# Topological Kondo effect with spinful Majorana fermions

Steffen Bollmann  Max-Planck-Institut für Festkörperforschung,
Heisenbergstraße 1, 70569 Stuttgart, Germany  Jukka I. Väyrynen  Department of
Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA.
Elio J. König  Max-Planck-Institut für Festkörperforschung, Heisenbergstraße
1, 70569 Stuttgart, Germany

###### Abstract

Motivated by the importance of studying topological superconductors beyond the
mean-field approximation, we here investigate mesoscopic islands of time
reversal invariant topological superconductors (TRITOPS). We characterize the
spectrum in the presence of strong order parameter fluctuations in the
presence of an arbitrary number of Kramers pairs of Majorana edge states and
study the effect of coupling the Coulomb blockaded island to external leads.
In the case of an odd fermionic parity on the island, we derive an
unconventional Kondo Hamiltonian in which metallic leads couple to both
topological Majorana degrees of freedom (which keep track of the parity in
different leads) and the overall spin-1/2 in the island. For the simplest case
of a single wire (two pairs of Majorana edge states), we demonstrate that
anisotropies are irrelevant in the weak coupling renormalization group flow.
This permits us to solve the Kondo problem in the vicinity of a Toulouse-like
point using Abelian Bosonization. We demonstrate a residual ground state
entropy of $\ln(2)$, which is protected by spin-rotation symmetry, but reduced
to $\ln(\sqrt{2})$ (as in the spinless topological Kondo effect) by symmetry
breaking perturbations. In the symmetric case, we further demonstrate the
simultaneous presence of both Fermi liquid and non-Fermi liquid like
thermodynamics (depending on the observable) and derive charge and spin
transport signatures of the Coulomb blockaded island.

##  I Introduction

Band structure topology has become a pillar of modern condensed matter
physics, with implications for both quantum technologies and quantum materials
[1]. In particular, unconventional spin triplet superfluids and
superconductors, as realized in 3He [2, 3], Uranium based heavy fermion
superconductors [4, 5, 6] and possibly in twisted van der Waals multilayers
[7], are important candidates to host topological fermionic boundary and low
energy states. While the emergent single particle band structure of these
phases is by now theoretically well understood, the additional complexity of
strong electronic correlations leads to much richer physics [8, 9] and is the
object of ongoing research. This particularly concerns the interplay of
fermionic boundary states (Majorana fermions) with quantum fluctuations of the
order parameter.

Figure 1:  a) Schematic of physical setup. The grey box represents the
Majorana Cooper pair box, which harbors a one-dimensional time-reversal
invariant SC (black line) that hosts four Majorana zero modes (yellow and red
dots). The device is coupled to leads (blue lines). b) Schematic
representation of the effective Kondo Hamiltonian (20). The even $\psi_{+}$
and odd $\psi_{-}$ superpositions of the lead electrons are coupled to an
effective quantum impurity with four internal states (square). The angular
momentum $\hat{\bm{J}}$ of the superconducting condensate (blue ellipse) acts
as an effective spin 1/2. The spin of the lead electrons (black arrows) is
coupled to the impurity via the orbitals formed by the MZMs (red and yellow
ellipses). The Kondo coupling to $\hat{\bm{J}}$ is effective only when the
orbital next to the lead electrons is occupied. c) Schematic Renormalization
Group flow diagram based in the plane of effective Kondo coupling $\lambda$
and relative strength of Cooper pair spin fluctuations $E_{s}$ and polarizing
field $E_{z}$. The parameter $g=e^{\mathcal{S}_{\rm imp}}$ is defined through
the zero temperature impurity entropy $\mathcal{S}_{\rm imp}$. d) Absolute
value of the charge conductance between the left and right wire in the limit
$E_{s}/E_{z}\gg 1$.

As a prime example, strong quantum fluctuations of the superconducting phase
in mesoscopic effectively spinless Majorana Cooper-pair boxes (MCPB) can be
induced by a large charging energy [10]. These devices are floating islands of
Josephson-connected spinless (i.e., spin-polarized) p-wave superconducting
chains [11, 12] and allow to implement a paradigmatic topological qubit, the
tetron [13]. While arrays of such tetrons emulate exotic fractionalized many-
body phases with topological order[14, 15, 16, 17, 13], a single spinless MCPB
coupled to metallic leads realizes a topological quantum impurity problem, the
topological Kondo effect [18, 19, 20, 21, 22, 23, 24, 25], hosting non-Fermi
liquid like low-energy characteristics and an irrational residual entropy
reminiscent of a non-trivial anyonic quantum dimension.

Exotic Kondo impurity problems of SU($N$) symmetry, most notably for $N=2$,
have been studied extensively for the past forty years both theoretically [26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36], most recently in the prospect of
quantum information [37, 38, 39, 40, 41], and experimentally, in particular in
mesoscopic quantum electronics devices [42, 43, 44]. For this symmetry group,
non-Fermi liquid behavior occurs only in the overscreened multichannel case
[45]. The Majorana based topological Kondo effect is special, inasmuch it
realizes an impurity spin transforming under O($M$) (where $M$ denotes the
number of Majorana zero modes coupled to external leads). Very recently,
Majorana-free setups of orthogonal [46, 47] and symplectic symmetry [48, 49,
50] were proposed, mathematically completing the set of possible classical Lie
groups. Interestingly, in the single channel case, both orthogonal and
symplectic Kondo models display exotic physics, including fingerprints of
bound anyons, already [51].

Within the superconducting Altland-Zirnbauer classes, two non-trivial
topological phases exist in one dimension. Apart from spinless p-wave
superconductors (e.g., the “Kitaev chain” [11], class D), there is a spinful
p-wave time reversal invariant topological superconductor (TRITOPS [52], class
DIII). In this work, we study TRITOPS islands in the Coulomb blockade regime
and, when the fermionic parity on the island is odd, uncover an unconventional
topological Kondo effect of spinful Majorana fermions. Using Abelian
Bosonization, we solve this problem in the simplest and most relevant case of
two pairs of Majorana edge states coupled to two spinful leads, Fig. 1. We
characterize the phase diagram and demonstrate that the topological Kondo
effect of spinful Majorana fermions is protected by spin-rotation symmetry,
but flows to the fixed point of the spinless topological Kondo effect in the
presence of symmetry breaking perturbations. Both fixed points display exotic
hallmarks of non-Fermi liquids. We make numerically and (potentially)
experimentally verifiable predictions for thermodynamic and transport
signatures of the non-trivial low-energy fixed point.

Single-particle physics in one-dimensional TRITOPS gained substantial
attention over the years [53, 54, 55, 56], in particular regarding its
unconventional transport through Josephson junctions [57, 58, 59, 60, 61, 62,
63]. Strong order parameter fluctuations of the superconducting phase, i.e.
spinful MCPBs in the Coulomb blockade regime, also gained some attention [54,
60], both in their context as topological qubits[64], topological Josephson
junction arrays [65], and mesoscopic Kondo impurities [66]. At the same time,
to the best of our knowledge, the impact of strong order parameter
fluctuations of the Cooper pair orientation $\hat{\bm{d}}$ in the spin sector
was considered only in Ref. [67], where it was uncovered that the
bandstructure topology induces a theta term[68] in the effective non-linear
sigma model of the $\hat{\bm{d}}$ vector, thereby stabilizing 2e-paired-
superconductivity. Here, similar fluctuations will be studied in zero-
dimensional mesoscopic islands.

The paper is organized as follows: In section II we introduce the spinful
Majorana Cooper-pair box and present the solution, finding the eigenfunctions
and spectrum. Building upon that result, we discuss in section III how a novel
topological Kondo effect can arise from coupling a Coulomb blockaded MCPB to
spinful normal metal leads. This is done in the cases of dominant and sub-
dominant spin interactions within the MCPB. In section IV, we discuss the
topological Kondo effect in more depth by means of Bosonization and poor man’s
scaling. This allows for constructing a schematic RG-flow diagram in the
parameter space of Kondo coupling and spin interaction strength, see Fig. 1c.
This is followed by presenting several observables, such as transport
coefficients and thermodynamic properties, in section V. We end the paper with
a conclusion and give an outlook to further research concerning MCPBs.

##  II Spinful MCPB

###  II.1 Setup

We consider a floating mesoscopic quantum device consisting of a one-
dimensional time-reversal invariant topological triplet superconductor in the
Cartan-Altland-Zirnbauer class DIII. We focus on the Coulomb blockade regime
where the charging energy $E_{c}$ is large enough that (thermal) fluctuations
of the total charge of the island can be neglected. Also, $E_{c}$ will be the
largest energy scale we assume in the system. In analogy to the charging
energy, we also consider an interaction ($E_{s}$) that punishes the formation
of a large total angular momentum within the island. The full Hamiltonian
reads

$\displaystyle\begin{split}\hat{H}&=E_{c}(2\hat{N}_{c}+\hat{n}_{f}-2N^{c}_{g})^{2}+E_{s}(\hat{\bm{L}}+\hat{\bm{S}})^{2}\\\
&+\hat{H}_{\text{BdG}}+\hat{H}_{z}\end{split}$ (1a) where
$\hat{H}_{z}=-E_{z}\hat{d}_{y}$ (1b) and
$\hat{H}_{\text{BdG}}=\int\text{d}x\left\\{\psi(x)^{\dagger}\epsilon(\hat{p})\psi(x)+[\psi(x)^{\text{T}}\hat{\Delta}\psi(x)+\text{H.c.}]\right\\},$
(1c) with
$\hat{\Delta}=u\sigma_{y}\hat{\bm{d}}\cdot\bm{\sigma}e^{i\hat{\varphi}}\partial_{x}.$
(1d)

This model can be thought of as a two-fluid model with the condensate of
Cooper-pairs and the electrons, described by spinor fields
$\psi=(\psi_{\uparrow},\psi_{\downarrow})^{\text{T}}$, where
$\psi^{(\dagger)}_{\sigma}(x)$ annihilates (creates) an electron of spin
$\sigma$ at position $x$; similar models have been discussed in the context of
one-dimensional superconductors [69, 70, 71, 67]. We chose the topological
triplet superconductor large enough to host independent zero energy edge
states (larger than the superconducting coherence length) but sufficiently
small that, in view of long-range interactions, the condensate incorporates
only quantum (i.e., temporal) fluctuations and no spatial fluctuation. The
assumptions of long-range interactions in the charge channel is realized by
Coulomb interactions, while the limit of long-range interactions in the spin
channel is met in the vicinity of magnetic phase transition. The problem will
thus eventually turn out to be 0+1 dimensional.

The first term in Eq. (1a) corresponds to the previously mentioned charging
energy where $\hat{N}_{c}=-i\partial_{\varphi}$ is the total number operator
of Cooper-pairs, and $\hat{n}_{f}=\int\text{d}x\,\psi^{\dagger}(x)\psi(x)$ is
the total number operator of electrons. Hence the total charge operator on the
island is [71] $\hat{N}_{\text{tot}}=2\hat{N}_{c}+\hat{n}_{f}$ and commutes
with $\hat{H}$. The constant $N^{c}_{g}=\frac{eV_{g}}{2}$ ($e$ being the
electron charge) can be tuned by changing the gate voltage $V_{g}$ and
determines the expected value of $\hat{N}_{\text{tot}}$.

The second term describes the spin interactions on the island. The vectors
$\hat{\bm{L}}=(\hat{L}_{x},\hat{L}_{y},\hat{L}_{z})$ and
$\hat{\bm{S}}=(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ are the canonically
conjugate operator to order parameter $\hat{\bm{d}}\in\mathbb{S}^{2}$ and the
total spin operator of the fermions
$\hat{S}_{i}=\frac{1}{2}\int\text{d}x\,\psi^{\dagger}(x)\sigma_{i}\psi(x)$,
respectively. In the following, we will refer to $\hat{\bm{L}}$ as the angular
momentum of the superconducting order parameter, which should not be confused
with the angular momentum of the individual Cooper-pair wavefunction (which is
obviously absent in the present problem of one-dimensional TRITOPS).

The third term, $\hat{H}_{\text{BdG}}$ describes the triplet superconductor.
The free dynamics of the electrons is governed by the dispersion relation
$\epsilon(\hat{p})=\frac{\hat{p}^{2}}{2m}-\mu$ where $\hat{p}$ is the momentum
operator. The pairing operator $\hat{\Delta}$ consists of the superconducting
phase $\hat{\varphi}$, the Cooper-pair orientation $\hat{\bm{d}}$ in spin
space, the pairing strength $u$ (units of velocity) and the partial derivative
$\partial_{x}$ reflecting the p-wave nature of the superconductor.
Furthermore, the Cooper-pair orientation can be parameterized by the two
operators $\hat{\theta}$ and $\hat{\phi}$, which eigenvalues can take the
values $\theta\in[0,\pi)$ and $\phi\in[0,2\pi)$. A possible parametrization in
terms of $\hat{\theta}$ and $\hat{\phi}$ is given as follows:

$\hat{\bm{d}}=\begin{pmatrix}\cos(\hat{\phi})\sin(\hat{\theta})\\\
\cos(\hat{\theta})\\\ \sin(\hat{\phi})\sin(\hat{\theta})\end{pmatrix}.$ (2a)
Another representation of $\hat{\bm{d}}$ that will be heavily used in this
work can be given in terms of a unitary operator, that is
$\hat{U}^{\dagger}\sigma_{y}\hat{U}=\hat{\bm{d}}\cdot\bm{\sigma},$ (2b) where
$\hat{U}=e^{i\frac{\hat{\theta}}{2}\sigma_{x}}e^{i\frac{\hat{\phi}}{2}\sigma_{y}}.$
(2c)

Finally, the term $\hat{H}_{z}$ in Eq. (1a) is a spin-rotation symmetry
breaking perturbation, which will be used to quantify the dominance of spin
fluctuations within the MCPB. This term can be understood as a Zeeman coupling
that favors a polarization of the $\hat{\bm{d}}$ vector in the y-direction.
Thus, there are two competing terms in the Hamiltonian. The first term,
$E_{s}\hat{\bm{J}}^{2}$, attempts to fix the angular momentum, resulting in
strong fluctuations in $\hat{\bm{d}}$, while $\hat{H}_{z}$ tries to localize
$\hat{\bm{d}}$, leading to strong fluctuations in $\hat{\bm{J}}$. The two
aforementioned limiting cases correspond to situations where either $E_{s}\gg
E_{z}$ or $E_{s}\ll E_{z}$. In the first case, $\hat{\bm{d}}$ is pinned while
$\hat{\bm{J}}$ fluctuates strongly, while in the other case, $\hat{\bm{J}}$ is
pinned, and $\hat{\bm{d}}$ freely fluctuates.

###  II.2 Spectrum of the spinful MCPB

In order to find the spectrum of Hamiltonian (1a), we transform the electrons
into a co-moving basis, which follows the fluctuations of the condensate.
Thus, the transformed electrons become essentially chargeless, and their spin
will be fixed to either point up or down without any fluctuation. The
canonical transformation that achieves this goal reads

$\mathscr{U}=\exp(-i\frac{\hat{\varphi}}{2}\hat{n}_{f})\exp(-i\hat{\phi}\hat{S}_{y})\exp(-i\hat{\theta}\hat{S}_{x}).$
(3)

This transformation applied to the Hamiltonian (1a) yields

$\displaystyle\mathscr{U}^{\dagger}\hat{H}\mathscr{U}|_{E_{z}=0}=4E_{c}(\hat{N}_{c}-N^{c}_{g})^{2}+E_{s}\hat{\bm{J}}^{2}$
$\displaystyle+\int\text{d}x\left\\{\psi(x)^{\dagger}\epsilon(\hat{p})\psi(x)+u[\psi(x)^{\text{T}}\partial_{x}\psi(x)+\text{H.c.}]\right\\},$
(4)

where
$\hat{\bm{J}}=\hat{\bm{L}}+\frac{\hat{S}_{y}}{\sin(\hat{\theta})}(\hat{\bm{d}}+\cot(\hat{\theta}){\bm{e}}_{\theta})$
with ${\bm{e}}_{\theta}$ being the unit vector in $\theta$ direction.

A few comments are in order: First, the transformation $\mathscr{U}$ decouples
the spin up and down sector of the fermions and diagonalized
$\hat{H}_{\text{BdG}}$ in spin space. Now, the superconducting part consists
of two copies of a one-dimensional p-wave superconductor of spinless fermions
per wire. Each system is a Kitaev topological superconductor and is known to
host Majorana zero modes (MZMs) at the edges. Thus, the MCPB can host up to
four MZMs, which we will call
$\gamma_{L\uparrow},\gamma_{L\downarrow},\gamma_{R\uparrow}$ and
$\gamma_{R\downarrow}$. Note that the symbols $\uparrow$ and $\downarrow$
should not be taken too seriously since the MZMs do not carry a spin anymore,
and the reader may think of them as mere labels. The labels $R$ and $L$ denote
whether an MZM is localized at the right or left edge of the wire (see Fig.
1a). Furthermore, we define non-local fermions

$\Gamma_{\sigma}=\frac{1}{2}(\gamma_{L\sigma}+i\gamma_{R\sigma}).$ (5)

Second, due to the canonical transformation, the total electron charge
vanished from the Hamiltonian. However, it enters implicitly through
$\hat{N}_{c}$, whose quantization properties have changed. The total number
operator of the Cooper-pair operator is usually integer quantized. However,
after the transformation $\hat{N}_{c}$ is half integer quantized, as it
physically describes the total charge on the island, including the
contribution of unpaired electrons. A technical explanation of the change in
the quantization condition is contained in appendix A.1.

Third, in contrast to the total charge, the spin operator $\hat{\bm{S}}$ did
not fully disappear since the spin is a vector quantity. Instead, the fermion
spin survives as a background field for the angular momentum $\hat{\bm{L}}$ of
the order parameter $\hat{\bm{d}}$. Suppose one understands $\hat{\bm{d}}$ as
the coordinate of a particle on a sphere. Then, due to the fermions, this
particle feels a magnetic monopole of charge $s_{y}$ (being an eigenvalue the
$\hat{S}_{y}$ operator) as can straightforwardly be calculated by taking the
rotation of $\hat{\bm{J}}-\hat{\bm{L}}$. Analogously, one can show that the
action corresponding to the Hamiltonian $E_{s}\hat{\bm{J}}^{2}$ takes the
shape of a non-linear sigma model with a topological Wess-Zumino-Witten term.
As a consequence of the background field, and somewhat in analogy to the
Cooper-pair charge $\hat{N}_{c}$, the angular momentum $\hat{\bm{L}}$ is
integer quantized while its transformed version $\hat{\bm{J}}$ is half odd
integer quantized if the fermion parity is odd.

Finally, we can find the spectrum of the Hamiltonian (4). The total Hilbert
space $\mathcal{H}$ of the MCPB can be decomposed as a tensor product of three
subspaces. That is,

$\mathcal{H}=\mathcal{H}_{n}\otimes\mathcal{H}_{j}\otimes\mathcal{H}_{f},$ (6)

where $\mathcal{H}_{n}$ is the Hilbert space spanned by the eigenstates of
$\hat{N}_{c}$, $\mathcal{H}_{j}$ is spanned by the eigenstates of
$\hat{J}_{y}$ and $\mathcal{H}_{f}$ is the Fock space of the fermionic neutral
excitations in the wire. Since we are interested in the low-temperature
behavior of the system, we assume that Bogoliubov quasi-particles in the
superconductor can not be created. However, the non-local electrons
$\Gamma_{\sigma}$ formed from the MZMs are accessible even at zero
temperature. Thus, we project $\mathcal{H}_{f}$ down to the Fock space, which
in the case of a single wire $w=1$ is

$P_{T=0}\mathcal{H}_{f}P_{T=0}=\text{span}(\ket{0},\Gamma_{\sigma}^{\dagger}\ket{0},\Gamma^{\dagger}_{\uparrow}\Gamma^{\dagger}_{\downarrow}\ket{0}),$
(7)

where the state $\ket{0}$ is the vacuum destroyed by the operators
$\Gamma_{\sigma}$.

