representation in Eq. (93), one encounters the matrix elements $\langle
a|(\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma^{\prime}}^{\,})_{\eta}|b\rangle\langle
b|(\hat{d}_{\sigma^{\prime}}^{\dagger}\hat{d}_{\sigma}^{\,})_{\eta^{\prime}}|a\rangle$.
For the case that $\eta=\eta^{\prime}$, i.e., intra-impurity spin
correlations, this product of matrix elements can be combined into $|\langle
a|..|b\rangle|^{2}\geq 0$, which is real. For the case of inter-impurity spin
correlations, $\eta=-\eta^{\prime}$, taking the complex conjugate and
inserting $\mathfrak{R}_{I}^{\dagger}\mathfrak{R}_{I}^{\,}$ twice,

$\displaystyle\langle
a|(\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma^{\prime}}^{\,})_{\eta}|b\rangle^{\ast}$
$\displaystyle=$ $\displaystyle\langle
b|(\hat{d}_{\sigma^{\prime}}^{\dagger}\hat{d}_{\sigma}^{\,})_{\eta}|a\rangle$
$\displaystyle=$ $\displaystyle\underbrace{\langle
b|(\mathfrak{R}_{I}^{\dagger}}_{\equiv\langle
b^{\prime}|}\underbrace{\mathfrak{R}_{I}^{\,})(\hat{d}_{\sigma^{\prime}}^{\dagger}\hat{d}_{\sigma}^{\,})_{\eta}(\mathfrak{R}_{I}^{\dagger}}_{=(\hat{d}_{\sigma^{\prime}}^{\dagger}\hat{d}_{\sigma}^{\,})_{\eta^{\prime}}}\underbrace{\mathfrak{R}_{I}^{\,})|a\rangle}_{\equiv|a^{\prime}\rangle}$
$\displaystyle=$ $\displaystyle\langle
b^{\prime}|(\hat{d}_{\sigma^{\prime}}^{\dagger}\hat{d}_{\sigma}^{\,})_{\eta^{\prime}}|a^{\prime}\rangle$

Now since $a$ and $b$ can be chosen to also be simultaneous eigenstates of
$\mathfrak{R}_{I}$, i.e., having $a=a^{\prime}$ and $b=b^{\prime}$ with the
same eigenvalue w.r.t. $\mathfrak{R}_{I}$, the Lehmann sum of the spin-spin
spectral functions, while having complex matrix elements, yields a purely real
result. In practice, $a$ and $b$ are not eigenstates of $\mathfrak{R}_{I}$.
Yet by explicitly resorting to complete many-body basis sets within the NRG
[47] evaluated within fmd-NRG [30], the spectral data has an imaginary
contribution with relative strength to the real part that is comparable to
numerical noise based on double precision accuracy, and hence can be ignored.

###  A.6 From Anderson to (anisotropic) Kondo type model

The entire discussion above assumed Anderson-type impurities that hybridize
with the helical edge mode. Now if the local Coulomb interaction $U$ with each
impurity is large, charge fluctuations get frozen out. Therefore in the low-
energy regime, charge fluctuations can be integrated out locally with each
impurity via Schrieffer Wolff transformation. Assuming half-filling of each
impurity with a single magnetic spin-half moment, second order perturbation
theory based on Eq. (42) yields the Kondo-type interaction

$\displaystyle\hat{H}_{\mathrm{K}}\sim\sum_{\eta\in
L,R}\tfrac{1}{-U}(\hat{H}_{\mathrm{hyb}}^{\eta})^{\dagger}\hat{H}_{\mathrm{hyb}}^{\eta}$
(98)

which is diagonal in $\eta$. When projecting the into the low-energy Kondo
regime of the Anderson model by scaling up local interactions, this leaves the
representation of the bath untouched. Therefore from the NRG point of view the
bath remains completely unaffected by whether one resorts to an Anderson-type
or low-energy Kondo-type impurity setup. Based on the coarse-grained version
in Eq. (72) then,

$\displaystyle\hat{H}_{\mathrm{K}}\sim 2J\sum_{\eta\in
L,R}\hat{\bf{S}}_{\eta}\cdot\hat{\bf{S}}_{0\eta}^{\,}\text{ ,}$ (99)

with $J$ the Kondo coupling, $\hat{\bf{S}}_{\eta}\equiv\hat{\bf{S}}_{d\eta}$
the spin operator of impurity $\eta$, and
$\hat{\bf{S}}_{0\eta}\equiv\sum_{\sigma\sigma^{\prime}}\tfrac{\boldsymbol{\tau}_{\sigma\sigma^{\prime}}}{2}\tilde{f}_{0\eta\sigma}^{\dagger}\tilde{f}_{0\eta\sigma^{\prime}}^{\phantom{\dagger}}$
the spin operator with respect to the bath site at the location of impurity
$\eta$. Therefore a single impurity interacting with in a helical edge mode is
identical to a regular 1-impurity Anderson or Kondo model without a helical
character. The dynamically generated low-energy Kondo scale
$T_{\mathrm{K}}\cong\sqrt{\rho_{0}J}e^{-\frac{1}{\rho_{0}J}}$ [42] for the
1-impurity problem with $\rho_{0}$ the one-particle density of states around
the Fermi level, however, also represents a relevant energy scale for the
2-impurity problem. Assuming the impurity distance on the grid (51), the local
bath operators $\tilde{f}_{0\eta\sigma}$ are already properly orthonormalized,
such that there are no issues with crosstalk between the impurities.

For spin-independent hybridization between the Anderson impurities and the
bath, the resulting Kondo coupling in Eq. (99) is SU(2) spin symmetric. While
bearing in mind that the helical bath mode itself already has the SU(2)
symmetry broken, the spin symmetry can also be broken at the level of the
Kondo Hamiltonian, giving rise to anisotropic local spin-spin interactions.
Assuming a single a preferred direction ($z$) with $J_{x}=J_{y}\equiv J\neq
J_{z}$, $J_{z}>J$ ($J_{z}<J$ describes an easy-axis (easy-plane) regime,
respectively. The global U(1) spin symmetry thus remains preserved.

###  A.7 Effects of finite bandwidth with Kondo interaction

Figure 11:  Static $\langle S_{L}^{z}S_{R}^{z}\rangle$ impurity correlations
for $T_{\rm K}\ll E_{\rm R}\sim\tfrac{J^{2}}{x}$ obtained by NRG towards the
wide-band limit $D\gg J(=J_{x}=J_{y})$ for $J_{z}=0$. (a) Deviations of the
computed $\langle S_{L}^{z}S_{R}^{z}\rangle$ from the expected RKKY value of
$-1/4$ on a loglog plot. Data is shown for various impurity distances $x$,
with the polynomial fit obtained for $x=10$ showing approximate quadratic
behavior (red dashed line). (b) Comparison of energy scales for the same
parameter range as in (a) where the temperature $T\sim 10^{-9}$ used in NRG
simulations in (a) is indicated by the horizontal gray line. The data for the
Kondo scale $T_{\rm K}$ was obtained via poor-man’s scaling [cf. App. B]. Red
line shows a simple estimate for the RKKY energy at $x=10$, demonstrating that
(a) is deeply in the RKKY regime, i.e., $E_{\rm R}\gg T_{\rm K},T$.

When starting from the Anderson model, one may scale the local Coulomb
interactions properly to infinity in relation to other parameters, with the
result that also in the numerical setting, one effectively arrives at the
Kondo model [48]. For the Anderson model the effective bandwidth relevant for
the impurities is cut off by $U$ if $U<D$. However, by taking $U\gg D$ when
transitioning towards the Kondo model, bandwidth keeps playing a considerable
role, and the universal wide-band limit is approached rather slowly.

Here for the anisotropic 2HKM in the present case, if the Kondo scale $T_{\rm
K}(J,J_{z})\ll E_{\rm R}\ll D=1$ for each impurity individually is just
several orders of magnitude smaller than all other energy scales, effects of
finite-bandwidth are still considerable. In order to reach the wide-band
scaling limit (here $J\lesssim 0.02$), the single-impurity Kondo scale is
already many many orders of magnitude lower than the band width $D=1$ ($T_{\rm
K}\lesssim 10^{-40}$), as demonstrated in Fig. 11, and also consistent with
the literature on the single-impurity case [48]. In the present case, we see
that similar arguments also carry over to the RKKY regime, even if Kondo
physics is irrelevant (in the sense that it sets in at much smaller energy
scales, even much smaller than any temperature of practical interest).

