Magnetic End States in a Strongly Interacting One-DimensionalTopological Kondo Insulator

Alejandro M. Lobos,1,2,* Ariel O. Dobry,1 and Victor Galitski2,31Facultad de Ciencias Exactas Ingeniera y Agrimensura, Universidad Nacional de Rosario and Instituto

de Fsica Rosario, Boulevard 27 de Febrero 210 bis, 2000 Rosario, Argentina2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,

University of Maryland, College Park, Maryland 20742-4111, USA3School of Physics, Monash University, Melbourne, Victoria 3800, Australia

(Received 4 December 2014; revised manuscript received 10 April 2015; published 22 May 2015)

Topological Kondo insulators are strongly correlated materials where itinerant electrons hybridize withlocalized spins, giving rise to a topologically nontrivial band structure. Here, we use nonperturbativebosonization and renormalization-group techniques to study theoretically a one-dimensional topologicalKondo insulator, described as a Kondo-Heisenberg model, where the Heisenberg spin-1=2 chain is coupled toa Hubbard chain through a Kondo exchange interaction in the p-wave channel (i.e., a strongly correlatedversion of the prototypical Tamm-Schockley model). We derive and solve renormalization-group equations attwo-loop order in the Kondo parameter, and find that, at half filling, the charge degrees of freedom in theHubbard chain acquire a Mott gap, even in the case of a noninteracting conduction band (Hubbard parameterU 0). Furthermore, at low enough temperatures, the system maps onto a spin-1=2 ladder with localferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-1 chainwith frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypicalinteracting topological spin model, and features two magnetic spin-1=2 end states for chains with openboundary conditions. Our analysis allows us to derive an insightful connection between topological Kondoinsulators in one spatial dimension and thewell-known physics of the Haldane chain, showing that the groundstate of the former is qualitatively different from the predictions of the naive mean-field theory.

DOI: 10.1103/PhysRevX.5.021017 Subject Areas: Condensed Matter Physics,Strongly Correlated Materials,Topological Insulators

I. INTRODUCTION

Starting with the pioneering works of Kane and Mele[1,2] and others [35], there has been a surge of interestin topological characterization of insulating states [68].It is now understood that there exist distinct symmetry-protected classes of noninteracting insulators, such thattwo representatives from different classes cannot beadiabatically transformed into one another (without closingthe insulating gap and breaking the underlying symmetryalong the way). A complete topological classificationof such band insulators has been developed in the formof a periodic table of topological insulators [9,10].Furthermore, it was realized that the nontrivial (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 [1316].The aforementioned classification, however, is limited to

noninteracting systems, and as such, it represents a clas-sification of single-particle band structures. Adding inter-actions to the theory leads to significant complications.Understanding and classifying strongly interacting topo-logical insulator phases in many-particle systems is afundamental open problem in condensed matter.A class of material that combines strong interactions and

nontrivial topology of emergent bands is topological Kondoinsulators (TKI) [17]. A basic model of these heavyfermion systems involves even-parity conduction electronshybridizing with strongly correlated f electrons. At lowtemperatures, a hybridization gap opens up and an insulat-ing state can be formed. Its simplified mean-field descrip-tion makes it amenable to a topological classificationaccording to the noninteracting theory, and a topologicallynontrivial state appears due to the opposite parities of thestates being hybridized. Although the mean-field descrip-tion (formally well controlled in the large-N approximation[1820]) does appear to correctly describe the nature of thetopological Kondo insulating states observed in bulk

*lobos@ifirconicet.gov.ar

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published articles title, journal citation, and DOI.

