Topological Kondo effect with spinful Majorana fermions

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 3 He [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.
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 pwave time reversal invariant topological superconductor (TRITOPS [52], class DIII).In this work, we study TRI-TOPS 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 lowenergy 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 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 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 A. 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 and This model can be thought of as a two-fluid model with the condensate of Cooper-pairs and the electrons, described by spinor fields ψ = (ψ ↑ , ψ ↓ ) T , where ψ ( †) σ (x) annihilates (creates) an electron of spin σ at position x; similar models have been discussed in the context of one-dimensional superconductors [67,[69][70][71].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 longrange 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 Nc = −i∂ φ is the total number operator of Cooper-pairs, and nf = ∫ dx ψ † (x)ψ(x) is the total number operator of electrons.Hence the total charge operator on the island is [71] Ntot = 2 Nc + nf and commutes with Ĥ.The constant (e being the electron charge) can be tuned by changing the gate voltage V g and determines the expected value of Ntot .
The second term describes the spin interactions on the island.The vectors L = ( Lx , Ly , Lz ) and Ŝ = ( Ŝx , Ŝy , Ŝz ) are the canonically conjugate operator to order parameter d ∈ S 2 and the total spin operator of the fermions Ŝi = 1 2 ∫ dx ψ † (x)σ i ψ(x), respectively.In the following, we will refer to 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, ĤBdG describes the triplet superconductor.The free dynamics of the electrons is governed by the dispersion relation ϵ(p) = p2 2m − µ where p is the momentum operator.The pairing operator ∆ consists of the superconducting phase φ, the Cooper-pair orientation d in spin space, the pairing strength u (units of velocity) and the partial derivative ∂ x reflecting the pwave nature of the superconductor.Furthermore, the Cooper-pair orientation can be parameterized by the two operators θ and φ, which eigenvalues can take the values θ ∈ [0, π) and ϕ ∈ [0, 2π).A possible parametrization in terms of θ and φ is given as follows: Another representation of d that will be heavily used in this work can be given in terms of a unitary operator, that is where Finally, the term Ĥz 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 d vector in the y-direction.Thus, there are two competing terms in the Hamiltonian.The first term, E s Ĵ 2 , attempts to fix the angular momentum, resulting in strong fluctuations in d, while Ĥz tries to localize d, leading to strong fluctuations in Ĵ .The two aforementioned limiting cases correspond to situations where either E s ≫ E z or E s ≪ E z .In the first case, d is pinned while Ĵ fluctuates strongly, while in the other case, Ĵ is pinned, and d freely fluctuates.

B. 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 This transformation applied to the Hamiltonian (1a) yields where Ĵ = L + Ŝy sin( θ) ( d + cot( θ)e θ ) with e θ being the unit vector in θ direction.
A few comments are in order: First, the transformation U decouples the spin up and down sector of the fermions and diagonalized ĤBdG in spin space.Now, the superconducting part consists of two copies of a onedimensional 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 γ L↑ , γ L↓ , γ R↑ and γ R↓ .Note that the symbols ↑ and ↓ 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 Second, due to the canonical transformation, the total electron charge vanished from the Hamiltonian.How-ever, it enters implicitly through Nc , whose quantization properties have changed.The total number operator of the Cooper-pair operator is usually integer quantized.However, after the transformation Nc 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 Ŝ did not fully disappear since the spin is a vector quantity.Instead, the fermion spin survives as a background field for the angular momentum L of the order parameter d.Suppose one understands 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 Ŝy operator) as can straightforwardly be calculated by taking the rotation of Ĵ − L. Analogously, one can show that the action corresponding to the Hamiltonian E s Ĵ 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 Nc , the angular momentum L is integer quantized while its transformed version Ĵ 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 H of the MCPB can be decomposed as a tensor product of three subspaces.That is, where H n is the Hilbert space spanned by the eigenstates of Nc , H j is spanned by the eigenstates of Ĵy and 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 quasiparticles in the superconductor can not be created.However, the non-local electrons Γ σ formed from the MZMs are accessible even at zero temperature.Thus, we project H f down to the Fock space, which in the case of a single wire w = 1 is where the state |0⟩ is the vacuum destroyed by the operators Γ σ .Furthermore, the Hamiltonian (4) commutes with the mutually commuting operators Nc and Ĵ .Therefore, we can construct the energies and energy eigenstates by solving the eigenvalue equation for Nc and Ĵ 2 separately.The low energy spectrum of the MCPB is summarized in the Tab.I for the case w = 1 and plotted in Fig. 2. In the case of multiple wires, w > 1, the low-energy spectrum of Tab.I is the same and characterized by the very same quantum numbers as displayed in the first four columns of Tab.I.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, . . .for w = 1, 2, 3, . . . .

