Uncovering non-Fermi-liquid behavior in Hund metals: conformal field theory analysis of a SU(2) $\times$ SU(3) spin-orbital Kondo model

When using dynamical mean field theory to study Hund metals, one arrives at self-consistent impurity models in which bath and impurity both have spin and orbital degrees of freedom. If these are screened at different energy scales, $T_\mathrm{sp}<T_\mathrm{orb}$, the intermediate energy window is governed by a novel non-Fermi-liquid (NFL) fixed point, involving screened orbital degrees of freedom weakly coupled to an unscreened local spin. Here we characterize the resulting NFL behavior in detail for a Kondo model with an impurity in a (3 $\times$ 3)-dimensional SU(2) $\otimes$ SU(3) spin-orbital multiplet, tuned such that the NFL energy window is very wide. We find excellent agreement between conformal field theory predictions and numerical renormalization group results.


I. INTRODUCTION
Hund metals are multi-orbital materials with broad bands which are correlated via the ferromagnetic Hund coupling J, rather than the Hubbard interaction U . The coupling J implements Hund's rule, favoring electronic states with maximal spin. Examples include transition metal oxides with partially filled d-shells, such as ruthenates, or iron-based superconductors [1][2][3][4][5][6].
In Hund metals the interplay between spin and orbital degrees of freedom can lead to spin-orbital separation (SOS): the energy scales below which spin and orbital degrees are screened differ, T sp < T orb [5][6][7][8][9]. The lowenergy regime below T sp shows Fermi-liquid (FL) behavior. The intermediate SOS window, [T sp , T orb ], by contrast, shows incoherent behavior, featuring almost fully screened orbital degrees of freedom coupled to almost free spin degrees of freedom. Experimentally, the incoherent regime shows bad-metal behavior [10,11], hence it is of great interest to understand it theoretically. It has been conjectured to have non-Fermi-liquid (NFL) properties [5,12], but the nature of the putative underlying NFL state has not yet been clarified.
Here we explore its properties within the context of a minimal 3-orbital Hubbard-Hund model for Hund metals, proposed in Ref. [5] and studied extensively in Refs. [6][7][8][9]13]. A treatment of this model by dynamical mean field theory (DMFT) at 1 3 filling yields a self-consistent threeorbital Anderson-Hund (3oAH) model, in which bath and impurity both have spin and orbital degrees of freedom. The impurity hosts two electrons forming an antisymmetric orbital triplet and a symmetric spin triplet (S = 1), reflecting Hund's rule. At energies so low that charge fluctuations can be treated by a Schrieffer-Wolff transformation [6], the 3oAH model maps onto a 3-channel spin-orbital Kondo (3soK) model whose impurity forms a 3 × 3-dimensional SU(2) × SU(3) spin-orbital multiplet.
The 3oAH model exhibits SOS [5][6][7][8][9]. However, its orbital and spin screening scales cannot be tuned independently. The SOS window turns out to be rather small, masking the NFL behavior expected to occur within it. In this paper, we sidestep this limitation by instead studying the 3soK model and treating its exchange couplings as independent parameters, freed from the shackles of their 3oAH origin. This enables us to characterize the NFL fixed point obtained for T sp = 0, which also governs the intermediate NFL window if T sp T orb . We compute fixed point spectra and the scaling behavior of dynamical spin and orbital susceptibilities using both the numerical renormalization group (NRG) and conformal field theory (CFT), with mutually consistent results. In particular, we find an analytical explanation for a peculiar power law behavior for the imaginary part of the dynamical spin susceptibility, χ imp sp ∼ ω −6/5 , found previously in the SOS regime in DMFT studies of the 3-orbital Hubbard-Hund model for Hund metals [7,13,14].
Our CFT analysis builds on that devised by Affleck and Ludwig (AL) [15][16][17][18][19] for the k-channel Kondo model, describing k spinful channels exchange-coupled to an impurity with spin S, but no orbital degrees of freedom. If k > 2S, the impurity spin is overscreened. AL described the corresponding NFL point using a charge-spin-orbital U(1) × SU(2) k × SU(k) 2 Kac-Moody decomposition of the bath states, and fusing the spin degrees of freedom of impurity and bath using SU(2) k fusion rules. Here we generalize this strategy to our situation, where the impurity has spin and orbital "isospin" degrees of freedom: the NFL fixed point at T sp = 0 can be understood by applying SU(3) 2 fusion rules in the orbital sector, leading to orbital overscreening. If T sp is nonzero (but T orb ), the overscreened orbital degrees of freedom couple weakly to the impurity spin, driving the system to a FL fixed point. There both spin and orbital degrees of freedom are fully screened, in a manner governed by SU(6) 1 fusion rules.
The paper is structured as follows. Section II defines the 3soK model and discusses its weak-coupling RG flow. Section III presents our NRG results. Section IV gives a synopsis of our CFT results, summarizing all essential insights and arguments, while Sec. V elaborates the corresponding CFT arguments in more detail. A concluding section Sec. VI puts our results in a general perspective. Appendix A discusses a hybrid Anderson-Hund model, App. B a two-orbital Kondo model studied by Ye [20].
The Hamiltonian has U(1) ch ×SU(2) sp ×SU(3) orb symmetry. We label its symmetry multiplets by Q = (q, S, λ), with q the bath particle number relative to half-filling (the 3soK impurity has no charge dynamics, hence we may choose q imp = 0), S the total spin and λ a Young diagram denoting an SU(3) representation. The values of the spin, orbital and spin-orbital exchange couplings, J 0 , K 0 , I 0 , can be derived from the 3oAH model [6]. Here, instead, we choose them independently to tune T sp T orb . Aron and Kotliar [6] have performed a perturbative analysis of the RG flow of the 3soK model. Their Eqs. (8)(9)(10) describe the flow of the coupling vector, c(D) = (J, K, I), upon reducing the half-bandwidth, D, starting from c 0 = (J 0 , K 0 , I 0 ) at D 0 . Fig. 1 illustrates the resulting RG flow. There are several fixed points. The free-impurity fixed point, c * FI = (0, 0, 0), is unstable: for any nonzero c 0 one or more couplings flow towards strong coupling, and the D-values where J or K become of order unity yield estimates of T sp and T orb , respectively. For c 0 = (0, K 0 = 0, 0) (black arrows in Fig. 1), the system flows towards a NFL fixed point, c * NFL = (0, 1, 0). It is unstable against nonzero J 0 or I 0 . For I 0 = 0 the flow equations for J and K are decoupled, such that for a small but nonzero J 0 K 0 (red arrows) the flow first closely approaches c * NFL , until J grows large, driving it towards a FL fixed point, c * FL . Fig. 1(b) shows that the NFL regime (J K) governed by c * NFL can be large. For I 0 = 0, the J-and K-flows are coupled, hence the growth of K triggers that of J, accelerating the flow towards c * FL . In this case, the NFL energy window is rather small [cf. Fig. 1(c)]. For example, for c 0 = (0.1, 0.3, 0.5) (light green arrows), typical for the values obtained through a Schrieffer-Wolff 3oAH to 3soK mapping, the RG flow does not approach c * NFL very closely, thus fully developed NFL behavior is not observed.