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2160-3308=15=5(2)=021017(13) 021017-1 Published by the American Physical Society

materials thus far [2125], it is interesting to see ifnonperturbative effects beyond mean field can qualitativelychange the mean-field picture.In contrast to higher dimensions, where reliable theo-

retical techniques to treat strong interactions are scarce,there exists a rich arsenal of such nonperturbative methodsfor one-dimensional systems, where strongly correlatednon-mean-field ground states abound. Since the Kondoinsulating Hamiltonian and its mean-field treatment arelargely dimension independent, it is interesting to considerthe one-dimensional model as a natural playground to studythe interplay between strong interactions and nontrivialtopology.With this motivation in mind, we study here a strongly

interacting model of a one-dimensional topological Kondoinsulator, i.e., a p-wave Kondo-Heisenberg model, intro-duced earlier by Alexandrov and Coleman [26], whotreated the problem in the mean-field approximation.Here, we go beyond the mean-field level and considerquantum fluctuations nonpeturbatively, using the Abelianbosonization technique. It is shown that a topologicalcoupling between the electrons in the Hubbard chain andspins in the Heisenberg chain gives rise to a charge gap athalf filling in the former. The relevant interaction betweenthe remaining spin-1=2 degrees of freedom in the chains iseffectively ferromagnetic, which locks them into a statequalitatively similar to the Haldane spin-1 chain. Theground state, therefore, is a strongly correlated topologicalinsulator, which exhibits neutral spin-1=2 end modes.While our main motivation is essentially theoretical

(i.e., to allow a deeper understanding of strongly interactingtopological matter), we believe our results might havedirect application in ultracold atom experiments, wheredouble-well optical superlattices loaded with atoms in s andp orbitals have been realized [27,28]. In addition, our workmight have some relevance in recent experimental results[2931], which suggest the existence of a ferromagneticphase transition and/or suppressed surface charge transportin samples of samarium hexaboride (SmB6a three-dimensional topological Kondo insulator).This article is organized as follows. In Sec. II, we specify

the model for a 1D TKI and introduce the Abelianbosonization description. In Sec. III, we present therenormalization-group (RG) analysis and discuss the quan-tum phase diagram of the system. In Sec. IV, we analyze thetopological aspects of the problem and explain the emer-gence of topologically protected magnetic edge states, andin Sec. V, we present a summary and discussion of theresults. Finally, in the Appendix, we present the technicalderivation of the renormalization-group equations.

II. MODEL

We start our theoretical description by consideringthe Hamiltonian of the system depicted in Fig. 1,H H1 H2 HK, where

H1 tXNs1j1;

cj;cj1; H:c: XNsj1;

nj;

UXNsj1

nj 1

2

nj 1

2

1

is a fermionic 1D Hubbard chain with Ns sites, wherenj; cj;cj; is the density of spin- electrons at site j, isthe chemical potential, and U is the Hubbard interactionparameter. In this work, we focus on only the half filledcase 0, where there is one electron per site. However,we expect our results to also remain valid for smalldeviations of half filling. The spin chain is described bythe spin-1=2 Heisenberg model,

H2 JXNs1j1

Sj Sj1; 2

with J > 0. Here, we assume the same lattice parameter afor both chains H1 and H2. Finally, motivated by the workby Alexandrov and Coleman [26], we assume the followingexchange coupling between the two chains,

HK JKXNsj1

Sj j; 3

where JK > 0 is the Kondo interaction between the jthspin (Sj) in the Heisenberg chain and the p-wave spindensity in the fermionic chain at site j, defined as

j pj;2

pj;; 4

where pj; cj1; cj1;=2

pis a linear combination

of orbitals with p-wave symmetry and is the vector ofPauli matrices. This model can be regarded as a stronglyinteracting version of the Tamm-Shockley model [3234].While for our present purposes this is an interesting

FIG. 1. Schematic representation of the 1D p-wave Kondo-Heisenberg model. The top chain corresponds to the Hubbardmodel and the bottom chain is the S 1=2 antiferromagneticHeisenberg model. The Kondo exchange (depicted pictorially asslanted bonds with odd inversion symmetry) is a nonlocalnearest-neighbor antiferromagnetic interaction JK > 0 thatcouples a spin Sj in the Heisenberg chain with the p-wave spindensity j in the Hubbard chain [see Eq. (4)].