Nc jy J P
Classification of the three lowest energy eigenstates of Hamiltonian (4) by their quantum numbers: Charge Nc (in units of 2e), magnetic quantum number jy, total angular momentum J as determind by the Ĵ 2 eigenvalue J(J + 1), fermion parity P f = (−1) n f and magnetic quantum number of the fermions sy.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.g where it becomes favorable to add one unit of charge (one Cooper pair) onto the island and corresponds to Ec where Ec 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 ∆E = Ec − 3Es/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: Here, Ĥdot denotes the Hamiltonian (4), and Ĥ0 is the Hamiltonian of free electrons ψ R/L of the left and right leads.The Hamiltonian Ĥt couples the electrons to the MZMs and is given by: Here, the operator ψ † i,σ (x) creates an electron in the ith lead with spin σ.To straighten up the notation, we define ψ iσ (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 = 1 2 such that the ground state of the quantum dot has an odd fermion parity, i.e., one non-local fermion Γ σ is occupied.Therefore, the ground state corresponds to the blue parabola in Fig. 2 and the second row in Table I.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: The objective is to develop an effective lowtemperature 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 ≫ E s .Thus, the timescale on which the superconducting phase φ fluctuates is much smaller compared to the timescale of the fluctuations of d ( Û ).Therefore, we first only perform a Schrieffer-Wolf transformation in the charge sector H n of the condensate Hilbert space.
While in all of the previous discussions, a single TRI-TOPS 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 where λ = 4t 2 Ec .This expression holds for an arbitrary number of wires (in which case i, j ∈ {1, . . ., 2w}).In the simplest case w = 1 (in this case we use labels i, j ∈ {L, R}) this expression can be further simplified as follows where with ψ L/R = (ψ L/R↑ , ψ L/R↓ ) T being spinors in spin space.
Since the spin quantum number of the MZMs behaves more like a static quantum number than an actual spin, we should rather think of Ŝ as an impurity that acts on the orbital (charge parity) space spanned by γ ↑/↓ than an actual spin.

A. Limit Ez ≫ Es
We take E s and E c of the same order of magnitude but also take the perturbation Ĥz into account where we work in the limit E z ≫ E s .Thus, the d vector is polarized and, again, fluctuates on a much larger time scale than the superconducting phase φ.Therefore, we can continue working with the equation (12).
In the case where E s /E z → 0, Û does not display any quantum fluctuations (it is a constant of motion).We can thus choose U = 1, and the Hamiltonian (11) becomes an O(4w) 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 It is well known that the two-channel Kondo model has a quantum critical point at a finite Kondo coupling λ [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 Û (i.e.finite E s /E z ), the situation becomes more complex.As discussed previously, in the limit E s /E z → 0 we expect twochannel 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 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: The d vector has been linearized with respect to the deviation angle θ.This linearization yields the effective two-dimensional vector δ 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 d.
To determine the spectrum of the vector d, we expand the Hamiltonian of the MCPB (including the E z term) to the first non-trivial order in θ.This leads to a Hamiltonian Ĥθ≪1 , in which the Schrödinger equation resembles the radial part of a harmonic oscillator in two dimensions (see App.A 2 a).The level spacing of the energy eigenstates is ∆E n = √ 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 = ±1/2.Furthermore, the ground state expectation value of δ d is 0. A comprehensive derivation of the spectrum can be found in appendix A 2 a.
To derive the effective Kondo Hamiltonian in the limiting case θ ≪ 1 (i.e.E z ≫ E s ), we use equation (12) as a starting point and, similarly, linearize U .Initially, we introduce a new parametrization Û = exp(i Ŵ 2 ), where Ŵ = ∂ ϕ δ d⋅σ.Expanding Û to first order in Ŵ , we arrive at a linearized version of the Hamiltonian (12), that is From this expression, it becomes evident that even at the first order in d fluctuations, the effective Hamiltonian exhibits a more intricate structure than a two-channel Kondo Hamiltonian.
However, if E z ≫ ( λ 4 Es ) 1 3 the system effectively stays in the ground state manifold.Consequently, we can replace the operator δ 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 d.