III. NRG RESULTS
To study the RG flow in a quantitatively reliable manner, we solve the 3soK model using NRG [21][22][23], exploiting non-abelian symmetries using QSpace [22]. The bath is discretized logarithmically and mapped to a semiinfinite "Wilson chain" with exponentially decaying hoppings, and the impurity coupled to site 0. The chain is diagonalized iteratively while discarding high-energy states, thereby zooming in on low-energy properties: the (finite-size) level spacing of a chain ending at site k is of order ω k ∝ Λ −k/2 , where Λ > 1 is a discretization parameter. The RG flow can be visualized using NRG eigenlevel spectra, showing how the chain's lowest-lying eigenenergies, E, evolve when k is increased by plotting The E-level flow is stationary (ω k -independent) while ω k traverses an energy regime governed by one of the system's fixed points, but changes during crossovers between fixed points.
To analyze the NFL regime in detail, we choose I 0 = 0 and J 0 K 0 , so that the SOS window becomes very large, with T sp n T orb . Fig. 2(a) shows the NRG eigenlevel flow diagram c 0 = (10 ≠4 , 0.3, 0). We discern four distinct regimes, separated by three scales, T sp , T ss , T orb : (i) The free-impurity (FI) regime, Ê k > T orb , involves an unscreened impurity, with ground state multiplet Q = (0, 1, ) (brown line).
(ii) In the NFL regime, T ss < Ê k < T orb , two degen-erate multiplets, (1, 1 2 , •) and (1, 3 2 , •) (dashed-green and red lines) become the new ground state multiplets. Below the scale T orb , the impurity orbital isopin is thus screened into an orbital singlet, • , by binding one bath electron, which couples to the impurity spin 1 to yield a total spin of 1 2 or 3 2 . (iii) In the spin splitting (SS) regime, T sp < Ê k < T ss , the e ects of nonzero J 0 becomes noticeable, splitting apart (1, 1 2 , •) and (1, 3 2 , •), the latter drifting down. (iv) In the FL regime, Ê k < T sp , (≠2, 0, •) becomes the new ground state multiplet. Below the scale T sp , the spin 3 2 is thus screened into a spin singlet by binding three bath holes, yielding a fully screened impurity. Note the equidistant level spacing, characteristic of a FL.
To further elucidate the consequences of orbital and spin screening, we computed the impurity's zerotemperature orbital and spin susceptibilities, and analogous susceptibilities, ‰ bath orb , ‰ bath sp (involving J orb , J sp ) for the bath site coupled to it. To this end we used full-density-matrix (fdm) NRG [24] and adaptive broadening of the discrete NRG data [25]. Figures 2(c,d) show these susceptibilities on a log-log scale. ‰ imp orb and ‰ imp sp each exhibit a maximum, at two widely di erent scales, T orb and T sp , coinciding with the onset of the stationary NFL or FL regimes in Fig. 2(a), respectively. Moreover, the four susceptibilities ‰ imp,bath orb,sp all exhibit kinks at a coinciding energy scale, T ss , matching the onset of the SS regime in Fig. 2(a). If Ê lies within one of the regimes NFL, SS or FL, the susceptibilities all show behavior consistent with power laws (grey lines). These power laws can all be explained by CFT, as discussed in Sec. IV. Here we focus on their qualitative features, which by themselves give striking clues about the nature of orbital and spin screening.
In the NFL regime, where ‰ imp orb decreases with decreasing Ê, it exhibits the same power law as ‰ bath orb . In this sense, the impurity's orbital pseudospin has taken on the same character as that of the bath site it couples to, indicative of orbital screening -in the parlance of AL's CFT analysis, it has been "absorbed" by the bath. This power law, Ê 1/5 , is non-trivial, di ering from the Ê 1 expected for fully screened local degree of freedom. This indicates that the local orbital degree of freedom, even while being screened, is still somehow a ected by the spin sector. The converse is also true: the onset of orbital screening at T orb is accompanied by a change in behavior for both spin susceptibilities, ‰ imp sp and ‰ bath sp . Both increase with decreasing Ê, with di erent powers, indicative of the absence of spin screening in the NFL regime. The exponent for the impurity spin susceptibility, ‰ imp sp ≥ Ê ≠11/6 , is remarkably large in magnitude. large, with T sp ≪ T orb . Fig. 2(a) shows the NRG eigenlevel flow diagram c 0 = (10 −4 , 0.3, 0). We discern four distinct regimes, separated by three scales, T sp , T ss , T orb : (i) The free-impurity (FI) regime, ω k > T orb , involves an unscreened impurity, with ground state multiplet Q = (0, 1, ) (brown line).
(ii) In the NFL regime, T ss < ω k < T orb , two degenerate multiplets, (1, 1 2 , •) and (1, 3 2 , •) (dashed-green and red lines) become the new ground state multiplets. Below the scale T orb , the impurity orbital isopin is thus screened into an orbital singlet, • , by binding one bath electron, which couples to the impurity spin 1 to yield a total spin of 1 2 or 3 2 . (iii) In the spin splitting (SS) regime, T sp < ω k < T ss , the effects of nonzero J 0 becomes noticeable, splitting apart (1, 1 2 , •) and (1, 3 2 , •), the latter drifting down. (iv) In the FL regime, ω k < T sp , (−2, 0, •) becomes the new ground state multiplet. Below the scale T sp , the spin 3 2 is thus screened into a spin singlet by binding three bath holes, yielding a fully screened impurity. Note the equidistant level spacing, characteristic of a FL.
To further elucidate the consequences of orbital and spin screening, we computed the impurity's zerotemperature orbital and spin susceptibilities, and analogous susceptibilities, χ bath orb , χ bath sp (involving J orb , J sp ) for the bath site coupled to it. To this end we used full-density-matrix (fdm) NRG [24] and adaptive broadening of the discrete NRG data [25]. Figures 2(c,d) show these susceptibilities on a log-log scale. χ imp orb and χ imp sp each exhibit a maximum, at two widely different scales, T orb and T sp , coinciding with the onset of the stationary NFL or FL regimes in Fig. 2(a), respectively. Moreover, the four susceptibilities χ imp,bath orb,sp all exhibit kinks at a coinciding energy scale, T ss , matching the onset of the SS regime in Fig. 2(a). If ω lies within one of the regimes NFL, SS or FL, the susceptibilities all show behavior consistent with power laws (grey lines). These power laws can all be explained by CFT, as discussed in Sec. IV. Here we focus on their qualitative features, which by themselves give striking clues about the nature of orbital and spin screening.
In the NFL regime, where χ imp orb decreases with decreasing ω, it exhibits the same power law as χ bath orb . In this sense, the impurity's orbital pseudospin has taken on the same character as that of the bath site it couples to, indicative of orbital screening -in the parlance of AL's CFT analysis, it has been "absorbed" by the bath. This power law, ω 1/5 , is non-trivial, differing from the ω 1 expected for fully screened local degree of freedom. This indicates that the local orbital degree of freedom, even while being screened, is still somehow affected by the spin sector. The converse is also true: the onset of orbital screening at T orb is accompanied by a change in behavior for both spin susceptibilities, χ imp sp and χ bath sp . Both increase with decreasing ω, with different powers, indicative of the absence of spin screening in the NFL regime. The exponent for the impurity spin susceptibility, χ imp sp ∼ ω −11/6 , is remarkably large in magnitude.
This highly singular behavior -our perhaps most unexpected result -indicates that the strength of spin fluctuations is somehow amplified by the onset of orbital screening. Our CFT analysis below will reveal the reason for this: orbital screening is accompanied be a renormalization of the local bath spin density at the impurity site.
Upon entering the SS regime, all susceptibility lines show a kink, i.e. change in power law, such that the impurity and bath exponents match not only in the orbital sector, χ imp orb ∼ χ bath orb , but now also in the spin sector, χ imp sp ∼ χ bath sp . The latter fact indicates clearly that bath and impurity spin degrees of freedom have begun to interact with each other. However, this is only a precursor to spin screening, since the spin susceptibilities still increase with decreasing ω, albeit with a smaller exponent, χ imp,bath sp ∼ ω −6/5 , than in the NFL regime. Full spin screening only sets in the FL regime, where the spin susceptibilities χ imp,bath sp show the ω 1 behavior characteristic of a FL. We expect this behavior also for the orbital susceptibilities, but have not been able to observe it directly, since our results for χ imp,bath orb become numerically unstable when dropping below 10 −5 [as indicated by dotted lines in Figs. 2(c,d)].
In the following two sections we explain how the above NRG results can be understood using CFT arguments.