Furthermore, the Hamiltonian (4) commutes with the mutually commuting
operators $\hat{N}_{c}$ and $\hat{\bm{J}}$. Therefore, we can construct the
energies and energy eigenstates by solving the eigenvalue equation for
$\hat{N}_{c}$ and $\hat{\bm{J}}^{2}$ separately. The low energy spectrum of
the MCPB is summarized in the Tab. 1 for the case $w=1$ and plotted in Fig. 2.
In the case of multiple wires, $w>1$, the low-energy spectrum of Tab. 1 is the
same and characterized by the very same quantum numbers as displayed in the
first four columns of Tab. 1. However, at $w>1$, the entries for $s_{y}$ and
for the degeneracy of the various states may be higher. In particular, the
ground state degeneracy is $4,16,204,\dots$ for $w=1,2,3,\dots$.

${N}_{c}$ | ${j}_{y}$ | $J$ | ${P}_{f}$ | ${s}_{y}$ | $E$ | $\\#$
---|---|---|---|---|---|---
0 | 0 | $0$ | 1 | 0 | $4E_{c}(N^{c}_{g})^{2}$ | 2
$\nicefrac{{1}}{{2}}$ | $\pm\nicefrac{{1}}{{2}}$ | $\nicefrac{{1}}{{2}}$ | $-1$ | $\pm\nicefrac{{1}}{{2}}$ | $4E_{c}(\nicefrac{{1}}{{2}}-N_{g}^{c})^{2}+E_{s}\nicefrac{{3}}{{4}}$ | 4
0 | $0,\pm 1$ | 1 | 1 | 0 | $4E_{c}(N^{c}_{g})^{2}+2E_{s}$ | 6
Table 1: Classification of the three lowest energy eigenstates of Hamiltonian
(4) by their quantum numbers: Charge ${N}_{c}$ (in units of $2e$), magnetic
quantum number ${j}_{y}$, total angular momentum $J$ as determind by the
$\hat{\bm{J}}^{2}$ eigenvalue $J(J+1)$, fermion parity ${P}_{f}=(-1)^{n_{f}}$
and magnetic quantum number of the fermions ${s}_{y}$. The second last column
shows the total energy of a particular state as a function of $N^{c}_{g}$
while the last column shows the corresponding degeneracy of each state. Figure
2: The spectrum of Hamiltonian (4) as a function of $N^{c}_{g}$ in the case
$E_{c}>3E_{s}/4$. The labeling of parabolas corresponds to eigenvalues
$(N_{c},J)$ of total charge and total angular momentum as defined in Tab. 1.
The crossing point of two red, solid parabolas denotes the value of
$N^{c}_{g}$ where it becomes favorable to add one unit of charge (one Cooper
pair) onto the island and corresponds to $E_{c}$ where $E_{c}$ is the charging
energy. The two black arrows denote a second-order process of virtual
transitions from a state with odd fermion parity to a state with even fermion
parity and reverse and determine the superexchange interaction The energy gap
$\Delta E=E_{c}-3E_{s}/4$ must be overcome for such a process.

##  III Spinful Topological Kondo Effect

This section explains the setup that gives rise to a topological Kondo-effect
by coupling leads to an MCPB. We model the system with two free electron gases
that are brought into the vicinity of an MCPB, where electrons from a left and
right lead can tunnel via the tunneling amplitude $t$ into the Majorana edge
states on the left and right sides of the superconductor, respectively. A
schematic of the setup for $w=1$ can be found in Fig. 1a. The Hamiltonian of
the full system reads:

$\hat{H}=\hat{H}_{\text{dot}}(\gamma)+\hat{H}_{0}(\psi)-t\,\hat{H}_{t}(\psi,\gamma)+\hat{H}_{z}.$
(8)

Here, $\hat{H}_{\text{dot}}$ denotes the Hamiltonian (4), and $\hat{H}_{0}$ is
the Hamiltonian of free electrons $\psi_{R/L}$ of the left and right leads.
The Hamiltonian $\hat{H}_{t}$ couples the electrons to the MZMs and is given
by:

$\hat{H}_{t}=\sum_{i=R/L}\left(\psi^{\dagger}_{i,\sigma}e^{-i\varphi/2}(U^{\dagger})^{\phantom{\dagger}}_{\sigma\sigma^{\prime}}\gamma^{\phantom{\dagger}}_{i\sigma^{\prime}}+\text{H.c.}\right),$
(9)

Here, the operator $\psi_{i,\sigma}^{\dagger}(x)$ creates an electron in the
$i$th lead with spin $\sigma$. To straighten up the notation, we define
$\psi^{(\dagger)}_{i\sigma}\equiv\psi^{(\dagger)}_{i\sigma}(0)$ where we
assume that the MCPB is located at $x=0$. This setup is readily generalized to
$w>1$ where we assume a spinful lead coupled to each of the $2w$ Kramers pairs
of Majoranas on the island.

We choose the gate charge $N^{c}_{g}=\frac{1}{2}$ such that the ground state
of the quantum dot has an odd fermion parity, i.e., one non-local fermion
$\Gamma_{\sigma}$ is occupied. Therefore, the ground state corresponds to the
blue parabola in Fig. 2 and the second row in Table 1. Thus, the ground state
is fourfold degenerate in the $w=1$ wire case. In this exemplary case, the
restriction to the odd fermion parity operator imposes a constraint on the
MZMs of the form:

$\gamma_{L\downarrow}\gamma_{L\uparrow}\gamma_{R\downarrow}\gamma_{R\uparrow}=-1$
(10)

The objective is to develop an effective low-temperature theory for the ground
state manifold spanned by the four degenerate states. Processes in which
fermions tunnel into and away from the MCPB change the fermion parity from
$P_{f}=-1$ to $P_{f}=1$. The first mentioned process adds charge, while the
second process removes charge from the box. These tunneling events correspond
to transitions in higher energy eigenstates (see Fig. 2).

We account for these processes in the low energy regime by treating them as
virtual and integrating out the higher excited states using a Schrieffer-Wolf
transformation [72]. To illustrate this procedure and as a preliminary
problem, we first apply it to the limiting case where $E_{c}\gg E_{s}$. Thus,
the timescale on which the superconducting phase $\hat{\varphi}$ fluctuates is
much smaller compared to the timescale of the fluctuations of $\hat{\bm{d}}$
($\hat{U}$). Therefore, we first only perform a Schrieffer-Wolf transformation
in the charge sector $\mathcal{H}_{n}$ of the condensate Hilbert space.

While in all of the previous discussions, a single TRITOPS wire on the island
is assumed, see Fig. 1, the problem is readily generalized to a spinful MCPB
with $w$ wires ($w>1$), all of which are coupled to two leads and to the same
order parameter field by introducing a wire index to the fermions and assuming
the mean field Hamiltonian is diagonal in wire-space (see [18] for the
discussion of the analogous situation without spin fluctuations). We will
suppress the wire index in what follows but occasionally comment on the case
$w>1$. The such derived, effective Hamiltonian reads

$\hat{H}_{K}=-\frac{\lambda}{2}\sum_{i,j}\sum_{\sigma,\rho}[\psi^{\dagger}_{i}\hat{U}^{\dagger}]_{\sigma}\gamma_{i,\sigma}\gamma_{j,\rho}[\hat{U}\psi_{j}]_{\rho},$
(11)

where $\lambda=\frac{4t^{2}}{E_{c}}$. This expression holds for an arbitrary
number of wires (in which case $i,j\in\\{1,\dots,2w\\}$). In the simplest case
$w=1$ (in this case we use labels $i,j\in\\{L,R\\}$) this expression can be
further simplified as follows

$\hat{H}_{K}=\lambda\left(\psi_{+}^{\dagger}\hat{U}^{\dagger}\bm{\sigma}\hat{U}\psi_{+}^{\phantom{\dagger}}+\psi^{\dagger}_{-}\sigma_{y}\hat{U}^{\dagger}\sigma_{y}\bm{\sigma}\sigma_{y}\hat{U}\sigma_{y}\psi_{-}^{\phantom{\dagger}}\right)\cdot\bm{\hat{S}},$
(12)

where

$\begin{split}\psi_{+}&=\frac{1}{\sqrt{2}}\left(\psi_{L}+i\psi_{R}\right)\\\
\psi_{-}&=\frac{i\sigma_{y}}{\sqrt{2}}\left(\psi_{L}-i\psi_{R}\right)\end{split}$
(13)

with $\psi_{L/R}=(\psi_{L/R\uparrow},\psi_{L/R\downarrow})^{\text{T}}$ being
spinors in spin space. The impurity spin
$\hat{\bm{S}}=(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})^{\text{T}}$ contains the
fermion spin operators
$\hat{S}_{i}=\frac{1}{2}\Gamma^{\dagger}\sigma_{i}\Gamma$, where
$\Gamma=(\Gamma_{\uparrow},\Gamma_{\downarrow})^{\text{T}}$. These spin
operators can be expressed in terms of the MZMs. Using the parity constraint
(10), we obtain

$\displaystyle\hat{S}_{x}=\frac{X}{2},\quad\quad
X=i\gamma_{L\downarrow}\gamma_{R\uparrow}=i\gamma_{L\uparrow}\gamma_{R\downarrow},$
(14) $\displaystyle\hat{S}_{y}=\frac{Y}{2},\quad\quad
Y=i\gamma_{L\downarrow}\gamma_{L\uparrow}=i\gamma_{R\downarrow}\gamma_{R\uparrow},$
(15) $\displaystyle\hat{S}_{z}=\frac{Z}{2},\quad\quad
Z=i\gamma_{L\uparrow}\gamma_{R\uparrow}=i\gamma_{R\downarrow}\gamma_{L\downarrow},$
(16)

where the operators $X$, $Y$, and $Z$ fulfill an SU(2) algebra. Since the spin
quantum number of the MZMs behaves more like a static quantum number than an
actual spin, we should rather think of $\hat{\bm{S}}$ as an impurity that acts
on the orbital (charge parity) space spanned by $\gamma_{\uparrow/\downarrow}$
than an actual spin.

###  III.1 Limit $E_{z}\gg E_{s}$

We take $E_{s}$ and $E_{c}$ of the same order of magnitude but also take the
perturbation $\hat{H}_{z}$ into account where we work in the limit $E_{z}\gg
E_{s}$. Thus, the $\hat{\bm{d}}$ vector is polarized and, again, fluctuates on
a much larger time scale than the superconducting phase $\hat{\varphi}$.
Therefore, we can continue working with the equation (12).

In the case where $E_{s}/E_{z}\rightarrow 0$, $\hat{U}$ does not display any
quantum fluctuations (it is a constant of motion). We can thus choose
$U=\mathds{1}$, and the Hamiltonian (11) becomes an O(4$w$) topological Kondo
model [18]. Moreover, in view of a well established equivalence between O(4)
topological Kondo effect and two-channel SU(2) Kondo effect (cf. App. B), the
$w=1$ Hamiltonian (12) becomes a two-channel Kondo Hamiltonian

$\hat{H}_{2CK}=\lambda\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi^{\phantom{\dagger}}_{a}\cdot\hat{\bm{J}}.$
(17)

It is well known that the two-channel Kondo model has a quantum critical point
at a finite Kondo coupling $\lambda$ [28, 29, 30]. Emery and Kivelson [73]
already formulated an exact solution to this problem with Bosonization methods
and found that fractionalized (Majorana fermions) excitations govern the
system.

However, in the case of fluctuating $\hat{U}$ (i.e. finite $E_{s}/E_{z}$), the
situation becomes more complex. As discussed previously, in the limit
$E_{s}/E_{z}\rightarrow 0$ we expect two-channel Kondo physics, Eq. (17), for
the $w=1$ spinful topological Kondo effect. This is illustrated as a red star
in Fig. 1c. Here, we consider small $E_{s}/E_{z}$ corrections and find that
the two-channel Kondo fixed point is stable concerning this perturbation.

We assume that the vector $\hat{\bm{d}}$ is predominantly polarized in the
y-direction of the “north-pole” and weakly fluctuates around this state.
Mathematically, this can be expressed as follows:

$\hat{\bm{d}}=\bm{e}_{y}+\delta\hat{\bm{d}},\\\
\quad\delta\hat{\bm{d}}=\begin{pmatrix}\delta\hat{d}_{x},\ 0,\
\delta\hat{d}_{z}\end{pmatrix}=\begin{pmatrix}\hat{\theta}\sin(\hat{\phi}),\
0,\ \hat{\theta}\cos(\hat{\phi})\end{pmatrix},$ (18)

The $\hat{\bm{d}}$ vector has been linearized with respect to the deviation
angle $\hat{\theta}$. This linearization yields the effective two-dimensional
vector $\delta\hat{\bm{d}}$ that characterizes the spin sector of the
condensate, confined to a tangent plane attached to the north pole of the
sphere encompassing all possible configurations of $\hat{\bm{d}}$.

To determine the spectrum of the vector $\hat{\bm{d}}$, we expand the
Hamiltonian of the MCPB (including the $E_{z}$ term) to the first non-trivial
order in $\theta$. This leads to a Hamiltonian $\hat{H}_{\theta\ll 1}$, in
which the Schrödinger equation resembles the radial part of a harmonic
oscillator in two dimensions (see App. A.2.1). The level spacing of the energy
eigenstates is $\Delta E_{n}=\sqrt{2E_{s}E_{z}}$. The four-fold degeneracy of
the MCPB ground state for $E_{z}=0$ gets lifted by the presence of a finite
$E_{z}$ with a new two-fold degenerate ground state manifold where
$s_{y}=j_{y}=\pm 1/2$. Furthermore, the ground state expectation value of
$\delta\hat{\bm{d}}$ is 0. A comprehensive derivation of the spectrum can be
found in appendix A.2.1.

To derive the effective Kondo Hamiltonian in the limiting case $\theta\ll 1$
(i.e. $E_{z}\gg E_{s}$), we use equation (12) as a starting point and,
similarly, linearize $U$. Initially, we introduce a new parametrization
$\hat{U}=\exp(i\frac{\hat{W}}{2})$, where
$\hat{W}=\partial_{\phi}\delta\hat{\bm{d}}\cdot\bm{\sigma}$. Expanding
$\hat{U}$ to first order in $\hat{W}$, we arrive at a linearized version of
the Hamiltonian (12), that is

$\hat{H}_{K}=\lambda\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi_{a}\cdot\left(1-a\,\partial_{\phi}\delta\hat{\bm{d}}\times\right)\hat{\bm{S}}.$
(19)

From this expression, it becomes evident that even at the first order in
$\hat{\bm{d}}$ fluctuations, the effective Hamiltonian exhibits a more
intricate structure than a two-channel Kondo Hamiltonian.

However, if $E_{z}\gg\left(\frac{\lambda^{4}}{E_{s}}\right)^{\frac{1}{3}}$ the
system effectively stays in the ground state manifold. Consequently, we can
replace the operator $\delta\hat{\bm{d}}$ with its ground-state expectation
value, which is zero. Second-order processes in the first excited state and
back into the ground state modify the Hamiltonian such that the isotropic 2CK
Hamiltonian becomes anisotropic in SU(2) spin space, but still preserves
channel isotropy. Furthermore, they add an interaction term coupling the
relative spin densities between the two types of lead electrons. However, the
interaction term has a scaling dimension of two and is deemed irrelevant in
the Renormalization Group (RG) sense. The anisotropic 2CK model flows towards
isotropy and, consequently, converges to the same fixed point as the isotropic
2CK model [73]. Therefore, the two-channel Kondo effect remains stable against
small fluctuations in $\hat{\bm{d}}$.

###  III.2 Limit $E_{z}\ll E_{s}$

If, conversely, the magnitude of $E_{s}$ greatly surpasses that of $E_{z}$,
the phase $\hat{\varphi}$ and $\hat{U}$ experiences rapid fluctuations and,
necessarily, need to be treated on equal footing. Within the low-energy
domain, the system is confined to the fourfold degenerate ground state
manifold in Tab. 1 (blue parabola in Fig. 2). Details concerning the precise
computation are presented in appendix B. Employing the Schrieffer-Wolf
transformation, we arrive at the effective Hamiltonian given by

$\hat{H}_{\text{eff}}=\lambda\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi^{\phantom{\dagger}}_{a}\cdot\frac{1-a\,Y}{2}\hat{\bm{J}}-\frac{E_{z}}{3}Z.$
(20)

The vector $\hat{\bm{J}}$ is projected onto the ground state manifold and can
be expressed as
$\hat{\bm{J}}=(\frac{J_{x}}{2},\frac{J_{y}}{2},\frac{J_{z}}{2})$, with $J_{i}$
denoting the Pauli matrices acting within the space spanned by the states
$\ket{j=\pm\nicefrac{{1}}{{2}},J=\nicefrac{{1}}{{2}}}$ in the Hilbert space
$\mathcal{H}_{j}$. The Kondo coupling parameter is now defined as
$\lambda=\frac{4t^{2}}{E_{c}-\nicefrac{{3}}{{4}}E_{s}}$. The disparity from
the Kondo coupling discussed in the previous section arises from including
virtual processes in the angular momentum sector. Of course, in the limit
$E_{c}\gg E_{s}$, which was assumed to derive Eq. (12), this disparity
vanishes.

At first glance, one might mistake the Hamiltonian (20) for twice a
conventional SU(2) Kondo Hamiltonian, given that the projector
$P_{a}=\frac{1-aY}{2}$ projects the impurity $\hat{\bm{J}}$ onto two
independent sectors. In other words, the two effective impurities
$\hat{\bm{J}}_{a}=P_{a}\hat{\bm{J}}$ commute with each other, i.e.,
$\left[\hat{J}_{+}^{i},\hat{J}_{-}^{j}\right]=0$. However, the distinction to
two SU(2) Kondo effects lies in the fact that the entire impurity transforms
under SU(2) $\oplus$ SU(2) rather than SU(2) $\otimes$ SU(2), as it is the
case for two instances of a standard SU(2) Kondo effect. To give Hamiltonian
(20) a physical meaning, one can understand the projector $\frac{1-aY}{2}$ as
an operator that distinguishes between the occupation of the two orbitals
formed by the four MZMs. The lead electrons $\psi_{a}$ can only interact with
the impurity if the orbital assigned by the projection operator is occupied.
Therefore, the MZMs act as gatekeepers, limiting access to the impurity for
the lead electrons, see Fig. 1 b) for an illustration.