In the RKKY regime and in the wide-band limit, the impurities are expected to
be well decoupled from the bath and described by $H_{\rm R}{=}-E_{\rm
R}\left(S^{+}_{L}S^{-}_{R}+\mathrm{H.c.}\right)$ [cf. App. D] with the
dynamically generated effective direct RKKY impurity coupling $E_{\rm R}$. The
impurities are thus expected in the $S_{z}^{\mathrm{tot}}{=}0$ triplet state
$|0\rangle\equiv\tfrac{1}{\sqrt{2}}|{\uparrow\downarrow}{+}{\downarrow\uparrow}\rangle$
which yields the low-energy static spin correlation $\langle
S_{L}^{z}S_{R}^{z}\rangle=-1/4$. In practice, however, one sees substantial
deviations from this expectation value up to nearly 20% for $J=0.1$ in the NRG
data even using $J_{z}=0$ as shown in Fig. 11(a). Hence these deviations must
derive from finite bandwidth, but also higher-order processes that go beyond
second order PT [see App. D]. These deviations can be reduced systematically
by lowering the Kondo coupling $J$, e.g., having a deviation already below 1%
for $J\lesssim 0.02$. As shown in Fig. 11(a) the static expectation value
$\langle S_{L}^{z}S_{R}^{z}\rangle$ approaches $-1/4$ in a polynomial fashion
as $J$ is lowered, down to the smallest $J$ considered. Therefore, indeed, the
deviations seen in Fig. 11(a) are clearly due to finite bandwidth. This
demonstrates that finite bandwidth does play an observable role in the Kondo
setting [48] when comparing numerical to analytical results if the latter
strictly assumed the wideband limit. If one considers the wide-band scaling
limit reached within deviations in observables of about $10^{-3}$, this
suggests approximately $J\lesssim 0.02$ in Fig. 11(a). On the single impurity
level, this already corresponds to astronomically small Kondo scales $T_{\rm
K}\ll 10^{-40}$ in Fig. 11(b) consistent with earlier NRG studies [48]. From a
physics point of view, however, we do not expect that the observed minor
variations change the overall physical picture in any significant qualitative
manner.

##  Appendix B Poor-mans scaling for anisotropic Kondo

The poor man’s scaling equation for the renormalization group (RG) flow of the
anistropic Kondo model are given by [42, 49, 50]

$\displaystyle\tfrac{\partial}{\partial\ln
D}(\rho_{0}J_{x})=-(\rho_{0}J_{y})(\rho_{0}J_{z})\text{ ,}$ (100)

and similarly for the other components using cyclic permutations of $(x,y,z)$.
In particular, with $x\equiv\rho_{0}J_{x}=\rho_{0}J_{y}$ and
$z\equiv\rho_{0}J_{z}$,

$\displaystyle\tfrac{\partial}{\partial\ln D}\,x$ $\displaystyle=$
$\displaystyle-xz$ $\displaystyle\tfrac{\partial}{\partial\ln D}\,z$
$\displaystyle=$ $\displaystyle-x^{2}\text{ ,}$ (101)

It follows from the above that

$\displaystyle-\tfrac{dz}{x^{2}}=d\ln D=-\tfrac{dx}{xz}\quad$
$\displaystyle\Rightarrow$ $\displaystyle z\,dz=x\,dx$ (102)
$\displaystyle\Rightarrow$ $\displaystyle z^{2}=x^{2}\pm a^{2}$

where $\pm a^{2}$ with $a\geq 0$ is some constant of integration. If the
starting point has $|z_{0}|>|x_{0}|$ (easy-axis), then the positive sign is
chosen for $a^{2}$, whereas the regime $|z_{0}|<|x_{0}|$ (easy-plane) has the
negative sign. The contours described by Eq. (102) exactly reflect the RG
paths of the anisotropic Kondo (parabolas or hyperbolas separated by
$|x_{0}|=|z_{0}|$, as shown by the gray lines in Fig. 12). It simply also
follows from Eq. (101) that for $x\to 0$, $(x,z)$ stops flowing. The model is
physically equivalent for $x_{0}\to-x_{0}$ (vertical flip in Fig. 12) as this
can be absorbed into a gauge transformation of the spin basis.

Figure 12: Kondo scale of the anisotropic Kondo model from poor Man’s scaling
[Eqs. (103) and (104)]. $J_{x}<0$ is physically equivalent to $J_{x}>0$, such
that the lower half of the panel is a mirror image of the upper half. The
thick black horizontal line ($J_{z}<0$ for $J_{x}=0$) represents a stable line
of fixed points without a Kondo scale. The entire white region below the
diagonal lines to the left flows towards it. The black dotted horizontal line
($J_{z}>0$ for $J_{x}=0$) represents a line of unstable fixed points that flow
to strong coupling for any small but finite $J_{x}$ (strictly at $J_{x}=0$ one
has $T_{\rm K}=0$, represented by white color). The gray lines represent RG
flow contours as in Eq. (102). The Kondo temperature is defined by the
starting point $(x_{0},z_{0})$ on such a contour. The Kondo scale increases
monotonically from left to right, and also with increasing $|J_{x}|$. The
green contours show lines of constant $T_{K}(J,J_{z})-(\rho_{0}J)^{2}\leq 0$.
The last thick green contour to the right describes
$T_{K}(J,J_{z})=(\rho_{0}J)^{2}=E_{\rm R}(x=1)$, and thus represents the
largest RKKY energy possible by putting two impurities right next to each
other, with the closest distance being one ‘lattice spacing’ [cf. Eq. (21)].

For $z_{0}>0$ and $|x_{0}|>|z_{0}|$, the anisotropic Kondo model flows to the
isotropic strong coupling regime. That is integrating out the bath starting
from large $D$ needs to be stopped at some $D^{\ast}>0$ where $x$ and $z$
diverge, which thus defines the dynamically generated low-energy Kondo scale
$T_{\rm K}:=D^{\ast}$.

In the easy-axis Kondo regime ($J_{z}>|J_{x}|$), with $z\geq a$, the RG
differential equations diverge at

$\displaystyle\ln\tfrac{D^{\ast}}{D_{0}}=\tfrac{1}{2a}\ln\tfrac{z_{0}-a}{z_{0}+a}\
,\qquad a=\sqrt{z_{0}^{2}-x_{0}^{2}}$ (103)

The Kondo scale vanishes to exponentially small energy scales for $J_{x}\to
0$, i.e., $a\to z_{0}^{-}$.

In the easy-plane regime ($|J_{x}|>|J_{z}|$), with $|x|\geq a$, the RG
differential equations diverge at

$\displaystyle\ln\tfrac{D^{\ast}}{D_{0}}=\tfrac{1}{a}\bigl{(}\tan^{-1}\tfrac{z_{0}}{a}-\tfrac{\pi}{2}\bigr{)}\
,\quad a=\sqrt{x_{0}^{2}-z_{0}^{2}}$ (104)

Both Kondo scales, Eq. (103) and Eq. (104) connect smoothly across
$J_{x}=J_{z}>0$, where

$\displaystyle\ln\tfrac{D^{\ast}}{D_{0}}=-\tfrac{1}{z_{0}}\qquad(a\to 0)$
(105)

thus having $T_{\rm K}^{\mathrm{isotropic}}\\!\simeq
D_{0}\,e^{-\frac{1}{\rho_{0}J}}$. Here the Kondo scale remains finite if
$J_{z}\to 0$,

$\displaystyle\ln\tfrac{D^{\ast}}{D_{0}}=-\tfrac{\pi}{2x_{0}}\qquad(a\to
x_{0})$ (106)

thus having $T_{\rm K}^{J_{z}=0}\simeq D_{0}\,e^{-\pi/2\rho_{0}J_{x}}$.

For $J_{z}<-|J_{x}|$ the system flows to the ferromagnetic regime with
$J_{x}\to 0$ without any Kondo physics. That is $D$ can be integrated out all
the way to zero, which thus implies $D^{\ast}=T_{\rm K}=0$.