IV. CFT ANALYSIS: SYNOPSIS
This section presents a synopsis of our CFT analysis. It aims to be accessible also to readers without in-depth knowledge of AL's CFT work on Kondo models. A more elaborate discussion of CFT details follows in Sec. V.

A. NFL regime
Following AL [15][16][17][18][19], the fixed points of the 3soK model can be described using (1 + 1)-dimensional boundary CFT. First U(1) × SU(2) 3 × SU(3) 2 non-abelian bosonization of H bath is used to decompose the bath into charge, spin and orbital excitations, with free-fermion eigenenergies E(q, S, λ) [Eq. (9a) in Sec. V A]. The NRG spectrum in the NFL fixed point regime can be reproduced analytically (see Table II below) by combining the bath and impurity degrees of freedom, with quantum numbers (q, S, λ) and Q imp = (0, 1, ), using the following "fusion hypothesis" (inspired by and generalizing that of AL [15][16][17][18][19]): The orbital degrees of freedom are combined, λ⊗λ imp = ⊕ λ , using the fusion rules of the SU(3) 2 Kac-Moody algebra, reflecting the screening of the impurity orbital isospin. By contrast, the spin degrees of freedom are combined, S ⊗ S imp = ⊕ S , using the fusion rules of the SU(2) Lie algebra, reflecting the fact that at c * NFL , where J 0 = I 0 = 0, the impurity spin is a spectator, decoupled from the bath. For the same reason, the set of excitations (q, S , λ ) so obtained have energies given by E(q, S, λ ), not E(q, S , λ ). Table I exemplifies a few many-body states obtained via this "single fusion" scheme. In particular, the degenerate ground state multiplets of c * NFL , (1, 1 2 , •) and (1, Fig. 2), arise via fusion of a one-particle bath excitation, (+1, 1 2 , ), with the impurity, (0, 1, ). The fixed point c * NFL is characterized by a set of local "boundary operators." These can be obtained [17][18][19] via a second fusion step ("double fusion"), which combines the single fusion excitations, (q, S , λ ), with the conjugate impurity representation,Q imp = (0, 1, ). Each resulting multiplet, (q, S , λ ), is associated with a boundary operatorÔ with the same quantum numbers, a scaling dimension given by ∆ = E(q, S, λ ) (cf. Table I), and correlators Ô (t)Ô(0) ∼ t −2∆ and Ô ||Ô ω ω 2∆−1 . Each boundary operator can be viewed as the renormalized version, obtained via orbital screening, of some bare local operator having the same quantum numbers.
To explain the power laws found in the NFL regime of Figs Here Φ orb has (0, 0, ) (same as T, J orb ) and dimension ∆ orb = 3 5 , while Φ sp has quantum numbers (0, 1, •) (same as S, J sp ) and ∆ sp = 2 5 , (cf. Table I). The local impurity and bath orbital susceptibilities thus both scale as and the bath spin susceptibility as By contrast, the impurity spin S is not renormalized, because at the fixed point c * NFL , where J 0 = 0, it is decoupled from the bath. Thus its scaling dimension is zero. The leading behavior of χ imp sp , obtained in second-order perturbation theory in the renormalized spin exchange interaction, is proportional to the Fourier transform of S(t)S(0)( dt J 0 S · Φ sp ) 2 , and power counting yields The above predictions are all borne out in Figs. 2(c,d).
The remarkably large negative exponent, − 11 5 , for χ imp sp reflects the fact that the renormalized spin exchange interaction J 0 S · Φ sp , with scaling dimension 2 5 < 1, is a relevant perturbation. Its strength, though initially miniscule if J 0 1, grows under the RG flow, causing a crossover away from c * NFL for ω T ss . This is reflected in the level crossings around T ss in the NRG eigenlevel flow of Fig. 2. In particular, the double-fusion parent multiplets for Φ orb and Φ sp , namely (0, 1, ) and (0, 0, ), Table I. Left: Five low-lying free-fermion multiplets (|FS denotes the Fermi sea), with quantum numbers (q, S, λ), multiplet dimensions d and energies E(q, S, λ). Center: "Single fusion" with an impurity Qimp = (0, 1, ) leads to multiplets with quantum numbers (q, S , λ ), dimensions d , eigenenergies E = E(q, S, λ ), and excitation energies δE = E − E min . Right: "Double fusion", which fuses multiplets from the middle column with an impurity in the conjugate [26] representationQimp = (0, 1, ), yields the multiplets (q, S , λ ). These characterize the CFT boundary operatorsÔ, with scaling dimensions ∆ = E(q, S, λ ). Φ orb and Φsp are the leading boundary operators in the orbital and spin sectors, respectively. In the spin-splitting regime, their roles are taken byΨ orb andΨsp, respectively. "Bare" free-fermion versions of these boundary operators, having the same quantum numbers, are listed on the very right. For clarity, not all possible multiplets arising from single and double fusion multiplets are shown. A more comprehensive list is given in Table II. free fermions single fusion double fusion undergo level crossings with the downward-moving multiplets (−1, 1 2 , ) and (−3, 1 2 , ), respectively. These in turn are double-fusion parent multiplets for the boundary operatorsΨ orb andΨ sp , with scaling dimensions ∆ orb =∆ sp = 9 10 (Table I). To explain the SS regime of Figs. 2(c,d), and particularly that there the power laws for χ imp and χ bath match in both the orbital and spin sectors, we posit the RG replacements Here S +Ψ sp is symbolic notation for some linear admixture of both operators, induced by the action of the renormalized spin exchange interaction. We thus obtain and the leading contribution to χ imp sp and χ bath sp , obtained by perturbing S(t)S(0) to second order in SΨ sp [27], is This reproduces the power laws found in Figs. 2(c,d).
Remarkably, χ imp sp ∼ ω −6/5 behavior has also been found in studies of the self-consistent 3oAH model arising in our DMFT investigations of the 3-orbital Hubbard-Hund model for Hund metals. For the 3oAH model the spin-orbital coupling I 0 in Eq. (1) is always nonzero, so that a fully-fledged NFL does not emerge -instead, T orb and T ss effectively coincide (as further discussed in App. A). However, the SS regime between T sp and T ss T orb can be quite wide, typically at least an order of magnitude. In Fig. 3(c) of Ref. [7], the behavior of χ imp sp in this regime (between the vertical solid and black lines there) is consistent with ω −6/5 behavior. Though this fact was not noted in Ref. [7], it was subsequently pointed out in Ref. [14] (see Fig. S1 of their supplement). Behavior consistent with χ imp sp ∼ ω −6/5 can also be seen in Figs. 5.1(c,d) of Ref. [13], as discussed on p. 152 there. The explanation for this behavior presented here, via a CFT analysis of the NFL and SS regimes, is one of the main results of this work, and the justification for the first line of the title of this paper.

