2}F_{1}\left(\frac{n}{2};-\frac{1}{2},-\frac{1}{2};\frac{n}{2}+1;-\frac{2\epsilon}{-2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}-1},\frac{2\epsilon}{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+1}\right)}{nn!\sqrt{\frac{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}-2\epsilon+1}{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+1}}\sqrt{\frac{-2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+2\epsilon-1}{-2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}-1}}}+\frac{\sin(\sqrt{\epsilon}r)}{\epsilon^{\frac{3}{2}}}+\frac{\sin(\sqrt{\epsilon}r)}{\sqrt{\epsilon}}..\end{split}$
(38)

The first term is a sum and can be shown as an odd and even
$J_{Y}=\sum^{\infty}_{n=0}=J^{odd}_{Y}+J^{even}_{Y}$ so therefore, all the odd
terms can be expressed as the elliptic functions graphically shown in the main
article in figure 10 which host exceptional points. Even terms consist of
transcendental functions

##  Appendix E Analytic Solution of RG equations

A natural simplification of $J_{k^{3}}$ and $J_{k}$ RG equation by eliminating
$J_{0}$ gives the following,

$\begin{split}\frac{dJ_{k^{3}}}{dl}-J^{2}_{k}=\frac{J_{k^{3}}}{J_{k}}\frac{dJ_{k}}{dl}\\\
\frac{1}{J_{k^{3}}}\frac{dJ_{k^{3}}}{dl}-\frac{J^{2}_{k}}{J_{k^{3}}}=\frac{1}{J_{k}}\frac{dJ_{k}}{dl}\end{split}$
(39)

Now with a substitution of $\frac{J^{2}_{k}}{J_{k^{3}}}=x$ which gives upon
differentiation
$\frac{dx}{dl}=2\frac{J_{k}}{J_{k^{3}}}\frac{dJ_{k}}{dl}-\frac{J^{2}_{k}}{J^{2}_{k^{3}}}\frac{dJ_{k^{3}}}{dl}$
using this and rearranging we get solution as
$J_{k}=-\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4mJ_{k^{3}}}$ this is related to
second solution by phase factors Now we can derive the full solution as the
following,

$\begin{split}J_{0}\frac{dJ_{0}}{dl}-J^{3}_{0}=n,\hskip
5.69054pt\bigg{(}\frac{J^{2}_{k}+J_{k}}{m}+J_{k}\frac{(J_{k}+1)^{2}}{m^{2}}+J_{k}\bigg{)}\frac{dJ_{k}}{dl}=n\end{split}$
(40)

This yield solution is as follows; One can show the m has complex roots when
the effective bandwidth vanishes.
$J^{*}_{k}\to\frac{\sqrt{3}e^{i\frac{\pi}{3}}}{2}$,$J^{*}_{k^{3}}\to\frac{9e^{i\frac{\pi}{3}}}{16}$
and $m=e^{i\frac{\pi}{3}}$ where the $J^{*}_{0}$ will have $\tan^{-1}(..)$
quantity vanishing for $J^{*}_{0}=\pm 1$ in ferromagnetic and
antiferromagnetic cases for $n\to 1$ see equations in 41, also log
contribution to scale vanish when $J^{*}_{0}=-2$.We get these FP’s when we add
the potential scattering terms.

$\begin{split}\log\bigg{(}\frac{[m^{2}+m(J_{k}-1)+(J_{k}-1)^{2}]^{3\gamma}}{J^{2\gamma}_{k}}\bigg{)}+\frac{(m-2)}{\sqrt{3}(m^{2}-m+1)}\tan^{-1}\left(\frac{m+2J_{k}-2}{\sqrt{3}m}\right)=n\log
D_{eff}\\\
\frac{\log\left(n^{2/3}+\sqrt[3]{n}(-J_{0})+J_{0}^{2}\right)-2\log\left(\sqrt[3]{n}+J_{0}\right)-2\sqrt{3}\tan^{-1}\left(\frac{1-\frac{2J_{0}}{\sqrt[3]{n}}}{\sqrt{3}}\right)}{6\sqrt[3]{n}}=\log
D_{eff}\\\ \text{where}\hskip
5.69054pt\gamma=\frac{-m}{6*(m^{2}-m+1)}\implies\gamma\to-\infty*\hskip
5.69054pt\text{for}\hskip 5.69054ptm=e^{\pm i\frac{\pi}{3}}\end{split}$ (41)