For further use in the remainder of the paper, we generalize the isotropic
$E_{z}=0$ Hamiltonian to an anisotropic model as follows:

$\displaystyle\hat{H}_{K}$
$\displaystyle=\sum_{a=\pm}\left(\sum_{i\in{x,z}}\psi^{\dagger}_{a}\sigma_{i}\psi^{\phantom{\dagger}}_{a}(\lambda^{1}_{\perp}-a\,\lambda^{Y}_{\perp}Y)\hat{J}_{i}\right.$
$\displaystyle\left.+\phantom{\sum_{a}}\psi^{\dagger}_{y}\sigma_{i}\psi^{\phantom{\dagger}}_{a}(\lambda^{1}_{y}-a\,\lambda^{Y}_{y})Y\hat{J}_{y}\right).$
(21)

We employ a poor man’s scaling [74] analysis (see App. C) for Eq. (21) and
obtain the following flow equations:

$\displaystyle\frac{\text{d}g_{\perp}}{\text{d}l}$
$\displaystyle=g_{\perp}\left(g^{1}_{y}+g^{Y}_{y}\right),$ (22a)
$\displaystyle\frac{\text{d}g^{1}_{y}}{\text{d}l}$
$\displaystyle=2g_{\perp}^{2},$ (22b)
$\displaystyle\frac{\text{d}g^{Y}_{y}}{\text{d}l}$
$\displaystyle=2g_{\perp}^{2},$ (22c)

where $l=\log\left(\frac{D_{0}}{D}\right)$ is the RG time and $D_{0}$ being
the bare bandwidth. We introduced the dimensionless
$g^{1/Y}_{\perp/y}=\nu\lambda^{1/Y}_{\perp/y}$ coupling constants, with $\nu$
denoting the density of states at the Fermi level and set
$g^{1}_{\perp}=g^{Y}_{\perp}=g_{\perp}$. Fig. 3 displays the RG flow created
by Eq. (22), where it is clearly visible that the couplings flow towards
strong coupling and isotropy. In particular, this is true for the red
trajectory, which starts at the parameter values corresponding to the Toulouse
point introduced in the next section. This result suggests that the Toulouse
point and weak isotropic coupling points in parameter space reside in the
basin of attraction of the same strong coupling fixed point. This observation
will have significance in the next section, where we present an exact solution
of the model at the Toulouse point: It can be expected that the Toulouse point
solution correctly describes the strong coupling physics of Eq. (20) and Eq.
(21), alike.

Figure 3: RG flow for anisotropic coupling constants. In the lower-left
corner, all couplings are zero. In contrast, the upper left corner symbolizes
the point where all coupling constants are infinity, which is the strong
coupling fixed point. The red trajectory reflects the RG flow for which the
Toulouse point is the starting point. A more detailed discussion on how this
figure is created can be found in App. C

##  IV Single wire at the Toulouse point

Generalizing the protocol of Emery and Kivelson [73], we can exactly solve
Hamiltonian Eq. (21) at a specific hyperplane in parameter space called
“Toulouse point”. For this, we bosonize the lead fermions, apply a non-local
canonical transformation $U_{\text{E.K.}}$, and refermionize the lead
electrons as well as expressing the impurity in terms of Majorana fermions.

###  IV.1 Bosonization and Refermionization

We take the usual Bosonization approach and decompose the fields
$\psi_{a\sigma}$, with respect to the lattice constant $a_{0}$, into slow-
varying right and left-moving fields:

$\psi_{a\sigma}(x)=\text{e}^{ik_{F}x}R_{a\sigma}(x)+\text{e}^{-ik_{F}x}L_{a\sigma}(x),$
(23)

where $k_{F}$ is the Fermi wavevector. Note that $\psi_{+\sigma}(x)$ and
$\psi_{-\sigma}(x)$ only have support on one half-axis. Thus, one can
represent the two fields by only one chiral (right-moving) field that extends
over the whole real axis [75]. This new chiral fermion, written in terms of
the chiral Bose fields $\phi_{a\sigma}(x)$, read

$\tilde{\psi}_{a\sigma}(x)=\frac{F_{a\sigma}}{\sqrt{2\pi
a_{0}}}\text{e}^{i\sqrt{4\pi}\phi_{a\sigma}(x)},$ (24)

where $F_{a\sigma}$ is a Klein factor that ensures the correct fermionic
statistics. The bosonic field follows the commutation relation

$[\partial_{x}\phi_{i}(x),\phi_{j}(y)]=\frac{i}{2}\delta(x-y)\delta_{ij},$
(25)

where $i,j$ are multindices of the shape $(a,\sigma)$. Again, we define
$\tilde{\psi}_{a\sigma}(0)\equiv\tilde{\psi}_{a\sigma}$ and
$\phi_{a\sigma}(0)\equiv\phi_{a\sigma}$.

It will be convenient to introduce a new basis of Bose fields that are related
by

$\begin{pmatrix}\phi_{s}\\\ \phi_{sf}\\\ \phi_{c}\\\
\phi_{cf}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1&-1&1&-1\\\ 1&-1&-1&1\\\
1&1&1&1\\\ 1&1&-1&-1\\\ \end{pmatrix}\begin{pmatrix}\phi_{+\uparrow}\\\
\phi_{+\downarrow}\\\ \phi_{-\uparrow}\\\ \phi_{-\downarrow}\end{pmatrix}.$
(26)

A similar basis transformation can also be done for the Klein factors with the
identities [47, 76, 77]

$\displaystyle
F^{\dagger}_{sf}F^{\dagger}_{s}=F^{\dagger}_{+\uparrow}F_{+\downarrow},\quad
F_{sf}F_{s}^{\dagger}=F^{\dagger}_{-\uparrow}F_{-\downarrow},$ $\displaystyle
F^{\dagger}_{sf}F^{\dagger}_{cf}=F^{\dagger}_{+\uparrow}F_{-\uparrow},\quad
F^{\dagger}_{c}F^{\dagger}_{s}=F^{\dagger}_{+\uparrow}F^{\dagger}_{-\uparrow}.$
(27)

Also, the Kondo Hamiltonian can be rewritten in the following form

$\displaystyle\hat{H}_{K}$
$\displaystyle=\frac{\lambda_{\perp}}{2}\left[(I^{+}_{+}+I^{+}_{-})\hat{J}_{-}+(I^{+}_{-}-I^{+}_{+})Y\hat{J}_{-}+\text{H.c.}\right]$
$\displaystyle\quad+\lambda^{1}_{y}(I^{y}_{+}+I^{y}_{-})\hat{J}_{y}+\lambda_{y}^{Y}(I^{y}_{-}-I^{y}_{+})Y\hat{J}_{y},$
(28)

where

$I^{+/-}_{a}=\tilde{\psi}^{\dagger}_{a\uparrow/\downarrow}\tilde{\psi}^{\phantom{\dagger}}_{a\downarrow/\uparrow}\quad\text{and}\quad
I^{y}_{a}=\frac{1}{2}(\tilde{\psi}^{\dagger}_{a\uparrow}\tilde{\psi}^{\phantom{\dagger}}_{a\uparrow}-\tilde{\psi}^{\dagger}_{a\downarrow}\tilde{\psi}^{\phantom{\dagger}}_{a\downarrow}).$
(29)

Note that we work in a basis in which $\sigma_{y}$ is diagonal and, thus, the
spin indices $\uparrow/\downarrow$ refer to spins quantized in the
$y$-direction. We used the notation $\hat{J}_{\pm}=\hat{J}_{z}\pm
i\hat{J}_{x}$. We next bosonize according to equation (24) and perform a non-
local unitary rotation (see App. D) to simplify the Hamiltonian. It is
convenient to define the new fermions

$\chi_{r}=\frac{F_{r}}{\sqrt{2\pi
a_{0}}}\text{e}^{i\sqrt{4\pi}\phi_{r}}\quad\text{and}\quad{d}=F^{\dagger}_{s}\hat{J}_{-},$
(30)

$(r=s,sf,c,cf)$ one can show that the effective Kondo Hamiltonian becomes

$\displaystyle\hat{H}^{\prime}_{K}$
$\displaystyle=\frac{\lambda\perp}{\sqrt{8\pi
a_{0}}}(\chi_{sf}^{\phantom{\dagger}}+\chi_{sf}^{\dagger})(d-d^{\dagger})$
$\displaystyle\quad-\frac{\lambda_{\perp}}{\sqrt{8\pi
a_{0}}}(\chi_{sf}^{\phantom{\dagger}}-\chi_{sf}^{\dagger})(d+d^{\dagger})Y$
$\displaystyle\quad+\delta\lambda_{y}:\chi_{s}^{\dagger}\chi_{s}^{\phantom{\dagger}}:(d^{\dagger}d-\frac{1}{2})$
$\displaystyle\quad+\lambda_{y}^{Y}:\chi_{sf}^{\dagger}\chi^{\phantom{\dagger}}_{sf}:Y(d^{\dagger}d-\frac{1}{2}),$
(31)

where $\delta\lambda_{y}=\lambda^{1}_{y}-2\pi v_{F}$ and $v_{F}$ is the Fermi
velocity.

The former Hamiltonian takes a more convenient shape if one rewrites the
fermions in terms of real Majorana fermions. Let us consider the following
decomposition

$\chi_{r}=\frac{1}{2}(\xi_{r}+i\zeta_{r}),\quad\text{and}\quad
d=\frac{1}{2}(\alpha+i\beta),$ (32)

of the complex fermions into real Majorana fermions. Note that in the present
basis, the number of Majorana fermions that describe the four-dimensional
impurity is represented by the six Majorana fermions and one constraint.
Namely, the four MZMs (i.e.
$\gamma_{L\uparrow},\gamma_{L\downarrow},\gamma_{R\uparrow},$ and
$\gamma_{R\downarrow}$) and the two Majorana fermions $\alpha$ and $\beta$
that describe ${\hat{\bm{J}}}$. It is useful to represent every term in
Hamiltonian (31) in terms of four composite Majorana fermions without
constraint, i.e.

$\eta=\beta,\quad\eta_{x}=\alpha X,\quad\eta_{y}=\alpha
Y,\quad\text{and}\quad\eta_{z}=\alpha Z.$ (33)

These four fermions allow for a faithful representation of the Hamiltonian
(31) that preserves all mutual commutation relations of the different
operators appearing in the Hamiltonian. Using this representation, the
Hamiltonian becomes

$\displaystyle\hat{H}^{\prime}_{\text{eff}}$
$\displaystyle=i\frac{\lambda_{\perp}}{\sqrt{8\pi
a_{0}}}\xi_{sf}\eta+i\frac{\lambda_{\perp}}{\sqrt{8\pi
a_{0}}}\zeta_{sf}\eta_{y}$
$\displaystyle\quad+i\frac{\delta\lambda_{y}}{4}\xi_{s}\zeta_{s}\eta_{x}\eta_{y}\eta_{z}\eta+\frac{\lambda^{Y}_{y}}{4}\xi_{sf}\zeta_{sf}\eta_{y}\eta+i\frac{E_{z}}{3}\eta_{y}\eta_{x}.$
(34)

Thus, the spinful topological Kondo effect for a single wire maps to a special
interacting resonant level model of Majorana fermions, see Fig. 4 for
illustration.

###  IV.2 Toulouse point and strong coupling fixed point

In this section, we investigate the Hamiltonian at the Toulouse point and the
stability of the emergent strong coupling fixed point by means of the
symmetry-breaking Zeeman term as well as small perturbations away from the
Toulouse point. The Toulouse point [78] is defined as the point in parameter
space where interactions in the resonant level Hamiltonian (34) are absent,
i.e., $\delta\lambda_{y}=\lambda^{Y}_{y}=0$. As was already discussed in the
previous section, the poor man’s scaling reveals that the different coupling
constants flow towards strong coupling and isotropy. This is also true if we
choose the Toulouse point as the starting point for the RG flow, see Fig. 3.
Thus, we expect that the physics that is present in the Hamiltonian at the
exact solvable Toulouse point extends towards the strong coupling fixed point.

Figure 4: Schematic representation of Hamiltonian (34). The red dots denote
the impurity Majorana fermions while the black symbols represent the lead
Majorana fermions. The hybridization terms are represented by colored lines
(solid magenta for $\lambda_{\perp}$, dashed blue for $E_{z}$). The
interaction terms are given by transparent rectangles (red for
$\lambda^{Y}_{y}$, green for $\delta\lambda_{y}$).

Fig. 4 shows a schematic of the Hamiltonian (34). At the Toulouse point and in
the case of $E_{z}=0$, only two impurity Majorana fermions hybridize with the
lead electrons, namely $\eta_{y}$ and $\eta$. Physically, that means that the
two sectors spanned by the MZMs are degenerate. Thus, there are two dangling
Majorana fermions. This translates into a twofold ground state degeneracy.
However, the moment one has a finite $E_{z}$, regardless of how small,
$\eta_{x}$ hybridizes with $\eta_{y}$ and there is only one Majorana fermion
dangling anymore. Thus, the twofold ground state degeneracy reduces to a non-
integer degeneracy of $\sqrt{2}$.

###  IV.3 Renormalization group flow

In the following, it is useful to invoke the impurity entropy defined by

$S_{\text{imp}}=S_{\text{tot}}-S_{\text{bulk}}\equiv\ln(g),$ (35)

where the quantity $g$ can be interpreted as a generalized ground state
degeneracy which can take non-integer values. We remind the reader that,
generally under RG unstable fixed points flow towards stable fixed points with
lower $g$ (“$g$-theorem” [79]).

We are now in the position to discuss the stability of the Toulouse point
solution in a renormalization group sense and to construct the schematic RG
flow diagram, Fig. 1c). For the case $E_{z}=0$ and at the Toulouse point, we
found a twofold degeneracy $g=2$ (represented by a blue star). We repeatedly
argued that anisotropy (i.e. unequal $\lambda_{y,\perp}^{1,Y}$) is irrelevant
within poor man’s scaling. The effective interacting resonant level model, Eq.
(34) corroborates this statement from the strong-coupling perspective, since
$\lambda_{y}^{Y}$, $\delta\lambda_{y}$ terms are RG irrelevant (the
corresponding operators have scaling dimension 4 and 2 respectively).

While the Toulouse point is stable towards restoring isotropy, it is unstable
towards the inclusion of $E_{z}$, which couples to an operator of scaling
dimension $1/2$ ($\eta_{y}$ acquires the scaling dimension of lead electrons
through $\lambda_{\perp}$ hybridization) and generates a state of
$g=\sqrt{2}$. We have thus demonstrated that the Kondo fixed point at infinite
$E_{s}/E_{z}$, blue star in Fig. 1c), is unstable and, given that our
Bosonization solution is valid for any $E_{z}$, flows towards a fixed point
with $g=\sqrt{2}$. As we had previously argued, this fixed point may be
interpreted as an O(4) topological Kondo effect (or equivalently a two-channel
Kondo effect).

##  V Observables

Within this section, we present a comprehensive overview of specific
observables derived from the application of the Hamiltonian (34) at the
Toulouse point, with a fixed value of $E_{z}=0$. Details are relegated to App.
E.

###  V.1 Thermodynamics

We first focus on the correction to the free energy denoted by $\delta F$. To
determine the scaling dimension of the interaction terms, we analyzed the
operators $O_{s}$ and $O_{sf}$ defined as follows:

$\displaystyle O_{s}$
$\displaystyle=i\frac{\delta\lambda_{y}}{4}\xi_{s}\zeta_{s}\eta_{x}\eta_{y}\eta_{z}\eta,$
(36) $\displaystyle O_{sf}$
$\displaystyle=\frac{\lambda^{Y}_{y}}{4}\xi_{sf}\zeta_{sf}\eta_{y}\eta.$ (37)

By calculating the expectation value

$\braket{O_{r}(\tau)O_{r}(0)}\sim\left(\frac{1}{\tau}\right)^{2\Delta_{x}},$
(38)

where $\Delta_{r}$ represents the scaling dimension, we found that the scaling
dimensions of $O_{s}$ and $O_{sf}$ are $\Delta_{s}=2$ and $\Delta_{sf}=4$,
respectively. Following [75], we inferred that the correction to the free
energy scales as:

$\delta F\sim\mathscr{O}(T^{2}),$ (39)

which relates to the thermodynamic entropy and specific heat, respectively:

$\displaystyle S=-\frac{\partial}{\partial T}F\sim\kappa
T,\quad\text{and}\quad C_{v}=T\frac{\partial}{\partial T}S\sim\kappa T,$ (40)

with $\kappa$ being some constant. This behavior resembles Fermi-liquid
characteristics.

Furthermore, we evaluated the susceptibilities at zero temperature for the
impurity, of which we have six different kinds, three in the orbital space and
three in the angular momentum space. The susceptibilities are as follows:

$\displaystyle\chi_{X}$
$\displaystyle\sim\ln\left(\nicefrac{{\Gamma}}{{T}}\right),\quad\quad\chi_{J_{x}}\sim\text{const.},$
(41) $\displaystyle\chi_{Y}$
$\displaystyle\sim\nicefrac{{1}}{{T}},\quad\quad\quad\quad\chi_{J_{y}}\sim\text{const.},$
(42) $\displaystyle\chi_{Z}$
$\displaystyle\sim\ln\left(\nicefrac{{\Gamma}}{{T}}\right),\quad\quad\chi_{J_{z}}\sim\text{const.},$
(43)

where $\Gamma=\frac{\lambda_{\perp}^{2}\nu}{2a_{0}}$. In the angular momentum
sector, we observed Pauli susceptibility with constant values akin to those in
a Fermi liquid. However, in the orbital space, the susceptibilities diverge as
$T\rightarrow 0$. Specifically, $\chi_{X,Z}$ diverges logarithmically, while
$\chi_{Y}$ exhibits an algebraic divergence. This anisotropic behavior is
possibly due to the explicit breaking of the SU(2) symmetry in the orbital
space by the Kondo Hamiltonian (20). These divergences signal non-Fermi liquid
behavior akin to the physics of the two-channel Kondo effect.

###  V.2 Transport Properties

Next, we consider various transport coefficients of our mesoscopic setup,
notably the DC conductance denoted as $G_{ij}^{c}$ and the spin current
conductance represented by $G_{ij}^{s}$. We remind the reader that the indices
$i,j\in\\{L,R\\}$ correspond to the left and right leads. At the smallest
temperatures, the DC conductance can be expressed as:

$G^{c}_{ij}=(2\delta_{ij}-1)\underbrace{(2+\mathscr{O}(T^{2}))}_{G(T)}G_{0}^{c},$
(44)

where $G_{0}^{c}=e^{2}/h$ is the perfect conductance. This equation reveals
that both channels within the impurity contribute to the charge transfer. In
parallel, when examining the spin conductance, we arrive at a comparable
outcome:

$G_{ij}^{s}=(2\delta_{ij}-1)\frac{1}{h}G(T).$ (45)

In the small coupling regime where the impurity is not screened (i.e. above
the Kondo temperature $T_{K}$) we find for both conductances a dependence on
the square of the Kondo coupling $\lambda$, which scales as

$\lambda\sim\frac{1}{\ln(\nicefrac{{T}}{{T_{K}}})}.$ (46)

The temperature dependence $G(T)$ of the transport coefficients is illustrated
in Fig. 1d).

##  VI Outlook and Conclusions

In summary, we have studied a floating mesoscopic topological superconductor
of symmetry class DIII, which realizes the spinful Majorana Cooper pair box.
In contrast to the more prominent spinless Majorana Cooper pair boxes, the
present system is subject to strong quantum fluctuations of the non-Abelian
order parameter describing the Cooper pair orientation $\hat{\bm{d}}$ in spin
space. We carefully characterized the spectrum of such a box. After coupling
the device to external leads, we uncovered a spinful topological Kondo problem
in the Coulomb blockade regime. This problem has an SU(2) $\oplus$ SU(2)
symmetry in the simplest situation of just two external leads.

We study the spinful topological Kondo problem in the isotropic case, but also
in the presence of anisotropies and in the presence of a perturbation
polarizing the $\hat{\bm{d}}$-vector. At weak coupling, we find that the
unperturbed anisotropic model flows to isotropy. We argue that this justifies
solving the problem at an anisotropic Toulouse point. We use the Abelian
Bosonization technique to solve the problem and demonstrate that the
unperturbed spinful topological Kondo problem realizes a non-Fermi liquid
fixed point and determines its thermodynamic and transport observables. We
also determine that this fixed point is unstable to the
$\hat{\bm{d}}$-polarizing perturbation, i.e. it relies on symmetry protection.