##  Appendix C Basic comparison Kondo vs. RKKY energy

The single-impurity Kondo scale from the previous section may be compared to
the RKKY energy $E_{\rm R}$ [cf. Eq. (21)]. The latter decays with distance,
hence is largest for short distances, with the shortest possible distance
given by one ‘lattice spacing’ [cf. Eq. (2)], hence $E_{\rm
R}\leq(\rho_{0}J)^{2}$ in the adopted units [Sec. II.1]. With this in mind,
Fig. 12 also includes contours of constant $T_{K}(J,J_{z})-(\rho_{0}J)^{2}\leq
0$ (green lines), with the thick line representing
$T_{K}(J,J_{z})=(\rho_{0}J)^{2}$. Therefore the parameter regime to the right
of the thick green contour always has $T_{K}>E_{\rm R}$ for any $x\geq 1$,
meaning that in this regime Kondo is always dominant over RKKY. In particular,
this includes the isotropic Kondo for $J\gtrsim 0.25$, corresponding to a
dimensionless coupling $j_{0}\equiv\rho_{0}J=0.125$. While still clearly below
1., this is on the upper end of what may be considered acceptable on physical
grounds for the Kondo model. For example, note that the poor-man’s scaling
approach for the Kondo model only includes second order virtual processes by
integrating out the bath which assumes (and thus is justified only if)
$\tfrac{J^{2}}{D}\ll D$, i.e., $j_{0}\ll 0.5$. Conversely, note that coming
from an Anderson model, one has $j_{0}^{\mathrm{eff}}\simeq\tfrac{4\Gamma}{\pi
U}$ with hybridization strength $\Gamma$ and onsite interaction $U$. So in
order for the Kondo model to be justified in the first place, the onsite
interaction $U$ needs to be large enough so that charge fluctuations can be
integrated out via virtual second order processes (cotunneling) in order to
obtain a pure effective spin Hamiltonian. This clearly requires $U\gg\Gamma$,
and thus also $j_{0}\ll 1$ on physical grounds.

Conversely, to the left of the thick green line in Fig. 12, RKKY can dominate
in the low-energy regime if a pair of impurities is just brought close enough
to each other. In particular, RKKY also occurs within the entire white region
to the left where $T_{\rm K}=0$. There with $E_{\rm R}>T_{\rm K}=0$, the
impurity distance $x$ can be taken to infinity while still seeing RKKY in the
low-energy regime (note that since RKKY is second order, with $E_{\rm
R}\propto J^{2}/x$, the sign of the Kondo couplings is irrelevant).

##  Appendix D Second order perturbation processes entering RKKY

Complimentary to the field theoretic approach in the main text, the RKKY
processes can also be analyzed from the point of view second order
perturbation theory (PT). Ultimately, this generates the same effective
Hamiltonian as in the field theoretic approach, but purely within the
Hamiltonian setting, e.g. via the Feshbach formalism [51, 52].

Let the full Hamiltonian of two Kondo-coupled impurity spins
$\eta\in\\{\mathcal{R},\mathcal{L}\\}\equiv\\{+1,-1\\}$ at location
$\frac{\eta x}{2}$, i.e., at mutual distance $x$, interacting with a one-
dimensional helical bath encoded by the Fermionic operators
$\hat{c}_{k\sigma}$ be given by

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\\!\\!\\!\sum_{\eta
a}2J_{a}\
(\hat{S}_{\eta}^{a})^{\dagger}\cdot\hat{s}_{\eta}^{a}+\sum_{k\sigma}\varepsilon_{k\sigma}^{\,}\hat{c}_{k\sigma}^{\dagger}\hat{c}_{k\sigma}^{\,}$
(107a) with $a\in x,y,z$, and bath spins (with $\tau_{a}$ the Pauli matrices),
$\displaystyle\hat{s}_{\eta}^{a}\equiv\tfrac{1}{2}\hat{\sigma}_{\eta}^{a}=\tfrac{1}{N}\\!\\!\\!\sum_{k\sigma,k^{\prime}\sigma^{\prime}}\\!\\!\\!\bigl{(}\hat{c}_{k\sigma}^{\dagger}\,\tfrac{1}{2}\tau_{a}^{\sigma\sigma^{\prime}}\hat{c}_{k^{\prime}\sigma^{\prime}}^{\,}\bigr{)}e^{\mathrm{i}\mkern
1.0mu(k^{\prime}-k)\frac{\eta x}{2}}\text{,}\quad$ (107b) assuming a total of
$N$ levels $k$ for each spin. Furthermore, the energies of the bath
$\varepsilon_{k\sigma}=\sigma vk$ are constrained to within a finite half-
bandwidth $|\varepsilon_{k\sigma}|\leq D$. Taking $J{\equiv}J_{x}{=}J_{y}$ and
$\hat{\sigma}_{\eta}^{\pm}\equiv\hat{s}_{\eta}^{\pm}\equiv\hat{s}_{\eta}^{x}\pm\mathrm{i}\mkern
1.0mu\hat{s}_{\eta}^{y}$, the Kondo interaction can be written from the point
of view of the helical Fermions as, $\displaystyle
2J_{a}(\hat{S}_{\eta}^{a})^{\dagger}\cdot\hat{s}_{\eta}^{a}$
$\displaystyle\equiv$ $\displaystyle
J\,\bigl{(}\hat{S}_{\eta}^{+}\hat{\sigma}_{\eta}^{-}+\mathrm{H.c.}\bigr{)}+J_{z}\
\hat{S}_{\eta}^{z}\hat{\sigma}_{\eta}^{z}$ $\displaystyle\equiv$
$\displaystyle\tfrac{J}{N}\\!\\!\\!\sum_{k\sigma,k^{\prime}\sigma^{\prime}}\\!\\!\\!\bigl{(}\hat{c}_{k\sigma}^{\dagger}\hat{T}^{\eta}_{\sigma\sigma^{\prime}}\,\hat{c}_{k^{\prime}\sigma^{\prime}}^{\,}\bigr{)}e^{\mathrm{i}\mkern
1.0mu(k^{\prime}-k)\frac{\eta x}{2}}$ with the matrix notation in
$\sigma{\in}\\{\uparrow,\downarrow\\}{\equiv}\\{1,2\\}$ for the operators of
each impurity [see also Eq. (13)], $\displaystyle\hat{T}^{\eta}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\Delta\hat{S}^{z}_{\eta}&\hat{S}^{-}_{\eta}\\\
\hat{S}^{+}_{\eta}&-\Delta\hat{S}^{z}_{\eta}\end{pmatrix}\equiv\hat{S}^{-}_{\eta}{\otimes}\tau_{+}+\hat{S}^{+}_{\eta}{\otimes}\tau_{-}+\Delta\hat{S}^{z}_{\eta}{\otimes}\tau_{z}\text{
,}$

with anisotropy $\Delta{\equiv}\tfrac{J_{z}}{J}$ and
$\tau_{\pm}=\tfrac{1}{2}(\tau_{x}\pm\mathrm{i}\mkern 1.0mu\tau_{y})$. The
isotropic case $\Delta{=}1$ just reduces to
$\hat{T}^{\eta}=\hat{S}^{\dagger}_{\eta}\cdot\tau$.

The effective Hamiltonian in between the impurities, i.e., their direct
interaction, is generated via second order PT in the Kondo setting
($4^{\mathrm{th}}$ order in the hybridization for an Anderson model). Then
with the shortcuts $1\equiv(k_{1}\sigma_{1})$, etc., a typical second order
process is given by

$\displaystyle\delta\hat{H}(x)$ $\displaystyle\sim$ $\displaystyle
P_{0}\,\bigl{(}\hat{c}_{1}^{\dagger}\hat{T}_{\sigma_{1}\sigma_{2}}^{\eta}\hat{c}_{2}^{\,}\,e^{\mathrm{i}\mkern
1.0mu(k_{2}-k_{1})\frac{\eta x}{2}}\bigr{)}\
\tfrac{(J/N)^{2}}{\mathrm{i}\mkern
1.0mu0^{+}-(\varepsilon_{3}-\varepsilon_{4})}\
\bigl{(}\hat{c}_{3}^{\dagger}\hat{T}_{\sigma_{3}\sigma_{4}}^{\eta^{\prime}}\hat{c}_{4}^{\,}\,e^{\mathrm{i}\mkern
1.0mu(k_{4}-k_{3})\frac{\eta^{\prime}x}{2}}\bigr{)}\,P_{0}$ (108)

with degrees of freedom such as impurity location
$\eta{\in}\\{L,R\\}{\equiv}\\{-1,1\\}$, or spin $\sigma$ or momenta $k$ to be
summed over eventually. $P_{0}$ is a projector into the target low-energy
regime of the bath without acting at the impurities. This is in the spirit of
generating the low-energy effective Hamiltonian such as the Feshbach-Fano
partitioning [51, 52]. There, by construction, one needs to start out of the
low-energy regime, say by considering some infrared cutoff energy $D^{\ast}\ll
D$. In the low-energy regime described by $P_{0}$ then, all bath levels with
energy $\varepsilon_{k\sigma}<-D^{\ast}$ are strictly occupied and all energy
levels $\varepsilon_{k\sigma}>+D^{\ast}$ are strictly empty. Now in Eq. (108),
the perturbation $\hat{T}$ on the r.h.s. generates a particle-hole (p/h) pair
with particle energy $\varepsilon_{3}>0$ and hole energy $\varepsilon_{4}<0$.
Therefore as long as either $\varepsilon_{3}>+D^{\ast}$ or
$\varepsilon_{4}<-D^{\ast}$ (or both), this represents state space outside the
low-energy regime $P_{0}$.