Due to the bath’s exceptional dispersion, there may be an unusual impurity
screening. The kondo destruction will be at the critical points found above,
leading to some complex FPs. The similarity between the RG of a single
impurity in edge states and two impurities in conventional Fermi gas. For this
reason, we solve two-impurity problem RG equations by limiting all couplings
to zero except $J_{0}\neq J_{Y}\neq K\neq 0$.We need to solve the equations
$\frac{dJ_{0}}{dl}=J^{2}_{0}+J_{Y}J_{0}+KJ_{0},\frac{dJ_{Y}}{dl}=J^{2}_{Y}+J^{2}_{0}+K^{2},\frac{dK}{dl}=K^{2}+KJ_{Y}$.
If we solve $J_{0}$ and K equations, we get $J_{0}=\frac{RK}{K-1}$ as a
solution where R is a constant or invariant under renormalization. Using this
solution, we solve $J_{Y}$ and K equations as follows,

$\begin{split}J_{Y}\frac{dJ_{Y}}{dl}-J^{3}_{Y}=R_{1},\frac{dK}{dl}-K^{2}=\frac{R_{1}}{\frac{R^{2}K}{(1-K)^{2}}+K}\\\
\frac{-2\log\left(J_{Y}+\sqrt[3]{R_{1}}\right)+\log\left(\sqrt[3]{R_{1}}\left(-J_{Y}\right)+J_{Y}^{2}+R_{1}^{2/3}\right)+2\sqrt{3}\tan^{-1}\left(\frac{\frac{2J_{Y}}{\sqrt[3]{R_{1}}}-1}{\sqrt{3}}\right)}{6\sqrt[3]{R_{1}}}=\log
D_{eff}\\\
\frac{-2\log\left(K+\sqrt[3]{R_{1}}\right)+\log\left(\sqrt[3]{R_{1}}\left(-K\right)+K^{2}+R_{1}^{2/3}\right)+2\sqrt{3}\tan^{-1}\left(\frac{\frac{2K}{\sqrt[3]{R_{1}}}-1}{\sqrt{3}}\right)}{6\sqrt[3]{R_{1}}}=\log
D_{eff},\hskip 5.69054pt\text{for}\hskip 5.69054ptK\to\infty\\\
\frac{-\frac{R^{2}}{K-1}+R^{2}\log(K-1)+\frac{1}{2}(K-1)^{2}+K}{R_{1}}=\log
D_{eff},\hskip 5.69054pt\text{for}\hskip 5.69054ptK\to 0\\\
(K-1)^{\frac{R}{R_{1}}}\exp{\frac{1}{R_{1}}\bigg{(}-\frac{R^{2}}{K-1}+\frac{1}{2}(K-1)^{2}+K\bigg{)}}=D_{eff}\\\
\implies(KR/J_{0})^{\frac{R}{R_{1}}}e^{\frac{1}{R_{1}}\bigg{(}-\frac{R^{2}}{KR/J_{0}}+\frac{1}{2}(KR/J_{0})^{2}+K\bigg{)}}=D_{eff}\\\
\frac{\sqrt[3]{R_{1}}\left(-J_{Y}\right)+J_{Y}^{2}+R_{1}^{2/3}}{(J_{Y}+\sqrt[3]{R_{1}})^{2}}e^{2\sqrt{3}\tan^{-1}\left(\frac{\frac{2J_{Y}}{\sqrt[3]{R_{1}}}-1}{\sqrt{3}}\right)}=D^{6R^{\frac{1}{3}}_{1}}_{eff}\end{split}$
(42)

Solving for $D_{eff}$ we can see how the two impurity kondo scale renormalized
in this problem. Similarly, We will solve the $J_{0}$ equation with $J_{Y}$ to
separate the solutions and for $R=R_{1}=1.0$, we show the flow and critical
points. We expect FP’s in different quadrants depending on the sign we choose
for these constants.

Figure 13: Plots above show the FP can be in different quadrants with
$R_{1}=-1$ and $R=1.0$. We get around the spiral FP’s we get sign reversion in
couplings.

##  Appendix F Impurity Transport Calculation

We follow the $\mathcal{T}_{kk^{\prime}}$ formalism (without expanding in
terms of $\Delta$ as we did for separating RG flow and also not including the
potential scatterings) for the derived kondo model after the k-dependent
unitary to compute the relaxation time as follows,