B. Limit Ez ≪ Es
If, conversely, the magnitude of E s greatly surpasses that of E z , the phase φ and Û 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.I (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 The vector Ĵ is projected onto the ground state manifold and can be expressed as Ĵ = ( Jx 2 , Jy 2 , Jz 2 ), with J i denoting the Pauli matrices acting within the space spanned by the states |j = ± 1 /2, J = 1 /2⟩ in the Hilbert space H j .The Kondo coupling parameter is now defined as λ = 4t 2 Ec− 3 /4Es .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 ≫ 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 = 1−aY 2 projects the impurity Ĵ onto two independent sectors.In other words, the two effective impurities Ĵa = P a Ĵ commute with each other, i.e., [ Ĵi + , Ĵj − ] = 0.However, the distinction to two SU(2) Kondo effects lies in the fact that the entire impurity transforms under SU(2) ⊕ SU(2) rather than SU(2) ⊗ 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 1−aY 2 as an operator that distinguishes between the occupation of the two orbitals formed by the four MZMs.The lead electrons ψ 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: We employ a poor man's scaling [74] analysis (see App. C) for Eq. ( 21) and obtain the following flow equations: where l = log ( D0 D ) is the RG time and D 0 being the bare bandwidth.We introduced the dimensionless g ⊥/y coupling constants, with ν denoting the density of states at the Fermi level and set g 1 ⊥ = g Y ⊥ = g ⊥ .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.

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 E.K. , and refermionize the lead electrons as well as expressing the impurity in terms of Majorana fermions.

A. Bosonization and Refermionization
We take the usual Bosonization approach and decompose the fields ψ aσ , with respect to the lattice constant a 0 , into slow-varying right and left-moving fields: where k F is the Fermi wavevector.Note that ψ +σ (x) and ψ −σ (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 where F aσ is a Klein factor that ensures the correct fermionic statistics.The bosonic field follows the commutation relation where i, j are multindices of the shape (a, σ).Again, we define ψaσ (0) ≡ ψaσ and ϕ aσ (0) ≡ ϕ aσ .It will be convenient to introduce a new basis of Bose fields that are related by A similar basis transformation can also be done for the Klein factors with the identities [47,76,77] Also, the Kondo Hamiltonian can be rewritten in the following form where Note that we work in a basis in which σ y is diagonal and, thus, the spin indices ↑ / ↓ refer to spins quantized in the y-direction.We used the notation Ĵ± = Ĵz ± i Ĵx .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 and (r = s, sf, c, cf ) one can show that the effective Kondo Hamiltonian becomes where δλ y = λ 1 y − 2πv 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 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.γ L↑ , γ L↓ , γ R↑ , and γ R↓ ) and the two Majorana fermions α and β that describe Ĵ .It is useful to represent every term in Hamiltonian (31) in terms of four composite Majorana fermions without constraint, i.e. η = β, η x = αX, η y = αY, and η z = αZ.
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 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.

B. 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., δλ y = λ 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.
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 η y and η.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, η x hybridizes with η y and there is only one Majorana fermion dangling anymore.Thus, the twofold ground state degeneracy reduces to a non-integer degeneracy of √ 2.

C. Renormalization group flow
In the following, it is useful to invoke the impurity entropy defined by S imp = S tot − S bulk ≡ ln(g), (35) where the quantity g can be interpreted as a generalized ground state degeneracy which can take non-integer val-ues.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 λ 1,Y 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 λ Y y , δλ 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 (η y acquires the scaling dimension of lead electrons through λ ⊥ hybridization) and generates a state of g = √ 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 = √ 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.