Beyond its apparent relevance in the context of realizing strongly correlated
phases in mesoscopic topological devices, we hope that this work with further
help understanding the non-trivial interplay of topology and strong
correlations in triplet superconductors. Future directions of research could
involve the spinful topological Kondo effect with more leads, multichannel
versions thereof [24], and arrays of spinful Majorana Cooper pair boxes. In
the long run, the latter could be valuable emulators of the phases of matter
in quantum materials with triplet pairing tendency.

###### Acknowledgements.

It is a pleasure to thank Erez Berg, Matan Lotem, Pavel Ostrovsky, Aline
Ramires, Pietro M. Bonetti, Nikolaos Parthenios, Raffaele Mazzilli, Lukas
Debbeler, Kirill Alpin und Robin Scholle for useful discussions on the
problem. JIV thanks the Max Planck Institute for Solid State Research for
hospitality. JIV was supported by the U.S. Department of Energy, Office of
Science, National Quantum Information Science Research Centers, Quantum
Science Center. EJK acknowledges hospitality by the Kavli Institute for
Theoretical Physics, where part of this work was completed. This research was
supported in part by the National Science Foundation under Grants No. NSF
PHY-1748958 and PHY-2309135.

##  Appendix A Spectrum of the spinful MCPB

In this appendix, we present the calculation of the spectrum of the spinful
MCPB. The starting point is Hamiltonian (4), where the model has already been
transformed into the co-moving frame of the condensate

$\displaystyle\hat{H}_{\text{dot}}=4E_{c}(\hat{N}_{c}-N^{c}_{g})^{2}+E_{s}\hat{\bm{J}}^{2}$
$\displaystyle+\int\text{d}x\left\\{\psi(x)^{\dagger}\epsilon(\hat{p})\psi(x)+u[\psi(x)^{\text{T}}\partial_{x}\psi(x)+\text{H.c.}]\right\\}.$
(47)

###  A.1 Boundary Condition and Quantization of Condensate Operators

First, we demonstrate how the quantization condition of the operator
$\hat{N}_{c}$ and $\hat{\bm{J}}$ change with respect to their transformed
counterparts. Let be the Cooper-pair number operator before the transformation
that is

$\mathscr{U}^{\dagger}\left(2\hat{N}_{c}+\hat{n}_{f}\right)\mathscr{U}=2\hat{N}_{c}.$
(48)

Before the transformation, the operator $\hat{N}_{c}$ is integer quantized and
has the eigenstates $\ket{\Phi}_{N}$ with eigenvalue $N$, which denotes the
number of Cooper-pairs. We can project the state into the basis of the
superconducting phase, that is $\braket{\varphi}{\Phi}_{N}=\Phi_{N}(\varphi)$.
Note that, due to the missing hat, $\varphi$ is a quantum number and not an
operator. Since the phase is only defined up to $2\pi$, the wavefunction has
to fulfill the boundary condition

$\Phi_{N}(\varphi)=\Phi_{N}(\varphi+2\pi).$ (49)

The transformed wavefunction reads

$\mathscr{U}^{\dagger}\Phi_{N}(\varphi)=\text{e}^{i\frac{\varphi}{2}\hat{n}_{f}}\Phi_{N}(\varphi).$
(50)

There, we can see that for an even number of electrons, the wave function is
$2\pi$ periodic, and for an odd number of electrons $4\pi$ periodic. Thus, the
operator $\hat{N}_{c}$ after the transformation is half-integer quantized
since we changed the boundary condition by $\mathscr{U}$.

A similar effect can also be observed in $\hat{\bm{J}}$. The operator
$\hat{L}_{y}=-i\partial_{\phi}$ has, since it is an angular momentum operator,
integer eigenvalues $l_{y}$ with eigenfunctions $\Psi_{l_{y}}(\phi)$. These
eigenfunctions are also $2\pi$ periodic. After the transformation, the new
eigenfunctions read

$\mathscr{U}^{\dagger}\Psi_{l_{y}}(\phi)=\text{e}^{i\phi\hat{S}_{y}}\Psi_{l_{y}}(\phi),$
(51)

where these are eigenfunctions of $\hat{J}_{y}=-i\partial_{\phi}$ and
$\hat{S}_{y}$ is the spin operator of the fermion excitation in the
y-direction. Similar to the superconducting phase and the Cooper-pair number,
in the case of an odd fermion number (i.e. $\hat{S}_{y}$ has half-integer
eigenvalues), the eigenfunction of $\hat{J}_{y}$ are $4\pi$ periodic and have
half-integer eigenvalues.

###  A.2 Spectrum and Eigenfunctions

As already discussed in the main text, the fermion sector does not contribute
to the energy and is spanned by a four-dimensional Hilbert space in the case
of a single wire $w=1$.

In the condensate sector, we note that the two operators $\hat{N}_{c}$ and
$\hat{\bm{J}}$ commute with the Hamiltonian, and hence, the spectrum can be
constructed using their eigenvalues and eigenfunctions. The charge
eigenfunctions are straightforwardly constructed by the eigenvalue equation:

$\hat{N}_{c}\ket{\Phi}_{N}=N\ket{\Phi}N\iff-i\partial_{\varphi}\Phi_{N}(\varphi)=N\Phi_{N}(\varphi),$
(52)

where the solutions are $\Phi_{N}(\varphi)=\text{e}^{iN\varphi}$, and $N$ is a
half-integer.

For the angular momentum sector, the energy eigenstates correspond to the
eigenstates of the operator $\hat{\bm{J}}^{2}$. Since $\hat{\bm{J}}$ is an
angular momentum/spin operator, we can construct the spectrum and the
eigenfunctions using standard methods. That is, we choose the operator
$\hat{J}_{y}$ to commute with $\hat{\bm{J}}^{2}$, and the eigenstates are
given by the following two eigenvalue equations:

$\displaystyle\hat{J}_{y}\ket{\Xi}$ $\displaystyle=j_{y}\ket{\Xi},$ (53)
$\displaystyle\hat{\bm{J}}^{2}\ket{\Xi}$ $\displaystyle=J(J+1)\ket{\Xi}.$ (54)

To construct the corresponding differential equation, we note that
$\hat{\bm{L}}$ is the angular momentum operator that is canonically conjugate
to $\hat{\bm{d}}$. Thus, in terms of coordinates, the $\hat{L}$ operator can
be written as:

$\hat{\bm{L}}=-i\hat{\bm{d}}\times\nabla=-i\hat{\bm{d}}\times\left(\bm{e}_{\theta}\partial_{\theta}+\frac{\bm{e}_{\phi}}{\sin(\theta)}\partial_{\phi}\right),$
(55)

where $\bm{e}_{i}$ are unit vectors. Note that in coordinate space,
$\hat{\bm{d}}\equiv\bm{e}_{r}$. From this, we can calculate:

$\mathscr{U}^{\dagger}\hat{\bm{L}}\mathscr{U}=\hat{\bm{L}}-\bm{e}_{\phi}\hat{S}_{x}+\cot(\phi)\bm{e}_{\theta}\hat{S}_{y}-\bm{e}_{\theta}\hat{S}_{z}.$
(56)

On the other hand, the fermion spin transforms as:

$\mathscr{U}^{\dagger}\hat{\bm{S}}\mathscr{U}=\bm{e}_{r}\hat{S}_{y}+\bm{e}_{\phi}\hat{S}_{x}+\bm{e}_{\theta}\hat{S}_{z}.$
(57)

Now, we can conclude:

$\mathscr{U}^{\dagger}\left(\hat{\bm{L}}+\hat{\bm{S}}\right)\mathscr{U}=\hat{\bm{L}}+\frac{\hat{S}_{y}}{\sin(\hat{\theta})}\begin{pmatrix}\sin(\hat{\phi})\\\
0\\\ \cos(\hat{\phi})\end{pmatrix}\equiv\hat{\bm{J}}.$ (58)

In order to solve the two eigenvalue equations (53) and (54), we use the
separation of variables as an Ansatz, that is,
$\braket{\theta,\phi}{\Xi}=\Xi(\theta,\phi)=\Theta(\theta)\Psi(\phi)$. This
Ansatz leads to the equation:

$-i\partial_{\phi}\Psi(\phi)=j_{y}\Psi(\phi)$ (59)

with the solution $\Psi(\phi)=\text{e}^{ij_{y}\phi}$. Note that $\hat{S}_{y}$
also commutes with the Hamiltonian. Therefore, we can replace the operator
$\hat{S}_{y}$ with its eigenvalue $s_{y}$. The second eigenvalue equation
yields the differential equation:

$\begin{split}\,&\left[(1-x^{2})\partial^{2}_{x}-2x\partial_{x}-\frac{(j_{y}-s_{y}x)^{2}}{1-x^{2}}\right.\\\
&+\left(J(J+1)-s_{y}^{2}\right)\biggr{]}P(x)=0,\end{split}$ (60)

where $x=\cos(\theta)$ and $P(x)=P(\cos(\theta))=\Theta(\theta)$. This
corresponds to the eigenvalue equation of “monopole harmonics”, i.e. the
generalization of spherical harmonics to $s_{y}\neq 0$.[80]

Assuming $N_{g}^{c}=\nicefrac{{1}}{{2}}$, the ground state manifold is given
by the second row in table 1. Inserting the corresponding quantum numbers in
equation (60), we find the normalizable solutions:

$\Theta(\theta)=\sqrt{1+4j_{y}s_{y}\cos(\theta)}.$ (61)

Thus, the full wave function of the condensate (charge and spin part) in the
ground state manifold reads:

$\Phi_{\frac{1}{2}}(\varphi)\Xi_{s_{y},j_{y}}(\phi,\theta)=\text{e}^{i\varphi/2}\text{e}^{ij_{y}\phi}\sqrt{1+4j_{y}s_{y}\cos(\theta)}.$
(62)

In the first excited states, the condensate wavefunction takes a trivial
shape:

$\Phi_{N={0,1}}(\varphi)\Xi_{0,0}(\phi,\theta)=\exp(iN\varphi)\cdot 1.$ (63)

####  A.2.1 Spectrum in the limit of a strong perturbation

We now consider a regime in which the perturbation $E_{z}$ is large and find
the spectrum in the limit $E_{z}\gg E_{s}$, and hence, $\theta\ll 1$. To do
so, we expand the Hamiltonian:

$\begin{split}\hat{H}&=E_{s}\hat{\bm{J}}^{2}-E_{z}\hat{d}_{y}\\\
&=E_{s}\left(\frac{-\partial_{\theta}(\sin(\theta)\partial_{\theta}\cdot)}{\sin(\theta)}+\frac{j_{y}^{2}-2s_{y}j_{y}\cos(\theta)+s_{y}^{2}}{\sin^{2}(\theta)}\right)\\\
&-E_{z}\cos(\theta)\\\ &\approx
E_{s}\left(-\frac{\partial_{\theta}(\theta\partial_{\theta}\cdot)}{\theta}+\frac{n^{2}}{\theta^{2}}\right)+\frac{1}{2}E_{z}\theta^{2},\end{split}$
(64)

up to the first non-trivial order in $\theta$ and ignoring the constant shift
in energy. Note that $n=j_{y}-s_{y}$. After applying the approximation
$\theta\ll 1$, the differential equation for the $\Theta(\theta)$ part of the
condensate wavefunction changed, which, therefore, needs to be adopted. The
Hamiltonian (64) resembles the radial part of a harmonic oscillator. The
spectrum for the 2-dimensional harmonic oscillator in polar coordinates is
well known and reads

$E_{n}=\omega(|n|+1+2n_{\theta}),$ (65)

with energy eigenstates:

$\Theta_{n,n_{\theta}}(\theta)=C_{n,n_{\theta}}\theta^{|n|}\exp\left(-\frac{m\omega}{2}\theta^{2}\right)L_{n_{\theta}}^{n}(m\omega\theta^{2}).$
(66)

In analogy to the harmonic oscillator we identify $\omega=\sqrt{2E_{z}E_{s}}$
and $m=\frac{1}{2E_{s}}$. The quantum number $n_{\theta}$ describes the
quantization of radius in the $x$-$z$-plane. However, in the low-energy
sector, we can choose $n_{\theta}=0$. $C_{n,n_{\theta}}$ is a normalization
constant, and $L_{n_{\theta}}^{n}$ are the generalized Laguerre polynomials.
The total condensate wave function is most conveniently expressed in terms of
$n$ and $s_{y}$ as

$\Xi_{n,s_{y}}(\theta,\phi)=\Theta_{n,0}(\theta)\text{e}^{i(n+s_{y})\phi}.$
(67)

The ground state has $n=0$, from which it follows that
$s_{y}=j_{y}=\pm\nicefrac{{1}}{{2}}$. Hence, the ground state manifold is
twofold degenerate. The first excited states have $n=\pm 1$ and are,
therefore, 4 states. Furthermore, it follows immediately that:

$\bra{\Xi_{0,s_{y}}}\delta\hat{d}_{i}\ket{\Xi_{0,s_{y}}}=0.$ (68)

##  Appendix B Second order perturbation theory and emergent Kondo effect

In this section, we describe how we obtained an effective Kondo Hamiltonian in
the low-energy sector. First, we demonstrate the application of the
Schrieffer-Wolf transformation to the warm-up problem in the limit where
$E_{c}\gg E_{s}$, and we also assume $E_{z}\gg E_{s}$. This will be followed
by a discussion of how to obtain the effective Kondo Hamiltonian in the limit
where $E_{c}\sim E_{s}$ and $E_{z}\ll E_{c}$.

To derive the effective low-energy Hamiltonian, we apply a Schrieffer-Wolf
transformation, treating the hopping parameter $t$ of lead electrons onto the
quantum dot as a perturbation, which implies that $E_{c}\gg t$.

###  B.1 Limit $E_{c}\gg E_{s}$

In this limiting case, the fluctuations in the superconducting phase
($\hat{\varphi}$) are much faster than the spin fluctuations ($\hat{U}$).
Thus, we absorb the $\hat{U}$ matrix into the lead electrons:

$\Psi_{i}=\hat{U}\psi_{i},$ (69)

where $\psi_{i}=(\psi_{i\uparrow},\psi_{i\downarrow})$ are spinors of the lead
electrons. The hopping Hamiltonian becomes:

$-t\hat{H}_{t}=-t\sum_{i}\left(\Psi^{\dagger}_{i}\text{e}^{-i\hat{\varphi}/2}\gamma_{i}+\text{H.c.}\right).$
(70)

The low-energy sector is mostly determined by the charge on the island. If we
choose $N_{g}^{c}=\frac{1}{2}$, the ground state manifold in the charge sector
can be classified by the total charge
$\ket{GS}=\ket{N_{c}=\nicefrac{{1}}{{2}}}$. The states with the next higher
energy are the states where a fermion takes some charge from the condensate
and tunnels into an electron state in the wire or an electron that tunnels
into the MZMs and donates its charge to the condensate. These states can be
written as $\ket{0}$ and $\ket{1}$, respectively.

Applying the Schrieffer-Wolf transformation, Hamiltonian
$\eqref{eq:fullBareHamiltonian}$ becomes:

$\displaystyle\hat{H}=\hat{H}_{\text{dot}}+\hat{H}_{0}+H_{z}\underbrace{-\frac{t^{2}}{E_{c}}\bra{\nicefrac{{1}}{{2}}}\sum_{n=0,1}\hat{H}_{t}\ket{n}\bra{n}\hat{H}_{t}\ket{\nicefrac{{1}}{{2}}}}_{\Delta\hat{H}}.$
(71)

The last term will become the effective Kondo Hamiltonian. Hence, we will
focus on this term and drop the other parts of the Hamiltonian. If we expand
the Hamiltonian, it becomes:

$\begin{split}\,&\Delta\hat{H}=-\frac{t^{2}}{E_{c}}\sum_{i,j}\bra{\frac{1}{2}}\left[\left(\Psi^{\dagger}_{i}\text{e}^{-i\hat{\varphi}}\gamma_{i}+\text{H.c.}\right)\ket{0}\bra{0}\left(\Psi^{\dagger}_{j}\text{e}^{-i\hat{\varphi}/2}\gamma_{j}+\text{H.c.}\right)\right.\\\
&+\left.\left(\Psi^{\dagger}_{i}\text{e}^{-i\hat{\varphi}}\gamma_{i}+\text{H.c.}\right)\ket{1}\bra{1}\left(\Psi^{\dagger}_{j}\text{e}^{-i\hat{\varphi}/2}\gamma_{j}+\text{H.c.}\right)\right]\ket{\frac{1}{2}}.\end{split}$
(72)

The operator $\text{e}^{\pm i\hat{\varphi}/2}$ are translation operators in
charge space that destroy (create) charge in the condensate. The matrix
elements of the charge translation operators are straightforwardly calculated
and read

$\begin{cases}\bra{1/2}e^{-i\hat{\varphi}/2}\ket{0}=0,\quad\quad\quad\bra{1/2}e^{i\hat{\varphi}/2}\ket{0}=1\\\
\bra{0}e^{-i\hat{\varphi}/2}\ket{1/2}=1,\quad\quad\quad\bra{0}e^{i\hat{\varphi}/2}\ket{1/2}=0\\\
\bra{1/2}e^{-i\hat{\varphi}/2}\ket{1}=1,\quad\quad\quad\bra{1/2}e^{i\hat{\varphi}/2}\ket{1}=0\\\
\bra{1}e^{-i\hat{\varphi}/2}\ket{1/2}=0,\quad\quad\quad\bra{1}e^{i\hat{\varphi}/2}\ket{1/2}=1.\end{cases}$
(73)

Applying these rules, the effective Hamiltonian collapses to:

$\displaystyle\Delta\hat{H}=$
$\displaystyle-\frac{t^{2}}{E_{c}}\sum_{i,j}\left(\gamma^{\text{T}}_{i}\Psi_{i}^{\phantom{\dagger}}\Psi_{j}^{\dagger}\gamma_{j}^{\phantom{T}}+\Psi^{\dagger}_{i}\gamma_{i}^{\phantom{\dagger}}\gamma_{j}^{\text{T}}\Psi^{\phantom{\dagger}}_{j}\right).$
(74)

This expression is the origin of Eq. (11) of the main text, note that
$i=1,\dots,w$.