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toy model Hamiltonian that allows us to extract a usefulinsight into strongly interacting topological phases, itcould, in principle, be realized in ultracold-atom experi-ments (see Sec. V for details). In the absence of interactionsin the fermionic chain (i.e., U 0) and in the large-Nmean-field approximation, Alexandrov and Coleman haveshown the emergence of topologically protected edge statesarising from the nontrivial form of the Kondo term [Eq. (3)][26]. In their mean-field approach, the effective descriptionof the system corresponds to noninteracting quasiparticlesfilling a strongly renormalized valence band with a non-trivial topology, stemming from the charge conjugation,time reversal, and charge U(1) symmetry of the effectivelynoninteracting Hamiltonian (see also Ref. [35] for adiscussion of a closely related system).In this paper, our goal is to understand the emergence of

topologically protected edge states without introducing anydecoupling of the Kondo interaction, including the inter-acting case, U 0. We consider the case of small U andJK . This is formally represented by linearizing the non-interacting spectrum k 2t cos ka in the fermionicchain H1 around the Fermi energy 0 and taking thecontinuum limit where the lattice constant a 0. Then, thefermionic operators admit the low-energy representation[36,37]

cj;a

p eikFxjR1;xj eikFxjL1;xj; 5

where R1;x and L1;x are right- and left-movingfermionic field operators, which vary slowly on the scaleof a. As we are interested in the edge-state physics, weconsider open boundary conditions c0; cNs1; 0,leading to the following constraints:

L1;0 R1;0; 6L1;Lc ei2kFLcR1;Lc; 7

where Lc Nsa is the length of the chain. We nextintroduce the Abelian bosonization formalism [36,37],

R1;x F1;2

p ei1;R;x;

L1;x F1;2

p ei1;L;x; 8

where is a short distance cutoff in the bosonizationprocedure (we take a hereafter). In Eq. (8), 1;x(with fR;Lg) are bosonic fields obeying the commu-tation relations 1;Rx;1;R0 y isgnx y;0 ,1;Lx;1;L0 y isgnx y;0 , and F1; areKlein operators, which obey anticommutation relationsfF1;; F1;0 g ;0 , and, therefore, ensure the correctanticommutation relations for fermions. Because of the

constraints (6) and (7) introduced by the open boundaryconditions, the right and left movers are not independent,and obey the constraints

1;L;0 1;R;0 ; 91;L;Lc 1;R;Lc 2kFLc 2q: 10

Here, q is an integer representing the occupation of thezero-mode excitations, i.e., particle-hole excitations withmomentum k 0 and total spin . Its presence in Eq. (10)can be understood recalling that the expression of thenonchiral bosonic field 1; 1;R; 1;L;=2 is [38]

1;x

2 kFLc q xLc

Xn1

sin knxn

p n; n;;

where kn n=Lc, with integer n > 0, and n; arebosonic operators obeying the commutation relationn;;m;0 n;m;0 (see Refs. [38,39] for details).From here, we obtain the additional commutationrelations [38]

1;Rx;1;L0 y

8>:i;0 for 0 < x; y < Lc0 for x y 02i;0 for x y Lc:

11

The bosonization procedure applied to the 1D Hubbardmodel is standard, and we refer the reader to textbooks fordetails [36,37]. Introducing charge and spin bosonic fields1; 1=

2

p 1c 1c 1s 1s (where the con-vention of signs fR;Lg f;g and f;g f;g is implied), the 1D Hubbard model at half filling(i.e., kF =2a) becomes [36]

H1 Xc;s

ZLc

0

dx

v1

2K1x12 v1K1

2x12

U

2a2Z

Lc

0

dxcos8

p1c cos

8

p1s; 12

where the new fields obey the boundary conditions

1s0 0; 1sLc 2

p q q; 13

1c0 2

p ;

1cLc 2

p kFLc

2

2p q q; 14

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and the commutation relation 1;x; 1;0 y i=2;0sgnx y. From here, we conclude that thefield 1=x1;x is the momentum canonically con-jugated to 1;x.As is well known, in 1D charge and spin excitations

generally decouple and the above Hamiltonian can be splitas H1 H1c H1s, with the first line describing indepen-dent Luttinger liquids for the charge and spin sectors,which are characterized by charge (spin) acoustic modeswith velocities v1c vF

1 Ua=vF

p v1s vF, andLuttinger parameter controlling the decay of the correlationfunctions K1c 1=