A. Thermodynamics
We first focus on the correction to the free energy denoted by δF .To determine the scaling dimension of the interaction terms, we analyzed the operators O s and O sf defined as follows: By calculating the expectation value where ∆ r represents the scaling dimension, we found that the scaling dimensions of O s and O sf are ∆ s = 2 and ∆ sf = 4, respectively.Following [75], we inferred that the correction to the free energy scales as: which relates to the thermodynamic entropy and specific heat, respectively: with κ 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: where Γ = 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 → 0. Specifically, χ X,Z diverges logarithmically, while χ 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.

B. Transport Properties
Next, we consider various transport coefficients of our mesoscopic setup, notably the DC conductance denoted as G c ij and the spin current conductance represented by G s ij .We remind the reader that the indices i, j ∈ {L, R} correspond to the left and right leads.At the smallest temperatures, the DC conductance can be expressed as: where G c 0 = 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: 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 λ, which scales as 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 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) ⊕ 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 dvector.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 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.
First, we demonstrate how the quantization condition of the operator Nc and Ĵ change with respect to their transformed counterparts.Let be the Cooper-pair number operator before the transformation that is Before the transformation, the operator Nc is integer quantized and has the eigenstates |Φ⟩ 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 ⟨φ|Φ⟩ N = Φ N (φ).Note that, due to the missing hat, φ is a quantum number and not an operator.Since the phase is only defined up to 2π, the wavefunction has to fulfill the boundary condition The transformed wavefunction reads There, we can see that for an even number of electrons, the wave function is 2π periodic, and for an odd number of electrons 4π periodic.Thus, the operator Nc after the transformation is half-integer quantized since we changed the boundary condition by U .A similar effect can also be observed in Ĵ .The operator Ly = −i∂ ϕ has, since it is an angular momentum operator, integer eigenvalues l y with eigenfunctions Ψ ly (ϕ).These eigenfunctions are also 2π periodic.After the transformation, the new eigenfunctions read where these are eigenfunctions of Ĵy = −i∂ ϕ and Ŝy is the spin operator of the fermion excitation in the y-direction.Similar to the superconducting phase and the Cooperpair number, in the case of an odd fermion number (i.e.Ŝy has half-integer eigenvalues), the eigenfunction of Ĵy are 4π periodic and have half-integer eigenvalues.

