Magnetic end-states in a strongly-interacting one-dimensional topological Kondo insulator

Topological Kondo insulators are strongly correlated materials, where itinerant electrons hybridize with localized spins giving rise to a topologically non-trivial band structure. Here we use non-perturbative bosonization and renormalization group techniques to study theoretically a one-dimensional topological Kondo insulator. It is described as a Kondo-Heisenberg model where the Heisenberg spin-1/2 chain is coupled to a Hubbard chain through a Kondo exchange interaction in the p-wave channel - a strongly correlated version of the prototypical Tamm-Shockley model. We derive and solve renormalization group equations at two-loop order in the Kondo parameter, and find that, at half-filling, the charge degrees of freedom in the Hubbard chain acquire a Mott gap, even in the case of a non-interacting conduction band (Hubbard parameter $U=0$). Furthermore, at low enough temperatures, the system maps onto a spin-1/2 ladder with local ferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-$1$ chain with frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypical interacting topological spin model, and features two magnetic spin-$1/2$ end states for chains with open boundary conditions. Our analysis allows to derive an insightful connection between topological Kondo insulators in one spatial dimension and the well-known physics of the Haldane chain, showing that the ground state of the former is qualitatively different from the predictions of the naive mean-field theory.

It is now understood that there exist distinct symmetryprotected classes of non-interacting insulators, such that two representatives from different classes can not be adiabatically transformed into one another (without closing the insulating gap and breaking the underlying symmetry along the way). A complete topological classification of such band insulators has been developed in the form of a "periodic table of topological insulators" [9,10]. Furthermore, it was realized that the non-trivial (topological) insulators from this table possess, as their hallmark features, gapless boundary modes. The latter have been spectacularly observed in a variety of experiments in both three [11,12] and two-dimensional systems [13][14][15][16].
The aforementioned classification however is limited to non-interacting systems and as such it represents a classification of single-particle band structures. Adding interactions to the theory leads to significant complications. To understand and classify strongly-interacting topological insulator phases in many-particle systems is a fundamental open problem in condensed matter.
A class of material that combines strong interactions and non-trivial topology of emergent bands are topological Kondo insulators. A basic model of these heavy , +J K -J K fermion systems involves even-parity conduction electrons hybridizing with strongly correlated f-electrons. At low temperatures, a hybridization gap opens up and an insulating state can be formed. Its simplified mean-field description makes it amenable to a topological classification according to the non-interacting theory, and a topologically-non-trivial state appears due to the opposite parities of the states being hybridized. Although, the mean-field description (formally well-controlled in the large-N approximation [17][18][19]) does appear to correctly describe the nature of the topological Kondo insulating states observed in bulk materials so far [20][21][22][23][24], it is interesting to see if non-perturbative effects beyond mean-field can qualitatively change the mean-field picture.
In contrast to higher dimensions, where reliable theoretical techniques to treat strong interactions are scarce, there exists a rich arsenal of such non-perturbative methods for one-dimensional systems, where stronglycorrelated "non-mean-field" ground states abound. Since the Kondo insulating Hamiltonian and its mean-field treatment are largely dimension-independent, it is interesting to consider the one-dimensional such model as a natural playground to study interplay between strong interactions and non-trivial topology.
With this motivation in mind, we study here a strongly-interacting model of a one-dimensional topological Kondo insulator, i.e., a "p-wave" Kondo-Heisenberg model, introduced earlier by Alexandrov and Coleman [25], who treated the problem in the mean-field approximation. Here, we go beyond the mean-field level and consider quantum fluctuations non-peturbatively using the Abelian bosonization technique. It is shown that a "topological coupling" between the electrons in the Hubbard chain and spins in the Heisenberg chain, gives rise to a charge gap at half-filling in the former. The relevant interaction between the remaining spin-1/2 degrees of freedom in the chains is effectively ferromagnetic, which locks them into a state qualitatively similar to the Haldane's spin-1 chain. The ground state therefore is a stronglycorrelated topological insulator, which exhibits neutral spin-1/2 end modes.