In the case $w=1$ we use the notation

$\displaystyle\Delta\hat{H}=$
$\displaystyle-\frac{2t}{E_{c}}\Psi^{\dagger}(\gamma\cdot\gamma^{\text{T}})\Psi,$
(75)

where we introduced the four-component spinor
$\Psi=(\Psi_{L\uparrow},\Psi_{L\downarrow},\Psi_{R\uparrow},\Psi_{R\downarrow})^{\text{T}}$
and the symbol $(\gamma\cdot\gamma^{\text{T}})$ that represents the matrix:

$\begin{split}(\gamma\cdot\gamma^{\text{T}})&=\begin{pmatrix}1&\gamma_{L\uparrow}\gamma_{L\downarrow}&\gamma_{L\uparrow}\gamma_{R\uparrow}&\gamma_{L\uparrow}\gamma_{R\downarrow}\\\
\gamma_{L\downarrow}\gamma_{L\uparrow}&1&\gamma_{L\downarrow}\gamma_{R\uparrow}&\gamma_{L\downarrow}\gamma_{R\downarrow}\\\
\gamma_{R\uparrow}\gamma_{L\uparrow}&\gamma_{R\uparrow}\gamma_{L\downarrow}&1&\gamma_{R\uparrow}\gamma_{R\downarrow}\\\
\gamma_{R\downarrow}\gamma_{L\uparrow}&\gamma_{R\downarrow}\gamma_{L\downarrow}&\gamma_{R\downarrow}\gamma_{R\uparrow}&1\end{pmatrix}\\\
&=-2\begin{pmatrix}0&-i\hat{S}_{y}&i\hat{S}_{z}&i\hat{S}_{x}\\\
i\hat{S}_{y}&0&i\hat{S}_{x}&-i\hat{S}_{z}\\\
-i\hat{S}_{z}&-i\hat{S}_{x}&0&-i\hat{S}_{y}\\\
-i\hat{S}_{x}&i\hat{S}_{z}&i\hat{S}_{y}&0\\\
\end{pmatrix}+\mathds{1}_{4x4}.\end{split}$ (76)

Decomposing this matrix further into Pauli matrices yields:

$(\gamma\cdot\gamma^{\text{T}})=-2\left(\mathds{1}_{\tau}\sigma_{y}\hat{S}_{y}-\tau_{y}\sigma_{z}\hat{S}_{z}-\tau_{y}\sigma_{x}\hat{S}_{x}\right)+\mathds{1}_{\tau}\mathds{1}_{\sigma},$
(77)

where $\sigma_{i}$ and $\tau_{i}$ act in the spin and left/right space,
respectively. In the next step, we change into the eigenbasis of $-\tau_{y}$
and obtain:

$\Delta\hat{H}=\frac{4t^{2}}{E_{c}}\begin{pmatrix}\Psi_{+}^{\dagger}&\Psi_{-}^{\dagger}\end{pmatrix}\begin{pmatrix}\bm{\sigma}&0\\\
0&\bm{\sigma}\end{pmatrix}\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix}\cdot\bm{\hat{S}}-\frac{2t^{2}}{E_{c}}\Psi^{\dagger}\Psi,$
(78)

where $\Psi_{+}=\frac{1}{\sqrt{2}}(\Psi_{L}+i\Psi_{R})$ and
$\Psi_{-}=\frac{i\sigma_{y}}{\sqrt{2}}(\Psi_{L}-i\Psi_{R})$. The vector
$\bm{\hat{S}}=(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ contains the bilinears of
the MZMs. The second term is nothing but a potential term that we will omit
from now on. Separating the $U$ matrix from the spinors, the Kondo Hamiltonian
evaluates to:

$\hat{H}_{K}=\lambda\left(\psi_{+}^{\dagger}\hat{U}^{\dagger}\bm{\sigma}\hat{U}\psi_{+}^{\phantom{\dagger}}+\psi^{\dagger}_{-}\sigma_{y}\hat{U}^{\dagger}\sigma_{y}\bm{\sigma}\sigma_{y}\hat{U}\sigma_{y}\psi_{-}^{\phantom{\dagger}}\right)\cdot\bm{\hat{S}},$
(79)

where $\lambda=\frac{4t^{2}}{E_{c}}$,
$\psi_{+}=\frac{1}{\sqrt{2}}\left(\psi_{R}+i\psi_{L}\right)$ and
$\psi_{-}=\frac{i\sigma_{y}}{\sqrt{2}}\left(\psi_{R}-i\psi_{L}\right)$.

###  B.2 Limit $E_{z}\gg E_{s}$

In this section, we build upon the derivations from the previous section,
assuming that $E_{z}\gg E_{s}$, and that $\hat{\bm{d}}$ only experiences weak
fluctuations around the y-axis. Our strategy is to expand the unitary matrix
$\hat{U}$ in terms of $\hat{\theta}$. We introduce a new representation of
$\hat{U}$ using Pauli matrices to achieve this. We utilize the gauge freedom
within $\hat{U}$ and redefine it as follows:

$\hat{U}=\text{e}^{-i\hat{\phi}/2\sigma_{y}}\text{e}^{i\hat{\theta}/2\sigma_{x}}\text{e}^{i\hat{\phi}/2\sigma_{y}}=\text{e}^{i\hat{W}/2},$
(80)

where

$\hat{W}=\underbrace{\hat{\theta}\cos(\hat{\phi})}_{\partial_{\phi}\delta\hat{d}_{x}}\sigma_{x}-\underbrace{\hat{\theta}\sin(\hat{\phi})}_{\partial_{\phi}\delta\hat{d}_{x}}\sigma_{z}$
(81)

We can now expand $\hat{U}$ in powers of $\hat{\theta}$ since the fluctuations
in this angle are small. Therefore, up to the first order, we obtain

$\begin{split}\hat{U}^{\dagger}\bm{\sigma}\hat{U}&\approx\left(\mathds{1}-\frac{i}{2}\hat{W}\right)\bm{\sigma}\left(\mathds{1}+\frac{i}{2}\hat{W}\right)\\\
&=\bm{\sigma}+\partial_{\phi}\delta\hat{\bm{d}}\times\bm{\sigma}+\mathscr{O}(\hat{\theta}^{2}),\end{split}$
(82)

where $\delta\hat{\bm{d}}=(\delta\hat{d}_{x},0,\delta\hat{d}_{z})^{\text{T}}$.
Similarly, we can calculate

$\hat{U}^{\dagger}\sigma_{y}\bm{\sigma}\sigma_{y}\hat{U}\approx\sigma_{y}(\bm{\sigma}-\partial_{\phi}\delta\hat{\bm{d}}\times\bm{\sigma})\sigma_{y}.$
(83)

The Hamiltonian can be expressed compactly as

$\hat{H}_{K}=\lambda\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi_{a}\cdot\left(1-a\,\partial_{\phi}\delta\hat{\bm{d}}\times\right)\hat{\bm{S}}.$
(84)

This is the origin of Eq. (19) of the main text.

###  B.3 Limit $E_{z}\ll E_{s}$

In this limit, we also consider transition in higher excited states induced by
angular momentum and orbital fluctuations. We chose the third row in Table 1
to be the ground state manifold (again setting $N_{g}^{c}=\frac{1}{2}$). The
energy ground states can be written as

$\displaystyle\begin{split}\ket{GS}&=\underbrace{\ket{N_{c}=\frac{1}{2},j_{y},J=3/4}_{s_{y}}}_{\text{condensate
sector}}\otimes\underbrace{\ket{s_{y}}}_{\text{fermion sector}}\\\
&\equiv\ket{\frac{1}{2},j_{y},s_{y}},\end{split}$ (85)

where $j_{y}=\pm\nicefrac{{1}}{{2}}$ and $s_{y}=\pm\nicefrac{{1}}{{2}}$ are
the eigenvalues of the two operator $\hat{J}_{y}$ and $\hat{S}_{y}$,
respectively. As it will be shown, the ground state wavefunction of the
condensate sector also depends on $s_{y}$ due to the implicit dependence of
$\hat{\bm{J}}$ on $\hat{S}_{y}$. The two excited states can be written in a
similar fashion as

$\ket{l,k=\\{0,2\\}}\equiv\ket{N_{c}=l,j_{y}=0,J=0}\otimes\\{\ket{0},\ket{\Gamma^{\dagger}_{\uparrow}}\Gamma^{\dagger}_{\downarrow}\ket{0}\\}.$
(86)

The effective Hamiltonian after the Schrieffer-Wolf transformation reads

$\Delta\hat{H}=-t^{2}\sum_{\xi,\eta}\ket{GS}_{\xi}\bra{GS}_{\xi}\sum_{l,k}\frac{\hat{H}_{t}\ket{l,k}\bra{l,k}\hat{H}_{t}}{\Delta
E_{l}}\ket{GS}_{\eta}\bra{GS}_{\eta},$ (87)

where $\Delta E_{l}=E_{l}-E_{0}$. $E_{l}$ is the energy of the first excited
state, and $E_{0}$ is the ground state energy. The projector onto a ground
state can also compactly be written as

$\begin{split}\ket{GS}_{s_{y}}\bra{GS}_{s_{y}}&=\ket{\frac{1}{2},j_{y},s_{y}}\bra{\frac{1}{2},j_{y},s_{y}}\\\
&=\ket{\frac{1}{2},j_{y}}_{s_{y}}\bra{\frac{1}{2},j_{y}}_{s_{y}}\otimes\underbrace{\ket{s_{y}}\bra{s_{y}}}_{\Pi_{s_{y}}}.\end{split}$
(88)

Now, we can re-write the effective Hamiltonian as

$\Delta\hat{H}=-t^{2}\sum_{m,m^{\prime}}\ket{\frac{1}{2},m}\Delta
H^{m,m^{\prime}}\bra{\frac{1}{2},m^{\prime}},$ (89)

where

$\displaystyle\begin{split}\,&\Delta
H^{m,m^{\prime}}=-t^{2}\sum_{l,k,\xi,\eta}\Pi_{\xi}\bra{\frac{1}{2},m}_{\xi}\frac{\hat{H}_{t}\ket{l,k}\bra{l,k}\hat{H}_{t}}{\Delta
E_{l}}\ket{\frac{1}{2},m^{\prime}}_{\eta}\Pi_{\eta}\\\
&=-t^{2}\sum_{l,\xi,\eta}\Pi_{\xi}\bra{\frac{1}{2},m}_{\xi}\frac{\hat{H}_{t}\ket{l}\left(\sum_{k}\ket{k}\bra{k}\right)\bra{l}\hat{H}_{t}}{\Delta
E_{l}}\ket{\frac{1}{2},m^{\prime}}_{\eta}\Pi_{\eta},\end{split}$ (90)

where $\sum_{k}\ket{k}\bra{k}=\mathds{1}$ in the subspace of even electron
parity. The sum over the charge sector is equivalent to what was done in the
last section. The effective Hamiltonian becomes

$\displaystyle\begin{split}\Delta H^{m,m^{\prime}}&=-\frac{t^{2}}{\Delta
E}\Pi_{\xi}\sum_{i,j}\left\\{\bra{m}_{\xi}\gamma_{j}^{\text{T}}\hat{U}\psi_{j}\ket{0}\bra{0}\psi_{i}^{\dagger}\hat{U}^{\dagger}\gamma_{i}\ket{m^{\prime}}_{\eta}\right.\\\
&\left.+\bra{m}_{\xi}\psi^{\dagger}_{i}\hat{U}^{\dagger}\gamma_{i}\ket{0}\bra{0}\gamma_{j}^{T}\hat{U}\psi_{j}\ket{m^{\prime}}_{\eta}\right\\}\Pi_{\eta},\end{split}$
(91)

where $\Delta E=E_{c}-E_{s}\frac{3}{4}$. In order to evaluate the matrix
elements that appear in equation (91) we introduce a partition of unity of the
form
$\mathds{1}=\int_{0}^{\theta}\text{d}\theta\int_{0}^{2\pi}\text{d}\phi\,\frac{\sin(\theta)}{4\pi}\,\ket{\phi,\theta}\bra{\phi,\theta}$.
Also we decompose the lead electrons in eigenvectors of the $\sigma_{y}$
matrix, that is $\psi_{i}=\sum_{n=\pm}\psi_{n}\bm{e}_{n}$, where
$\bm{e}_{n}=\frac{1}{\sqrt{2}}(1,in)^{\text{T}}$ and $n$ being the eigenvalue
$n=\pm 1$. Applying these two steps the Hamiltonian (91) becomes

$\displaystyle\begin{split}\,\Delta H^{m,m^{\prime}}&=-\frac{t^{2}}{\Delta
E}\Pi_{\xi}\sum_{ij}\left\\{\gamma_{i}^{T}U_{\theta}^{\xi
m}\bm{e}^{\phantom{\dagger}}_{m}\psi^{\phantom{\dagger}}_{i,m}\psi^{\dagger}_{j,m}\bm{e}_{m^{\prime}}^{\dagger}\left(U^{\eta
m^{\prime}}_{\theta}\right)^{\dagger}\gamma_{im^{\prime}}\right.\\\
&+\left.\psi^{\dagger}_{i\bar{m}}\bm{e}_{\bar{m}}^{\dagger}\left(U^{\xi
m}_{\theta}\right)^{\dagger}\gamma_{i}^{\phantom{\text{T}}}\gamma_{j}^{\text{T}}U^{\eta
m^{\prime}}_{\theta}\bm{e}_{\bar{m^{\prime}}}\psi_{j,\bar{m^{\prime}}}\right\\}\Pi_{\eta},\end{split}$
(92)

where $\bar{m}=-m$ and

$U_{\theta}^{\xi
m}=\braket{m}{{}_{\xi}\hat{U}}{0}=\int\frac{\text{d}\theta\sin(\theta)}{2}\braket{m}{{}_{\xi}\theta}\text{e}^{i\theta/2\sigma_{x}}.$
(93)

The projector in the orbital/Majorana space can explicitly written in terms of
Majorana fermions as

$\Pi_{\xi}=\frac{1+2\xi\hat{S}_{y}}{2}=\frac{1}{2}\left(1+\xi\frac{i}{2}\gamma^{\text{T}}_{i}\sigma_{y}\gamma_{i}\right),$
(94)

where $i$ can either be $R$ or $L$. Due to the parity constraint, both are
equally good. As we did for the lead electrons, we expand the Majorana
fermions in terms of eigenvectors of $\sigma_{y}$. It is straightforward to
show that

$\displaystyle\bm{e}^{\dagger}_{n}\cdot\gamma_{i}^{\phantom{\dagger}}\Pi_{\xi}$
$\displaystyle=\delta_{\xi n}\gamma_{in},$ (95)
$\displaystyle\bm{e}^{\dagger}_{n}\cdot\Pi_{\xi}\gamma_{i}$
$\displaystyle=\delta_{\bar{\xi}n}\gamma_{in},$ (96)
$\displaystyle\gamma_{i}^{\text{T}}\Pi_{\xi}\cdot\bm{e}_{n}^{\phantom{\dagger}}$
$\displaystyle=\delta_{\xi\bar{n}}\gamma_{i\bar{n}},$ (97)
$\displaystyle\Pi_{\xi}\gamma_{i}^{\text{T}}\cdot\bm{e}_{n}$
$\displaystyle=\delta_{\bar{\xi}\bar{n}}\gamma_{i\bar{n}}.$ (98)

Now, we are in a position to evaluate the sums of the shape

$\displaystyle\begin{split}\,&\sum_{\xi}\Pi_{\xi}\gamma^{\text{T}}_{in}\bm{e}_{n}^{\phantom{\dagger}}\bm{e}^{\dagger}_{n^{\prime}}U^{\xi
m}_{\theta}\bm{e}_{m}\\\
&=\sum_{\xi}\int_{0}^{\pi}\frac{\sin(\theta)}{2}\braket{m}{{}_{\xi}\theta}\underbrace{\bm{e}^{\dagger}_{n}\text{e}^{i\sigma_{x}\nicefrac{{\theta}}{{2}}}\bm{e}_{m}^{\phantom{\dagger}}}_{\cos(\theta/2)\delta_{nm}-2m\sin{\theta/2}\delta_{n\bar{m}}}\gamma_{i\bar{n}}\delta_{\bar{\xi}\bar{n}}\\\
&=\int_{0}^{\pi}\frac{\sin(\theta)}{2}\Big{(}\bra{m}_{m}\ket{\theta}\cos(\theta/2)\delta_{nm}\\\
&-2m\bra{m}_{\bar{m}}\ket{\theta}\sin(\theta/2)\delta_{n\bar{m}}\Big{)}\gamma_{i\bar{n}}.\end{split}$
(99)

The two integrals are

$\int_{0}^{\pi}\text{d}\theta\,\frac{\sin(\theta)}{2}\sqrt{1+\cos{\theta}}\cos(\theta/2)=\frac{1}{\sqrt{2}}$
(100)

and

$\int_{0}^{\pi}\text{d}\theta\,\frac{\sin(\theta)}{2}\sqrt{1-\cos{\theta}}\sin(\theta/2)=\frac{1}{\sqrt{2}}.$
(101)

After calculating the other three sums of the shape of sum (99) the effective
Hamiltonian becomes

$\displaystyle\begin{split}\,&\Delta H^{m.m^{\prime}}=-\frac{t^{2}}{2\Delta
E}\sum_{i,j}\Big{(}\psi^{\phantom{\dagger}}_{im}\psi^{\dagger}_{im^{\prime}}(\gamma_{i\bar{m}}-2m\gamma_{im})\\\
&(\gamma_{jm^{\prime}}-2m^{\prime}\gamma_{j\bar{m^{\prime}}})\\\
&+\psi^{\dagger}_{i\bar{m}}\psi^{\phantom{\dagger}}_{j\bar{m^{\prime}}}(\gamma_{i\bar{m}}+2m\gamma_{im})(\gamma_{jm^{\prime}}+2m^{\prime}\gamma_{j\bar{m^{\prime}}})\Big{)}.\end{split}$
(102)

In the next step, we evaluate each matrix element in angular momentum space
separately. That is, we set up the matrix

$\Delta H=\begin{pmatrix}\Delta H^{++}&\Delta H^{+-}\\\ \Delta H^{-+}&\Delta
H^{--}\end{pmatrix}.$ (103)

The matrix elements can be written in a compact way as

$\displaystyle\begin{split}\Delta H^{++}&=-\frac{2t^{2}}{\Delta
E}+\frac{t^{2}}{\Delta
E}\psi^{\dagger}\sigma_{y}\psi+\frac{t^{2}}{\Delta}\psi^{\dagger}\sigma_{y}\tau_{y}\psi
Z,\\\ \Delta H^{--}&=-\frac{2t^{2}}{\Delta E}-\frac{t^{2}}{\Delta
E}\psi^{\dagger}\sigma_{y}\psi+\frac{t^{2}}{\Delta
E}\psi^{\dagger}\sigma_{y}\tau_{y}\psi Z,\\\ \Delta
H^{+-}&=\frac{2t^{2}}{\Delta E}\left(\psi^{\dagger}\sigma_{-}\psi
Y-i\psi^{\dagger}\tau_{y}\sigma_{-}\psi\right),\\\ \Delta
H^{-+}&=\frac{2t^{2}}{\Delta E}\left(\psi^{\dagger}\sigma_{+}\psi
Y+i\psi^{\dagger}\tau_{y}\sigma_{+}\psi\right),\end{split}$ (104)

where we introduced the spinor
$\psi=(\psi_{L\uparrow},\psi_{L\downarrow},\psi_{R\uparrow},\psi_{R\downarrow})^{\text{T}}$
and $\tau_{i}$ are Pauli matrices that act on the $R/L$-space. Furthermore, we
introduced the matrices $\sigma_{\pm}=\frac{1}{2}(\sigma_{z}\pm i\sigma_{x})$
The Kondo Hamiltonian can be found by decomposing the matrix in Pauli-
matrices, which correspond to the components of $\bm{J}$ projected down onto
the ground state manifold, that is

$\Delta\hat{H}=\sum_{i=0}^{3}\frac{1}{2}\text{Tr}\left[\Delta
HJ_{i}\right]J_{i}.$ (105)

Such a decomposition leads to (neglecting inessential constants)

$\displaystyle\begin{split}\hat{H}_{K}&=\frac{t^{2}}{\Delta
E}\left(\psi^{\dagger}\tau_{y}\sigma_{y}\psi Z+\psi^{\dagger}\sigma_{y}\psi
J_{y}\right.\\\ &+\left(\psi^{\dagger}\sigma_{x}\psi
Y+\psi^{\dagger}\tau_{y}\sigma_{z}X\right)J_{x}\\\
&+\left.\left(\psi^{\dagger}\sigma_{z}\psi
Y+\psi^{\dagger}\tau_{y}\sigma_{x}\psi X\right)J_{z}\right)\\\
&=\frac{t^{2}}{\Delta
E}\left(\psi\bm{\sigma}\psi+\psi^{\dagger}\bm{\sigma}\tau_{y}\psi
ZJ_{y}\right)\cdot\mathscr{S},\end{split}$ (106)

where

$\mathscr{S}=\begin{pmatrix}YJ_{x}\\\ J_{y}\\\ YJ_{z}\end{pmatrix}$ (107)

and $\bm{\sigma}$ is a vector with the Pauli matrices as components. Note that
we chose the y-axis as the quantization axis, which means that in angular
momentum space

$J_{y}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad
J_{z}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\quad\text{and}\quad
J_{x}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}.$ (108)

Further progress can be made by changing to a basis in which

$-\tau_{y}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ (109)

leaving us with the Hamiltonian

$\hat{H}_{K}=\frac{2t^{2}}{\Delta
E}\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi^{\phantom{\dagger}}_{a}\cdot\frac{1-aZJ_{y}}{2}\mathscr{S},$
(110)

where $\psi_{+}=\nicefrac{{1}}{{\sqrt{2}}}\left(\psi_{L}+i\psi_{R}\right)$ and
$\psi_{-}=\nicefrac{{i\sigma_{y}}}{{\sqrt{2}}}\left(\psi_{L}-i\psi_{R}\right)$.
Finally, we diagonalize the matrix structure of $\hat{H}_{K}$ with respect to
projector $\frac{1-aZJ_{y}}{2}$ by applying the transformation
$T=J_{y}\text{e}^{i\frac{\pi}{4}XJ_{y}}$ which yields the Kondo part of
Hamiltonian (20)

$\hat{H}_{K}=\frac{4t^{2}}{\Delta
E}\sum_{a=\pm}\psi^{\dagger}_{a}\bm{\sigma}\psi_{a}^{\phantom{\dagger}}\frac{1-aY}{2}\cdot\hat{\bm{J}},$
(111)

where
$\hat{\bm{J}}=(\nicefrac{{J_{x}}}{{2}},\nicefrac{{J_{y}}}{{2}},\nicefrac{{J_{z}}}{{2}})^{\text{T}}$
is a vector of angular momentum operator.