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 Nc and Ĵ 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: where the solutions are Φ N (φ) = e iN φ , and N is a halfinteger.
For the angular momentum sector, the energy eigenstates correspond to the eigenstates of the operator Ĵ 2 .Since Ĵ is an angular momentum/spin operator, we can construct the spectrum and the eigenfunctions using standard methods.That is, we choose the operator Ĵy to commute with Ĵ 2 , and the eigenstates are given by the following two eigenvalue equations: To construct the corresponding differential equation, we note that L is the angular momentum operator that is canonically conjugate to d.Thus, in terms of coordinates, the L operator can be written as: where e i are unit vectors.Note that in coordinate space, d ≡ e r .From this, we can calculate: On the other hand, the fermion spin transforms as: Now, we can conclude: In order to solve the two eigenvalue equations (A7) and (A8), we use the separation of variables as an Ansatz, that is, ⟨θ, ϕ|Ξ⟩ = Ξ(θ, ϕ) = Θ(θ)Ψ(ϕ).This Ansatz leads to the equation: with the solution Ψ(ϕ) = e ijyϕ .Note that Ŝy also commutes with the Hamiltonian.Therefore, we can replace the operator Ŝy with its eigenvalue s y .The second eigenvalue equation yields the differential equation: where x = cos(θ) and P (x) = P (cos(θ)) = Θ(θ).This corresponds to the eigenvalue equation of "monopole harmonics", i.e. the generalization of spherical harmonics to s y ≠ 0. [80] Assuming N c g = 1 /2, the ground state manifold is given by the second row in table I. Inserting the corresponding quantum numbers in equation (A14), we find the normalizable solutions: Thus, the full wave function of the condensate (charge and spin part) in the ground state manifold reads: In the first excited states, the condensate wavefunction takes a trivial shape: (A17) a. 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 ≫ E s , and hence, θ ≪ 1.To do so, we expand the Hamiltonian: up to the first non-trivial order in θ and ignoring the constant shift in energy.Note that n = j y − s y .After applying the approximation θ ≪ 1, the differential equation for the Θ(θ) part of the condensate wavefunction changed, which, therefore, needs to be adopted.The Hamiltonian (A18) resembles the radial part of a harmonic oscillator.The spectrum for the 2-dimensional harmonic oscillator in polar coordinates is well known and reads with energy eigenstates: In analogy to the harmonic oscillator we identify ω = √ 2E z E s and m = 1 2Es .The quantum number n θ describes the quantization of radius in the x-z-plane.However, in the low-energy sector, we can choose n θ = 0. C n,n θ is a normalization constant, and L n n θ are the generalized Laguerre polynomials.The total condensate wave function is most conveniently expressed in terms of n and s y as Ξ n,sy (θ, ϕ) = Θ n,0 (θ)e i(n+sy)ϕ . (A21) The ground state has n = 0, from which it follows that s y = j y = ± 1 /2.Hence, the ground state manifold is twofold degenerate.The first excited states have n = ±1 and are, therefore, 4 states.Furthermore, it follows immediately that: 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 ≫ E s , and we also assume E z ≫ E s .This will be followed by a discussion of how to obtain the effective Kondo Hamiltonian in the limit where E c ∼ E s and E z ≪ E c .
To derive the effective low-energy Hamiltonian, we apply a Schrieffer-Wolf transformation, treating the hop-ping parameter t of lead electrons onto the quantum dot as a perturbation, which implies that E c ≫ t.

Limit Ec ≫ Es
In this limiting case, the fluctuations in the superconducting phase ( φ) are much faster than the spin fluctuations ( Û ).Thus, we absorb the Û matrix into the lead electrons: where ψ i = (ψ i↑ , ψ i↓ ) are spinors of the lead electrons.
The hopping Hamiltonian becomes: The low-energy sector is mostly determined by the charge on the island.If we choose N c g = 1 2 , the ground state manifold in the charge sector can be classified by the total charge |GS⟩ = |N c = 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 |0⟩ and |1⟩, respectively.
Applying the Schrieffer-Wolf transformation, Hamiltonian (8) becomes: 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: The operator e ±i φ/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 Applying these rules, the effective Hamiltonian collapses to: This expression is the origin of Eq. ( 11) of the main text, note that i = 1, . . ., w.
In the case w = 1 we use the notation where we introduced the four-component spinor Ψ = (Ψ L↑ , Ψ L↓ , Ψ R↑ , Ψ R↓ ) T and the symbol (γ ⋅ γ T ) that represents the matrix: Decomposing this matrix further into Pauli matrices yields: where σ i and τ i act in the spin and left/right space, respectively.In the next step, we change into the eigenbasis of −τ y and obtain: where The vector Ŝ = ( Ŝx , Ŝy , Ŝz ) 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:

Limit Ez ≫ Es
In this section, we build upon the derivations from the previous section, assuming that E z ≫ E s , and that d only experiences weak fluctuations around the y-axis.Our strategy is to expand the unitary matrix Û in terms of θ.We introduce a new representation of Û using Pauli matrices to achieve this.We utilize the gauge freedom within Û and redefine it as follows: Û = e −i φ/2σy e i θ/2σx e i φ/2σy = e i Ŵ /2 , (B12) where Ŵ = θ cos( φ) We can now expand Û in powers of θ since the fluctuations in this angle are small.Therefore, up to the first order, we obtain where δ d = (δ dx , 0, δ dz ) T .Similarly, we can calculate The Hamiltonian can be expressed compactly as This is the origin of Eq. ( 19) of the main text.