This article is organized as follows: in Section II we specify the model for a 1D TKI and introduce the Abelian bosonization description. In Section III we present the renormalization group analysis and discuss the quantum phase diagram of the system. In Section IV we analyze the topological aspects of the problem and explain the emergence of topologically protected magnetic edgestates and in Section V we present a summary and discussion of results. Finally, in the Appendix A we present the technical derivation of the renormalization group equations.

II. MODEL
We start our theoretical description by considering the Hamiltonian of the system depicted in Fig. 1, is a fermionic 1D Hubbard chain with N s sites, where n j,σ = c † j,σ c j,σ is the density of spin-σ electrons at site j, µ is the chemical potential, and U is the Hubbard interaction parameter. In this work we will only focus on the half-filled case µ = 0, where there is one electron per site. However, we expect our results to remain also valid for small deviations of half-filling. The spin chain is described by the spin-1/2 Heisenberg model with J > 0. Here we assume the same lattice parameter a for both chains H 1 and H 2 . Finally, motivated by the work by Alexandrov and Coleman [25], we assume the following exchange coupling between the two chains where J K > 0 is the Kondo interaction between the j−th spin (S j ) in the Heisenberg chain and the p-wave spin density in the fermionic chain at site j, defined as where p j,α ≡ (c j+1,α − c j−1,α ) / √ 2 is a linear combination of orbitals with p-wave symmetry, and σ αβ is the vector of Pauli matrices. This model can be regarded as a strongly interacting version of the Tamm-Shockley model [26][27][28]. In the absence of interactions in the fermionic chain (i.e., U = 0) and in the large−N mean-field approximation, Alexandrov and Coleman have shown the emergence of topologically-protected edge states arising from the non-trivial form of the Kondo term (3) [25]. In their mean-field approach, the effective description of the system corresponds to non-interacting quasiparticles filling a strongly renormalized valence band with a nontrivial topology, stemming from the charge-conjugation, time-reversal and charge U(1) symmetry of the effectively non-interacting Hamiltonian (see also Ref. [29] for a discussion of a closely related system).
In this paper, our goal is to understand the emergence of topologically protected edge-states without introducing any decoupling of the Kondo interaction, including the interacting case, U = 0. We consider the case of small U and J K . This is formally represented by linearizing the non-interacting spectrum k = −2t cos ka in the fermionic chain H 1 around the Fermi energy µ = 0, and taking the continuum limit where the lattice constant a → 0. Then, the fermionic operators admit the low-energy representation [30,31] where R 1,σ (x) and L 1,σ (x) are right and left moving fermionic field operators, which vary slowly on the scale of a. As we are interested in the edge-state physics, we consider open boundary conditions c 0,σ = c Ns+1,σ = 0, leading to the following constraints: where L = N s a. We next introduce the Abelian bosonization formalism [30,31] where α is a short distance cutoff in the bosonization procedure (we will take α = a hereafter). In Eq.
are Klein operators which obey anticommutation relations {F 1,σ , F 1,σ } = δ σ,σ , and therefore ensure the correct anticommutation relations between fermions of different spin species. Due to the constraints (6) and (7) introduced by the open boundary conditions, the right and left movers are not independent, and obey the constraints where q σ is an integer representing the occupation of the zero-mode. From here we obtain the additional commutation relations [32] [ The bosonization procedure applied to the 1D Hubbard model is standard and we refer the reader to textbooks for details [30,31]. Introducing charge and spin bosonic fields φ 1,λσ = 1 where the new fields obey the boundary conditions As is well-known, in 1D charge and spin excitations generally decouple and the above Hamiltonian can be split as H 1 = H 1c + H 1s , with the first line describing independent Luttinger liquids for the charge and spin sectors, which are characterized by charge (spin) acoustic modes with velocities and Luttinger parameter controlling the decay of correlation functions K 1c = 1/ 1 + U a/ (πv F ) (K 1s = 1) . The presence of the cosine terms in the second line of (12) changes the physics qualitatively. In the present work, we restrict our focus to the case U ≥ 0, where the term ∼ cos √ 8φ 1c is marginally relevant in the renormalization-group sense, and opens a Mott gap in the charge sector. At the same time, the term ∼ cos √ 8φ 1s is marginally irrelevant at the SU(2) symmetric point, and the spin sector remains gapless [30,31].