If $D^{\ast}\ll D$, the overwhelming number of processes will involve both,
particle and hole at energy cost above $D^{\ast}$. There in order to return to
the low energy regime, exactly the same p/h pair must be destroyed again. This
constrains the second processes above to $\delta_{14}\delta_{23}$, with the
shortcut notation
$\delta_{ij}\equiv\delta_{k_{i}k_{j}}\delta_{\sigma_{i}\sigma_{j}}$. Actually,
the processes $\delta_{14}\delta_{23}$ can be included all the way down to
zero energy, i.e., there is apriori no need for an infrared energy cutoff
$D^{\ast}$, so one can assume $D^{\ast}\to 0^{+}$ as long as the sum is well-
defined (which it turns out it is). With the bath projected via $P$ onto the
exactly filled helical Fermi sea, this generates the effective, purely local
inter-impurity Hamiltonian

$\displaystyle\hat{H}^{\mathrm{eff}}_{\mathrm{imp}}(x)$ $\displaystyle=$
$\displaystyle\sum_{\eta\eta^{\prime}}\sum_{k_{1}\sigma_{1}}^{\varepsilon_{1}<0}\sum_{k_{2}\sigma_{2}}^{\varepsilon_{2}>0}\bigl{(}\hat{T}_{\sigma_{1}\sigma_{2}}^{\eta}\,e^{\mathrm{i}\mkern
1.0mu(k_{2}-k_{1})x_{\eta}}\bigr{)}\ \tfrac{(J/N)^{2}}{\mathrm{i}\mkern
1.0mu0^{+}-(\varepsilon_{2}-\varepsilon_{1})}\
\bigl{(}\hat{T}_{\sigma_{2}\sigma_{1}}^{\eta^{\prime}}\,e^{\mathrm{i}\mkern
1.0mu(k_{1}-k_{2})x_{\eta^{\prime}}}\bigr{)}$ (109) $\displaystyle=$
$\displaystyle-(\rho_{0}J)^{2}\sum_{\begin{subarray}{c}\eta\eta^{\prime}\\\
\sigma_{1}\sigma_{2}\end{subarray}}\hat{T}_{\sigma_{1}\sigma_{2}}^{\eta}\hat{T}_{\sigma_{2}\sigma_{1}}^{\eta^{\prime}}\underbrace{\int\limits_{-D}^{0}d\varepsilon_{1}\int\limits_{0}^{D}d\varepsilon_{2}\,\tfrac{e^{\mathrm{i}\mkern
1.0mu(\sigma_{2}\varepsilon_{2}-\sigma_{1}\varepsilon_{1})\frac{1}{v}(x_{\eta}-x_{\eta^{\prime}})}}{-\mathrm{i}\mkern
1.0mu0^{+}+(\varepsilon_{2}-\varepsilon_{1})}}_{\equiv
I^{\eta\eta^{\prime}}_{\sigma_{1}\sigma_{2}}}$

having taken the continuum limit
$i\equiv(k_{i}\sigma_{i})\to(\varepsilon_{i}\sigma_{i})$, with $\rho_{0}$ the
constant one-particle density of states of the helical edge for a given spin
at the Fermi level as in Eq. (2). As an aside, we note that the energy
denominator of $I^{\eta\eta^{\prime}}_{\sigma_{1}\sigma_{2}}$ can be rewritten
into an integral over imaginary frequencies,

$\displaystyle\int\tfrac{d\omega}{2\pi\mathrm{i}\mkern
1.0mu}\,\tfrac{1}{\mathrm{i}\mkern
1.0mu\omega-\varepsilon_{1}}\tfrac{1}{\mathrm{i}\mkern
1.0mu\omega-\varepsilon_{2}}=\tfrac{\theta(-\varepsilon_{1}\varepsilon_{2})}{|\varepsilon_{2}-\varepsilon_{1}|}\text{
,}$ (110)

which resembles a Matsubara summation at zero temperature. Via contour
integral, the result is non-zero only if $\varepsilon_{1}$ and
$\varepsilon_{2}$ have opposite sign as encoded into the Heaviside step
function on the r.h.s. This already reflects the particle-hole nature of the
excitations in Eq. (109) where $\varepsilon_{1}<0$ and $\varepsilon_{2}>0$.
The case with opposite signs is included in Eq. (109) via the overall sum.
Assuming $\eta{\neq}\eta^{\prime}$, then the case
$\eta\leftrightarrow\eta^{\prime}$ gives the additional contribution above,
while also exchanging the labels $1\leftrightarrow 2$, such that the sign in
the phase factor is properly restored, while also having
$[\hat{T}^{\eta},\hat{T}^{\eta^{\prime}}]{=}0$ for $\eta{\neq}\eta^{\prime}$.
Hence by summing over all second order processes, one can connect the second
order perturbative approach here to the double-propagator structure in Eq.
(III.1) in the main text obtained from the field-theoretic approach.
Collecting phase factors in $k_{1}$ and $k_{2}$, the latter thus permits the
interpretation that any second order process in the effective impurity
Hamiltonian requires the free propagator of two particles shuttling back and
forth in between the impurities (one particle needs to propagate the distance
$+x$, and another the distance $-x$). Here in the helical setting, however,
the directions that particles can move are constrained depending on their
spin. This manifests itself in the overall structure of the resulting RKKY
Hamiltonian [1].

The integral $I^{\eta\eta^{\prime}}_{\sigma_{1}\sigma_{2}}$ in Eq. (109) is
generally well defined when both, $\varepsilon_{1}$ and $\varepsilon_{2}$
approach zero energy, irrespective of the phase factor since

$\displaystyle\bigl{|}I^{\eta\eta^{\prime}}_{\sigma_{1}\sigma_{2}}\bigr{|}$
$\displaystyle\leq$
$\displaystyle\int_{-D}^{0}\\!\\!\\!d\varepsilon_{1}\int_{0}^{D}\\!\\!\\!d\varepsilon_{2}\
\bigl{|}\tfrac{1}{-\mathrm{i}\mkern
1.0mu0^{+}+(\varepsilon_{2}-\varepsilon_{1})}\bigr{|}=2D\ln 2\text{ .}\qquad$
(111)

The imaginary part form $\mathrm{i}\mkern 1.0mu0^{+}$ does not contribute, as
by the integral limits, it can only contribute at
$\varepsilon=\varepsilon^{\prime}=0$, where by the double integral the
integrated weight vanishes.

When $\eta{=}\eta^{\prime}$, the complex phases drop out, and with
$S_{\eta}^{2}\propto 1\\!\\!1$ the integral in Eq. (109) just generates a
plain irrelevant shift in the global energy reference as estimated above (the
generation of the single impurity Kondo couplings needs to consider finite
$D^{\ast}$ and relax the condition $\delta_{14}\delta_{23}$ above). Hence the
following discussion focusses on $\eta{\neq}\eta^{\prime}$, i.e.,
$\eta^{\prime}{=}-\eta$, in which case the phase factors in the enumerator are
non-trivial and reflect the underlying helical physics. In contrast to the
energy denominator, however, the phase factors carry momenta as arguments.
Converting these into energies, in the helical setting with
$\varepsilon_{k\sigma}=\sigma vk$, this involves signs depending on the spin
as already indicated with the integral
$I^{\eta\eta^{\prime}}_{\sigma_{1}\sigma_{2}}$ in Eq. (109).