1Ua=vF

p(K1s 1). The pres-

ence of the cosine terms in the second line of Eq. (12)changes the physics qualitatively. In the present work, werestrict our focus to the case U 0, where the term cos 8p 1c is marginally relevant in the renormaliza-tion-group sense and opens a Mott gap in the charge sector.At the same time, the term cos 8p 1s is marginallyirrelevant at the SU(2) symmetric point, and the spin sectorremains gapless [36,37].The bosonization of the Heisenberg chain H2 is also

quite standard, and we refer the reader to the above-mentioned textbooks [36,37]. A usual trick consists ofrepresenting the spin operators Sj by auxiliary fermionicoperators in a half filled Hubbard model with interactionparameter U0 U. Therefore, while technically the pro-cedure is identical to Eq. (12), the charge degrees offreedom in the bosonized Hamiltonian, H2 can be assumedto be absent at the relevant energy scales of the problem dueto the Mott gap U0. Then, ignoring the charge degrees of

freedom and irrelevant operators in Eq. (12), and replacingthe chain label 1 2, we obtain

H2 v2s2

ZLc

0

dxx2s2 x2s2: 15

Finally, we bosonize the Kondo Hamiltonian. Thep-wave spin density in the fermionic chain and the spindensity in the Heisenberg chain are, respectively,

ja 2J1Rxj J1Lxj 1jN1xj; 16Sja J2Rxj J2Lxj 1jN2xj; 17

where JaRx Ra;x;=2Ra;x and JaLx La;x;=2La;x (with a 1; 2) are thesmooth components of the spin density, with bosonicrepresentation

Jxax 1

2acosf

2

pa;sx a;sxg; 18

Jyax 1

2asinf

2

pa;sx a;sxg; 19

Jzax 18

pxa;sx xa;sx; 20

where RL corresponds to the plus (minus) sign, andwhereNax Ra;x;=2La;x H:c: are the stag-gered components

N1x cos 2p 1cx

afcos

2

p1sx; sin

2

p1sx; cos

2

p1sxg; 21

N2x m2a

fcos2

p2sx; sin

2

p2sx; cos

2

p2sxg; 22

with m2 hcos 2

p2cxi resulting from the integration

of the gapped charge degrees of freedom in the Heisenbergchain. Therefore, although the spin densities (16) and (17)look similar in the bosonized language, they actually differin two crucial aspects. (1) While in Eq. (22) the chargedegrees of freedom are absent in the expression of thestaggered magnetization, they are still present in Eq. (21) inthe term cos 2p 1cx and we need to consider them.(2) Comparing Eqs. (16) and (17), we note a sign differencein the staggered components. This sign is related to thep-wave nature of the operators pj; , and is, therefore,intimately connected to the topology of the Kondo inter-action. The role of this sign turns out to be crucial in the restof the paper.

Replacing the above results into Eq. (3), and taking thecontinuum limit, the Kondo interaction becomes, in thebosonic language,

HK 2JKaZ

Lc

0

dx

X;0L;R

J1x J20 x

N1x N2x; 23

where the sign in the second line is a consequence of theabove-mentioned sign in the staggered part of x.Note that this model is reminiscent of the (nontopolog-

ical) 1D Kondo-Heisenberg model, which has recently

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received much attention in the context of pair-densitywave-ordered phases in high-Tc cuprate physics [4046],and of the Hamiltonian of a spin-1=2 ladder [38,4749].However, a crucial difference from those works is thenontrivial structure of the Kondo interaction, whichdiffers from the usual coupling JKSj sj, where sj cj;=2cj; is the standard (i.e., s wave in this context)spin density in the fermionic chain. The first line in Eq. (23)is, in fact, closely related to the model considered inRefs. [4145]. In the half filling situation we are analyzinghere, however, the most relevant part of HK (in the RGsense) is given by the product N1x N2x, whichdominates the physics at low energies [38,4749]. Theterm N1x N2x survives when the system is at (or closeenough to) half filling, and when both chains have the samelattice parameter (in other situations, the oscillatory factorsei2kFx suppress this term, and the situation corresponds tothe case analyzed in Refs. [4145]). Therefore, for ourpresent purposes, we can neglect the first term in Eq. (23)and focus on the second term:

HK 2JKaZ

Lc

0

dxN1x N2x;

2JKm22a

ZLc

0

dx cos2

p1c

cos2

p1s cos

2

p2s sin

2

p1s

sin2

p2s sin

2

p1s sin

2

p2s: 24

At this point, we note that the problem is reminiscent of thewell-known case of S 1=2 ladders with open boundaryconditions [38,48], with the important difference that herethere is an extra factor cos 2p 1c.The physics of the spin sector [i.e., the term in square

brackets in Eq. (24)] is quite nontrivial due to the presenceof both the canonically conjugate fields a;sx and a;sx,which cannot be simultaneously stabilized. However,the analysis of the charge sector is simpler, as only thefield 1cx appears in the expression. This means thatin the limit JK , the system can gain energy byfreezing out the charge degrees of freedom, i.e., m1 hcos 2p 1cxi, as there is no other competing mecha-nism. In the next section, we substantiate these ideas byproviding a rigorous analysis.

III. RENORMALIZATION-GROUP ANALYSIS

Based on the similarity with the physics of spin ladders,we introduce symmetric and antisymmetric fields 1= 2p 1s 2s and 1= 2p 1s 2s, in termsof which the Hamiltonian becomes

HK JKm22aZ

Lc

0

dx cos2

p1c

cos 2 cos 2 2 cos 2: 25In what follows, we assume identical spinon dispersionv1s v2s vs. Although this assumption is certainly anidealization, one can show that the asymmetry vv1s v2s is a marginal perturbation in the renormaliza-tion-group sense, and therefore we do not expect that smallasymmetries will have a qualitative effect on our results.We now write the Euclidean action of the system usingcomplex space-time coordinates z vF ix and z vF ix, with it the imaginary time, and the leftand right fields LR=2, where f;;1cg.The Euclidean action becomes

S S0 SU SK; 26

S0 14

Zd2r

zcL2 zcR2

X

zR2 zR2; 27

SU G2cZ

d2rO2cr G3Z

d2rO3r; 28

SK GKZ

d2ra

p OKr; 29

with d2r vFdxd, and where we have defined thedimensionless couplings,

G02c G03 UavF

; G0K JKm2avF

; 30

and the scaling operators,

O2c 1

4z1cL z1cR; 31

O3 2L2 cos22

p1c; 32

OK 2

p

L3=2 cos

2

p1c cos 2

cos 2 2 cos 2; 33

where we have explicitly normal ordered the operators. S0corresponds to a free fixed-point action, and SU and SK areperturbations arising from the Hubbard repulsion in chain 1and the Kondo interchain interaction, respectively. Weneglect all the perturbations in the spin sector generatedby U, as they renormalize to zero along the SU(2)-invariantline in the parameter space. We also neglect the less

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relevant terms coming from the product of the smooth partof the currents in Eqs. (16) and (17).Expanding the generating functional Z R Qif1c;gDi; ieSi;i perturbatively at second order

in Gi, we obtain the product of the different operators [i.e.,the operator product expansion (OPE)] of Eqs. (31)(33).Importantly, the OPE of the Kondo interactionOKz0; z0OKz; z gives rise to operators O3 and O2c,which are already present in the charge sector of chain 1.This corresponds to an effective dynamically generatedHubbard repulsion originated by the interchain Kondocoupling (see the Appendix). Therefore, even for aninitially noninteracting chain (i.e., U 0), this emergentrepulsive interaction induces the opening of a Mott insu-lating gap in the charge sector of the half filled conductionband. At energies below this gap, the field 1c becomespinned to the degenerate values 0 or =

2

p. Note that only

the latter is consistent with the boundary conditions givenin Eq. (14). Therefore, this analysis suggests that the energyis minimized by a uniform configuration of the field 1c,which freezes at the bulk value =