$\begin{split}\frac{1}{\tau(k)}\propto(1-2J\tilde{g}_{\alpha}(\epsilon_{k})-2J\tilde{g}_{\alpha}^{*}(\epsilon_{k})-2J_{k^{3}}\tilde{g}_{\alpha
k^{3}}(\epsilon_{k})-2J_{k^{3}}\tilde{g}_{\alpha
k^{3}}^{*}(\epsilon_{k})-2J_{k}\tilde{g}_{\alpha
k}(\epsilon_{k})-2J_{k}\tilde{g}_{\alpha k}^{*}(\epsilon_{k}))\end{split}$
(43)
$\begin{split}\tilde{g}_{\alpha}(\epsilon)=\int^{2\pi}_{0}\int^{\pi}_{0}\int^{k_{f}}_{0}\frac{k^{2}sin\theta
dkd\theta
d\phi}{k^{2}+\alpha\sqrt{k^{6}\beta^{2}\cos^{2}3\theta+k^{2}}-\epsilon}\end{split}$
(44)

Solving the $\theta$ integral first, we get two pieces as follows,

$\begin{split}\tilde{g}_{\alpha}(\epsilon)=2\pi\int_{k}k^{2}\oint\frac{dt}{k^{2}+\alpha\sqrt{k^{6}\beta^{2}(4t^{3}-3t)^{2}+k^{2}}-\epsilon}\end{split}$
(45)

we rationalize the above integral and write as the following by introducing
$q=\frac{\sqrt{(k^{2}-\epsilon)^{2}-k^{2}}}{\beta k^{3}}$,

$\begin{split}\tilde{g}_{\alpha}(\epsilon)&=2\pi\int_{k}\frac{1}{k}\oint\frac{q+\bar{\alpha}\sqrt{(4t^{3}-3t)^{2}+\frac{1}{\beta^{2}k^{4}}}dt}{q^{2}-(4t^{3}-3t)^{2}}\\\
&=\pi\int_{k}\frac{1}{kq}\oint\frac{q+\bar{\alpha}\sqrt{(4t^{3}-3t)^{2}+\frac{1}{\beta^{2}k^{4}}}dt}{q-(4t^{3}-3t)}+\pi\int_{k}\frac{1}{kq}\oint\frac{q+\bar{\alpha}\sqrt{(4t^{3}-3t)^{2}+\frac{1}{\beta^{2}k^{4}}}dt}{q+(4t^{3}-3t)}\\\
&=\int_{k}\frac{\pi}{kq}\bigg{(}\sum_{res}f(t,q)+\sum_{res}f(t,-q)\bigg{)}\end{split}$
(46)

For finding these residues, we will use the roots of the cubic equations
$\beta(4t^{3}-3t)\pm q=0$. We collect positive q roots and negative as follows
for doing contour integrals,

$\begin{split}t^{+}=\begin{cases}\frac{1}{2}\left(\sqrt[3]{\sqrt{q^{2}-1}+q}+\frac{1}{\sqrt[3]{\sqrt{q^{2}-1}+q}}\right),\\\
-\frac{1}{4}\left(1-i\sqrt{3}\right)\sqrt[3]{\sqrt{q^{2}-1}+q}-\frac{1+i\sqrt{3}}{4\sqrt[3]{\sqrt{q^{2}-1}+q}}\\\
-\frac{1}{4}\left(1+i\sqrt{3}\right)\sqrt[3]{\sqrt{q^{2}-1}+q}-\frac{1-i\sqrt{3}}{4\sqrt[3]{\sqrt{q^{2}-1}+q}}\end{cases}t^{-}=\begin{cases}\frac{1}{2}\left(\sqrt[3]{\sqrt{q^{2}-1}-q}+\frac{1}{\sqrt[3]{\sqrt{q^{2}-1}-q}}\right),\\\
-\frac{1}{4}\left(1-i\sqrt{3}\right)\sqrt[3]{\sqrt{q^{2}-1}-q}-\frac{1+i\sqrt{3}}{4\sqrt[3]{\sqrt{q^{2}-1}-q}}\\\
-\frac{1}{4}\left(1+i\sqrt{3}\right)\sqrt[3]{\sqrt{q^{2}-1}-q}-\frac{1-i\sqrt{3}}{4\sqrt[3]{\sqrt{q^{2}-1}-q}}\end{cases}\end{split}$
(47)