Contributions that involve a spin flip ($\sigma_{2}{=}-\sigma_{1}$), will have
opposite relative sign of $\varepsilon_{1}$ vs. $\varepsilon_{2}$ in the phase
factor in Eq. (109) as compared to the energy denominator, whereas in the
absence of a spin flip the relative sign is the same. To evaluate this
integral including the phases, it is therefore convenient to change variables
to symmetric and antisymmetric combinations

$\displaystyle\begin{pmatrix}\varepsilon^{\prime}\\\
\varepsilon\end{pmatrix}\equiv\underbrace{\begin{pmatrix}\ 1&1\\\
\\!\\!-1&1\end{pmatrix}}_{\equiv U}\begin{pmatrix}\varepsilon_{1}\\\
\varepsilon_{2}\end{pmatrix}\text{ .}$ (112)

With $\varepsilon_{1}<0$ and $\varepsilon_{2}>0$, by construction,
$0\leq\varepsilon\leq 2D$ and
$\varepsilon^{\prime}\in[-\varepsilon,\varepsilon]$. The integral will
converge with large $D$, such that the upper integral limit can be taken more
loosely by deforming the integration area to
$\varepsilon\lesssim\tilde{D}=\frac{3D}{2}$, with the upper integral limit
$\tilde{D}$ assumed large, and eventually taken to infinity if well-defined.
With this one obtains for the case including a spin flip, with
$\tilde{\tau}\equiv\frac{\sigma_{2}}{v}(x_{\eta}-x_{\eta^{\prime}})$, and thus
$\tau\equiv|\tilde{\tau}|=\tfrac{x}{v}$ having $\eta\neq\eta^{\prime}$, with
$x=|x_{\eta}-x_{\eta^{\prime}}|$,

$\displaystyle
I^{\eta\neq\eta^{\prime}}_{\sigma_{1}\neq\sigma_{2}}=-\int\limits_{-D}^{0}\\!d\varepsilon_{1}\int\limits_{0}^{D}\\!d\varepsilon_{2}\,\tfrac{e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}(\varepsilon_{2}+\varepsilon_{1})}}{\mathrm{i}\mkern
1.0mu0^{+}-(\varepsilon_{2}-\varepsilon_{1})}$ (113)
$\displaystyle\simeq-\underbrace{\tfrac{1}{\det
U}}_{=\frac{1}{2}}\int\limits_{0}^{\tilde{D}}\\!d\varepsilon\underbrace{\int_{-\varepsilon}^{\varepsilon}\\!d\varepsilon^{\prime}\,\tfrac{e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon^{\prime}}}{\mathrm{i}\mkern
1.0mu0^{+}-\varepsilon}}_{=\tfrac{1}{\mathrm{i}\mkern
1.0mu\tilde{\tau}}\tfrac{2\mathrm{i}\mkern
1.0mu\sin\tilde{\tau}\varepsilon}{\mathrm{i}\mkern
1.0mu0^{+}-\varepsilon}}=\tfrac{1}{\tilde{\tau}}\underbrace{\int\limits_{0}^{\tilde{D}}\\!d\varepsilon\,\tfrac{\sin\tilde{\tau}\varepsilon}{\varepsilon}}_{\simeq\
\frac{\mathrm{sgn}\tilde{\tau}}{2}\pi}\simeq\tfrac{\pi}{2\tau}=\tfrac{\pi
v}{2x}$

where the contribution from the imaginary part $\mathrm{i}\mkern 1.0mu0^{+}$
in the denominator vanishes, as this gives $\int
d\varepsilon\,\delta(\varepsilon)\sin\tau\varepsilon\to 0$. The remaining
integral is well-defined and converges for $|\tau D|=\frac{xD}{v}\gg 1$ to
$\pi/2$ as indicated. Combining all $\sigma_{1}=-\sigma_{2}$ and
$\eta=-\eta^{\prime}$ contributions, where the latter just duplicates entries,
then generates the RKKY Hamiltonian for the helical setting,

$\displaystyle\hat{H}_{\rm R}(x)$ $\displaystyle=$
$\displaystyle-\underbrace{(\underbrace{\rho_{0}J}_{\equiv
j_{0}})^{2}\\!\\!\underbrace{\tfrac{\pi v}{x}}_{=\pi E_{x}}}_{=E_{\rm
R}}\bigl{(}\hat{S}^{+}_{\mathcal{L}}\hat{S}^{-}_{\mathcal{R}}+\hat{S}^{-}_{\mathcal{L}}\hat{S}^{+}_{\mathcal{R}}\bigr{)}$
(114)

with the dimensionless Kondo coupling strength $j_{0}$. Overall, this
generated direct impurity interaction is ferromagnetic and non-oscillatory
with a plain decay with inverse distance $x$, having the RKKY energy scale

$\displaystyle E_{\rm R}=\pi j_{0}^{2}E_{x}\text{ ,}$ (115)

as already encountered in Eq. (20) in the main text. The RKKY Hamiltonian is
independent of the bandwidth $D$, and aside from the dimensionless Kondo
coupling strength $j_{0}$, only references the coherence scale
$E_{x}\equiv\tfrac{1}{\tau}$ with $\tau=x/v$ the time required for a helical
particle to travel from one impurity to the other. With the lattice spacing in
Eq. (2), nevertheless, this energy scale may be rewritten as
$E_{x}=\tfrac{D/\pi}{x/a}$. That is when measuring distance on the grid (51),
by definition, this involves a finite bandwidth, such that the bandwidth does
appear in the definition of $E_{x}$. In the continuum wide-band limit,
however, the natural way to think about the coherence scale is $E_{x}=1/\tau$
without any reference to bandwidth.

In the absence of a spin flip, i.e., $\sigma_{1}=\sigma_{2}$ the integral can
be similarly evaluated,

$\displaystyle I^{\eta\neq\eta^{\prime}}_{\sigma_{1}=\sigma_{2}}$
$\displaystyle=$
$\displaystyle-\int\limits_{-D}^{0}\\!d\varepsilon_{1}\int\limits_{0}^{D}\\!d\varepsilon_{2}\,\tfrac{e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}(\varepsilon_{2}-\varepsilon_{1})}}{\mathrm{i}\mkern
1.0mu0^{+}-(\varepsilon_{2}-\varepsilon_{1})}$ $\displaystyle\simeq$
$\displaystyle-\underbrace{\tfrac{1}{\det
U}}_{=1/2}\int_{0}^{\tilde{D}}\\!d\varepsilon\underbrace{\int_{-\varepsilon}^{\varepsilon}\\!d\varepsilon^{\prime}\,}_{=2\varepsilon}\tfrac{e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon}}{\mathrm{i}\mkern
1.0mu0^{+}-\varepsilon}=\int\limits_{0}^{\tilde{D}}\\!d\varepsilon\,e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon}\text{ ,}$

where again the contribution from the imaginary part $\mathrm{i}\mkern
1.0mu0^{+}$ in the denominator vanishes, since $\int
d\varepsilon\,\delta(\varepsilon)\,\varepsilon e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon}\to 0$. As already expected from Eq. (111), the
integral in Eq. (D) is also well-defined for small $\varepsilon$. By summing
over spin or location, with $\tilde{\tau}\propto\eta\sigma_{2}$, only the real
part remains,
$\sum_{\eta\sigma_{1}}I^{\eta=-\eta^{\prime}}_{\sigma_{1}=\sigma_{2}}\simeq\tfrac{4}{\tau}\sin\tau\tilde{D}$
which seems to suggest $1/x$ behavior similar in magnitude to Eq. (114). In
the present case, however, the integral remains sensitive on the bandwidth
$D$. Therefore the rather crudely adjusted upper limit in Eq. (D) by
introducing a simple averaged upper bound $\tilde{D}\sim\tfrac{3D}{2}$ needs
to be elaborated more carefully. The exact representation of the integral in
the first line of Eq. (D) yields with $z\equiv\tfrac{1}{D}(\varepsilon-D)$ and
$\lambda\equiv\tilde{\tau}D$,