Limit Ez ≪ Es
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 I to be the ground state manifold (again setting N c g = 1 2 ).The energy ground states can be written as where j y = ± 1 /2 and s y = ± 1 /2 are the eigenvalues of the two operator Ĵy and Ŝy , 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 Ĵ on Ŝy .The two excited states can be written in a similar fashion as (B18) The effective Hamiltonian after the Schrieffer-Wolf transformation reads where ∆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 Now, we can re-write the effective Hamiltonian as where where ∑ k |k⟩ ⟨k| = 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 where ∆E = E c − E s where m = −m and The projector in the orbital/Majorana space can explicitly written in terms of Majorana fermions as 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 σ y .It is straightforward to show that Now, we are in a position to evaluate the sums of the shape The two integrals are and After calculating the other three sums of the shape of sum (B31) the effective Hamiltonian becomes In the next step, we evaluate each matrix element in angular momentum space separately.That is, we set up the matrix The matrix elements can be written in a compact way as where we introduced the spinor ψ = (ψ L↑ , ψ L↓ , ψ R↑ , ψ R↓ ) T and τ i are Pauli matrices that act on the R/L-space.Furthermore, we introduced the matrices σ ± = 1 2 (σ z ± iσ x ) The Kondo Hamiltonian can be found by decomposing the matrix in Pauli-matrices, which correspond to the components of J projected down onto the ground state manifold, that is Tr Such a decomposition leads to (neglecting inessential constants) where and σ 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 Further progress can be made by changing to a basis in which leaving us with the Hamiltonian where Finally, we diagonalize the matrix structure of ĤK with respect to projector by applying the transformation T = J y e i π 4 XJy which yields the Kondo part of Hamiltonian (20 where Ĵ = ( Jx /2, Jy /2, Jz /2) T is a vector of angular momentum operator.
The operator Ĥz = −E z ny can be projected onto the ground state manifold as well.Again, we introduce the 4π |ϕ, θ⟩ ⟨ϕ, θ|.Applying one identity from both sides on Ĥz gives the matrix where Applying the transformation T yields Adding the Hamiltonians (B46) and (B43) gives rise the the effective Hamiltonian (20).