B. Fermi liquid regime
As mentioned above, the low-energy regime below T sp is a FL. The fixed-point spectrum at c * FL can be obtained by fusing a free-fermion spectrum with an impurity with , representing the effective local degree of freedom obtained after completion of orbital screening (see Table III). Since the ground state describes a fully screened orbital and spin singlet, it actually is the singlet of a larger symmetry group, U(1) × SU (6). Indeed, the fixed-point spectrum at c * FL matches that of the U(1) × SU(6) symmetric Kondo model. We demonstrate this, using both NRG and CFT with SU(6) 1 fusion rules, in Sec. V E (see Table IV). The FL nature of the ground state is also borne out by the ω 1 scaling of χ imp,bath sp in the FL regime of Figs. 2(c,d).

V. CFT ANALYSIS: DETAILS
We now provide technical details for our CFT analysis of the NFL and FL fixed points of the three-orbital Kondo (3soK) model discussed in Sections III and IV. We closely follow the strategy devised by Affleck and Ludwig (AL) for their pioneering treatment of the strongcoupling fixed points of Kondo models [15][16][17][18][19] (for pedagogical reviews, see [28,29] and Appendices A-D of [30]). In a series of works, they considered a variety of Kondo models of increasing complexity. These include the standard 1-channel, SU(2) spin Kondo model with a spin exchange interaction between bath and impurity with U(1) × SU(2) 1 symmetry; a spinful k-channel bath coupled to an SU(2) impurity (U(1) × SU(2) k × SU(k) 2 symmetry), and an SU(N ) k-channel bath coupled to an Our 3soK model features a spinful 3-channel bath and an SU (2) The impurity multiplet is a direct product of a spin triplet (S = 1) and an orbital triplet (λ = ). Its direct-product structure is more general than any of the cases considered by AL. (A 2-channel version of our model, with U(1) × SU(2) 2 × SU(2) 2 symmetry, has been studied by Ye [20], which we discuss in App. B below.) However, at the NFL fixed point c * NFL of our model, where J 0 = I 0 = 0, the impurity's SU(2) spin is a decoupled, three-fold degenerate spectator degree of freedom. Hence AL's analysis [19] can be employed, with N = 3 and k = 2 channels, modulo some minor changes to account for the impurity spin.
By contrast, in the spin-splitting (SS) crossover regime the spin exchange interaction comes to life, so that the impurity's SU(2) spin degrees of freedom cease to be mere spectators. This regime thus lies outside the realm of cases studied by AL; in particular, it is not manifestly governed by the NFL fixed point c * NFL , or other welldefined fixed point. Correspondingly, our discussion of this crossover regime in Sec. V C 2 is more speculative than that of the NFL regime, though our heuristic arguments are guided by and consistent with our NRG results.
Finally, for our model's FL fixed point, c * FL , we are again in well-chartered territory: it can be understood by applying AL's strategy to an SU(6) 1-channel bath coupled to an SU(6) impurity (U(1)×SU(6) 1 symmetry).
Below we assume the reader to be familiar with AL's work and just focus on documenting the details of our analysis. Section V A describes how the free-fermion bath spectrum is decomposed into charge, spin and orbital excitations using U(1) × SU(2) 3 × SU(3) 2 non-abelian bosonization. Section V B derives the finite-size spectrum and boundary operators of the NFL fixed point via single and double fusion, using the fusion rules of the SU(3) 2 Kac-Moody algebra in the orbital sector and the SU(2) Lie algebra in the spin sector. Section V C describes the computation of the spin and orbital susceptibilities in the NFL and SS regimes, linking AL's strategy for computing such quantities to the compact scaling arguments used in Section IV. Section V D presents our results for the impurity spectral function in the NFL regime. Finally, Sec. V E, devoted to the FL regime, shows how its spectrum can be derived using either SU(2) 3 fusion rules in the spin sector, or SU(6) 1 fusion rules in the flavor (combined spin+orbital) sector.

A. Non-abelian U(1) × SU(2)3 × SU(3)2 bosonization
The first step of AL's CFT approach for multi-channel Kondo models is to use non-abelian bosonization to decompose the bath degrees of freedom into charge, spin and orbital excitations in a manner respecting the symmetry of the impurity-bath exchange interactions. Our 3soK model features a spinful 3-channel bath, with We assume a linear dispersion, ε p = v F p, with v F = 1. Using non-abelian bosonization with the U(1)×SU(2) 3 ×SU(3) 2 Kac-Moody (KM) current algebra, the spectrum of bath excitations can be expressed as (see [15,17], or Appendix A of [30]) Here κ 2 (S) and κ 3 (λ) are the eigenvalues of the quadratic Casimir operators of the SU(2) and SU (3) Lie algebras, respectively [31]. q ∈ Z is the U(1) charge quantum number, S ∈ 1 2 Z the SU(2) spin quantum number and λ = (λ 1 , λ 2 ) the SU(3) orbital quantum number, denoting a Young diagram with λ j j-row columns: Finally, ∈ Z counts higher-lying "descendent" excitations; for present purposes it suffices to set = 0. The free-fermion spectrum of H bath is recovered from Eq. (9a) by imposing free-fermion "gluing conditions", allowing only those combinations of quantum numbers (q, S, λ) for which E(q, S, λ) is an integer multiple of 1 2 . The resulting multiplets are listed in the left column ("free fermions") of Table II below.