The bosonization of the Heisenberg chain H 2 is also quite standard and we refer the reader to the abovementioned textbooks [30,31]. A usual trick consists in representing the spin operators S j by auxiliary fermionic operators in a half-filled Hubbard model with interaction parameter U U . Therefore, while technically the procedure is identical to Eq. (12), the charge degrees of freedom in the bosonized Hamiltonian, H 2 , can be assumed to be absent at the relevant energy scales of the problem due to the Mott gap ∼ U . Then, ignoring the charge degrees of freedom and irrelevant operators in Eq. (12), and replacing the chain label 1 → 2, we obtain Finally, we bosonize the Kondo Hamiltonian. The pwave spin density in the fermionic chain and the spin density in the Heisenberg chain are where L a,β (x) (with a = 1, 2) are the smooth components of the spin density, with bosonic representation and where N a (x) = R † a,α (x) σ α,β 2 L a,β (x) + H.c., are the staggered components with m 2 = cos √ 2φ 2c (x) resulting from the integration of the gapped charge degrees of freedom in the Heisenberg chain. Therefore, although the spin densities (16) and (17) look similar in the bosonized language, they actually differ in two crucial aspects: 1) while in Eq. (22) the charge degrees of freedom are absent in the expression of the staggered magnetization, they are still present in (21) in the term cos √ 2φ 1c (x) and we need to consider them. 2) Comparing Eqs. (16) and (17), we note a sign difference in the staggered components. This sign is related to the p-wave nature of the operators p j,σ , and is therefore intimately connected to the topology of the Kondo interaction. The role of this sign turns out to be crucial in the rest of the paper.
Replacing the above results into Eq. (3), and taking the continuum limit, the Kondo interaction becomes in the bosonic language where the sign in the second line is a consequence of the above mentioned sign in the staggered part of π (x). Note that this model is reminiscent of the (nontopological) 1D Kondo-Heisenberg model, which has recently received much attention in the context of pair-density wave ordered phases in high-T c cuprate physics [33][34][35][36][37][38], and to the Hamiltonian of a spin-1/2 ladder [32,39,40]. However, a crucial difference with those works is the non-trivial structure of the Kondo interaction, which differs from the usual coupling c j,β is the standard (i.e., s-wave in this context) spin density in the fermionic chain. The first line in (23) is in fact closely related to the model considered in Refs. [34][35][36][37][38]. In the half-filling situation we are analyzing here, however, the most relevant part of H K (in the RG sense) is given by the product , which dominates the physics at low energies [32,39,40]. Note that this term only survives at half filling, and when both chains have the same lattice parameter. Therefore, for our present purposes, we can neglect the first term in Eq. (23) and focus on the second term At this point we note that the problem is reminiscent of the well-known case of S = 1/2 ladders with open boundary conditions [32,40], with the important difference that here there is an extra term ∼ cos √ 2φ 1c . The physics of the spin sector [i.e., term in square brackets in (24)] is quite non-trivial due to the presence of both the canonically conjugate fields φ a,s (x) and θ a,s (x), which cannot be simultaneously stabilized. However, the analysis of the charge sector is simpler, as only the field φ 1c (x) appears in the expression. This means that in the limit J K → ∞ the system can gain energy by "freezing out" the charge degrees of freedom, i.e., , as there is no other competing mechanism. In the next Section we substantiate these ideas by providing a rigorous analysis.