$\displaystyle\sum_{\eta\sigma_{1}}I^{\eta\neq\eta^{\prime}}_{\sigma_{1}=\sigma_{2}}$
$\displaystyle=$ $\displaystyle
4{{\,\rm{Re}\,}}\,\Bigl{(}\int\limits_{0}^{D}\\!d\varepsilon\,e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon}+\int\limits_{D}^{2D}\\!d\varepsilon\,\underbrace{\tfrac{2D-\varepsilon}{\varepsilon}}_{=\frac{D-(\varepsilon-D)}{D+(\varepsilon-D)}}e^{\mathrm{i}\mkern
1.0mu\tilde{\tau}\varepsilon}\Bigr{)}$ (117) $\displaystyle=$
$\displaystyle\underbrace{\tfrac{4}{\tau}\sin\tau D}_{\to\
0}\,+\,4{{\,\rm{Re}\,}}\,D\\!\underbrace{\int\limits_{0}^{1}\\!\\!dz\,\tfrac{1-z}{1+z}e^{\mathrm{i}\mkern
1.0mu\lambda(1+z)}}_{\equiv I_{\lambda}}\text{.}\qquad$

On the grid (51), with $|\tilde{\tau}|=\pi x/a$, the first term vanishes, and
the second term can be evaluated by repeated integration by parts

$\displaystyle I_{\lambda}$ $\displaystyle=$
$\displaystyle-e^{\mathrm{i}\mkern
1.0mu\lambda}\sum_{n=0}^{\infty}(\tfrac{1}{-\mathrm{i}\mkern
1.0mu\lambda})^{n+1}\Bigl{(}e^{\mathrm{i}\mkern 1.0mu\lambda
z}\Bigl{.}\tfrac{d^{n}}{dz^{n}}\tfrac{1-z}{1+z}\Bigr{)}\Bigr{|}_{0}^{1}$ (118)
$\displaystyle=$ $\displaystyle\tfrac{e^{\mathrm{i}\mkern
1.0mu\lambda}}{\mathrm{i}\mkern 1.0mu\lambda}(e^{\mathrm{i}\mkern
1.0mu\lambda}-1)+\mathcal{O}(\tfrac{1}{\lambda^{2}})$

where the real part of the leading contribution
$\propto\tfrac{1}{\lambda}\sim\tfrac{1}{x}$ also vanishes on the grid (51),
since $\sin\lambda=\sin 2\lambda=0$ (only the imaginary part has a non-zero
leading term $\propto\tfrac{1}{\lambda}$, but these cancel after summation of
all terms). Hence only subleading terms survive, having
${{\,\rm{Re}\,}}\,I_{\lambda}\sim\tfrac{1}{D^{2}\tau^{2}}$. Thus, as expected
the $ZZ$ contribution to the RKKY Hamiltonian properly vanishes for
$D\to\infty$ even for finite $J_{z}$. At finite bandwidth, however, there is a
finite return probability, resulting in a small but finite
$S_{\mathcal{L}}^{z}S_{\mathcal{R}}^{z}$ interaction strength across the
impurities. This may be considered acceptable on physical grounds, bearing in
mind that for a true 2D model the bulk states also may offer a small but
finite return probability [see also end of Sec. A.2 in the main text for a
related discussion].

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$J(z):=(j(z),j^{\prime}(z),j^{\prime\prime}(z))$.

###  6.1. Definitions

We start by setting up some notation. Define
$\mathrm{Y}_{2}(1):=\mathrm{Y}(1)\times\mathbb{C}^{2}$. Let
$\mathrm{E}_{J}^{n}:=\left\\{(\mathbf{z},J(\mathbf{z})):\mathbf{z}\in\mathbb{H}^{n}\right\\}\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$.

Let $n,\ell$ be positive integers with $\ell\leq n$ and
$\mathbf{i}=(i_{1},\ldots,i_{\ell})$ in $\mathbb{N}^{\ell}$ with $1\leq
i_{1}<\ldots<i_{\ell}\leq n$. Define
$\mathrm{PR}_{\mathbf{i}}:\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}\rightarrow\mathbb{C}^{\ell}\times\mathrm{Y}_{2}(1)^{\ell}$
by

$\mathrm{PR}_{\mathbf{i}}:(\mathbf{x},\mathbf{y}_{0},\mathbf{y}_{1},\mathbf{y}_{2})\mapsto(\mathrm{pr}_{\mathbf{i}}(\mathbf{x}),\mathrm{pr}_{\mathbf{i}}(\mathbf{y}_{0}),\mathrm{pr}_{\mathbf{i}}(\mathbf{y}_{1}),\mathrm{pr}_{\mathbf{i}}(\mathbf{y}_{2})).$

We also define
$\pi_{\mathbb{C}}:\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}\to\mathbb{C}^{n}$
and
$\pi_{\mathrm{Y}}:\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}\to\mathrm{Y}(1)^{n}$
as the coordinate projections. Notice that $\pi_{\mathrm{Y}}$ still maps onto
$\mathrm{Y}(1)^{n}$, not to $\mathrm{Y}_{2}(1)^{n}$.

######  Definition.

An algebraic set $V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ is said
to be $J$-_broad_ if for any $\mathbf{i}=(i_{1},\ldots,i_{\ell})$ in
$\mathbb{N}^{\ell}$ with $1\leq i_{1}<\ldots<i_{\ell}\leq n$ we have
$\dim\mathrm{PR}_{\mathbf{i}}(V)\geq 3\ell$. In particular, if $V$ is
$J$-broad then $\dim V\geq 3n$.

We say $V$ is _strongly $J$-broad_ if the strict inequality
$\dim\mathrm{PR}_{\mathbf{i}}(V)>3\ell$ holds for every $\mathbf{i}$.

######  Definition.

A subvariety $T\subseteq\mathrm{Y}_{2}(1)^{n}$ is called a _special
subvariety_ of $\mathrm{Y}_{2}(1)^{n}$ if there is a Möbius subvariety
$M\subseteq\mathbb{C}^{n}$ defined over $\mathbb{Q}$ such that $T$ is the
Zariski closure over $\overline{\mathbb{Q}}$ of the set
$J(M\cap\mathbb{H}^{n})$. We will say that $T$ is _weakly special_ if there is
a Möbius subvariety $M\subseteq\mathbb{C}^{n}$ such that $T$ is the Zariski
closure over $\mathbb{C}$ of the set $J(M\cap\mathbb{H}^{n})$.

######  Definition.

We will say that an irreducible constructible set
$V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ is $J$-_free_ if no
coordinate of $V$ is constant, and $V$ is not contained in any subvariety of
the form $M\times\mathrm{Y}_{2}(1)^{n}$ or $\mathbb{C}^{n}\times T$, where
$M\subset\mathbb{C}^{n}$ is a proper Möbius subvariety defined over
$\mathbb{Q}$, and $T\subset\mathrm{Y}(1)^{n}$ is a proper special subvariety.

A constructible subset $V\subseteq\mathbb{C}^{n}\times\mathrm{Y}(1)^{n}$ is
$J$-_free_ if every irreducible component of $V$ is free.

######  Definition.

We say that an algebraic variety
$V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ satisfies the
_Existential Closedness condition for $J$_, or $\mathrm{(EC)}_{J}$, if the set
$V\cap\mathrm{E}_{J}^{n}$ is Zariski dense in $V$.

We say that $V$ satisfies the _Strong Existential Closedness condition for
$J$_, or $\mathrm{(SEC)}_{J}$, if for every finitely generated field
$K\subset\mathbb{C}$ over which $V$ can be defined, there exists
$(\mathbf{z},J(\mathbf{z}))\in V$ such that $(\mathbf{z},J(\mathbf{z}))$ is
generic in $V$ over $K$.

######  Conjecture 6.1.

For every positive integer $n$, every algebraic variety
$V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ which is $J$-broad and
$J$-free satisfies $\mathrm{(EC)}_{J}$. Even more, such $V$ satisfy
$\mathrm{(SEC)}_{J}$.

As explained in [Asl22a, §5.2], Conjecture 6.1 can be reduced to the case
where $\dim V=3n$ by doing an obvious adaptation of Proposition 4.4.

Before continuing to the results with derivatives, in the next few sections we
will go over the key ingredients that we need.

###  6.2. Convenient tuples for $J$

We could start this section by giving a natural definition of _convenient
generators for $J$_, following what we did in §4.2. However this definition
would be exactly the same as the definition of convenient generators for $j$.
To see this we recall that [AEK21b, Theorem 1.2] and the results in [AEK21a,
§5] already include the derivatives of $j$. This is manifested in the fact
that the various inequalities for $j$ we showed in §4.2 were all proven by
first proving the statement with derivatives. So we already have all the
transcendence inequalities we need.