B. Non-Fermi-liquid fixed point
We now focus on the NFL fixed point of the 3soK model, at c * NFL , where (J 0 , K 0 , I 0 ) = (0, 1, 0). According to AL's general strategy, the orbital isospin T can be then "absorbed" by the bath through the substitution J orb,n → J orb,n = J orb,n + T .
Here J orb,n and J orb,n are Fourier components (n being a Fourier index) of the bare and bulk orbital isospin currents, respectively, defined for a bath in a finitesized box. (The local bath operator J orb is proportional to n∈Z J orb,n .) The right-hand side of Eq. (10) is reminiscent of the addition of Lie algebra generators, S = S +S, when performing a direct product decomposition, S ⊗S = ⊕ S , of SU(2) multiplets. The terms Figure 3. Schematic depiction of single fusion (left) and double fusion (right), for the four multiplets giving rise to the boundary operators Φ orb , Φsp,Ψ orb ,Ψsp discussed in Section IV (corresponding to rows 1,3,4,5 in Table I). Filled arrows represent electrons, empty arrows represent holes. An electron with spin ↑ and a hole with spin ⇓ (missing electron with spin ↑) can be combined to annihilate each other, as indicated by small dashed circles in the last column. Our cartoons depict the impurity using a fermionic representation, as would be appropriate for the 3oAH model, even though the 3soK impurity has no charge dynamics. In the 'single fusion' column, excitations of the free bath are fused with the impurity, Qimp = (0, 1, ), to obtain the eigen-multiplets of the full system at the NFL fixed point. In the 'double fusion' column (right), the single fusion results are fused with the conjugate [26] impurity representation,Qimp = (0, 1, ). Each of the resulting multiplets is associated with a boundary operator having the same quantum numbers. Colors relate the multiplets obtained after single fusion to the corresponding lines in Fig. 2.
added in Eq. (10), however, generate two different algebras: J orb,n are generators of the SU(3) 2 KM algebra, T of the SU(3) Lie algebra. AL proposed a remarkable "fusion hypothesis" for dealing with such situations (and confirmed its veracity by detailed comparisons to Bethe ansatz and NRG computations). For the present context their fusion hypothesis states: the eigenstates of the combined bath+impurity system can be obtained by combining (or "fusing") their orbital degrees of freedom, Having discussed orbital fusion, we now turn to the spin sector -how should the impurity's spectator spin be dealt with? This question goes beyond the scope of AL's work, who did not consider impurities with spectator degrees of freedom. We have explored several spin fusion strategies and concluded that the following one yields spectra consistent with NRG: In parallel to orbital fusion, the bath and impurity spin degrees should be combined too, as S ⊗ S imp = ⊕ S , but using the fusion rules of the SU(2) Lie algebra, not the SU(2) 3 KM algebra. Heuristically, the difference -KM vs. Lie -between the algebras governing orbital and spin fusion reflects the fact that the bath and impurity are coupled in the orbital sector, where the bath "absorbs" the impurity orbital isospin, but decoupled in the spin sector, where the impurity spin remains a spectator.
The fusion of bath and impurity degrees of freedom, called "single fusion" by AL, is illustrated schematically in the left part of Fig. 3 for four selected multiplets. Table II gives a comprehensive list of low-lying multiplets obtained in this manner. On the left it enumerates the 14 lowest-lying multiplets, (q, S, λ), of the free bath, with dimensions d and energies E(q, S, λ). Fusing these with a Q imp = (0, 1, ) impurity yields the multiplets, (q, S , λ ), listed in the center. Their energies are given by E = E(q, S, λ ), not E(q, S , λ ), since at the NFL fixed point, where J 0 = I 0 = 0, the impurity spin is decoupled from the bath.
The single-fusion excitation energies, δE = E − E min , relative to the lowest-lying multiplet (E min = 7/30) are in good agreement (deviations 10%) with the values, E NRG , found by NRG (for K 0 = 0.3, J 0 = I 0 = 0) for multiplets with corresponding quantum numbers. The agreement improves upon decreasing the NRG discretization parameter Λ (here Λ = 2.5 was used). This remarkable agreement between CFT predictions and NRG confirms the applicability of the SU(2) ⊗ SU(3) 2 fusion hypothesis proposed above.  E(q, S, λ), computed using Eqs. (9) and Table S1 of [32]. Center: "Single fusion" with an Qimp = (0, 1, ) impurity, using SU(2) fusion rules in the spin sector and SU(3)2 fusion rules (listed in Table S2 of [32]) in the orbital sector. This yields multiplets (q, S , λ ), with dimensions d , energies E = E(q, S, λ ) and excitation energies δE = E − E min . These are compared to the values, E NRG , computed by NRG for (J0, K0, I0) = (0, 0.3, 0). The NRG energies have been shifted and rescaled such that the lowest energy is zero and the second-lowest values for E NRG and δE match. The single fusion and NRG spectra agree well (deviations 10%). Right: "Double fusion", which fuses multiplets from the middle column with an impurity in the conjugate [26] representationQimp = (0, 1, ), yields the quantum numbers (q, S , λ ). These characterize the CFT boundary operatorsÔ, with scaling dimensions ∆ = E(q, S, λ ). free fermions single fusion, with Qimp = (0, 1, ) NRG double fusion, withQimp = (0, 1, )       As mentioned in Section IV, the fixed point c * NFL is characterized by a set of local operators, called "boundary operators" by AL (since they live at the impurity site, i.e. at the boundary of the two-dimensional spacetime on which the CFT is defined). These can be obtained by a second fusion step, called "double fusion" by AL: the multiplets (q, S , λ ) obtained from single fusion are fused with the conjugate [26] impurity representation,Q imp = (0, 1, ), to obtain another set of multiplets, (q, S , λ ), listed on the right of Fig. 3 and Table II. Each such multiplet is associated with a boundary operatorÔ with the same quantum numbers and scaling dimension ∆ = E = E(q, S, λ ). The operators called Φ orb and Φ sp are the leading boundary operators (with smallest scaling dimension) in the orbital and spin sectors, respectively. They determine the behavior of the orbital and spin susceptibilities in the NFL regime (see Sec. V C) below. In the spin-splitting (SS) regime, their role is taken by the operatorsΨ orb andΨ sp , respectively, as discussed in Section IV.

C. Scaling behavior of the susceptibilities
In this subsection, we compute the leading frequency dependence of the dynamical spin and orbital susceptibilities. We begin with the NFL regime, where we directly follow the strategy used by AL in Sec. 3.3 of [17] and show how it reproduces the results presented in Section IV. Thereafter we discuss the SS regime, which has no analogue in AL's work, using somewhat more heuristic arguments.

NFL regime
At the NFL fixed point, the impurity's orbital isospin, T, has been fully absorbed into the bath orbital current, J orb [cf. Eq. (10)]. From this perspective, the impurity orbital susceptibility, χ imp orb , is governed by the leading local perturbation of the bulk orbital susceptibility, χ bulk orb ∼ J bulk orb ||J bulk orb ω , where J bulk orb (t) = ∞ −∞ dxJ orb (t, x) ∼ J orb,n=0 is the bulk orbital current. The leading local perturbations are those combinations of boundary operators (found via double fusion, Table II) having the smallest scaling dimensions and the same symmetry as the bare Hamiltonian [16][17][18].
In the orbital sector, the leading boundary operator is Φ orb , with quantum numbers (0, 0, ) and scaling dimension ∆ orb = 3 5 (cf . Tables I and II). The orbital current, J orb , has the same quantum numbers. Its first descendant, J orb,−1 , can be combined with Φ orb to obtain an orbital SU(3) singlet boundary operator, H orb = J orb,−1 · Φ orb , with scaling dimension 1 + ∆ orb = 1 + 3 5 . This is the leading irrelevant (dimension > 1) boundary perturbation to the fixed-point Hamiltonian in the orbital sector. Its contribution to the impurity orbital susceptibility, χ imp orb ∼ χ bulk orb , evaluated perturbatively to second order, is The last line follows by power counting (J bulk orb has di-mension 0, each time integral dimension −1). The local bath site coupled to the impurity will show the same behavior, χ bath orb ∼ ω 1/5 , since the orbital exchange interaction strongly couples its orbital isospin, J orb , to T -indeed, J orb is constructed from a linear combination of both these operators [cf. Eq. (10)].
The above results can be obtained in a more direct way by positing that at the NFL fixed point, orbital screening causes both T and J orb to be renormalized into the same boundary operator, Φ orb . We then obtain reproducing Eq. (11). This is the argument presented in Section IV. We next turn to the spin sector. Exactly at the NFL fixed point, where J 0 = I 0 = 0, the impurity spin S is decoupled from the bath. At c * NFL it hence has no dynamics, scaling dimension 0, and a trivial spin susceptibility, χ imp sp (ω) ∼ δ(ω). By contrast, χ bath sp , the susceptibility of J sp , the local bath spin coupled to the impurity, does show non-trivial dynamics at the fixed point. The reason is that some of the boundary operators induced by orbital screening actually live in the spin sector (a highly non-trivial consequence of non-abelian bosonization and orbital fusion). The leading boundary operator in the spin sector is Φ sp , with quantum numbers (0, 1, •) and scaling dimension ∆ sp = 2 5 (cf . Tables I and II). It can be combined with the first descendant of the (bare, unshifted) spin current to obtain a spin SU(2) singlet boundary operator, H sp = J sp,−1 · Φ sp , with scaling dimension 1 + ∆ sp = 1 + 2 5 . This is the leading irrelevant boundary perturbation to the fixed-point Hamiltonian in the spin sector. Its contribution to the local bath spin susceptibility, χ bath sp ∼ χ bulk sp , evaluated to second order, is This result, too, can be obtained more directly, by positing that J sp is renormalized to Φ sp , with as argued in Section IV.
If the system is tuned very slightly away from the NFL fixed point, J 0 1, I 0 = 0, the impurity spin does acquire non-trivial dynamics, due to the action of the spin exchange interaction, J 0 S · J sp . According to the above argument, orbital screening renormalizes it to J 0 S · Φ sp . Its second-order contribution to the impurity spin sus-ceptibility is The occurrence of such a large, negative exponent for the spin susceptibility is very unusual. It reflects the fact that near (but not at) the NFL fixed point the impurity spin is almost (but not fully), decoupled from the bath, and hence able to "sense" that orbital screening modifies the bath spin current in a non-trivial manner.