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 δD.
In second-order perturbation theory, the Hamiltonian, after integrating out the high-energy modes, takes the form: where Here, Ĥ0 is the free Hamiltonian of the lead electrons, and ĤK is the Kondo Hamiltonian (B43).The states |a/b⟩ are energy eigenstates with energies C3) 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 where PH is a projector into the Hilbert space of energetically high-lying states (i.e., ϵ ∈ [D − δD, D] or ϵ ∈ [−D, −D + δD]), and Ĝ0 (E) = (E − Ĥ0 ) −1 .The transition matrix 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 (B43), which reads: where we use the Einstein convention for an implicit sum over i ∈ x, y, z.The sum of both diagrams is where we approximate the integrals over energies that are integrated out by the factor − νδD D .Using the relation the result simplifies to From equation (C8), we can see that the renormalized coupling constants take the form: These equations can be rewritten as a differential equation: Setting λ = λ ⊥ and defining the RG-times l = ln(D 0 /D), where D 0 is the initial bandwidth, the flow equations read 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 ∞.
First, we define the spin operator of the lead electrons in terms of the chiral fermions ψR/L as follows: Additionally, it is worth noting that Now, the Hamiltonian can be rewritten in terms of these spin operators as follows: Here, we introduce an anisotropy in the Kondo coupling constants.The lead electrons are bosonized in the following way: and (D5) 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: Here, we introduce the new fermion χ a = Fs √ 2πa0 e i √ 4πϕa .
We apply an Emery-Kivelson transformation U E.K. = e i √ 4πϕs Ĵy which manages to decouple the sf from the s bosons/fermions.The operators affected by the Emery-Kivelson transformation transform as follows: (D7) The transformed and fully fermionized Hamiltonian reads as follows: where ) and δλ y = λ Y y − 2πv F .The contribution of the Fermi velocity enters v F since the free Hamiltonian of the ϕ s gets also transformed.We also used the fact that the derivative of the Bose fields becomes in fermionic language the following: where ∶ ... ∶ denotes normal ordering.This is the origin of Eq. (31).
Appendix E: Observables 1. Green's function at the Toulouse point In this section, we calculate the Green's functions of the constituents in Hamiltonian (34) at the Toulouse point, which involves setting the coupling constants of the interaction terms to zero.In our case, this means we choose λ Y y = 0 and λ 1 y = 2πv F (i.e.δλ 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 and respectively.The transformation from imaginary time to Matsubara frequencies at zero temperature is given by: and where β is the inverse temperature.The retarded and imaginary time correlation functions are related by analytical continuation, which means: Now we calculate the free local Green's function of the lead fermions: G (0) (τ, x) = − ⟨T τ χ a (τ, x)χ † a (0, 0)⟩.In a diagonal basis, the Green's function reads: The local Green's function at x = 0 is: where ρ(ϵ) = ν is the density of states, which can be well modeled as being constant within the bandwidth 2D, leading to: ) .(E8) In the limit ω m ≪ D, the asymptotic behavior of the Green's function becomes: To calculate the Green's function for the Majorana fermions η and η x,y,z we introduce complex fermions which can be chosen in a very suggestive and physical way, that is (E10) The f electron can be understood as a ladder operator within the Hilbert space that belongs to the order parameter angular momentum Ĵ .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 where we ignored the Zeeman term for a moment.One can observe that in the case of no interaction (i.e., λ y = δλ y = 0), the Hamiltonian (E11) 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 (E11) at the Toulouse point in terms of the new fermions: where λ = λ⊥ √ 2πa0 .From the action, we can read off the Green's function: (E13) where Γ = λ2 νπ.Collecting everything leads to the fol-lowing list of Green's functions: We also find the Green's functions in imaginary time by applying the Fourier transform (E3).If we assume that |ω m | ≪ Γ, the imaginary time Green's functions behave asymptotically at large τ as follows: Furthermore, we deduce the Majorana Green's function from the fermionic one.We note that where it has been used that ⟨T τ η(τ )η y (0)⟩ = ⟨T τ η y (τ )η(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 ⟨T τ η y (τ )η y (0)⟩ = ⟨T τ η(0)η(τ )⟩.Hence, to fulfill equation (E24) must hold.It is straightforward to verify that similar relations also hold for ξ s,sf,c,cf , ζ s,sf,c,cf , η x and η y .

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 (E19)-(E23)).Using conformal transformations, the zero temperature correlation functions can be mapped onto their finite temperature counterparts as ) where ∆ is the scaling dimension of the operator O(τ ) and T is the temperature.Integrals over these correlation functions can be expressed in a more convenient form by the coordinate transformation x = tan(πT τ ) and become , which can be done exactly and evaluates to where x 0 = tan(πT τ 0 ) and 2 F 1 is the hypergeometric function.We have introduced a regularization τ 0 = 1 Γ since the correlation functions are only valid for long imaginary times.For further reference, we list a couple of special cases for T ≪ Γ (i.e.x 0 ≈ πT τ 0 ) to leading order in T : The local impurity susceptibility of V ∈ {X, Y, Z, Ĵx , Ĵx , Ĵx } is defined as where T and β are the temperature and inverse temperature, respectively.Later on, we are only interested in the static susceptibility However, for further reference, it will be beneficial to calculate the dynamical susceptibility for the Ĵy 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 η, η y , η x , η z .One finds Thus, the time-dependent correlation function evaluates to where we used η(τ ) = η τ as a shortcut notation.Using the former result, we obtain the static susceptibility where we introduced the cut of τ 0 = 1 /Γ to regularize the integral.
The dynamical susceptibility can be calculated along the same line, except that the integral reads Hence, the full dynamical susceptibility evaluates In the following, we only consider the static susceptibilities.For Ĵx and Ĵz , the calculation is slightly more complicated since these operators are affected by the Emery-Kivelson transformation.According to equation (D7), we find that: where we inserted the identity 1 = F s F † s .A similar calculation yields: To calculate the susceptibility, we need the correlator of the transformed spin operators: Thus, the susceptibility can be calculated by 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.