Similarly, the other contributions are as follows,

$\begin{split}\tilde{g}_{\alpha
k^{3}}(\epsilon)&=2\pi\beta\int_{k}k^{2}\oint\frac{((4t^{3}-3t)^{2})(q+\bar{\alpha}\sqrt{(4t^{3}-3t)^{2}+\frac{1}{\beta^{2}k^{4}}})}{q^{2}-(4t^{3}-3t)^{2}}dt\\\
\tilde{g}_{\alpha
k}(\epsilon)&=\int_{k}\oint\frac{(q+\bar{\alpha}\sqrt{(4t^{3}-3t)^{2}+\frac{1}{\beta^{2}k^{4}}})}{q^{2}-(4t^{3}-3t)^{2}}dt\end{split}$
(48)

after summing over the $\alpha=\pm$ bands we get following simplifications,

$\begin{split}\tilde{g}_{\alpha}(\epsilon)&=2\pi\int_{k}\frac{1}{k}\oint\frac{qdt}{q^{2}-(4t^{3}-3t)^{2}},\tilde{g}_{\alpha
k^{3}}(\epsilon)=2\pi\beta\int_{k}k^{2}\oint\frac{(4t^{3}-3t)^{2}qdt}{q^{2}-(4t^{3}-3t)^{2}},\tilde{g}_{\alpha
k}(\epsilon)=\int_{k}\oint\frac{qdt}{q^{2}-(4t^{3}-3t)^{2}}\end{split}$ (49)

After contour integrals we can show each integral contributes as
$\oint\frac{dt}{q^{2}-(4t^{3}-3t)^{2}}=q$ hence sum of all $(g_{\alpha})$
contribution as following,

$\begin{split}\tilde{g}_{\epsilon}=\int(\beta qk^{2}-\beta
q^{4}k+\frac{q}{k}-q)dk\end{split}$ (50)

We know from derivation $\epsilon\approx k^{2}$ and the density of states in
3D as $\rho(\epsilon)\approx\sqrt{\epsilon}$ then above integral yields,

$\begin{split}\tilde{g}_{\epsilon^{\prime}}=\int\bigg{(}\frac{\sqrt{(\epsilon-\epsilon^{\prime})^{2}-\epsilon}}{\sqrt{\epsilon}}-\frac{((\epsilon-\epsilon^{\prime})^{2}-\epsilon)^{2}}{\beta^{3}\epsilon^{6-\frac{1}{2}}}+\frac{\sqrt{(\epsilon-\epsilon^{\prime})^{2}-\epsilon}}{\beta\epsilon^{2}}-\frac{\sqrt{(\epsilon-\epsilon^{\prime})^{2}-\epsilon}}{\beta\epsilon^{\frac{3}{2}}}\bigg{)}d\epsilon\end{split}$
(51)

Where in above $\epsilon^{\prime}$ is the chemical potential can take any
values around fermi energy. We perform this above integral exactly in terms of
special functions to extract contribution to $\frac{1}{\tau}$, which indeed
scales with RG invariant in the elliptic functions.

$\begin{split}&\tilde{g}_{\epsilon}\propto\frac{P_{1}(\epsilon)+P_{\frac{3}{2}}(\epsilon)+T_{1}(\epsilon)+E_{1}(\epsilon)+E_{2}(\epsilon)+\log(\epsilon)}{\beta^{3}}\\\
&P_{1}(\epsilon)=-\frac{3\epsilon^{\prime 4}-16\epsilon^{\prime
3}\epsilon+4\epsilon^{\prime
2}\epsilon(9\epsilon-2)+2\epsilon^{2}\left(2\epsilon\left(\beta^{2}\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}\left(2\sqrt{\epsilon}(\beta-\beta\epsilon+3)+3\right)-6\right)+3\right)}{12\epsilon^{4}}\\\
&P_{\frac{3}{2}}(\epsilon)=\frac{8\epsilon^{\prime}\epsilon^{2}\left(2\beta^{3}\epsilon^{3/2}\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}-6\epsilon+3\right)}{12\epsilon^{4}}\\\
&T_{1}(\epsilon)=\beta^{2}\tanh^{-1}\left(\frac{-2\epsilon^{\prime}+2\epsilon-1}{2\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}}\right)+\frac{(2\epsilon^{\prime}+1)\beta^{2}\coth^{-1}\left(\frac{2\epsilon^{\prime}\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}}{2\epsilon^{\prime}(\epsilon^{\prime}-\epsilon)-\epsilon}\right)}{2\epsilon^{\prime}}\\\
&E_{1}(\epsilon)=\frac{i\beta^{2}\epsilon\sqrt{\frac{-2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+2\epsilon-1}{\epsilon}}\sqrt{-\frac{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}-2\epsilon+1}{\epsilon}}\left(AE\left(i\sinh^{-1}\left(\frac{\sqrt{-2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}-1}}{\sqrt{2}\sqrt{\epsilon}}\right)|\frac{2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}+1}{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+1}\right)\right)}{3\sqrt{2}\sqrt{-2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}-1}\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}}\\\
&A=\left(2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+1\right)(2\epsilon^{\prime}\beta+\beta+6)\\\
&E_{2}(\epsilon)=-\frac{B\left(\left(\sqrt{4\epsilon^{\prime}+1}+2\epsilon^{\prime}\left(\sqrt{4\epsilon^{\prime}+1}+2\right)+1\right)\beta+6\sqrt{4\epsilon^{\prime}+1}\right)F\left(i\sinh^{-1}\left(\frac{\sqrt{-2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}-1}}{\sqrt{2}\sqrt{\epsilon}}\right)|\frac{2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}+1}{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+1}\right)}{3\sqrt{2}\sqrt{-2\epsilon^{\prime}-\sqrt{4\epsilon^{\prime}+1}-1}\sqrt{(\epsilon^{\prime}-\epsilon)^{2}-\epsilon}}\\\
&B=i\beta^{2}\epsilon\sqrt{\frac{-2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}+2\epsilon-1}{\epsilon}}\sqrt{-\frac{2\epsilon^{\prime}+\sqrt{4\epsilon^{\prime}+1}-2\epsilon+1}{\epsilon}}\end{split}$
(52)