Spin-slitting regime
The renormalized exchange interaction J 0 S · Φ sp is a relevant perturbation, with scaling dimension 2 5 < 1. It grows under the RG flow, eventually driving the system away from the NFL fixed point and into a crossover regime, T sp < ω < T ss , called spin-splitting (SS) regime in Section III. In the NRG flow diagram of Fig. 2(a), this regime is characterized by level crossings, extending over several orders of magnitude in energy, rather than a stationary level structure. Hence the SS regime cannot be characterized by proximity to some well-defined fixed point. (A stationary level structure, characteristic of a FL fixed point, emerges only after another crossover, setting in at the scale T sp .) Nevertheless, Figs. 2(c,d) show that the local orbital and spin susceptibilities do exhibit well-defined power law behavior in the SS regime, We define the width of the SS regime as the energy range showing this behavior. It extends over about three orders of magnitude, independent of J 0 and I 0 -increasing either of these couplings rigidly shifts the SS regime to larger energies without changing its width (see Fig. 4), i.e. the ratio T sp /T ss is independent of these couplings. The latter fact leads us to conjecture that the NFL fixed point does, after all, govern the SS regime too, though "from afar" rather than from up close. In technical terms, we conjecture that the leading behavior in the SS regime is governed by two different boundary operators,Ψ orb andΨ sp , with scaling dimensions, ∆ orb =∆ sp = 9 10 (cf. Tables I and II) instead of the boundary operators Φ orb and Φ sp governing the NFL regime. This conjecture is encoded in the equation above Eq. (7). It states that J orb and T are both renormalized toΨ orb , causing χ bath orb and χ imp orb to scale with the same power, 10 !12 10 !9 10 !6 10 !3 10 0 ! 10 !5 10 0 10 5 10 !12 10 !9 10 !6 10 !3 10 0 10 !12 10 !9 10 !6 10 !3 10 0 J 0 = 10 !2 (c) 10 !12 10 !9 10 !6 10 !3 10 0   and that J orb and T are both renormalized to S +Ψ sp , causing χ bath sp and χ imp sp to scale with the same power, The latter result is obtained in a manner analogous to Eq. (15), with S · Φ replaced by SΨ sp [27].

D. Impurity spectral function
We next consider the leading frequency dependence of the impurity spectral function in the NFL regime. For a Kondo-type impurity, this function is given by [33].
For ω > 0 our NRG results are consistent with ImT ∼ ω 3/5 (cf. Fig. 5). This suggests that the prefactor of H orb is much larger than that of H sp , presumably because the computation was done for J 0 = I 0 = 0. For ω < 0, by contrast, our numerical results do no exhibit clear power law behavior for small |ω|, implying that ImT does not have particle-hole symmetry. This is not surprising: the 3soK model itself breaks particle-hole symmetry, since under a particle-hole transformation, the impurities orbital multiplet is mapped to . We suspect that the prefactor of the |ω| ∆ orb contribution to ImT vanishes for ω < 0 for the impurity orbital representation , such that only subleading boundary operators, with dimensions ∆ ≥ 9/10 (cf . Table II), determine the small-ω scaling behavior. However, a detailed understanding of this matter is still lacking.

E. Fermi-liquid fixed point
In this section we show how the FL spectrum at the fixed point c * FL can be derived analytically. This can be done in two complementary ways. The first uses SU(2) 3 fusion in the spin sector, the second SU(6) 1 fusion in the flavor (combined spin+orbital) sector.

Fermi-liquid spectrum via SU(2)3 fusion
It is natural to ask whether the FL spectrum at c * FL can be derived from the NFL spectrum of c * NFL via some type of fusion in the spin sector, reflecting spin screening induced by the spin exchange interaction. For example, we have tried the following simple strategy ("naive spin fusion"): when setting up the fusion table (Table  II), the bath and impurity spin degrees of freedom are combined, S ⊗ S imp = ⊕ S , using the fusion rules  Table II, but here single fusion of bath and impurity multiplets in the charge and spin sectors is performed using U(1) × SU(2)3 fusion rules (listed in Table S3 of [32]). Moreover, we choose Qimp = (1, 3 2 , •) for the impurity, representing the effective local degree of freedom obtained after the completion of orbital screening. The resulting multiplets (q , S , λ) have eigenenergies E = E(q , S , λ) and excitation energies δE = E − E min . The NRG energies, computed for (J0, K0, I0) = (10 −4 , 0.3, 0), have been shifted and rescaled such that the lowest energy is zero and the second-lowest values for E NRG and δE match. The single fusion and NRG spectra agree very well (deviations 2%). free fermions single fusion, with Qimp = (1, 3 2   We suspect that this failure is due to the fact that the RG flow does not directly pass from the NFL regime into the FL regime, but first traverses the intermediate SS regime. In the latter, the degeneracy between the two degenerate ground state multiplets of the NFL regime, (1, 1 2 , •) and (1, 3 2 , •), is lifted, in a manner that seems to elude a simple description via a modified spin fusion rule.
Instead, the FL spectrum can be obtained via the following arguments. The ground state multiplet of the SS regime, (1, 3 2 , •), describes an effective local degree of freedom coupled to a bath in such a manner that one bath electron fully screens the impurity orbital isospin, while their spins add to a total spin of 1 2 + 1 = 3 2 [see Fig. 2(b)]. Let us view this as an effective impurity with Q imp = (1, 3 2 , •). If we combine its charge and spin degrees of freedom with those of a free bath, using q + q imp = q and S ⊕ S imp = ⊕ S , fused according to the SU(2) 3 KM algebra, the resulting single-fusion spectrum fully reproduces the FL spectrum found by NRG, as shown in see Table III.