2

p. While other

configurations with kink excitations connecting the differ-ent minima are certainly possible, these configurationsare more costly energetically speaking and do not belong tothe ground state. Therefore, energy minimization preventsthe ground state from developing kink excitations in thecharge sector and, consequently, we can exclude thepresence of localized charge edge states. Then, at lowenough energies, the system becomes a Mott insulator inthe bulk due to the electronic correlations, and no topo-logical effects arise in the charge sector.A more quantitative study can be done by analyzing the

two-loop RG-flow equations (see the Appendix for details):

dG2cdl

G23 3

4G2K; 34

dG3dl

G2cG3 3

4G2K; 35

dGKdl

GK2

14G2cGK G3GK; 36

where l ln a=a0 is the logarithmic RG scale. WhenGK 0, these equations reduce to the Kosterlitz-ThoulessRG equations corresponding to the charge sector of theHubbard model. They predict a charge gap that is expo-nentially small in U [36].Let us now analyze the case of an initially U 0. In this

situation, only the linear term survives in the right-handside of Eq. (36), which expresses the relevance of theoperator OK and gives rise to an exponential increase ofGKl with the RG scale l as GKl G0Kel=2. As aconsequence, the coupling G3l in Eq. (35), representingthe Hubbard repulsion, also increases exponentially

G3l 34G0K2el 1, from its initial value G30 0.This produces the anticipated Mott insulating gap c in thecharge sector. Its dependence with the parameters can beobtained by the procedure described in Ref. [36] (p. 65).We obtain c G0K2 for small enough JK . We canenvisage that if U and JK would be of the same order,the dominant contribution to c would come from theinterchain Kondo coupling.Following previous references [38,4749], we can refer-

mionize Eq. (25), noting that the scaling dimension of thecosines in the square bracket exactly corresponds to thefree-fermion point and therefore they can be written interms of right- and left-moving free Dirac fermion fields;Rx and ;Lx as

cos 2 iaRL LR; 37cos 2 iaRL LR: 38

For later purposes, it is more convenient to introduce aMajorana-fermion representation of the fields ; 1=

2

p 2 i1, ; 1=2

p 3 i0; the Hamiltoniancan be compactly written as H H0 HK , where

H0 v1c2

ZLc

0

dx

x1c2K1c

K1cx1c2 U cos8

p1c

v1ca2

ivs2

X3a0

ZLc

0

dxaRxaR aLxaL; 39

HK iJKm22

ZLc

0

dx cos2

p1c

30R

0L

X3a1

aRaL

;

40

where the Majorana fields obey the boundary conditions

aR0 aL0; 41

aRLc aLLc: 42

The uniform symmetric and antisymmetric spin densities inthe ladder become [47]

Ma Ja1; Ja2; i2 abcb

c; 43

Ka Ja1; Ja2; i2 0

a ; 44

with a f1; 2; 3g and fR;Lg. This is a well-knownrepresentation of two independent SU21 Kac-Moodycurrents J1; and J2; in terms of four Majorana fields[47]. In our case, these four degrees of freedom, resultingfrom the combination of the two SU(2) spin-density fields

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in the two chains, are expressed in terms of a singlet 0 andtriplet a Majorana fields.From the previous analysis, we conclude that at temper-

atures T c, the charge and spin degrees of freedombecome effectively decoupled, and the low-energyHamiltonian of the system can be written as H ~H ~Hc ~Hs, with

~Hc v1c2

ZLc

0

dx

x1c2K1c

K1cx1c2

Ueffm21

v1ca21c2

; 45

where, based on the discussion above Eq. (34), we haveexpanded the charge field near the value 1c =

2

p, and

~Hs ivs2

X3a0

ZLc

0

dxaRxaR aLxaL

i JKm2m12

ZLc

0

dx30R

0L

X3a1

aRaL

; 46

where Ueff UJK is the renormalized Coulomb repul-sion parameter and where the charge degrees of freedomin the fermionic chain develop a gap m1 m1JK hcos 2p 1ci. Note that, in this form, the spinHamiltonian ~Hs is similar to that of a spin-1=2 ladder[38,47,48]. The quantitative determination of the parameterm1 requires a self-consistent calculation, which is beyondthe scope of the present work. Nevertheless, the previousanalysis allows us to understand two important aspects ofthe topological Kondo-Heisenberg chain: (a) the generationof an insulating state in the bulk (necessary to reproduce thebulk insulating state of a TKI) and (b) the emergence ofmagnetic edge states, which is the subject of the nextsection.