In the above solution, F is an incomplete elliptic function of the first kind,
and E is an elliptic function of the second kind. P and T are polynomial and
transcendental functions, respectively.Similar elliptic functions are found to
solve nonhermitian problem [34] to connect the nonequilibrium self-energy of
an open system.

##  Appendix G Numerical Solutions

###  G.1 Solutions for RG ODE’s

Figure 14: We plot the solutions of the RG equations around the critical
points from the flow diagram. The x-axis is the bandwidth diverging points
corresponding to the Kondo scale, and cusp-like behaviors indicate the two
impurity model FP’s. The inset graph shows the non-interacting FP we got when
we set either $J_{Y}$ or $K$ to negative otherwise, we have two impurities
Kondo effect, where all couplings diverge. SR in both can be seen for the
couplings $J_{0},K$ when for both $J_{k^{3}},J_{k}$ given -ve initial
conditions(IC) are given. Otherwise, only the $K$ reverses sign with cusp
$-\infty\to\infty$.With various IC’s, we verified it is necessary to have
$J_{k^{3}}\neq 0$ to get cusp-like sign reversion in the couplings. IC’s are
mentioned above each plot above.

###  G.2 Single Impurity Case

The solutions are examined for the spiral FP’s whether we have SR in the
imaginary part of the RG beta functions $\beta_{n}$. Indeed we verify FP’s
corresponding to two spiral points and one marginal point, all of them a
reverse sign, whereas we had negative $\beta_{n}$ when all the couplings are
very small except $J_{0}$ shown in figure 15. From the invariant derived
analytically $J_{k}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4mJ_{k^{3}}}$ for
$J_{k^{3}}=0.5$(that is where we got FP see fig 5 of main paper) and $m=4$ we
get two roots as $J_{k}=\frac{1}{2}\pm\frac{3}{2}=2,-1$ exactly these were
chosen as initial conditions and shown $\beta_{n}\to\beta^{*}_{n}$.We also
have various 3-pole and 2-pole regimes where we don’t have FP’s.

Figure 15: We plot the solutions of the RG equations around the critical
points from the flow diagram for the single impurity case($K=J_{Y}=0$). plots
(c) and (d) correspond to spiral FP’s and plot (b) is marginal point case when
$J_{k^{3}}=0,J_{k}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4mJ_{k^{3}}}=0,1$. plot
(a) is usual Kondo when homogeneous bath case so that all set zero except
$J_{0}\neq 0$.Plots (e) and (f) are the solutions where flow is divergent with
no FP.

###  G.3 Two Impurity Case

Unlike single impurity cases, we do not have the flow FP’s in the TIK problem;
hence we need to generate many data files to identify where the SR regime
exists. We can set $J_{k^{3}}=J_{k}=0$ and start analyzing, and we find the SR
regime for the spiral points. But in general, this is a rich problem and
requires more analysis, but here we show a critical $J_{k^{3}}$ required to
get SR in figure 14. Having the SR scenarios in RG indicate there is the
emergence of complex solutions as found by analytically and numerically in a
single impurity.