Fermi-liquid spectrum via SU(6)1 fusion
The FL ground state of the 3soK model is a fully screened spin and orbital singlet. It is therefore natural to expect that the FL spectrum has a higher symmetry, namely that of the group U(1) × SU(6), which treats spin and orbital excitations on an equal footing. This is indeed the case: we will now show that the FL spectrum of the 3soK model discussed above matches that of an SU(6) Kondo model which does not distinguish between spin and orbital degrees of freedom. We consider a bath with six flavors of electrons, H bath = p 6 ν=1 ε p ψ † νσ ψ νσ and an impurity-bath coupling of the form J U U · J fl . Here J fl is the flavor density at the im- where ⁄ a are SU(6) matrices in the fundamental representation, and U describes the impurity's SU(6) flavor isospin, chosen in the fully antisymmetric representation, . The latter has dimension 15, representing the ! 6 2 " ways of placing two identical particles into six available states. Fig. 6 shows the NRG finite-size eigenlevel flow for this model. It exhibits a single crossover from a free impurity (FI) fixed point, with ground state multiplet (q, ⁄) = (0, ), to a FL fixed point, whose ground state multiplet (≠2, •) involves complete screening of the impurity's flavor isospin degree of freedom. This crossover can be described analytically by using non-abelian bosonization followed by flavor fusion. We begin by using non-abelian bosonization with the U(1) ◊ SU(6) 1 KM current algebra to express the bath excitation spectrum in the form E(q, ⁄) = 1 12 q 2 + 1 7 Ÿ 6 (⁄) +¸, (19a) Ÿ 6 (⁄) = 1 12 (5⁄ 2 1 + 8⁄ 2 2 + 9⁄ 2 3 + 8⁄ 2 4 + 5⁄ 2 5 ) (19b) with¸oe , where Ÿ 6 (⁄) is the quadratic Casimir for the representation ⁄ = (⁄ 1 , ⁄ 2 , ⁄ 3 , ⁄ 4 , ⁄ 5 ) of the SU(6) Lie algebra [31]. (The contributions from the two terms of Eq. (19a) are listed in Table S4 in [32] for all q and ⁄ values needed in Table. IV.) The few lowest-lying (q, ⁄) multiplets of the free bath, having E(q, ⁄) oe 1 2 , are listed on the left of Table S5. The strong-coupling FL spectrum can be obtained by combining the bath and impurity flavor degrees of freedom, ⁄ ¢ ⁄ imp = q ü ⁄ Õ , using the fusion rules of the SU(6) 1 KM algebra (see Table S5 in [32]). The resulting multiplets (q, ⁄ Õ ) are listed in the center of Table S5. Their eigenenergies fully match those from NRG.  The table has the same  structure as the left and center parts of Table II, but here the free bath excitations are labeled (q, ⁄), their energies are computed using Eqs. (19) and Table S4 of [32], and flavor fusion with Qimp = (0, ) is performed using SU(6)1 fusion rules (listed in Table S5 of [32]). The resulting multiplets (q, ⁄ Õ ) have eigenenergies E Õ = E(q, ⁄ Õ ), degeneracies d Õ and excitation energies, "E Õ = E Õ ≠ E Õ min . The FL spectrum, obtained by U(1) ◊ SU(6) NRG calculations (Fig. 6) for JU = 0.1, is shown on the right. It has been shifted and rescaled such that the lowest energy is zero and the second-lowest values for E NRG and "E Õ match. The single fusion and NRG spectra agree very well (deviations . 1%).

VI. CONCLUSION
While this work was motivated by DMFT studies of Hund metals, as indicated in the introduction, it has much wider implications. Let us assess these from several perspectives of increasing generality.
(i) We have used NRG and CFT to elucidate the NFL regime of a 3soK model, fine-tuned such that spin screening sets in at very much lower energies than orbital screening.

VI. CONCLUSION
While this work was motivated by DMFT studies of Hund metals, as indicated in the introduction, it has much wider implications. Let us assess these from several perspectives of increasing generality.
(i) We have used NRG and CFT to elucidate the NFL regime of a 3soK model, fine-tuned such that spin screening sets in at very much lower energies than orbital screening.
(ii) This allowed us to uncover the origin of hints of NFL behavior found previously for a 3oAH model and related models [1][2][3][4][5][7][8][9]. There the spin-orbital coupling I 0 is always nonzero, preventing RG trajectories from closely approaching the NFL fixed point. Nevertheless, even if they pass it "at a distance", it still leaves traces of NFL behavior for various observables, such as χ imp sp ∼ ω −6/5 behavior for the imaginary part of the impurity's dynamical spin susceptibility. We elaborate this further in App. A, showing how NFL emerges if the 3oAH model is "deformed" by additionally turning on the spin and orbital exchange couplings of the 3soK model. (iii) Taking a broader perspective, we have provided an analytic solution of a paradigmatic example of a "Hund impurity problem", involving a multi-orbital quantum impurity featuring a non-zero Hund coupling. This fundamental type of problem dates back to the 1950's, when it was observed that the Kondo scale for impurities in transition metals decreases exponentially as the shell filling approaches 1 2 . Early theoretical work by Coqblin and Schrieffer [34] focused only on the spin degrees of freedom. Okada and Yoshida [35] pointed out the importance of orbital degrees of freedom for non-half-filled shells, but theoretical tools for analyzing such cases were lacking at the time. Our work fills this long-standing void by combining state-of-the-art multi-orbital NRG with a suitable generalization of Affleck and Ludwig's CFT approach [15][16][17][18][19].
(iv) Regarding experimental relevance, Hund impurities appear in a multitude of contexts. First, as mentioned in the introduction, they are of central importance for understanding Hund metals. This is a large class of materials, including almost all 4d and 5d materials, and even in the 5f actinides Hund's coupling is the main cause for electronic correlations. Our work illustrates paradigmatically why hints of NFL physics can generically be expected to arise in such systems. Second, tunable Hund impurities can be realized using magnetic molecules on substrates [36] or multi-level quantum dots, raising hopes of tuning Hund impurities in such a way that truly welldeveloped NFL behavior can be observed experimentally.
Note added.-After this research had been completed, a paper closely related to ours appeared [37], with similar goals, a complementary analysis (using NRG but not CFT), and conclusions consistent with ours.
We thank I. Affleck 2 τ a mm f m σ . We treat J 0 and K 0 as free parameters and use them to "deform" the 3oAH model in way that widens the SOS regime between T sp and T orb . Fig. 7(a-d) show how the spin and orbital susceptibilities change upon increasing |J 0 | and |K 0 |, with J 0 < 0 and K 0 > 0. A pure 3oAH model, with (J 0 , K 0 ) = (0, 0), clearly shows spin-orbital separation (SOS), but T sp and T orb differ by less than two decades ( Fig. 7(a), see also [7]). Though the SOS window is too small to reveal a true power law for χ imp sp , the hints of ω −6/5 behavior are already discernable. Turning on the additional exchange coupling terms, with J 0 < 0 and K 0 > 0, causes T sp to decrease and T orb to increase, respectively, widening the SOS regime [ Fig. 7(b-d)]. For (J 0 , K 0 ) = (−0.5, 0.5) it spans more than six orders of magnitude, so that clear power laws, χ imp sp ∼ ω −6/5 and χ imp orb ∼ ω 4/5 , become accessible [ Fig. 7(d)]. These power laws are consistent with our findings for the spin-splitting (SS) regime in the main text. This scenario is evidently smoothly connected to that of the pure 3soK model [ Fig. 2(c)]. There the absence of charge fluctuations makes it possible to fully turn off the I 0 -contribution implicitly present in the 3oAH model, thereby widening the SOS regime even further and allowing the true NFL regime to be analyzed in detail.