IV. S 1=2 MAGNETIC EDGE MODESAND TOPOLOGICAL INVARIANT

The previous analysis shows that, at low temperaturesT c, the 1D p-wave Kondo-Heisenberg chain at halffilling can be effectively mapped onto a spin ladderproblem, which is dominated by the staggered componentsof the spin densities. Interestingly, for an antiferromagneticKondo coupling JK > 0, the negative sign emerging fromthe structure of the nonlocal Kondo interaction (24)effectively induces a local ferromagnetic interaction [seevertical dashed lines in Fig. 2(a)]. It is well known that thespin ladder with ferromagnetic exchange coupling J < 0along the rungs has a low-energy triplet sector [38,47,48],which maps onto the Haldane spin-1 chain [50,51].Therefore, at temperatures T J, our model describesthe physics of the Haldane spin-1 chain, which is known to

host symmetry-protected topological spin-1=2modes at theboundaries [5260]. This situation is also very reminiscentof the case of the ferromagnetic Kondo lattice [6164].To see how these spin-1=2 boundary modes emerge inour low-energy Hamiltonian (46), we consider solutionsof the eigenvalue equation ~Hsax 0, where ax aR; aLT is a Majorana spinor, and a 0; 1; 2; 3 [38].In matrix form, this equation is

ivs3x ma2ax 0; 47with ma 3JKm2m1=2, for a 0 [ma JKm2m1=2for a 1; 2; 3] and i the Pauli matrices acting on theright- or left-moving space. Equation (47) admits solutionsof the form ax exp ma1x=vsa0, and onewould think that, in principle, two normalizable solutionsin the limit x could arise: (1) the choicea0 1;1T , with ma < 0, and (2) a0 1;1T ,with ma > 0. However, note that only the last choice iscompatible with the boundary condition (41). Then, theonly physical solution localized around x 0 correspondsto the choice ma > 0, which in our case corresponds toa 1; 2; 3:

ax aemax=vs1

1

; a 1; 2; 3; 48

with a being a localized Majorana fermion. Using theexpression (43) for the smooth part of the spin density, wecan physically associate the presence of the localizedMajorana bound state with a localized spin-1=2 magneticedge state. This is consistent with the results in Ref. [26],where, in the case of a uniform Kondo interaction,

FIG. 2. (a) At temperatures T c J2K , the dynamicallyinduced Mott gap, the charge degrees of freedom in the Hubbardchain become frozen and the system maps onto a ladder where thenonlocal antiferromagnetic Kondo coupling effectively induceslocal ferromagnetic correlations between the spins in the ladder(vertical dashed lines). (b) The system depicted in (a) can bemapped onto a S 1 Haldane chain, which supports topologi-cally protected S 1=2 end states [38,47,48,5256].

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a magnetic mode with no admixture with charge degrees offreedom emerges at the boundaries. However, note that theorigin of these edge states is quite different: while in themean-field regime they emerge as a consequence of Kondo-unscreened end spins in the Heisenberg chain, in ourcase they are intimately related to the physics of theHaldane chain.We now derive a topological invariant to characterize the

presence of the edge states, using a suitable generalizationof the concept of electrical polarization in 1D insulators[6570]. We, therefore, focus on the uniform magnetizationEq. (43). Although the original Hamiltonian has spin-rotational symmetry, the Abelian bosonization is not anexplicitly SU(2)-invariant formalism. Therefore, while thechoice of axes is arbitrary due to the spin-rotation sym-metry of the problem, once we define the z direction as thespin-quantization axis, the perpendicular components ofthe magnetization Mxx and Myx acquire a morecomplicated mathematical form. For that reason, it isconvenient to focus only on the z component of thesymmetric spin density Eq. (43):

Mzx XL;R

Mz1;x Mz2;x;