Figure 16: We plot the RG solutions by allowing the complex solutions of RG
nonlinear equations. The dotted and thick lines correspond to real and
imaginary coupling values. Sign reversion in all couplings is captured when we
have a critical value of the nonlinear coupling $J_{k^{3}}>2.0$. Between the
two sign-reversing phases in plots $a\to f$, there are various 2-pole(b,c,e)
and 3-pole(d) regimes. This confirms the spiral FP’s we got are due to the
sign reversion in couplings. This happens only at two initial conditions.
Scales above the $10^{3}$ are not labeled in y - the axis and all plots’
x-axis is a flow parameter which is either in log/linear scale.The sign-
reversing phase happens in imaginary values of couplings equivalently RG beta
functions $\beta_{n}\to\beta_{n}^{*}$ also mean a renormalized self-energy
$\tilde{\Sigma}(l)\to\tilde{\Sigma}(l)^{*}$ which is equivalent of getting EP
in odd coupling in this problem. The central idea of this analysis is that
there is an exceptional point in $J^{odd}_{Y}$ and in bath integrals
$E_{1}+E_{2}$ to verify if at all there exist $J_{k^{3}}$ values for which
this transition occurs. There are $10^{5}$ data files for 10 IC’s hence it
requires further analysis to find other values at which this happens.