III. RENORMALIZATION-GROUP ANALYSIS
Based on the similarity with the physics of spin ladders, we introduce symmetric and antisymmetric fields φ ± = 1 √ 2 (φ 1s ± φ 2s ) and θ ± = 1 √ 2 (θ 1s ± θ 2s ), in terms of which the Hamiltonian becomes In what follows, we assume identical spinon dispersion v 1s = v 2s = v s . Although this assumption is certainly an idealization, one can show that the asymmetry δv ≡ v 1s − v 2s is an irrelevant perturbation in the renormalizationgroup sense, and therefore we do not expect that small asymmetries will have a qualitative effect on our results. We now write the Euclidean action of the system using complex space-time coordinates z = v F τ + ix and z = v F τ − ix, with τ = it the imaginary time, and the left and right fields φ ν = (φ νL + φ νR ) /2, where ν = +, −, 1c.
The action becomes: with d 2 r = v F dxdτ , and where we have defined the dimensionless couplings: and the scaling operators: where we have explicitly normal-ordered the operators. S 0 corresponds to a free fixed-point action, and S U and S K are perturbations arising from the Hubbard repulsion in chain 1 and the Kondo inter-chain interaction, respectively. We have neglected all the perturbations in the spin sector generated by U , as they renormalize to zero along the SU(2)-invariant line in the parameter space.
We have also neglected the less relevant terms coming from the product of the smooth part of the currents in equations (16) and (17). φi,θi] perturbatively at second order in G i , we obtain the product of the different operators (i.e., the operator product expansion or OPE) of Eqs. (31), (32) and (33). Importantly, the OPE of the Kondo interaction O K (z , z ) O K (z, z) gives rise to operators O 3 and O 2c , which are already present in the charge sector of chain 1. This corresponds to an effective dynamically generated Hubbard repulsion originated by the inter-chain Kondo coupling (see Appendix A). Therefore, even for an initially non-interacting chain (i.e., U = 0), this emergent repulsive interaction induces the opening of a Mott insulating gap in the charge sector of the half-filled conduction band. At energies below this gap, the field φ 1c becomes pinned to the degenerate values 0 or π √ 2 . Note that only the latter is consistent with the boundary conditions given in Eq. (14). Therefore, this analysis suggests that the energy is minimized by a uniform configuration of the field φ 1c , which "freezes" at the bulk value π √ 2 . Note that this situation prevents the ground state from developing "kink" excitations and, consequently, we can exclude the presence of localized charge edge-states. Then, at low enough energies, the system becomes a Mott insulator in the bulk due the electronic correlations, and no topological effects arise in the charge sector.
A more quantitative study can be done by analyzing the two-loop RG-flow equations (see Appendix A for details): where l = ln (a/a 0 ) is the logarithmic RG scale. When G K = 0 these equations reduce to the Kosterlitz-Thouless ones corresponding to the charge sector of the Hubbard model. They predict a charge gap which is exponentially small in U [30]. Let us now analyze the case of an initially U = 0. In this situation, only the linear term survives in the right hand side of Eq. (36), which expresses the relevance of the operator O K , and give rise to an exponential increase of G K (l) with the RG scale l as G K (l) = G 0 K e l 2 . As a consequence, the coupling G 3 (l) in Eq. (35), representing the Hubbard repulsion, also increases exponentially G 3 (l) = 3 4 G 0 K 2 e l − 1 , from its initial value G 3 (0) = 0. This produces the anticipated Mott insulating gap ∆ c in the charge sector. Its dependence with the parameters can be obtained by the procedure described in page 65 of Ref. [30]. We obtain ∆ c ∼ G 0 K 2 for small enough J K . We could envisage that if U and J K would have the same order, the dominant contribution to ∆ c would come from the inter-chain Kondo coupling. Following previous references [32,39,40], we can refermionize Eq. (25) noting that the scaling dimension of the cosines in the square bracket exactly corresponds to the free-fermion point and therefore they can be written in terms of right and left-moving free Dirac fermion fields η ±,R (x) and η ±,L (x) as For later purposes, it is more convenient to introduce a Majorana-fermion representation of the fields η +,λ = 1 √ where the Majorana fields obey the boundary conditions The uniform symmetric and antisymmetric spin densities in the ladder become with a = {1, 2, 3} and λ = {R, L}. This is a well-known representation of two independent SU (2) 1 Kac-Moody currents J 1,λ and J 2,λ in terms of four Majorana fields. In our case, these four degrees of freedom, resulting from the combination of the two SU (2) spin density fields in the two chains, are expressed in terms of a singlet ξ 0 λ and triplet ξ a λ Majorana fields. From the previous analysis we conclude that at temperatures T ∆ c , the charge and spin degrees of freedom become effectively decoupled, and the low-energy Hamiltonian of the system can be written as H →H =H c +H s , with: Figure 2: (a) At temperatures T ∆c ∼ J 2 K , the dynamically induced Mott gap, the charge degrees of freedom in the Hubbard chain become frozen and the system maps onto a ladder where the non-local antiferromagnetic Kondo coupling effectively induces local ferromagnetic correlations between the spins in the ladder (vertical dashed lines). (b) The system depicted in (a) can be mapped onto a S = 1 Haldane chain, which supports topologically protected S = 1/2 end-states [32,[39][40][41][42][43][44][45].