###  6.3. Weak Zilber–Pink with derivatives

Here we recall one of the main results of [Asl22b]. Abusing notation, we use
$\pi_{\mathrm{Y}}$ to also denote the projection
$\pi_{\mathrm{Y}}:\mathrm{Y}_{2}(1)^{n}\to\mathrm{Y}(1)^{n}$.

######  Definition.

Given a special subvariety $S$ of $\mathrm{Y}_{2}(1)^{n}$ and a special
subvariety $T$ of $\mathrm{Y}(1)^{n}$, we say that $S$ _is associated with_
$T$ if $\pi_{\mathrm{Y}}(S)=T$.

######  Definition.

Let $V$ be an algebraic subvariety of $\mathrm{Y}_{2}(1)^{n}$. An _atypical
component_ of $V$ is an irreducible component $X$ of the intersection between
$V$ and a special subvariety $T$ of $\mathrm{Y}_{2}(1)^{n}$ such that

$\dim X>\dim V+\dim T-3n.$

Furthermore, we say that $X$ is a _strongly atypical component_ of $V$ if $X$
is an atypical component of $V$ and no coordinate is constant on
$\pi_{\mathrm{Y}}(X)$.

For the definition of _upper triangular $D$-special subvariety_ used in the
following theorem, see [Asl22b, §6.1]. In particular, we can choose
$S=\mathrm{Y}_{2}(1)^{n}$.

######  Theorem 6.2 (Uniform weak MZP with derivatives, see [Asl22b, Theorem
7.9]).

Let $S$ be an upper-triangular $D$-special subvariety of
$\mathrm{Y}_{2}(1)^{n}$. Given a parametric family
$(V_{\mathbf{q}})_{\mathbf{q}\in Q}$ of algebraic subvarieties of
$\mathrm{Y}_{2}(1)^{n}$, there is a finite collection $\mathscr{S}$ of proper
special subvarieties of $\mathrm{Y}(1)^{n}$ such that for every $\mathbf{q}\in
Q$ we have that for every strongly atypical component $X$ of $V$ there is a
special subvariety $S$ of $\mathrm{Y}_{2}(1)^{n}$ such that $X\subseteq S$ and
$\pi_{\mathrm{Y}}(S)=T$ for some $T\in\mathscr{S}$.

######  Theorem 6.3.

Given a parametric family $(U_{\mathbf{q}})_{\mathbf{q}\in Q}$ of
constructible subsets of $\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$, there is
a finite collection $\mathscr{S}$ of of proper special subvarieties of
$\mathrm{Y}(1)^{n}$ such that for every $\mathbf{q}\in Q$ we have that for
every strongly atypical component $X$ of $V_{\mathbf{q}}$ there is a special
subvariety $S$ of $\mathrm{Y}_{2}(1)^{n}$ such that
$X\subseteq\mathbb{C}^{n}\times S$ and $\pi_{\mathrm{Y}}(S)=T$ for some
$T\in\mathscr{S}$.

We can now go through the same sequence of steps as in §3.4 to obtain the
following analogue of Proposition 3.21.

######  Proposition 6.4.

Given a parametric family $(U_{\mathbf{q}})_{\mathbf{q}\in Q}$ of
constructible subsets of $\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$, there is
a positive integer $N$ such that for every $\mathbf{q}\in Q$, for every weakly
special subvariety $S\subset\mathrm{Y}_{2}(1)^{n}$ and for every atypical
component $X$ of $U_{\mathbf{q}}\cap(\mathbb{C}^{n}\times S)$, there is a
proper weakly special subvariety $S_{0}\subset\mathrm{Y}_{2}(1)^{n}$ with
$\Delta(\pi_{\mathrm{Y}}(S_{0}))\leq N$ such that
$X\subseteq\mathbb{C}^{n}\times S_{0}$.

###  6.4. Main Result

Here we present and prove an analogue of Theorem 1.2 which includes
derivatives.

######  Definition.

Let $L$ be an algebraically closed subfield of $\mathbb{C}$. A $J$-broad
algebraic variety $V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ is
said to have _no $L$-factors_ if for every choice of indices $1\leq
i_{1}<\ldots<i_{k}\leq n$ we have that either
$\dim\mathrm{PR}_{\mathbf{i}}(V)>3k$, or $\dim\mathrm{PR}_{\mathbf{i}}(V)=3k$
and $\mathrm{PR}_{\mathbf{i}}(V)$ is not definable over $L$.

######  Theorem 6.5.

Let $V\subseteq\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ be a $J$-broad and
$J$-free variety with no $C_{j}$-factors satisfying $\mathrm{(EC)}_{J}$. Then
for every finitely generated field $K$ over which $V$ can be defined, there
exists $\mathbf{z}\in\mathbb{H}^{n}$ such that $(\mathbf{z},J(\mathbf{z}))\in
V$ is generic over $K$.

We will set up very similar notation to the one used in §4.3. Let
$V\subset\mathbb{C}^{n}\times\mathrm{Y}_{2}(1)^{n}$ be a $J$-broad and
$J$-free variety with no $C_{j}$-factors. Let $K\subset\mathbb{C}$ be a
finitely generated subfield such that $V$ is defined over $K$. Let
$t_{1},\ldots,t_{m}\in\mathbb{H}\setminus C_{j}$ be given by Theorem 4.12
applied to $K$. By Lemma 4.6, we also assume that
$\dim_{G}(\mathbf{t}|C_{j})=m$.

Choose $(\mathbf{x},\mathbf{y}_{0},\mathbf{y}_{1},\mathbf{y}_{2})\in V$
generic over $C_{j}(\mathbf{t},J(\mathbf{t}))$. Let
$W\subseteq\mathbb{C}^{n+m}\times\mathrm{Y}_{2}(1)^{n+m}$ be the
$C_{j}$-Zariski closure of the point
$(\mathbf{x},\mathbf{t},\mathbf{y}_{0},\mathbf{y}_{1},\mathbf{y}_{2},J(\mathbf{t}))$.

######  Lemma 6.6.

$W$ is $J$-broad, $J$-free and $\dim W=3\dim V+3m+\dim^{j}(\mathbf{t})$.
Furthermore, if $\dim^{j}(\mathbf{t})>0$, then $W$ is strongly $J$-broad.

###### Proof.

Repeat the proof of Lemma 5.1. ∎

###### Proof of Theorem 6.5.

We assume that $\dim V=3n$. We then know that $V$ is not definable over
$C_{j}$, so $V(C_{j})$ is contained in a proper subvariety of $V$. In
particular, there is a Zariski open subset $V_{0}\subseteq V$ such that if
$(\mathbf{z},j(\mathbf{z}))\in V_{0}$, then some of the coordinates of
$j(\mathbf{z})$ are not in $C_{j}$. Also, in this case $K\not\subset C_{j}$ so
$\dim^{j}(\mathbf{t})\geq 1$.

Consider the parametric family of subvarieties
$(W_{\mathbf{q}})_{\mathbf{q}\in Q}$ of $W$ such that for every $\mathbf{q}\in
Q$ there is a Möbius subvariety $M_{\mathbf{q}}\subseteq\mathbb{C}^{n}$ such
that $W_{\mathbf{q}}:=W\cap(M_{\mathbf{q}}\times\mathrm{Y}_{2}(1)^{2})$. Let
$\mathscr{S}$ be the finite collection of special subvarieties of
$\mathrm{Y}_{2}(1)^{n+m}$ given by Theorem 6.3. Let $N$ be given by
Proposition 6.4 applied to $(W_{\mathbf{q}})_{\mathbf{q}\in Q}$.

Let $W_{0}\subseteq W$ be a Zariski open subset defined over $C_{j}$ such that
the points $(\mathbf{a},\mathbf{b}_{0},\mathbf{b}_{1},\mathbf{b}_{2})$ of
$W_{0}$ satisfy all of the following conditions:

  1. (a)

The point $\mathbf{b}_{0}$ does not lie in any $T\in\mathscr{S}$. As $W$ is
free, this condition defines a Zariski open subset of $W$.

  2. (b)

The coordinate of $\mathbf{b}_{0}$ do not satisfy any of the modular relations
$\Phi_{1},\ldots,\Phi_{N}$. As $W$ is free, this condition defines a Zariski
open subset of $W$.