## References

  * Lü  _et al._ [2013] H.-F. Lü, H.-Z. Lu, S.-Q. Shen, and T.-K. Ng, Quantum impurity in the bulk of a topological insulator, Phys. Rev. B 87, 195122 (2013).
  * Mitchell  _et al._ [2013] A. K. Mitchell, D. Schuricht, M. Vojta, and L. Fritz, Kondo effect on the surface of three-dimensional topological insulators: Signatures in scanning tunneling spectroscopy, Phys. Rev. B 87, 075430 (2013).
  * Dey  _et al._ [2018] U. Dey, M. Chakraborty, A. Taraphder, and S. Tewari, Bulk band inversion and surface dirac cones in lasb and labi: Prediction of a new topological heterostructure, Scientific Reports 8, 14867 (2018).
  * Li  _et al._ [2009] J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Topological anderson insulator, Phys. Rev. Lett. 102, 136806 (2009).
  * Kuzmenko  _et al._ [2014] I. Kuzmenko, Y. Avishai, and T. K. Ng, Anderson impurity in the bulk of topological insulators, Phys. Rev. B 89, 035125 (2014).
  * Boettcher [2020] I. Boettcher, Interplay of topology and electron-electron interactions in rarita-schwinger-weyl semimetals, Phys. Rev. Lett. 124, 127602 (2020).
  * Sierra  _et al._ [2018] M. A. Sierra, R. López, and J. S. Lim, Thermally driven out-of-equilibrium two-impurity kondo system, Phys. Rev. Lett. 121, 096801 (2018).
  * Affleck and Ludwig [1992] I. Affleck and A. W. Ludwig, Exact critical theory of the two-impurity kondo model, Physical review letters 68, 1046 (1992).
  * Mitchell  _et al._ [2012] A. K. Mitchell, E. Sela, and D. E. Logan, Two-channel kondo physics in two-impurity kondo models, Physical review letters 108, 086405 (2012).
  * Silva  _et al._ [1996] J. Silva, W. Lima, W. Oliveira, J. Mello, L. N. d. Oliveira, and J. Wilkins, Particle-hole asymmetry in the two-impurity kondo model, Physical review letters 76, 275 (1996).
  * O’Bannon  _et al._ [2016] A. O’Bannon, I. Papadimitriou, and J. Probst, A holographic two-impurity kondo model, Journal of High Energy Physics 2016, 1 (2016).
  * Sela  _et al._ [2011] E. Sela, A. K. Mitchell, and L. Fritz, Exact crossover green function in the two-channel and two-impurity kondo models, Physical review letters 106, 147202 (2011).
  * Gan [1995a] J. Gan, Solution of the two-impurity kondo model: Critical point, fermi-liquid phase, and crossover, Physical Review B 51, 8287 (1995a).
  * Gan [1995b] J. Gan, Mapping the critical point of the two-impurity kondo model to a two-channel problem, Physical review letters 74, 2583 (1995b).
  * Bayat  _et al._ [2012] A. Bayat, S. Bose, P. Sodano, and H. Johannesson, Entanglement probe of two-impurity kondo physics in a spin chain, Physical Review Letters 109, 066403 (2012).
  * Mross and Johannesson [2009a] D. F. Mross and H. Johannesson, Two-impurity kondo model with spin-orbit interactions, Physical Review B 80, 155302 (2009a).
  * Cho and McKenzie [2006] S. Y. Cho and R. H. McKenzie, Quantum entanglement in the two-impurity kondo model, Physical Review A 73, 012109 (2006).
  * Sire  _et al._ [1993] C. Sire, C. M. Varma, and H. Krishnamurthy, Theory of the non-fermi-liquid transition point in the two-impurity kondo model, Physical Review B 48, 13833 (1993).
  * Campo Jr and Oliveira [2004] V. L. Campo Jr and L. N. Oliveira, Thermodynamics for the two-impurity kondo model, Physical Review B 70, 153401 (2004).
  * Hallberg and Egger [1997] K. Hallberg and R. Egger, Two-impurity kondo problem for correlated electrons, Physical Review B 55, R8646 (1997).
  * Principi  _et al._ [2015] A. Principi, G. Vignale, and E. Rossi, Kondo effect and non-fermi-liquid behavior in dirac and weyl semimetals, Phys. Rev. B 92, 041107 (2015).
  * Liu  _et al._ [2010] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Model hamiltonian for topological insulators, Phys. Rev. B 82, 045122 (2010).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in bi2se3, bi2te3 and sb2te3 with a single dirac cone on the surface, Nature physics 5, 438 (2009).
  * Lü  _et al._ [2019] H.-F. Lü, Y.-H. Deng, S.-S. Ke, Y. Guo, and H.-W. Zhang, Quantum impurity in topological multi-weyl semimetals, Phys. Rev. B 99, 115109 (2019).
  * Mitchell and Fritz [2016] A. K. Mitchell and L. Fritz, Signatures of weyl semimetals in quasiparticle interference, Phys. Rev. B 93, 035137 (2016).
  * Zarea  _et al._ [2012] M. Zarea, S. E. Ulloa, and N. Sandler, Enhancement of the kondo effect through rashba spin-orbit interactions, Phys. Rev. Lett. 108, 046601 (2012).
  * Mastrogiuseppe  _et al._ [2014] D. Mastrogiuseppe, A. Wong, K. Ingersent, S. E. Ulloa, and N. Sandler, Kondo effect in graphene with rashba spin-orbit coupling, Phys. Rev. B 90, 035426 (2014).
  * Zarea and Sandler [2009] M. Zarea and N. Sandler, Rashba spin-orbit interaction in graphene and zigzag nanoribbons, Phys. Rev. B 79, 165442 (2009).
  * Žitko [2010] R. Žitko, Quantum impurity on the surface of a topological insulator, Phys. Rev. B 81, 241414 (2010).
  * Lourenço  _et al._ [2018] J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, Kondo effect in a $\mathcal{PT}$-symmetric non-hermitian hamiltonian, Phys. Rev. B 98, 085126 (2018).
  * Nakagawa  _et al._ [2018] M. Nakagawa, N. Kawakami, and M. Ueda, Non-hermitian kondo effect in ultracold alkaline-earth atoms, Phys. Rev. Lett. 121, 203001 (2018).
  * Michishita and Peters [2020] Y. Michishita and R. Peters, Equivalence of effective non-hermitian hamiltonians in the context of open quantum systems and strongly correlated electron systems, Phys. Rev. Lett. 124, 196401 (2020).