where the charge degrees of freedom in the fermionic chain develop a gap m 1 ≡ m 1 (J K ) = cos √ 2φ 1c . Note that, in this form, the spin HamiltonianH s is similar to that of a spin-1/2 ladder [32,39,40]. The quantitative determination of the parameter m 1 requires a self-consistent calculation, which is beyond the scope of the present work. Nevertheless, the previous analysis allows to understand two important aspects of the topological Kondo-Heisenberg chain: a) the generation of an insulating state in the bulk (necessary to reproduce the bulk insulating state of a TKI), and b) the emergence of magnetic edge-states, which is the subject of the next section.

IV. S = 1/2 MAGNETIC EDGE-MODES AND TOPOLOGICAL INVARIANT
The previous analysis shows that, at low temperatures T ∆ c , the 1D "p−wave" Kondo-Heisenberg chain at half-filling can be effectively mapped onto a spin ladder problem, which is dominated by the staggered components of the spin densities. Interestingly, for an antiferromagnetic Kondo coupling J K > 0, the negative sign emerging from the structure of the non-local Kondo interaction (24), effectively induces a local ferromagnetic interaction [see vertical dashed lines in Fig. 2(a)]. It is well-known that the spin ladder with ferromagnetic exchange coupling J ⊥ < 0 along the rungs, has a triplet low-energy sector and singlet excited states [32,39,40]. Therefore, at temperatures T J ⊥ , it describes the physics of the Haldane spin-1 chain [46,47], which is known to host symmetry-protected topological spin-1/2 modes at the boundaries [41][42][43][44][45][48][49][50][51]. To see how these excitations emerge in our low-energy Hamiltonian (46), we consider solutions of the eigenvalue equatioñ T is a Majorana spinor, and a = (0, 1, 2, 3) [32]. In matrix form, this equation is with ξ a being a localized Majorana fermion. Using the expression (43) for the smooth part of the spin density, we can physically associate the presence of the localized Majorana bound-state with a localized spin-1/2 magnetic edge-state. This is consistent with the results in Ref. [25] where, in the case of a uniform Kondo interaction, a magnetic mode with no admixture with charge degrees of freedom emerges at the boundaries. However, note that the origin of these edge-states is quite different: while in the mean-field regime they emerge as a consequence of Kondo-unscreened end-spins in the Heisenberg chain, in our case they are intimately related to the physics of the Haldane chain. We now derive a topological invariant to characterize the presence of the edge-states. To that end, we focus on the uniform magnetization Eq. (43). Although the original Hamiltonian has spin-rotational symmetry, the Abelian bosonization is not an explicitly SU(2)-invariant formalism. Therefore, while the choice of axes is arbitrary due to the spin rotation symmetry of the problem, once we define the z−direction as the spin-quantization axis, the perpendicular components of the magnetization M x (x) and M y (x) acquire a more complicated mathematical form. For that reason, it is more convenient to focus only on the z component of the symmetric spin density Eq. (43), which is conveniently simpler In the expression above, we have used Eq. (20) and the definition of φ . We now define the total magnetization at one end (for concreteness, the left end) of the chain as M z T ≡´x b 0 dx M z (x), where x b is an unspecified position in the interior of the chain where the magnetization reaches the value in the bulk. In bosonic language, it acquires the compact form From the expression of the bosonic Hamiltonian Eq. (25) in the limit J K → ∞, we see that the system minimizes the energy in the bulk by pinning the field φ + (x) to one of the degenerate values On the other hand, from the definition of φ ± (x) = and Eq. (13), the boundary condition φ + (0) = 0 is obtained. Replacing these values into Eq. (50), we obtain the following quantized values of the magnetization at the left end This magnetization at the end of the chain is an analog of the electrical polarization [52,53] or the time-reversal polarization [54] in 1D insulators. Frome here, we can define a Z 2 −topological invariant which characterizes the topological phase of the Kondo-Heisenberg chain which in the limit of an infinite system L → ∞ is Q = −1 in the topological phase (J K > 0), and Q = 1 in the trivial one (J K < 0). The bosonic representation also provides an alternative way to demonstrate the existence of magnetic edge modes. Since none of the degenerate values (51) of φ + (x) in the bulk satisfy the boundary condition at x = 0, we conclude that a kink excitation necessarily must emerge near the boundary in order to connect those values: precisely this kink excitation gives rise to the spin-1/2 endstate, upon use of Eq. (49). We remind the reader that in Section III, using similar arguments, we demonstrated the absence of kink configurations in the charge field φ 1c (x), and the fact that there are no charge edge-states in the ground state.

V. CONCLUSIONS
We have studied theoretically a model for a topological 1D Kondo insulator, using the Abelian bosonization formalism. We analyzed the 1D Kondo-Heisenberg model, coupled in the p-wave channel and with an on-site Hubbard interaction U in the conduction band, and derived the two-loop RG flow equations for the system at halffilling. Our RG analysis shows that the system develops a Mott-insulating gap at low enough temperatures, even if U = 0. Moreover, the remaining spin degrees of freedom are effectively described by a ferromagnetic spin-1/2 ladder, which in turn maps onto a spin-1 Haldane chain with topologically protected spin-1/2 magnetic edge-modes.
In contrast to three-dimensional bulk topological Kondo insulators, where the mean-field approximation is well justified and the system can be effectively described in terms of non-interacting quasiparticles opening a (renormalized) hybridization gap near the Fermi surface [18,55,56], in one spatial dimension the presence of strong quantum fluctuations cannot be ignored, and forces to use different approaches. The Abelian bosonization method allows to obtain a description of the 1D TKI which is fundamentally different from the mean-field picture. In the first place, the system develops a Mott gap (instead of a hybridization gap) in the spectrum of charge excitations when the conduction band is half-filled (small deviations from half-filling do not affect this scenario qualitatively [30]). This Mott gap arises from umklapp processes at second order in the Kondo interaction. Physically, this can be understood as a dynamically-induced effective interaction term, which appears at order J 2 K in the conduction band by integrating out perturbatively short-time spin excitations in the Heisenberg chain. In contrast to the mean-field description, where the hybridization gap depends exponentially on the microscopic Kondo coupling ∆ c ∼ exp (−1/J K ), the integration of the RG Eqs. (34)-(36) in the limit J K → 0 results in ∆ c ∼ J 2 K . Our RG analysis indicates that J K is a relevant perturbation and flows to strong coupling, dominating the physics at low temperatures. In particular, at temperatures below the Mott gap, the charge degrees of freedom are frozen and system effectively behaves as a ferromagnetic spin-1/2 ladder, which is known to map onto the spin-1 Haldane chain. Therefore, our work allows to make an insightful connection between two a priori unrelated physical models. Interestingly, exploiting this connection, we predict the existence of topologically protected spin-1/2 edge states. This seems to correspond to the "magnetic phase" found by Alexandrov and Coleman [25], which for a uniform Kondo coupling J K , is characterized by Kondo-unscreened spins at the end of the Heisenberg chain. However, the emergence of these edge states, again corresponds to a very different mechanism than the one provided by the mean-field theory. Interestingly, within the bosonization framework, we have been able to obtain a Z 2 topological invariant [see Eq. (53)] in terms of the magnetization at an end of the chain.