  3. (c)

For every $1\leq i_{1}<\cdots<i_{\ell}\leq n$, and letting
$\mathbf{i}=(i_{1},\ldots,i_{\ell})$, we have that

$\dim\left(W\cap\mathrm{PR}_{\mathbf{i}}^{-1}((\mathbf{a},\mathbf{b}_{0},\mathbf{b}_{1},\mathbf{b}_{2}))\right)=\dim
W-\dim\mathrm{PR}_{\mathbf{i}}(W).$

By the fibre-dimension theorem and the fact there are only finitely many
tuples $\mathbf{i}$ to consider, this defines a Zariski open subset of $W$.

By the construction of $W$, there is a Zariski open subset $V_{1}\subseteq V$
such that if $(\mathbf{x},\mathbf{y}_{0},\mathbf{y}_{1},\mathbf{y}_{2})$ is
any point of $V_{1}$, then
$(\mathbf{x},\mathbf{t},\mathbf{y}_{0},\mathbf{y}_{1},\mathbf{y}_{2},J(\mathbf{t}))$
is a point in $W_{0}$.

Choose $(\mathbf{z},J(\mathbf{z}))\in V_{0}\cap V_{1}$, so that
$(\mathbf{z},\mathbf{t},J(\mathbf{z}),J(\mathbf{t}))\in W_{0}$. We will show
that $\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})=n+m$. For this we proceed by
contradiction, so suppose that $\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})<n+m$.
Let $S$ be the weakly special subvariety of $\mathrm{Y}_{2}(1)^{n+m}$ of
minimal dimension defined over $C_{j}$ for which
$(J(\mathbf{z}),J(\mathbf{t}))\in T$. Observe that
$\dim(T)=3\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})$.

Let $M\subset\mathbb{C}^{n+m}$ be the Möbius subvariety of smallest dimension
defined by Möbius relation over $\mathbb{Q}$ and/or setting some coordinates
to be a constant in $C_{j}$, which satisfies $(\mathbf{z},\mathbf{t})\in M$.
Then $W\cap(M\times\mathrm{Y}_{2}(1)^{n+m})$ is an element of the family
$(W_{\mathbf{q}})_{\mathbf{q}\in Q}$, call it $W_{M}$. We remark that $W_{M}$
is defined over $C_{j}$.

Now let $X$ be the irreducible component of $W_{M}\cap(\mathbb{C}^{n+m}\times
S)$ containing $(\mathbf{z},\mathbf{t},J(\mathbf{z}),J(\mathbf{t}))$. Observe
that $X$ is defined over $C_{j}$. Then by Proposition 4.7 we get

(6.1) $\dim
X\geq\mathrm{tr.deg.}_{C_{j}}C_{j}(\mathbf{z},\mathbf{t},J(\mathbf{z}),J(\mathbf{t}))\geq
3\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})+\dim^{j}(\mathbf{z},\mathbf{t}).$

On the other hand, as $W$ is free, then $\dim W_{M}<\dim W$, so using Lemma
5.1 we have

(6.2) $\begin{array}[]{ccl}\dim W_{M}+\dim\mathbb{C}^{n+m}\times
S-\dim\mathbb{C}^{n+m}\times\mathrm{Y}_{2}(1)^{n+m}&<&\dim W+\dim T-n-m\\\
&=&\dim^{j}(\mathbf{t})+\dim S\\\
&=&\dim^{j}(\mathbf{t})+3\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})\\\ &\leq&\dim
X.\end{array}$

This shows that $X$ is an atypical component of
$W_{M}\cap(\mathbb{C}^{n+m}\times S)$ in
$\mathbb{C}^{n+m}\times\mathrm{Y}_{2}(1)^{n+m}$. If $\pi_{\mathrm{Y}}(X)$ has
no constant coordinates, then by 6.3 there exists a special subvariety
$S\subseteq\mathrm{Y}_{2}(1)^{n+m}$ and $T_{0}\in\mathscr{S}$ such that
$X\subset W_{M}\cap(\mathbb{C}^{n+m}\times S)$ and
$\pi_{\mathrm{Y}}(S)=T_{0}$. However, this would contradict condition (a) in
the definition of $W_{0}$.

So $\pi_{\mathrm{Y}}(X)$ has some constant coordinates. Then, as $X$ is
defined over $C_{j}$, those constant coordinates must be given by elements of
$C_{j}$. Since no element of $J(\mathbf{t})$ is in $C_{j}$ (by Lemma 4.6), the
constant coordinates of $\pi_{\mathrm{Y}}(X)$ must be found among the
coordinates of $J(\mathbf{z})$. Let $1\leq i_{1}<\cdots<i_{\ell}\leq n$ denote
all the coordinates of $j(\mathbf{z})$ which are in $C_{j}$. Recall that by
Lemma 4.6, if some coordinate of $(z,j(z),j^{\prime}(z),j^{\prime\prime}(z))$
is in $C_{j}$, then they all are. Since $(\mathbf{z},J(\mathbf{z}))\in V_{0}$,
then $\ell<n$. We also remark that at this point we have already proven the
theorem for the case $n=1$.

By Proposition 3.21 we know that there is a weakly special subvariety
$S_{0}\subset\mathrm{Y}_{2}(1)^{n+m}$ such that
$\Delta(\pi_{\mathrm{Y}}(S_{0}))\leq N$, $X\subseteq\mathbb{C}^{n+m}\times
S_{0}$, and

(6.3) $\dim X\leq\dim W_{M}\cap(\mathbb{C}^{n+m}\times S_{0})+\dim T\cap
S_{0}-\dim S_{0}.$

By condition (b) in the definition of $W_{0}$ we know that
$\Delta(\pi_{\mathrm{Y}}(S_{0}))=0$, which means that
$\pi_{\mathrm{Y}}(S_{0})$ is completely defined by setting certain coordinates
to be constant. As the constant coordinates of $\pi_{\mathrm{Y}}(X)$ are in
$C_{j}$, then $S_{0}$ is defined over $C_{j}$, and $\dim S_{0}\geq
3(n+m-\ell)$ (again by Lemma 4.6). But $S$ is, by definition, the smallest
weakly special subvariety of $\mathrm{Y}_{2}(1)^{n+m}$ which is defined over
$C_{j}$ and contains the point $(J(\mathbf{z}),J(\mathbf{t}))$. So $S\cap
S_{0}=S$. Combining (6.1) and (6.3) we get

(6.4) $\dim^{j}(\mathbf{t})+\dim S_{0}\leq\dim
W_{M}\cap(\mathbb{C}^{n+m}\times S_{0}).$

Set
$\Theta:=W\cap\mathrm{PR}_{\mathbf{i}}^{-1}\left(\mathrm{PR}_{\mathbf{i}}(\mathbf{z},\mathbf{t},j(\mathbf{z},\mathbf{t}))\right)$.
By the fibre dimension theorem, condition (c) of the definition of $W_{0}$ and
the fact that $W$ is strongly broad (Lemma 6.6) we know that

(6.5) $\dim\Theta\leq\dim W-\dim\mathrm{PR}_{\mathbf{i}}(W)<3(n+m-\ell).$

Observe that $W_{M}\cap(\mathbb{C}^{n+m}\times S_{0})=W\cap(M\times
S_{0})\subseteq\Theta$, so combining (6.4) and (6.5) gives

$\dim S_{0}<3(n+m-\ell)$

which is a contradiction.

We deduce from this that $\dim_{G}(\mathbf{z},\mathbf{t}|C_{j})=3(n+m)$. By
[AEK21a, Lemma 5.2] this implies that $(\mathbf{z},J(\mathbf{z}))$ is generic
in $V$ over $C_{j}(\mathbf{t},J(\mathbf{t}))$. Furthermore, by Lemma 4.1 and
the freeness of $V$, no coordinate of $(\mathbf{z},J(\mathbf{z}))$ is in
$\overline{C_{j}(\mathbf{t},J(\mathbf{t}))}$. ∎

We have not added here an analogue of Theorem 1.3 including derivatives
because [AK22, Theorem 1.8], which is the available result on EC for the
blurring of $J$, requires one to use a group larger than $G$ to define the
blurring. So the result we can obtain would not be as close an approximation
to Conjecture 6.1 as Theorem 1.3 is to Conjecture 4.2.

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