The operator $\hat{H}_{z}=-E_{z}\hat{n}_{y}$ can be projected onto the ground
state manifold as well. Again, we introduce the partition unity
$\mathds{1}=\int_{0}^{\pi}\text{d}\theta\int_{0}^{2\pi}\text{d}\phi\frac{\sin(\theta)}{4\pi}\ket{\phi,\theta}\bra{\phi,\theta}$.
Applying one identity from both sides on $\hat{H}_{z}$ gives the matrix

$\hat{H}_{z}=-E_{z}\begin{pmatrix}I_{1}&I_{3}&0&0\\\ I_{3}&I_{2}&0&0\\\
0&0&I_{2}&I_{3}\\\ 0&0&I_{3}&I_{1}\\\ \end{pmatrix}=-\frac{E_{z}}{3}J_{y}Y,$
(112)

where

$\begin{cases}I_{1}=\int_{0}^{\pi}\frac{\text{d}\theta}{2}(1+\cos(\theta))\cos{\theta}=\frac{1}{3},&m=\xi=\eta\\\
I_{2}=\int_{0}^{\pi}\frac{\text{d}\theta}{2}(1-\cos(\theta))\cos{\theta}=-\frac{1}{3},&m\neq\xi=\eta\\\
I_{3}=\int_{0}^{\pi}\frac{\text{d}\theta}{2}\sin(\theta)\cos{\theta}=0,&m=\xi\neq\eta.\end{cases}$
(113)

Applying the transformation $T$ yields

$\hat{H}_{z}=-\frac{E_{z}}{3}Z.$ (114)

Adding the Hamiltonians (114) and (111) gives rise the the effective
Hamiltonian $\eqref{eq:KondoHamEzSmall}$.

##  Appendix C Poor Man’s Scaling

We employ a small coupling Renormalization Group (RG) scheme to Eq. (21). The
high-energy states are integrated out in second-order perturbation theory. We
assume the lead electrons can only access states with energies $D$ above and
below the Fermi energy. One RG step reduces the bandwidth $D$ by $\delta D$.

In second-order perturbation theory, the Hamiltonian, after integrating out
the high-energy modes, takes the form:

$\hat{H}=\hat{H}_{0}+\hat{H}_{K}+\Delta\hat{H},$ (115)

where

$\Delta\hat{H}=\ket{a}\underbrace{\bra{a}\Delta\hat{H}\ket{b}}_{\Delta
H_{ab}}\bra{b}.$ (116)

Here, $\hat{H}_{0}$ is the free Hamiltonian of the lead electrons, and
$\hat{H}_{K}$ is the Kondo Hamiltonian (111). The states $\ket{a/b}$ are
energy eigenstates with energies $E_{a/b}\in[-D+\delta D,D-\delta D]$. The
matrix

$\Delta
H_{ab}=\bra{a}\frac{1}{2}\left(\hat{T}(E_{a})+\hat{T}(E_{b})\right)\ket{b}\stackrel{{\scriptstyle
E_{a}\approx E_{b}=E}}{{=}}\bra{a}\hat{T}(E)\ket{b}$ (117)

incorporates the T-matrix that describes the scattering events from low-lying
energy states into the high-energy (integrated out) states. It can formally be
written as

$\hat{T}(E)=\hat{H}_{K}\hat{P}_{H}\hat{G}_{0}(E)\hat{P}_{H}\hat{H}_{K},$ (118)

where $\hat{P}_{H}$ is a projector into the Hilbert space of energetically
high-lying states (i.e., $\epsilon\in[D-\delta D,D]$ or
$\epsilon\in[-D,-D+\delta D]$), and $\hat{G}_{0}(E)=(E-\hat{H}_{0})^{-1}$. The
transition matrix $\hat{T}$ can be diagrammatically calculated by adding the
two diagrams in Fig. 5.

We will follow the strategy outlined above with an anisotropic version of
Hamiltonian (111), which reads:

$\hat{H}_{K}=\frac{1}{2}\sum_{a}\left(\psi^{\dagger}_{a}\sigma_{i}\psi^{\phantom{\dagger}}_{a}\left(\lambda^{1}_{i}-a\lambda^{Y}_{i}Y\right)\hat{J}_{i}\right),$
(119)

where we use the Einstein convention for an implicit sum over $i\in{x,y,z}$.
The sum of both diagrams is

$\begin{split}\Delta\hat{H}_{ab}&=\hat{T}^{(I)}_{ab}+\hat{T}^{(II)}_{ab}\\\
&=-\frac{\nu\delta
D}{2D}\sum_{a}\left\\{\lambda_{i}^{1}\lambda_{j}^{j}[\sigma_{i},\sigma_{j}]\hat{J}_{i}\hat{J}_{j}-a\lambda_{i}^{1}\lambda_{j}^{Y}Y[\sigma_{i},\sigma_{j}]\hat{J}_{i}\hat{J}_{j}\right.\\\
&\left.-a\lambda^{Y}_{i}\lambda^{1}_{j}[\sigma_{i},\sigma_{j}]Y\hat{J}_{i}\hat{J}_{j}+\lambda_{i}^{Y}\lambda_{j}^{Y}[\sigma_{i},\sigma_{j}]\hat{J}_{i}\hat{J}_{j}\right\\},\end{split}$
(120)

where we approximate the integrals over energies that are integrated out by
the factor $-\frac{\nu\delta D}{D}$. Using the relation

$[\sigma_{i},\sigma_{j}]\hat{J}_{i}\hat{J}_{j}=-2\sigma_{k}\hat{J}_{k}\quad\text{with}\quad
k\neq i,j,$ (121)

the result simplifies to

$\begin{split}\Delta\hat{H}&=-\frac{\nu\delta
D}{D}\sum_{a}\left\\{(\lambda^{1}_{i}\lambda_{j}^{1}+\lambda_{i}^{Y}\lambda_{j}^{Y})\sigma_{k}\hat{J}_{k}\right.\\\
&\left.-a(\lambda_{i}^{1}\lambda_{j}^{Y}+\lambda^{Y}_{i}\lambda^{1}_{j})\sigma_{k}Y\hat{J}_{k}\right\\}.\end{split}$
(122)

From equation (122), we can see that the renormalized coupling constants take
the form:

$\begin{split}\tilde{\lambda}^{1}_{k}&=\lambda_{k}^{1}+\frac{\nu\delta
D}{D}(\lambda^{1}_{i}\lambda_{j}^{1}+\lambda_{i}^{Y}\lambda_{j}^{Y})\\\
\tilde{\lambda}^{Y}_{k}&=\lambda_{k}^{Y}+\frac{\nu\delta
D}{D}(\lambda_{i}^{1}\lambda_{j}^{Y}+\lambda^{Y}_{i}\lambda^{1}_{j}).\end{split}$
(123)

These equations can be rewritten as a differential equation:

$\begin{split}\frac{d\lambda_{k}^{1}}{dD}&=\lim_{\delta D\rightarrow
0}\frac{\tilde{\lambda}k^{1}-\lambda^{1}_{k}}{-\delta
D}=-\frac{\nu}{D}(\lambda^{1}_{i}\lambda_{j}^{1}+\lambda_{i}^{Y}\lambda_{j}^{Y})\\\
\frac{d\lambda_{k}^{Y}}{dD}&=\lim_{\delta D\rightarrow
0}\frac{\tilde{\lambda}_{k}^{Y}-\lambda^{Y}_{k}}{-\delta
D}=-\frac{\nu}{D}(\lambda_{i}^{1}\lambda_{j}^{Y}+\lambda^{Y}_{i}\lambda^{1}_{j}).\end{split}$
(124)

Setting $\lambda^{1/Y}_{x}=\lambda^{1/Y}_{z}=\lambda_{\perp}$ and defining the
RG-times $l=\ln(D_{0}/D)$, where $D_{0}$ is the initial bandwidth, the flow
equations read

$\displaystyle\frac{dg_{\perp}}{dl}$
$\displaystyle=g_{\perp}\left(g_{y}^{1}+g_{y}^{Y}\right),$ (125)
$\displaystyle\frac{dg^{1}_{y}}{dl}$ $\displaystyle=2g_{\perp}^{2},$ (126)
$\displaystyle\frac{dg^{Y}_{y}}{dl}$ $\displaystyle=2g_{\perp}^{2}.$ (127)

Here, $g^{1/Y}_{\perp/y}=\nu\lambda^{1/Y}_{\perp/y}$ are dimensionless
coupling constans. In order to visualize the tree dimensional RG flow, we
define new parameters of the shape

$y^{1/Y}_{\perp,y}=\frac{g^{1/Y}_{\perp,y}}{1+g^{1/Y}_{\perp,y}},$ (128)

such that the $g^{1/Y}_{\perp,y}\rightarrow\infty$ corresponds to
$y^{1/Y}_{\perp,y}\rightarrow 1$. The flow equation for the new parameters
follows from equations (125) - (127) and read

$\displaystyle\frac{\text{d}y_{\perp}}{\text{d}l}$
$\displaystyle=\frac{(y_{\perp}-1)y_{\perp}\left(y^{1}_{y}(2y^{Y}_{y}-1)-y^{Y}_{y}\right)}{(y^{1}_{y}-1)(y^{Y}_{y}-1)}$
(129) $\displaystyle\frac{\text{d}y_{y}^{1}}{\text{d}l}$
$\displaystyle=\frac{2(y^{1}_{y}-1)^{2}y_{\perp}^{2}}{(y_{\perp}-1)^{2}}$
(130) $\displaystyle\frac{\text{d}y_{y}^{Y}}{\text{d}l}$
$\displaystyle=\frac{2(y^{Y}_{y}-1)^{2}y_{\perp}^{2}}{(y_{\perp}-1)^{2}}.$
(131)

This equation has been used to create Fig. 3 where we identified the point
$(1,1,1)$ with the strong coupling fixed point, which we called $\infty$.

Figure 5: Scattering processes that contribute to the renormalization of the
Kondo Hamiltonian $\Delta\hat{H}$ are as follows: (I) Particle excitation: In
this process, a low-energy electron state $\ket{b}$ scatters at the impurity,
denoted by $\hat{\bm{J}}$, into a high-energy state $\ket{c}$ which, in turn,
scatters back into a low-energy state $\ket{a}$. (II) Hole excitation: In this
scenario, a low-energy state $\ket{b}$ scatters into a high-energy hole state
$\ket{c}$. The hole then scatters back into a low-energy electron state
$\ket{a}$.

##  Appendix D Bosonization and Refermionization

This Appendix contains details for the Emery-Kivelson solution of Eq. (21) and
thereby is a supplement to Sec. IV.1.

First, we define the spin operator of the lead electrons in terms of the
chiral fermions $\tilde{\psi}_{R/L}$ as follows:

$\bm{I}_{a}=\frac{\tilde{\psi}_{a}^{\dagger}\bm{\sigma}\tilde{\psi}_{a}^{\phantom{\dagger}}}{2},\quad\quad
I^{\pm}_{a}=I^{z}_{a}\pm iI^{x}_{a}.$ (132)

Additionally, it is worth noting that

$\begin{split}\bm{I}\cdot\hat{\bm{J}}=\frac{1}{2}\left(I_{a}^{+}\hat{J}_{-}+\text{H.c.}\right)+I^{y}_{a}\hat{J}_{y}.\end{split}$
(133)

Now, the Hamiltonian can be rewritten in terms of these spin operators as
follows:

$\displaystyle\begin{split}\hat{H}_{K}&=\frac{\lambda_{\perp}}{2}\left(\left(I^{+}_{+}+I^{+}_{-}\right)\hat{J}_{-}+\left(I^{+}_{-}-I^{+}_{+}\right)Y\hat{J}_{-}+\text{H.c.}\right)\\\
&+\lambda^{1}_{y}\left(I^{y}_{+}+I^{y}_{-}\right)\hat{J}_{y}+\lambda^{Y}_{y}(I^{y}_{-}-I^{y}_{+})Y\hat{J}_{y}.\end{split}$
(134)

Here, we introduce an anisotropy in the Kondo coupling constants. The lead
electrons are bosonized in the following way:

$\begin{split}\tilde{\psi}_{a\sigma}&=\frac{F_{a\sigma}}{\sqrt{2\pi
a_{0}}}\text{e}^{i\sqrt{4\pi}\phi_{a\sigma}}\\\
\Rightarrow&I^{+}_{a}=\psi^{\dagger}_{a\uparrow}\psi^{\phantom{\dagger}}_{a\downarrow}=\frac{F^{\dagger}_{a\uparrow}F^{\dagger}_{a\downarrow}}{2\pi
a_{0}}\text{e}^{i\sqrt{4\pi}(\phi_{a\downarrow}-\phi_{a\uparrow})},\end{split}$
(135)

and

$I^{y}_{a}=\frac{1}{2}\left(\psi^{\dagger}_{a\uparrow}\psi^{\phantom{\dagger}}_{a\uparrow}+\psi^{\dagger}_{a\downarrow}\psi^{\phantom{\dagger}}_{a\downarrow}\right)=\frac{1}{2\sqrt{\pi}}\partial_{x}(\phi_{a\uparrow}(x)+\phi_{a\downarrow}(x))|_{x=0}.$
(136)

Using the rules to transform the operators into the $s/sf/c/cf$ basis as
outlined in equations (26) and (27), the bosonized Hamiltonian reads as
follows:

$\displaystyle\begin{split}\hat{H}_{K}&=\frac{\lambda_{\perp}}{\sqrt{8\pi
a_{0}}}\left(\left(\chi_{sf}^{\dagger}+\chi^{\phantom{\dagger}}_{sf}\right)F^{\dagger}_{s}\text{e}^{-i\sqrt{4\pi}\phi_{s}}\hat{J}_{-}\right.\\\
+&\left.\left(\chi^{\phantom{\dagger}}_{sf}-\chi^{\dagger}_{sf}\right)F^{\dagger}_{s}\text{e}^{-i\sqrt{4\pi}\phi_{s}}Y\hat{J}_{+}+\text{H.c.}\right)\\\
+&\frac{\lambda_{y}^{1}}{\sqrt{\pi}}\partial\phi_{s}(x)|_{x=0}\hat{J}_{y}+\frac{\lambda_{y}^{Y}}{\sqrt{\pi}}\partial_{x}\phi_{sf}(x)|_{x=0}Y\hat{J}_{y}.\end{split}$
(137)

Here, we introduce the new fermion $\chi_{a}=\frac{F_{s}}{\sqrt{2\pi
a_{0}}}\text{e}^{i\sqrt{4\pi}\phi_{a}}$. We apply an Emery-Kivelson
transformation $U_{\text{E.K.}}=\text{e}^{i\sqrt{4\pi}\phi_{s}\hat{J}_{y}}$
which manages to decouple the $sf$ from the $s$ bosons/fermions. The operators
affected by the Emery-Kivelson transformation transform as follows:

$\displaystyle\begin{split}U^{\dagger}_{\text{E.K.}}\hat{J}_{-}U_{\text{E.K.}}&=\hat{J}_{-}\text{e}^{i\sqrt{4\pi}\phi_{s}},\\\
U^{\dagger}_{\text{E.K.}}\partial_{x}\phi_{s}(x)U_{\text{E.K.}}&=\partial_{x}\phi_{s}(x)-\sqrt{\pi}\hat{J}_{y}\delta(x).\end{split}$
(138)

The transformed and fully fermionized Hamiltonian reads as follows:

$\displaystyle\begin{split}\hat{H}^{\prime}_{K}&=\frac{\lambda_{\perp}}{\sqrt{4\pi
a_{0}}}\left((\chi_{sf}^{\dagger}+\chi^{\phantom{\dagger}}_{sf})(d-d^{\dagger}))+(\chi_{sf}^{\phantom{\dagger}}-\chi^{\dagger}_{sf})(d+d^{\dagger})Y\right)\\\
&+\delta\lambda_{y}:\chi^{\dagger}_{sf}\chi^{\phantom{\dagger}}_{sf}:(d^{\dagger}d-\frac{1}{2})-\lambda^{Y}_{y}:\chi^{\dagger}_{sf}\chi^{\phantom{\dagger}}_{sf}:Y(d^{\dagger}d-\frac{1}{2}),\end{split}$
(139)

where $F^{\dagger}_{s}\hat{J}_{-}=d$, $\hat{J}_{y}=(d^{\dagger}d-\frac{1}{2})$
and $\delta\lambda_{y}=\lambda_{y}^{Y}-2\pi v_{F}$. The contribution of the
Fermi velocity enters $v_{F}$ since the free Hamiltonian of the $\phi_{s}$
gets also transformed. We also used the fact that the derivative of the Bose
fields becomes in fermionic language the following:

$\displaystyle\partial_{x}\phi_{a}(x)=:\chi^{\dagger}_{x}\chi^{\phantom{\dagger}}_{x}:,$
(140)

where $:...:$ denotes normal ordering. This is the origin of Eq. (31).

##  Appendix E Observables

###  E.1 Green’s function at the Toulouse point

In this section, we calculate the Green’s functions of the constituents in
Hamiltonian $\eqref{eq:HamMajoranas}$ at the Toulouse point, which involves
setting the coupling constants of the interaction terms to zero. In our case,
this means we choose $\lambda_{y}^{Y}=0$ and $\lambda_{y}^{1}=2\pi v_{F}$
(i.e. $\delta\lambda_{y}=0$).