Our work opens the possibility to explore the physics of hidden Z 2 × Z 2 symmetry and the existence of a non-vanishing string order parameter O α string ≡ lim |l−m|→∞ S α l e iπ l≤j<m S α j S α m = 0 (which are wellknown features of the Haldane phase [41,42,57]) in the p−wave Kondo-Heisenberg model. In particular, note the close relation between the Z 2 topological invariant (53) and the string-order parameter in bosonized form [see Eq. (83) in Ref. [39]].
Furthermore, we reiterate that the model studied here can be viewed as a non-trivial strongly-correlated generalization of the old Tamm-Shockley model [26,27]. The latter is a prototypical one-dimensional model that exhibits a topological phase transition and it can be used to construct high-dimensional topological band insulators [28]. Likewise, the strongly-correlated topological Kondo-Heisenberg model could potentially become a building block in constructing higher-dimensional strongly-interacting topological states -not adiabatically connected to "simple" topological band insulators. Finally, we mention in this context recent experimental results [58][59][60], which suggest the existence of a magnetic phase transition and/or suppressed surface charge transport in select samples of Samarium hexaboride (SmB 6 -a three-dimensional topological Kondo insulator). It is possible that these phenomena, which remain unexplained at this stage, involve in a crucial way an interplay between band topology and strong correlations, which conceivably may lead to the formation of non-trivial magnetic topological surface modes reminiscent to the edge states found here.
AML acknowledges support from NSF-JQI-PFC. AOD is partially supported by PIP 11220090100392 of CON-ICET (Argentina). VG was supported by DOE-BES DESC0001911 and Simons Foundation. The authors would like to thank Dmitry Efimkin for useful discussions.
indexes α and β run on 2c, 3 and K. Here, ∆ α is the scaling dimension of the operator O α defined in Eqs. (31)(32)(33) (∆ 3 = ∆ 2c = 2, ∆ K = 3 2 ), Z 0 =´ i={1c,±} D [φ i , θ i ] e −S0 [φi,θi] is the generating function of the free theory, and the mean values . . . 0 correspond also to that theory. This formalism is standard in the analysis of 1D quantum systems, and has been applied in several previous works (see for example Ref. [63] where the method is explained in detail).
The third term of the r.h.s. in Eq. (A1) takes the same form as the second one if we assume that for r → r the product of two operators fulfills the following operator product expansion (OPE) property [61,62] : where O γ includes all the operators generated from each OPE. Let us focus on the OPE between two O K operators, which is the most relevant perturbation in the RG sense, and is precisely the contribution that leads to the umklapp processes we are trying to describe. To simplify the discussion, here we return to the representation of the bosonic fields in terms of left and right movers φ L (z) and φ R (z) [see Eq. (8)], with z = v F τ + ix andz = v F τ − ix. We now assume to be sufficiently deep in the bulk of the 1D system and far away from the boundaries. In these conditions, the boundary conditions (9) and (10), and the commutation relation (11) can be effectively neglected, and the fields φ L (z) and φ R (z) become independent (i.e., they do not mix). This allows us to focus only on the processes involving the left-moving field φ L (z)(for right-moving fields we just need to change L → R and z →z) . The basic OPE we need is : e i(λ+λ )φL(z ) 1 idea is that the theory defined with a microscopic cutoff a should remain invariant under a scaling transformation a → a (1 + dl), where dl is a dimensionless infinitesimal. Therefore, the couplings G α (a) in Eq. (A1) must be changed in such a way that they preserve the generating functional, i.e., Z [a] = Z [a (1 + dl)]. The method is standard and we refer the reader to Ref. [61] for details. The renormalization group flow equations can be written in terms of the coefficients C γ αβ as where the coefficients C 2c KK = C 3 KK = − 3 4π are extracted from Eq. (A4). The remaining coefficients are obtained by the OPEs between the corresponding operators. Following the lines of Ref. [64] we obtain straightforwardly: Inserting these values in Eq. (A6) we obtain Eqs. (34)- (36) in the main text.