(E47)
χ X (T ) can be calculated along the same line, which yields the same result as for χ Z (T ) b. 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: and where we used ξ τ s = ξ s (τ ) as a short-hand notation.Equations (E48) and (E49) show that the scaling dimensions of O s and O sf are ∆ s = 2 and ∆ 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: Thus, we obtain the final result: The correction leads directly to the expression for the thermodynamic entropy and the specific heat: 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: Here, the symbol I represents the current operator, defined as: 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: 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: ∫ dx ( ψ † (x)σ y ψ(x) ± ψ † τy ψ(x)) ,

(E61)
where we changed into the ± basis after the second equal sign.The first part becomes: and, thus, is affected by the Emery-Kivelson transformation.The total magnetization becomes after the transformation: With the corresponding current: where x ij = (1−2δ ij ).Thus, the current correlation function expressed in Matsubara frequencies reads where ω m and ϵ n are bosonic and fermionic Matsubara frequencies, respectively.The main task will be to evaluate the Mastubara sum Note that this sum doesn't converge.However, we can regularize the sum by subtracting the ω m = 0 part, which gives an imaginary contribution and will, thus, vanish in the conductance anyway.Hence, we are left with the sum ) where ψ is the digamma function and ψ ′ is it's first derivative.Since we are interested in the lowtemperature correction, we expand the result up to the first leading order in the temperature and obtain )) ω m .(E69) Therefore, we find the current correlation function to be Inserting this result into the Kubo formula (E54) leads to the final result for the DC conductance is where G c 0 = e 2 /h is the perfect conductance.Note that ̵ h = 1 and, thus, h = 2π.

c. 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: Here we implicitly subtracted the C Jy Jy (0) by using δχ Ĵy (iω m ) instead of χ Ĵy (iω m ).The third equality is achieved by partial integration.We see that the C Jy Jy correlator is of third order in ω and, thus, does not contribute to the DC conductance since the term vanishes in the limit ω → 0.

d. 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: where ψ = (ψ + , ψ − ) T .The charge current expressed in terms of the ± fermions is given by: The current correlator evaluates as follows: ⟨T τ I ic (τ )I jc (0)⟩ = 3 x ij (λe) 2 G 0 (τ )G 0 (−τ ), (E77) where the free Green's function of the electrons at finite temperature is: which is valid for small ω.This result yields the conductance at weak coupling: The spin current in the basis of the ± lead fermions is defined as: (E81) For this case, the current-current correlator reads: This results in the spin conductance:

FIG. 1
FIG. 1. a) Schematic of physical setup.The grey box represents the Majorana Cooper pair box, which harbors a onedimensional 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 ψ+ and odd ψ− superpositions of the lead electrons are coupled to an effective quantum impurity with four internal states (square).The angular momentum Ĵ 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 Ĵ 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 λ and relative strength of Cooper pair spin fluctuations Es and polarizing field Ez.The parameter g = e S imp is defined through the zero temperature impurity entropy Simp.d) Absolute value of the charge conductance between the left and right wire in the limit Es/Ez ≫ 1.

FIG. 2 .
FIG. 2. The spectrum of Hamiltonian (4) as a function of N c g in the case Ec > 3Es/4.The labeling of parabolas corresponds to eigenvalues (Nc, J) of total charge and total angular momentum as defined in Tab.I.The crossing point of two red, solid parabolas denotes the value of N cg where it becomes favorable to add one unit of charge (one Cooper pair) onto the island and corresponds to Ec where Ec 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 ∆E = Ec − 3Es/4 must be overcome for such a process.

FIG. 3 .
FIG.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

FIG. 4 .
FIG. 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 λ⊥, dashed blue for Ez).The interaction terms are given by transparent rectangles (red for λ Y y , green for δλy).

FIG. 5 .
FIG. 5. Scattering processes that contribute to the renormalization of the Kondo Hamiltonian ∆ Ĥ are as follows: (I) Particle excitation: In this process, a low-energy electron state |b⟩ scatters at the impurity, denoted by Ĵ , into a high-energy state |c⟩ which, in turn, scatters back into a low-energy state |a⟩.(II) Hole excitation: In this scenario, a low-energy state |b⟩ scatters into a high-energy hole state |c⟩.The hole then scatters back into a low-energy electron state |a⟩.