First, we introduce some conventions. The retarded and imaginary time
correlation functions of two observables, denoted as $A$ and $B$, are defined
as

$C^{R}_{AB}(t)=-i\theta(t)\braket{\left[A(t),B(t)\right]}$ (141)

and

$C_{AB}(\tau)=-\braket{T_{\tau}A(\tau)B(\tau)},$ (142)

respectively. The transformation from imaginary time to Matsubara frequencies
at zero temperature is given by:

$\displaystyle\begin{split}C_{AB}(\tau)&=\frac{1}{\beta}\sum_{m}\text{e}^{-i\omega_{m}\tau}C_{AB}(i\omega_{m})\\\
&\stackrel{{\scriptstyle T\rightarrow
0}}{{\longrightarrow}}\frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}\omega_{m}\text{e}^{-i\omega_{m}\tau}C_{AB}(i\omega_{m})\end{split}$
(143)

and

$\displaystyle\begin{split}\,C_{AB}(i\omega_{m})&=\int_{0}^{\beta}\text{d}\tau\text{e}^{i\omega_{m}\tau}C_{AB}(\tau)\\\
&\stackrel{{\scriptstyle T\rightarrow
0}}{{\longrightarrow}}\frac{1}{2}\int_{-\infty}^{\infty}\text{d}\tau\text{e}^{i\omega_{m}\tau}C_{AB}(\tau),\end{split}$
(144)

where $\beta$ is the inverse temperature. The retarded and imaginary time
correlation functions are related by analytical continuation, which means:

$C^{R}_{AB}(\omega)=\lim_{i\omega_{m}\rightarrow\omega+i0^{+}}C_{AB}(i\omega_{m})$
(145)

Now we calculate the free local Green’s function of the lead fermions:
$G^{(0)}(\tau,x)=-\braket{T_{\tau}\chi^{\phantom{\dagger}}_{a}(\tau,x)\chi_{a}^{\dagger}(0,0)}$.
In a diagonal basis, the Green’s function reads:

$G^{(0)}(i\omega_{m})_{k}=\frac{1}{i\omega_{m}-\epsilon_{k}}.$ (146)

The local Green’s function at $x=0$ is:

$g^{(0)}(i\omega_{m},x=0)=\int\frac{\text{d}k}{2\pi}\frac{1}{i\omega_{m}-\epsilon_{k}}=\int\text{d}\epsilon\frac{\rho(\epsilon)}{i\omega_{m}-\epsilon},$
(147)

where $\rho(\epsilon)=\nu$ is the density of states, which can be well modeled
as being constant within the bandwidth $2D$, leading to:

$g^{(0)}(i\omega_{m})=\nu\int_{-D}^{D}\text{d}\epsilon\frac{1}{i\omega_{m}-\epsilon}=-i2\nu\arctan\left(\frac{D}{\omega}\right).$
(148)

In the limit $\omega_{m}\ll D$, the asymptotic behavior of the Green’s
function becomes:

$g^{(0)}(i\omega_{m})=-i\nu\pi\text{sign}(\omega_{m}).$ (149)

To calculate the Green’s function for the Majorana fermions $\eta$ and
$\eta_{x,y,z}$ we introduce complex fermions which can be chosen in a very
suggestive and physical way, that is

$f=\frac{i}{2}(\eta+i\eta_{y}),\quad
s=\frac{1}{2}(\eta_{z}+i\eta_{x})=\frac{\alpha}{2}(Z+iX)=\alpha\hat{S}^{+}.$
(150)

The $f$ electron can be understood as a ladder operator within the Hilbert
space that belongs to the order parameter angular momentum $\hat{\bm{J}}$.
However, it acts differently on the subspaces defined by the MZMs. On the
other hand, the $s$ fermion acts as a ladder operator in the orbital space.

The Hamiltonian (34) can be expressed in terms of these new fermions and
becomes

$\displaystyle\hat{H}_{\text{eff}}$
$\displaystyle=\frac{\lambda_{\perp}}{\sqrt{2\pi
a_{0}}}(\chi_{sf}^{\dagger}f+\text{H.c.})+\lambda_{y}:\chi^{\dagger}_{sf}\chi_{sf}:(f^{\dagger}f-\frac{1}{2})$
$\displaystyle+2\delta\lambda_{y}:\chi^{\dagger}_{s}\chi_{s}:(f^{\dagger}f-\frac{1}{2})(s^{\dagger}s-\frac{1}{2}),$
(151)

where we ignored the Zeeman term for a moment. One can observe that in the
case of no interaction (i.e., $\lambda_{y}=\delta\lambda_{y}=0$), the
Hamiltonian (151) collapses to a resonant level model that describes a regular
Kondo effect. However, the presence of the orbital space in which the $f$
fermion acts differently distinguishes our system from “just” a regular Kondo
effect.

Now we can write down the local action for the Kondo Hamiltonian (151) at the
Toulouse point in terms of the new fermions:

$\displaystyle\begin{split}S_{\text{K}}&=\int\text{d}\omega_{m}\begin{pmatrix}\bar{\chi}_{sf}&\bar{f}\end{pmatrix}\begin{pmatrix}-g^{(0)}(i\omega_{m})&\tilde{\lambda}\\\
\tilde{\lambda}&-i\omega_{m}\end{pmatrix}\begin{pmatrix}\chi_{sf}\\\
f\end{pmatrix}\\\
&+\int\text{d}\omega_{m}\bar{s}\,(-i\omega_{m})\,s,\end{split}$ (152)

where $\tilde{\lambda}=\frac{\lambda_{\perp}}{\sqrt{2\pi a_{0}}}$. From the
action, we can read off the Green’s function:

$\mathcal{G}=\left[\begin{pmatrix}\frac{i}{\nu\pi}\text{sign}{\omega_{m}}&-\tilde{\lambda}\\\
\tilde{\lambda}&i\omega_{m}\end{pmatrix}\right]^{-1}=\frac{-1}{|\omega_{m}|+\Gamma}\begin{pmatrix}i\omega_{m}\pi\nu&\tilde{\lambda}\pi\nu\\\
\tilde{\lambda}\pi\nu&i\text{sign}(\omega_{m})\end{pmatrix}$ (153)

where $\Gamma=\tilde{\lambda}^{2}\nu\pi$. Collecting everything leads to the
following list of Green’s functions:

$\displaystyle G_{0}(i\omega_{m})$
$\displaystyle=-i\pi\nu\text{sign}(\omega_{m})\quad\text{(Green's function
$\chi_{s,c,cf}$)}$ (154) $\displaystyle G(i\omega_{m})$
$\displaystyle=\frac{-i\omega_{m}\pi\nu}{|\omega_{m}|+\Gamma}\quad\text{(Green's
function $\chi_{sf}$)}$ (155) $\displaystyle D_{0}(i\omega_{m})$
$\displaystyle=\frac{1}{i\omega_{m}}\quad\text{(Green's function of $s$)}$
(156) $\displaystyle D(i\omega_{m})$
$\displaystyle=\frac{-i\text{sign}(\omega_{m})}{|\omega_{m}|+\Gamma}\quad\text{(Green's
function $f$)}$ (157) $\displaystyle F(i\omega_{m})$
$\displaystyle=\frac{-\tilde{\lambda}\pi\nu}{|\omega_{m}|+\Gamma}\quad\text{($f\leftrightarrow\chi_{sf}$
matrix element)}\,.$ (158)

We also find the Green’s functions in imaginary time by applying the Fourier
transform (143). If we assume that $|\omega_{m}|\ll\Gamma$, the imaginary time
Green’s functions behave asymptotically at large $\tau$ as follows:

$\displaystyle G_{0}(\tau)$ $\displaystyle\approx-\frac{\nu}{\tau},$ (159)
$\displaystyle G(\tau)$
$\displaystyle\approx-\frac{2\nu}{\Gamma^{2}\tau^{3}},$ (160) $\displaystyle
D_{0}(\tau)$ $\displaystyle\approx-\frac{\text{sign}(\tau)}{2},$ (161)
$\displaystyle D(\tau)$ $\displaystyle\approx-\frac{1}{\pi\Gamma\tau},$ (162)
$\displaystyle F(\tau)$
$\displaystyle\approx-\frac{\tilde{\lambda}\nu}{\Gamma^{2}\tau^{2}}\,.$ (163)

Furthermore, we deduce the Majorana Green’s function from the fermionic one.
We note that

$\begin{split}\braket{T_{\tau}f(\tau)f(0)}&=\frac{1}{4}\braket{(\eta(\tau)+i\eta_{y}(\tau))(\eta(0)-i\eta_{y}(0))}\\\
&=\frac{1}{4}\left(\braket{T_{\tau}\eta(0)\eta(\tau)}+\braket{T_{\tau}\eta_{y}(\tau)\eta_{y}(0)}\right)\\\
&\equiv-D(\tau),\end{split}$ (164)

where it has been used that
$\braket{T_{\tau}\eta(\tau)\eta_{y}(0)}=\braket{T_{\tau}\eta_{y}(\tau)\eta(0)}=0$
since that the Hamiltonian at the Toulouse point does not has terms which
enable such processes. Also, the phase of the $f$ fermion is a Gauge degree of
freedom from which we conclude that
$\braket{T_{\tau}\eta_{y}(\tau)\eta_{y}(0)}=\braket{T_{\tau}\eta(0)\eta(\tau)}$.
Hence, to fulfill equation (164)

$\braket{T_{\tau}\eta_{y}(\tau)\eta_{y}(0)}=\braket{T_{\tau}\eta(0)\eta(\tau)}=-2D(\tau)$
(165)

must hold. It is straightforward to verify that similar relations also hold
for $\xi_{s,sf,c,cf},\zeta_{s,sf,c,cf},\eta_{x}$ and $\eta_{y}$.

###  E.2 Correlation Functions at finite temperature

Correlation functions of the operator at zero temperature with gap-less
constituents show an algebraic decay in conformal field theories (e.g.
equations (159)-(163)). Using conformal transformations, the zero temperature
correlation functions can be mapped onto their finite temperature counterparts
as

$\braket{T_{\tau}O(\tau)O(0)}\sim\left(\frac{1}{\tau}\right)^{2\Delta}\stackrel{{\scriptstyle\text{conf.
trafo.}}}{{\longrightarrow}}\left(\frac{\pi T}{\sin(\pi
T\tau)}\right)^{2\Delta},$ (166)

where $\Delta$ is the scaling dimension of the operator $O(\tau)$ and $T$ is
the temperature. Integrals over these correlation functions

$\begin{split}I(\Delta)=\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\,\left(\frac{\pi
T}{\sin(\pi
T\tau)}\right)^{2\Delta}=2\int_{\tau_{0}}^{\beta/2}\text{d}\tau\left(\frac{\pi
T}{\sin(\pi T\tau)}\right)^{2\Delta}\\\ \end{split}$ (167)

can be expressed in a more convenient form by the coordinate transformation
$x=\tan(\pi T\tau)$ and become

$I(\Delta)=2(\pi
T)^{2\Delta-1}\int_{x_{0}}^{\infty}\text{d}x\frac{(1+x^{2})^{\Delta-1}}{x^{2\Delta}}$
(168)

, which can be done exactly and evaluates to

$I(\Delta)=\frac{2(\pi
T)^{2\Delta-1}}{x_{0}}\,{}_{2}F_{1}\left(\frac{1}{2},1-\Delta,\frac{3}{2},-\frac{1}{x_{0}^{2}}\right),$
(169)

where $x_{0}=\tan(\pi T\tau_{0})$ and ${}_{2}F_{1}$ is the hypergeometric
function. We have introduced a regularization $\tau_{0}=\frac{1}{\Gamma}$
since the correlation functions are only valid for long imaginary times. For
further reference, we list a couple of special cases for $T\ll\Gamma$ (i.e.
$x_{0}\approx\pi T\tau_{0}$) to leading order in $T$:

  1. 1.

$\Delta=\nicefrac{{1}}{{2}}$

$I\left(\frac{1}{2}\right)=2\left(\ln\left(\frac{T}{\Gamma}\right)+\ln\left(\frac{2}{\pi}\right)\right),$
(170)

  2. 2.

$\Delta=1$

$I(1)=2\Gamma,$ (171)

  3. 3.

$\Delta=\nicefrac{{3}}{{2}}$

$I\left(\frac{3}{2}\right)=\Gamma^{2}-\pi^{2}\ln(T)T^{2}+\mathscr{O}(T^{2}),$
(172)

  4. 4.

$\Delta=2$

$I(2)=\frac{2}{3}\Gamma^{3}+2\pi^{2}\Gamma T^{2}+\mathscr{O}(T^{3}),$ (173)

  5. 5.

$\Delta>2$

$I(\Delta)=\text{const.}+\mathscr{O}(T^{2}).$ (174)

###  E.3 Thermodynamics

####  E.3.1 Susceptibilities

The local impurity susceptibility of
$\hat{V}\in\\{X,Y,Z,\hat{J}_{x},\hat{J}_{x},\hat{J}_{x}\\}$ is defined as

$\chi_{\hat{V}}(T,i\omega_{m})=-\int^{\beta}_{0}\text{d}\tau\braket{T_{\tau}\hat{V}(\tau)\hat{V}(0)}\text{e}^{i\omega_{m}\tau},$
(175)

where $T$ and $\beta$ are the temperature and inverse temperature,
respectively. Later on, we are only interested in the static susceptibility

$\chi_{\hat{V}}(T)=\lim_{\omega_{m}\rightarrow
0}\chi_{\hat{V}}(T,\omega_{m}).$ (176)

However, for further reference, it will be beneficial to calculate the
dynamical susceptibility for the $\hat{J_{y}}$ operator. First, we decompose
the susceptibility in the static and dynamical part as

The calculation of the static susceptibilities is straightforward and relies
on the decomposition into the four Majorana fermions
$\eta,\eta_{y},\eta_{x},\eta_{z}$. One finds

$\hat{J}_{y}=\frac{i\alpha\beta}{2}=\frac{1}{2}\eta_{x}\eta_{y}\eta_{z}\eta.$
(177)

Thus, the time-dependent correlation function evaluates to

$\begin{split}\braket{T_{\tau}\hat{J}_{y}(\tau)\hat{J}_{y}(0)}&=\frac{1}{4}\braket{T_{\tau}\wick{\c{1}\eta_{x}^{\tau}\c{2}\eta_{y}^{\tau}\c{3}\eta_{z}^{\tau}\c{4}\eta^{\tau}\c{1}\eta_{x}^{0}\c{2}\eta_{y}^{0}\c{3}\eta_{z}^{0}\c{4}\eta^{0}}}\\\
&=\frac{1}{4}(-2D_{0}(\tau))^{2}(-2D(\tau))^{2}=D(\tau)^{2},\end{split}$ (178)

where we used $\eta(\tau)=\eta^{\tau}$ as a shortcut notation. Using the
former result, we obtain the static susceptibility

$\begin{split}\chi_{\hat{J}_{y}}(T)&=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau
D(\tau)^{2}=-\left(\frac{1}{\pi\Gamma}\right)^{2}I(1)=-\frac{2}{\pi^{2}\Gamma}\end{split},$
(179)

where we introduced the cut of $\tau_{0}=\nicefrac{{1}}{{\Gamma}}$ to
regularize the integral.

The dynamical susceptibility can be calculated along the same line, except
that the integral reads

$\begin{split}\delta\chi_{\hat{J}_{y}}(T,i\omega_{m})&=-\int_{0}^{\beta}\text{d}\tau
D(\tau)^{2}(\text{e}^{i\omega_{m}\tau}-1)\\\
&=-\left(\frac{1}{\pi\Gamma}\right)^{2}\int_{0}^{\beta}\text{d}\tau(\pi
T)^{2}\frac{\cos(\omega_{m}\tau)-1}{\sin^{2}(\pi T\tau)}\\\
&=\frac{|\omega_{m}|}{2\pi\Gamma^{2}}.\end{split}$ (180)

Hence, the full dynamical susceptibility evaluates

$\chi_{\hat{J}_{y}}(T,i\omega_{m})=-\frac{1}{\pi\Gamma}\left(\frac{2}{\pi}-\frac{|\omega_{m}|}{2\Gamma}\right).$
(181)

In the following, we only consider the static susceptibilities. For
$\hat{J}_{x}$ and $\hat{J}_{z}$, the calculation is slightly more complicated
since these operators are affected by the Emery-Kivelson transformation.
According to equation (138), we find that:

$\begin{split}U^{\dagger}_{\text{E.K.}}\hat{J}_{z}U^{\phantom{\dagger}}_{\text{E.K.}}&=\frac{1}{2}U^{\dagger}_{\text{E.K.}}\left(J^{+}+J^{-}\right)U^{\phantom{\dagger}}_{\text{E.K.}}\\\
&=\frac{1}{2}\left(\text{e}^{-i\sqrt{4\pi}\phi_{s}}\hat{J}^{+}+\text{e}^{i\sqrt{4\pi}\phi_{s}}\hat{J}^{-}\right)\\\
&=\sqrt{\frac{\pi a_{0}}{2}}\left(d^{\dagger}\chi_{s}+\text{H.c.}\right)\\\
&=\sqrt{\frac{\pi
a_{0}}{8}}\left(i\xi_{s}\eta+\zeta_{s}\eta_{x}\eta_{y}\eta_{z}\right),\end{split}$
(182)

where we inserted the identity
$\mathds{1}=F^{\phantom{\dagger}}_{s}F^{\dagger}_{s}$. A similar calculation
yields:

$U^{\dagger}_{\text{E.K.}}\hat{J}_{x}U_{\text{E.K.}}^{\phantom{\dagger}}=\sqrt{\frac{\pi
a_{0}}{8}}\left(i\zeta_{s}\eta_{y}+\xi_{s}\eta_{x}\eta_{y}\eta_{z}\right).$
(183)

To calculate the susceptibility, we need the correlator of the transformed
spin operators:

$\begin{split}\braket{T_{\tau}U^{\dagger}_{\text{E.K.}}\hat{J}_{x/z}(\tau)U^{\phantom{\dagger}}_{\text{E.K.}}U^{\dagger}_{\text{E.K.}}\hat{J}_{z}(0)U^{\phantom{\dagger}}_{\text{E.K.}}}=\pi
a_{0}G_{0}(\tau)D(\tau).\end{split}$ (184)

Thus, the susceptibility can be calculated by

$\begin{split}\chi_{\hat{J}_{x,z}}&=-\pi
a_{0}\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau
G_{0}(\tau)D(\tau)=\frac{a_{0}\nu}{\Gamma}I(1)=2a_{0}\nu.\end{split}$ (185)

The second part of the impurity is given by the orbital degrees of freedom
$X,Y,Z$ and follows the same strategy as for the condensate. Note that none of
these operators are affected by the Emery-Kivelson transformation.