  * Oguri and Sakano [2013] A. Oguri and R. Sakano, Exact interacting green’s function for the anderson impurity at high bias voltages, Phys. Rev. B 88, 155424 (2013).
  * Kulkarni [2022] V. M. Kulkarni, Derivation of $\mathcal{PT}$-symmetric sine-gordon model and its relevance to non-equilibrium, arXiv preprint arXiv:2211.00333 (2022).
  * Nagai  _et al._ [2020] Y. Nagai, Y. Qi, H. Isobe, V. Kozii, and L. Fu, Dmft reveals the non-hermitian topology and fermi arcs in heavy-fermion systems, Phys. Rev. Lett. 125, 227204 (2020).
  * Fring and Taira [2020] A. Fring and T. Taira, Complex bps solitons with real energies from duality, Journal of Physics A: Mathematical and Theoretical 53, 455701 (2020).
  * Zhen  _et al._ [2015] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Spawning rings of exceptional points out of dirac cones, Nature 525, 354 (2015).
  * Betancourt  _et al._ [2016] J. Betancourt, S. Li, X. Dang, J. Burton, E. Tsymbal, and J. Velev, Complex band structure of topological insulator bi2se3, Journal of Physics: Condensed Matter 28, 395501 (2016).
  * Affleck and Ludwig [1993] I. Affleck and A. W. Ludwig, Exact conformal-field-theory results on the multichannel kondo effect: single-fermion green’s function, self-energy, and resistivity, Physical Review B 48, 7297 (1993).
  * Kulkarni [2021] V. M. Kulkarni, Functional renormalization analysis of bose-einstien condensation through complex interaction in harmonic oscillator; can bendixson criteria be extended to complex time?, arXiv preprint arXiv:2112.03035 (2021).
  * Han  _et al._ [2023] S. Han, D. J. Schultz, and Y. B. Kim, Complex fixed points of the non-hermitian kondo model in a luttinger liquid, arXiv preprint arXiv:2302.07883 (2023).
  * Nándori  _et al._ [2004] I. Nándori, U. D. Jentschura, K. Sailer, and G. Soff, Renormalization-group analysis of the generalized sine-gordon model and of the coulomb gas for $d\geq 3$dimensions, Phys. Rev. D 69, 025004 (2004).
  * Cheng and Liu [2009] W. W. Cheng and J.-M. Liu, Decoherence from spin environment: Role of the dzyaloshinsky-moriya interaction, Phys. Rev. A 79, 052320 (2009).
  * Nagaosa and Tokura [2013] N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature nanotechnology 8, 899 (2013).
  * Mross and Johannesson [2009b] D. F. Mross and H. Johannesson, Two-impurity kondo model with spin-orbit interactions, Phys. Rev. B 80, 155302 (2009b).
  * Jones and Varma [1987] B. A. Jones and C. M. Varma, Study of two magnetic impurities in a fermi gas, Phys. Rev. Lett. 58, 843 (1987).
  * Jones  _et al._ [1988] B. A. Jones, C. M. Varma, and J. W. Wilkins, Low-temperature properties of the two-impurity kondo hamiltonian, Phys. Rev. Lett. 61, 125 (1988).
  * Mitchell and Fritz [2015] A. K. Mitchell and L. Fritz, Kondo effect in three-dimensional dirac and weyl systems, Phys. Rev. B 92, 121109 (2015).
  * Wójcik and Kroha [2023a] K. P. Wójcik and J. Kroha, Quantum spin liquid in an rkky-coupled two-impurity kondo system, Phys. Rev. B 107, L121111 (2023a).
  * Wójcik and Kroha [2023b] K. P. Wójcik and J. Kroha, Asymmetry effects on the phases of rkky-coupled two-impurity kondo systems, Phys. Rev. B 107, 125146 (2023b).
  * Hewson [1997] A. C. Hewson,  _The Kondo problem to heavy fermions_ , Vol. 2 (Cambridge university press, 1997).
  * Yoshimura  _et al._ [2020] T. Yoshimura, K. Bidzhiev, and H. Saleur, Non-hermitian quantum impurity systems in and out of equilibrium: Noninteracting case, Physical Review B 102, 125124 (2020).
  * Kulkarni  _et al._ [2022] V. M. Kulkarni, A. Gupta, and N. S. Vidhyadhiraja, Kondo effect in a non-hermitian $\mathcal{PT}$-symmetric anderson model with rashba spin-orbit coupling, Phys. Rev. B 106, 075113 (2022).
  * Zhang  _et al._ [2021] W. Zhang, X. Ouyang, X. Huang, X. Wang, H. Zhang, Y. Yu, X. Chang, Y. Liu, D.-L. Deng, and L.-M. Duan, Observation of non-hermitian topology with nonunitary dynamics of solid-state spins, Phys. Rev. Lett. 127, 090501 (2021).
  * Harari  _et al._ [2018] G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Topological insulator laser: theory, Science 359, eaar4003 (2018).
  * Qin  _et al._ [2023] F. Qin, R. Shen, and C. H. Lee, Non-hermitian squeezed polarons, Phys. Rev. A 107, L010202 (2023).
  * Li  _et al._ [2021] L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Communications Physics 4, 42 (2021).
  * Abrikosov [1965] A. A. Abrikosov, Electron scattering on magnetic impurities in metals and anomalous resistivity effects, Physics Physique Fizika 2, 5 (1965).
  * Han  _et al._ [2022] S. Han, D. J. Schultz, and Y. B. Kim, Non-fermi liquid behavior and quantum criticality in cubic heavy fermion systems with non-kramers multipolar local moments, Phys. Rev. B 106, 155155 (2022).
  * Anderson [1970] P. Anderson, A poor man’s derivation of scaling laws for the kondo problem, Journal of Physics C: Solid State Physics 3, 2436 (1970).
  * Sólyom [1974] J. Sólyom, Renormalization and scaling in the x-ray absorption and kondo problems, Journal of Physics F: Metal Physics 4, 2269 (1974).
  * Sólyom and Zawadoswki [1974] J. Sólyom and A. Zawadoswki, Are the scaling laws for the kondo problem exact?, Journal of Physics F: Metal Physics 4, 80 (1974).
  * Szałowski and Balcerzak [2008] K. Szałowski and T. Balcerzak, Rkky interaction with diffused contact potential, Phys. Rev. B 78, 024419 (2008).
  * Okui [1974] S. Okui, Complete elliptic integrals resulting from integrals of bessel functions, Journal of Research of the National Bureau of Standards: Mathematical sciences. B , 113 (1974).