Appendix B: Ye's SU(2)×SU(2) spin-orbital Kondo model
In this appendix we revisit a SU(2)×SU(2) spin-orbital Kondo (2soK) model studied in a pioneering paper by Ye in 1997 [20]. It is a simpler cousin of our 3soK model, having a Hamiltonian of precisely the same form, with the following differences: the orbital channel index takes only two values, m = 1, 2; the local orbital current J orb is defined using Pauli (not Gell-Mann) matrices; and the impurity spin and orbital isospin operators, S and T, are both SU(2) generators, in the representation s = λ = 1 2 . In the context of the present study, Ye's paper is of interest because his Kondo impurity likewise features both spin and orbital degrees of freedom. From a conceptual perspective, his and our models differ only in the symmetry group, SU(2) vs. SU(3) in the orbital sector, and the choice of impurity multiplet, Q imp = ( 1 2 , 1 2 ) vs. (1, ). Moreover, he was able to obtain exact results for his model using abelian bosonization. Below we verify that when the NRG and CFT methodology used in the main text is applied to Ye's 2soK model, the results are consistent with his conclusions. For I 0 = 0, the 2soK model obeys particle-hole symmetry. Fig. 8(a) shows the finite-size eigenlevel flow computed by NRG for c 0 = (J 0 , K 0 , I 0 ) = (0.1, 0.3, 0). The low-energy fixed point spectrum features equidistant levels, but nevertheless has NFL properties, as predicted by Ye, in that it can not be understood in terms of combi-nations of single-particle excitations. Remarkably, this fixed point spectrum can be reproduced by CFT arguments. Using non-abelian bosonization according to the U(1)×SU(2) 2 ×SU(2) 2 KM algebra, the spectrum of free bath excitations can be expressed as with ∈ Z, while κ 2 (S), κ 2 (λ) are the quadratic SU (2) Casimirs in the spin and orbital sectors, respectively. We now combine bath and impurity degrees of freedom using simultaneous fusion in the spin and orbital sectors, S ⊗ S imp = ⊕ S and λ ⊗ λ imp = ⊕ λ , employing the fusion rules of the SU(2) 2 ×SU(2) 2 KM algebra (Table S7 in [32]). This reproduces the NFL fixed point spectrum found by NRG, as shown in Table V.
By contrast, we recall that for the 3soK model our attempts to use simultaneous spin and orbital fusion to obtain the FL ground state for 0 = J 0 K 0 , I 0 = 0, were unsuccessful (cf. Sec. V E 1). Thus the 2soK and 3soK models provide an example and a counterexample for the success of simultaneous spin and orbital fusion, succeeding or failing for a NFL or FL fixed point spectrum, respectively.
We have also computed the imaginary parts of spin and orbital susceptibilities, using full-density-matrix (fdm) NRG [24] and adaptive broadening of the discrete NRG data [25]. Fig. 8(b) shows the results. Both functions approach constants in the zero-frequency limit, i.e. scale as ω 0 . This can be understood in terms of the scaling dimensions of the leading boundary operators in the spin and orbital sectors, Φ sp and Φ orb , which have dimensions ∆ sp = ∆ orb = 1 2 (Table V). By the arguments of Sec. V C we thus obtain as predicted by Ye. This resembles the behavior observed for the celebrated two-channel Kondo model, featuring a spin-1 2 impurity having no orbital isospin (obtained from Ye's model by using λ = • for the impurity orbital pseudospin, and setting K 0 = I 0 = 0).  shown at the top. The low-energy fixed points in (a) and (c) exhibit a NFL or FL spectrum, respectively, reproduced analytically in Table V Table II for the 3soK model, but here the free bath excitations are computed using Eqs. (B1) and Table S6 of [32], and single fusion of bath and impurity degrees of freedom is performed simultaneously in the spin and orbital sectors, using SU(2)2 ◊ SU(2)2 fusion rules (listed in Table S7   shown at the top. The low-energy fixed points in (a) and (c) exhibit a NFL or FL spectrum, respectively, reproduced analytically in Table V Table II for the 3soK model, but here the free bath excitations are computed using Eqs. (B1) and Table S6 of [32], and single fusion of bath and impurity degrees of freedom is performed simultaneously in the spin and orbital sectors, using SU(2)2 × SU(2)2 fusion rules (listed in Table S7    For I 0 = 0, particle-hole symmetry is broken. Fig. 8(c) shows the eigenlevel flow computed by NRG for c 0 = (0, 0.3, 0.05). The low-energy fixed point is a FL, as predicted by Ye. Its spectrum shows the same equidistant set of energies as the NFL spectrum of I 0 = 0 [ Fig. 8(a)], but the degeneracies are different. This fixed point can not be understood by simultaneous fusion in the spin and orbital sector. However, it agrees with the FL spectrum of a SU(4) Kondo model with the higher symmetry U(1) ch × SU(4) fl , defined in analogy to the SU(6) Kondo model from Sec. V E 2, with a flavor index ν = 1, . . . , 4 encoding both spin and orbital degrees of freedom. Using non-abelian bosonization according to the U(1) × SU(4) 1 KM algebra, the free bath spectrum can be expressed as κ 4 (λ) = 1 8 (3λ 2 1 + 4λ 2 2 + 3λ 2 3 + 4λ 1 λ 2 + 2λ 2 λ 3 + 4λ 1 λ 3 +12λ 1 + 16λ 2 + 12λ 3 ) .
with ∈ Z, where κ 4 (λ) is the quadratic Casimir for the λ = (λ 1 , λ 2 , λ 3 ) representation of the SU(4) Lie algebra. (The contributions from the two terms of Eq. (B4a) are listed in Table VI for the lowest few q and λ values.) Combining the flavor degrees of freedom of bath and impurity, λ ⊗ λ imp = ⊕ λ , using the fusion rules of the SU(4) 1 KM algebra, we recover the FL fixed point spectrum found by NRG. This is shown in Table VI. In the FL regime, the spin and orbital susceptibilities scale as χ imp sp,orb ∼ ω 1 [ Fig. 8(d)], as expected for a Fermi liquid and predicted by Ye. Below we provide a number of tables needed for various non-abelian bosonization and Kac-Moody fusion schemes used in the main text: U(1) × SU(2) 3 × SU(3) 2 , U(1) × SU(6) 1 , U(1) × SU(2) 2 × SU(2) 2 , and U(1) × SU(4) 1 .

Supplemental
The fusion rules for the SU(N ) k Kac-Moody (KM) algebra differ from those of the SU(N ) Lie algebra in that some Young diagrams arising for the latter are forbidden for the former (such as Young diagrams with more than k columns, reflecting the fact that only two distinct spin species are available when constructing SU(N ) k representations). We constructed the KM fusion tables given below using a general recipe due to Cummins [38], explained in pedagogical detail in Sec. 16 Table S3. SU(2)3 fusion rules, listing various direct product decompositions of the form S ⊗ S = ⊕ S . Crossed-out numbers denote additional irreps occurring when considering direct product decompositions for SU(2) instead of SU(2)3.   Table S5. SU(6)1 fusion rules, listing some direct product decompositions λ ⊗ λ = ⊕ λ , with λ = . Crossed-out diagrams denote additional irreps occurring when considering direct product decompositions for SU(6) instead of SU(6)1. (λ) (λ1, λ2, λ3, λ4, λ5) λ  Table S7. SU(2)2 fusion rules, listing various direct product decompositions of the form S ⊗ S = ⊕ S . Crossed-out numbers denote additional irreps occurring when considering direct product decompositions for SU(2) instead of SU(2)2.   Table S9. SU(4)1 fusion rules, listing some direct product decompositions λ ⊗ λ = ⊕ λ , with λ = . Crossed-out diagrams denote additional irreps occurring when considering direct product decompositions for SU(4) instead of SU(4)1. (λ) (λ1, λ2, λ3)