$\begin{split}\chi_{Y}(T)&=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\braket{T_{\tau}Y(\tau)Y(0)}\\\
&=\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\braket{T_{\tau}\eta_{x}(\tau)\eta_{z}(\tau)\eta_{x}(0)\eta_{z}(0)}\\\
&=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau(-2D_{0})^{2}=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\sim\frac{1}{T},\end{split}$
(186)

where we introduced the cut of $\tau_{0}={\nicefrac{{1}}{{\Gamma}}}$ to
regularize the integral. Furthermore:

$\begin{split}\chi_{Z}(T)&=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\braket{T_{\tau}Z(\tau)Z(0)}\\\
&=\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\braket{T_{\tau}\eta_{y}(\tau)\eta_{x}(\tau)\eta_{y}(0)\eta_{x}(0)}\\\
&=-\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau(-2D(\tau))(-2D_{0}(\tau))\\\
&=-\frac{1}{\pi\Gamma}I(\nicefrac{{1}}{{2}})=\frac{2}{\pi\Gamma}\ln\left(\frac{\Gamma}{T}\right)-\frac{2}{\pi\Gamma}\ln\left(\frac{2}{\pi}\right).\end{split}$
(187)

$\chi_{X}(T)$ can be calculated along the same line, which yields the same
result as for $\chi_{Z}(T)$

####  E.3.2 Finite Temperature Corrections to the Free Energy

The scaling dimension of the interaction operators $O_{sf}$ and $O_{s}$ is
determined by investigating the algebraic decay of their correlations:

$\begin{split}&\braket{T_{\tau}O_{s}(\tau)O_{s}(0)}\\\
&=\braket{T_{\tau}\wick{\c{1}\xi_{s}^{\tau}\c{2}\zeta_{s}^{\tau}\c{3}\eta_{x}^{\tau}\c{4}\eta_{y}^{\tau}\c{5}\eta_{z}^{\tau}\c{6}\eta^{\tau}\c{1}\xi_{s}^{0}\c{2}\zeta_{s}^{0}\c{3}\eta_{x}^{0}\c{4}\eta_{y}^{0}\c{5}\eta_{z}^{0}\c{6}\eta^{0}}}\\\
&=(-2G_{0})^{2}(-2D_{0})^{2}(-2D)^{2}\sim\left(\frac{1}{\tau}\right)^{4}\end{split}$
(188)

and

$\begin{split}&\braket{T_{\tau}O_{sf}(\tau)O_{sf}(0)}=\braket{T_{\tau}\xi_{sf}^{\tau}\zeta_{sf}^{\tau}\eta_{y}^{\tau}\eta^{\tau}\xi_{sf}^{0}\zeta_{sf}^{0}\eta_{y}^{0}\eta^{0}}\\\
&=\braket{T_{\tau}\wick{\c{1}\xi_{sf}^{\tau}\c{2}\zeta_{sf}^{\tau}\c{3}\eta_{y}^{\tau}\c{4}\eta^{\tau}\c{1}\xi_{sf}^{0}\c{2}\zeta_{sf}^{0}\c{3}\eta_{y}^{0}\c{4}\eta^{0}}}+\braket{T_{\tau}\wick{\c{1}\xi_{sf}^{\tau}\c{2}\zeta_{sf}^{\tau}\c{3}\eta_{y}^{\tau}\c{4}\eta^{\tau}\c{4}\xi_{sf}^{0}\c{3}\zeta_{sf}^{0}\c{2}\eta_{y}^{0}\c{1}\eta^{0}}}\\\
&=(-2G)^{2}(-2D)^{2}-(-2F)^{4}\sim\left(\frac{1}{\tau}\right)^{8},\end{split}$
(189)

where we used $\xi_{s}^{\tau}=\xi_{s}(\tau)$ as a short-hand notation.
Equations (188) and (189) show that the scaling dimensions of $O_{s}$ and
$O_{sf}$ are $\Delta_{s}=2$ and $\Delta_{sf}=4$, respectively.

Since both operators are irrelevant in an RG sense, we can incorporate them
perturbatively into the calculation of the finite temperature corrections to
the free energy. Since the operator $O_{s}$ has the lower scaling dimension,
we will focus on that one.

In the leading order, the corrections to the free energy are given by the
expression:

$\begin{split}\delta
F(T)&=-\frac{1}{2}\int_{\tau_{0}}^{\beta-\tau_{0}}\text{d}\tau\braket{O_{s}(\tau)O_{s}(0)}\\\
&=-\delta\lambda_{y}^{2}\left(\frac{\nu}{\pi\Gamma}\right)^{2}I(2)\\\
&=-\frac{(\delta\lambda_{y}\nu)^{2}}{\Gamma}T^{2}-\frac{1}{3}\left(\frac{\delta\lambda_{y}\nu}{\pi}\right)^{2}\Gamma.\end{split}$
(190)

Thus, we obtain the final result:

$\delta F(T)-\delta F(0)=-\frac{(\delta\lambda_{y}\nu)^{2}}{\Gamma}T^{2}.$
(191)

The correction leads directly to the expression for the thermodynamic entropy
and the specific heat:

$S(T)=-\frac{\partial}{\partial
T}F(T)=2\frac{(\delta\lambda_{y}\nu)^{2}}{\Gamma}T,$ (192)

and

$C_{v}(T)=T\frac{\partial}{\partial
T}S(T)=2\frac{(\delta\lambda_{y}\nu)^{2}}{\Gamma}T.$ (193)

###  E.4 Transport

In the following chapter, we aim to calculate the conductance for both spin
and charge currents. To achieve this, we employ the Kubo formula, which is
expressed as:

$G=\lim_{\omega\to 0}\text{Re}\frac{i}{\omega}C^{R}_{II},$ (194)

Here, the symbol $I$ represents the current operator, defined as:

$I=\partial_{t}Q=-i[Q,H],$ (195)

where the variable $Q$ can represent either the total charge or spin of the
lead electrons. When expressed in terms of Matsubara frequencies, the Kubo
formula takes the following form:

$G=\lim_{\omega\to
0}\text{Re}\frac{1}{i\omega}\lim_{i\omega_{m}\to\omega+i0^{+}}\int_{-\beta/2}^{\beta/2}d\tau\text{e}^{i\omega_{m}\tau}\braket{T_{\tau}I(\tau)I(0)}.$
(196)

####  E.4.1 Charge and Spin current

The total charge in the left/right wire is given by:

$Q_{L/Rc}=-e\int\text{d}x\,\tilde{\psi}^{\dagger}_{L}(x)\tilde{\psi}^{\phantom{\dagger}}_{L}(x)\,$
(197)

where,
$\tilde{\psi}_{L}=(\tilde{\psi}_{L\uparrow},\tilde{\psi}_{L\downarrow})^{\text{T}}$
is a right-moving field that has been extended to the whole real axis. A basis
change into the $\pm$ basis results in:

$Q_{L/Rc}=-\frac{e}{2}\int\text{d}x\left(\tilde{\psi}^{\dagger}(x)\tilde{\psi}(x)\pm\tilde{\psi}^{\dagger}(x)\sigma_{y}\tilde{\tau}_{y}\tilde{\psi}(x)\right),$
(198)

where, $\tilde{\psi}=(\tilde{\psi}_{+},\tilde{\psi}_{-})^{\text{T}}$, and
$\tilde{\tau}_{i}$ acts in $\pm$ space. After Bosonization and
refermionization, the total charge expressed in terms of Majorana fermions
becomes:

$Q_{L/Rc}=-\frac{e}{2}\int\text{d}x\,(i\xi_{c}\zeta_{c}\pm
i\xi_{sf}\xi_{cf}).$ (199)

Equation (195) for the charge current yields:

$\displaystyle I_{L/Rc}=-\frac{e}{2}\left(\pm
i\tilde{\lambda}\xi_{cf}\eta+\frac{\lambda^{Y}_{y}}{2}\xi_{cf}\xi_{sf}\eta_{y}\eta\right)\stackrel{{\scriptstyle\text{T.P.}}}{{=}}\mp
i\frac{\tilde{\lambda}e}{2}\xi_{cf}\eta,$ (200)

where we used that we are evaluating the conductance at the Toulouse point
(T.P.).

In full analogy to the charge current, we express the total spin of one wire
first in chiral fermions with support on the whole axis. Thus, we have:

$\begin{split}Q_{L/Rs}&=\int\text{d}x\,\tilde{\psi}^{\dagger}_{L}(x)\sigma_{y}\tilde{\psi}^{\phantom{\dagger}}_{L}(x)\\\
&=\frac{1}{2}\int\text{d}x\left(\tilde{\psi}^{\dagger}(x)\sigma_{y}\tilde{\psi}(x)\pm\tilde{\psi}^{\dagger}\tilde{\tau}_{y}\tilde{\psi}(x)\right),\end{split}$
(201)

where we changed into the $\pm$ basis after the second equal sign. The first
part becomes:

$\tilde{\psi}^{\dagger}(x)\sigma_{y}\tilde{\psi}(x)=\frac{2}{\sqrt{\pi}}\partial_{x}\phi_{s}(x)$
(202)

and, thus, is affected by the Emery-Kivelson transformation. The total
magnetization becomes after the transformation:

$\begin{split}Q_{L/Rs}&=\frac{1}{2}\int\text{d}x\left(\frac{2}{\sqrt{\pi}}\partial_{x}\phi_{s}(x)\pm\tilde{\psi}^{\dagger}(x)\tilde{\tau}\tilde{\psi}(x)\right)-J^{y}\\\
&=\frac{1}{2}\int\text{d}x\left(i\xi_{s}\zeta_{s}\pm
i\zeta_{sf}\zeta_{cf}\right)-\frac{\eta_{x}\eta_{y}\eta_{z}\eta}{2}.\end{split}$
(203)

With the corresponding current:

$I_{L/Rs}\stackrel{{\scriptstyle\text{T.P.}}}{{=}}\frac{\tilde{\lambda}}{2}\left(\pm
i\zeta_{cf}\eta_{y}+\eta_{x}\eta_{z}\left(\xi_{sf}\eta_{y}-\zeta_{sf}\eta\right)\right),$
(204)

where the second term corresponds to $\dot{J}_{y}$.

####  E.4.2 Charge Conductance in Strong Coupling Limit

Knowing the charge currents, we can calculate the conductances $G_{ij}^{c}$,
where $i,j\in{R,L}$. The current-current correlator is expressed as:

$\begin{split}&\braket{T_{\tau}I_{ic}(\tau)I_{jc}(0)}=-x_{ij}\left(\frac{\tilde{\lambda}e}{2}\right)^{2}\braket{T_{\tau}\wick{\c{1}\xi_{cf}(\tau)\c{2}\eta(\tau)\c{1}\xi_{cf}(0)\c{2}\eta(0)}}\\\
&=x_{ij}\left(\frac{\tilde{\lambda}e}{2}\right)^{2}\left(-2G_{0}(\tau)\right)\left(-2D(\tau)\right)=-x_{ij}(\tilde{\lambda}e)^{2}G_{0}(\tau)D(\tau),\end{split}$
(205)

where $x_{ij}=(1-2\delta_{ij})$. Thus, the current correlation function
expressed in Matsubara frequencies reads

$C_{II}(i\omega_{m})=-x_{ij}\frac{(\tilde{\lambda}e)^{2}}{\beta}\sum_{n}G_{0}(i\omega_{m}-i\epsilon_{n})D(i\epsilon_{n}),$
(206)

where $\omega_{m}$ and $\epsilon_{n}$ are bosonic and fermionic Matsubara
frequencies, respectively. The main task will be to evaluate the Mastubara sum

$\begin{split}\Sigma(i\omega_{m})&=\frac{1}{\beta}\sum_{n}G_{0}(i\omega_{m}-i\epsilon_{n})D(i\epsilon_{n})\\\
&=\frac{-i\nu\pi}{\beta}\sum_{n}\frac{\text{sign}(\omega_{m}-\epsilon_{n})}{i\epsilon_{m}+i\Gamma\text{sign}(\epsilon_{n})}.\end{split}$
(207)

Note that this sum doesn’t converge. However, we can regularize the sum by
subtracting the $\omega_{m}=0$ part, which gives an imaginary contribution and
will, thus, vanish in the conductance anyway. Hence, we are left with the sum

$\begin{split}\delta\Sigma(i\omega_{m})&=\frac{-i\nu\pi}{\beta}\sum_{n}\frac{\text{sign}(\omega_{m}-\epsilon_{n})+\text{sign}(\epsilon_{n})}{i\epsilon_{n}+i\Gamma\text{sign}(\epsilon_{n})}\\\
&=\frac{-2\nu\pi}{\beta}\sum_{n=0}^{m-1}\frac{1}{\epsilon_{n}+\Gamma}\\\
&=-\nu\left(\psi\left(\frac{1}{2}+\frac{\omega_{m}+\Gamma}{2\pi
T}\right)-\psi\left(\frac{1}{2}+\frac{\Gamma}{2\pi T}\right)\right)\\\
&=-\frac{\nu}{2\pi T}\psi^{\prime}\left(\frac{1}{2}+\frac{\Gamma}{2\pi
T}\right)\omega_{m}+\mathscr{O}(\omega_{m}^{2}),\end{split}$ (208)

where $\psi$ is the digamma function and $\psi^{\prime}$ is it’s first
derivative. Since we are interested in the low-temperature correction, we
expand the result up to the first leading order in the temperature and obtain

$\delta\Sigma(i\omega_{m})\approx-\frac{\nu}{\Gamma}\left(1-\frac{\pi^{2}}{3}\left(\frac{T}{\Gamma}^{2}\right)\right)\omega_{m}.$
(209)

Therefore, we find the current correlation function to be

$\begin{split}C_{II}(i\omega_{m})&=-x_{ij}(\tilde{\lambda}e)^{2}\delta\Sigma(i\omega_{m})\\\
&=x_{ij}\frac{e^{2}}{\pi}\left(1-\frac{\pi^{2}}{3}\left(\frac{T}{\Gamma}\right)^{2}\right)\omega_{m}.\end{split}$
(210)

Inserting this result into the Kubo formula (194) leads to the final result
for the DC conductance is

$\begin{split}G^{c}_{ij}&=\text{Re}\lim_{\omega\rightarrow
0}\frac{1}{i\omega}\lim_{i\omega_{m}\rightarrow\omega+i0^{+}}x_{ij}\frac{e^{2}}{\pi}\omega_{m}\left(1-\frac{\pi}{3}\left(\frac{T}{\Gamma}\right)\right)\\\
&=\text{Re}\lim_{\omega\rightarrow
0}\frac{1}{i\omega}x_{ij}\frac{e}{\pi}(-i\omega)\times\left(1-\frac{\pi}{3}\left(\frac{T}{\Gamma}\right)\right)\\\
&=-x_{ij}G_{0}^{c}\underbrace{2\left(1-\frac{\pi}{3}\left(\frac{T}{\Gamma}\right)^{2}\right)}_{G(T)},\end{split}$
(211)

where $G^{c}_{0}=e^{2}/h$ is the perfect conductance. Note that $\hbar=1$ and,
thus, $h=2\pi$.

####  E.4.3 Spin Conductance

The calculation of the spin conductance is analogous to that of charge
conductance, with the key difference being the more intricate shape of the
spin current. To compute $G^{s}_{ij}$, we first introduce a new notation:

$I_{L/Rs}=i_{L/Rs}-\dot{J}_{y},$ (212)

where $i_{L/Rs}=\pm i\frac{\tilde{\lambda}}{2}\zeta_{cf}\eta_{y}$. The
current-current correlator essentially involves a sum over three distinct
correlators: $\braket{T_{\tau}i_{L/Rs}(\tau)i_{L/Rs}(0)}$,
$\braket{T_{\tau}i_{L/Rs}(\tau)\dot{J}y(0)}$, and
$\braket{T\tau\dot{J}_{y}(\tau)\dot{J}_{y}(0)}$. It is straightforward to show
that:

$\braket{T_{\tau}i_{is}(\tau)i_{js}(0)}=\frac{\braket{T_{\tau}I_{ic}(\tau)I_{jc}(0)}}{e^{2}}$
(213)

and that $\braket{T_{\tau}i_{L/Rs}(\tau)\dot{J}_{y}(0)}=0$. Thus, only the
last correlator $\braket{T_{\tau}\dot{J}_{y}(\tau)\dot{J}_{y}(0)}$ remains to
be calculated.

First, we express the correlator in terms of Matsubara frequencies

$\begin{split}C_{\dot{J}_{y}\dot{J}_{y}}(i\omega_{m})&=-\int_{0}^{\beta}\text{d}\tau\text{d}\tau^{\prime}\text{e}^{i\omega_{m}(\tau-\tau^{\prime})}\braket{\dot{J}_{y}(\tau)\dot{J}_{y}(\tau^{\prime})}\\\
&=-\int_{0}^{\beta}\text{d}\tau\text{d}\tau^{\prime}\text{e}^{i\omega_{m}(\tau-\tau^{\prime})}\partial_{\tau}\partial_{\tau^{\prime}}\delta\chi_{\hat{J_{y}}}(\tau,\tau^{\prime})\\\
&=-\omega_{m}^{2}\delta\chi_{\hat{J}_{y}}(i\omega_{m})=\frac{\omega_{m}^{2}|\omega_{m}|}{2\pi\Gamma^{2}}.\end{split}$
(214)

Here we implicitly subtracted the $C_{\dot{J}_{y}\dot{J}_{y}}(0)$ by using
$\delta\chi_{\hat{J}_{y}}(i\omega_{m})$ instead of
$\chi_{\hat{J}_{y}}(i\omega_{m})$. The third equality is achieved by partial
integration. We see that the $C_{\dot{J}_{y}\dot{J}_{y}}$ correlator is of
third order in $\omega$ and, thus, does not contribute to the DC conductance
since the term vanishes in the limit $\omega\rightarrow 0$.

####  E.4.4 Conductance at Weak Coupling

Finally, we aim to determine the temperature dependence of conductance at weak
coupling (i.e., temperatures above the Kondo temperature). To achieve this, we
express the Kondo Hamiltonian in a slightly different form:

$\hat{H}_{K}=\frac{\lambda}{2}\psi^{\dagger}\left(\mathds{1}-\tilde{\tau}_{z}Y\right)\bm{\sigma}\psi\cdot\hat{\bm{J}},$
(215)

where $\psi=(\psi_{+},\psi_{-})^{\text{T}}$. The charge current expressed in
terms of the $\pm$ fermions is given by:

$I_{L/Rc}=\mp\frac{e\lambda}{2}\psi^{\dagger}\left(\varepsilon_{2nm}\tilde{\tau}_{y}\sigma_{m}-\delta_{2n}\tilde{\tau}_{x}Y\right)\psi\hat{J}_{n}$
(216)

The current correlator evaluates as follows:

$\braket{T_{\tau}I_{ic}(\tau)I_{jc}(0)}=3\,x_{ij}\,(\lambda
e)^{2}G_{0}(\tau)G_{0}(-\tau),$ (217)

where the free Green’s function of the electrons at finite temperature is:

$G_{0}(\tau)=-\braket{\psi_{\gamma}^{\phantom{\dagger}}(\tau)\psi^{\dagger}_{\gamma}(0)}=-\nu\frac{\pi
T}{\sin(\pi T\tau)}.$ (218)

Note that
$\braket{\psi_{\gamma}^{\phantom{\dagger}}(\tau)\psi^{\dagger}_{\sigma}(0)}=0$
if $\gamma\neq\sigma$. Inserting the current-current correlator into the Kubo
formula leads to the integral:

$\lim_{i\omega_{n}\rightarrow\omega+i0^{+}}\int_{0}^{\beta}\text{d}\tau\,G_{0}(\tau)G_{0}(-\tau)\text{e}^{i\omega_{n}\tau}=-i\pi\nu^{2}\omega,$
(219)

which is valid for small $\omega$. This result yields the conductance at weak
coupling:

$G_{ij}^{c}=-6x_{ij}\,\frac{e^{2}}{h}(\lambda\pi\nu)^{2}.$ (220)

The spin current in the basis of the $\pm$ lead fermions is defined as:

$I_{L/Rs}=\frac{\lambda}{2}\left\\{\varepsilon_{2nm}\psi^{\dagger}\sigma_{m}\psi\hat{J}_{n}-\psi^{\dagger}(\varepsilon_{2nm}\tilde{\tau}_{z}\sigma_{m}\pm\tilde{\tau}_{x}\sigma_{n})\psi
Y\hat{J}_{n}\right\\}.$ (221)

For this case, the current-current correlator reads:

$\braket{T_{\tau}I_{is}(\tau)I_{js}(0)}=-\lambda^{2}(4-3x_{ij})G_{0}(\tau)G_{0}(-\tau).$
(222)

This results in the spin conductance:

$G^{s}_{ij}=(4-3x_{ij})\frac{2}{h}(\lambda\pi\nu)^{2}.$ (223)

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