1

p xx: 49

In the expression above, we use Eq. (20) and the definitionof x 1=

2

p 1x 2x. We now define thetotal magnetic moment along the z axis at one end(for concreteness, the left end) of the chain asmzT 1= p R xb0 dxMzx, where xb is an unspecifiedposition in the interior of the chain where the magnetizationreaches the value in the bulk. In bosonic language, itacquires the compact form

mzT 1 xb 0: 50

From the expression of the bosonic Hamiltonian Eq. (25) inthe limit JK , we see that the system minimizes theenergy in the bulk by pinning the field x to one of thedegenerate values:

xb

2: 51

On the other hand, from the definition of x 1= 2p 1sx 2sx and Eq. (13), the boundary con-dition 0 0 is obtained. Replacing these values intoEq. (50), we obtain the following quantized values of themagnetic moment at the left end:

mzT 1

2: 52

This magnetic moment at the end of the chain is analogousto the electrical polarization [6569] or the time-reversalpolarization [70] in 1D insulators. In particular, we note theclose relation between our Eq. (50) and the expressions forthe displacement operator appearing in Eq. (23) ofRef. [67], and for the time-reversal polarization appearingin Eq. (4.8) in Ref. [70], both given in bosonic language.From here, we can define a Z2 topological invariant thatcharacterizes the topological phase of the Kondo-Heisenberg chain,

Q 12mzT= ei2mzT ; 53which in the limit of an infinite system L is Q 1in the topological phase (JK > 0) and Q 1 in the trivialphase (JK < 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 x in thebulk satisfy the boundary condition at x 0, we concludethat a kink excitation necessarily must emerge near theboundary in order to connect those values: precisely thiskink excitation gives rise to the spin-1=2 end state, uponuse of Eq. (49). We remind the reader that in Sec. III, usingsimilar arguments, we demonstrate the absence of kinkconfigurations in the charge field 1cx, and the fact thatthere are no charge edge states in the ground state.

V. CONCLUSIONS

We study theoretically a model for a topological 1DKondo insulator (the 1D Kondo-Heisenberg model coupledin the p-wave channel, with an on-site Hubbard interactionU in the conduction band) using the Abelian bosonizationformalism and derive the two-loop RG flow equations forthe system at half filling. Our RG analysis shows that thesystem develops a Mott insulating gap at low enoughtemperatures, even if U 0. Moreover, the remaining spindegrees of freedom are effectively described by a ferro-magnetic spin-1=2 ladder, which in turn maps onto a spin-1Haldane chain with topologically protected spin-1=2 mag-netic edge modes. This situation is reminiscent of thephysics of the ferromagnetic Kondo necklace, which alsomaps onto the spin-1 Haldane chain [6164], although inour case it arises as a result of the nontrivial structure of theKondo coupling.In contrast to three-dimensional bulk topological Kondo

insulators, where the mean-field approximation is welljustified and the system can be effectively described interms of noninteracting quasiparticles opening a (renor-malized) hybridization gap near the Fermi surface[17,19,71], in one spatial dimension the presence of strongquantum fluctuations cannot be ignored, and one is forcedto use different approaches. The Abelian bosonizationmethod allows us to obtain a description of the 1D TKI,which is fundamentally different from the mean-field

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picture. In the first place, the system develops a Mott gap(instead of a hybridization gap) in the spectrum of chargeexcitations when the conduction band is half filled (smalldeviations from half filling do not affect this scenarioqualitatively [36]). This Mott gap arises from umklappprocesses at second order in the Kondo interaction.Physically, this can be understood as a dynamically inducedeffective interaction term, which appears at order J2K in theconduction band by integrating out perturbatively short-time spin excitations in the Heisenberg chain. In contrast tothe mean-field description, where the hybridization gapdepends exponentially on the microscopic Kondocoupling c exp 1=JK, the integration of the RGEqs. (34)(36) in the limit JK 0 results in c J2K .Our RG analysis indicates that JK is a relevant perturbationand flows to strong coupling, dominating the physics at lowtemperatures. In particular, at temperatures below the Mottgap, the charge degrees of freedom are frozen and thesystem effectively behaves as a ferromagnetic spin-1=2ladder, which is known to map onto the spin-1 Haldanechain. Therefore, our work allows us to make an insightfulconnection between two a priori unrelated physical mod-els. Interestingly, exploiting this connection, we predict theexistence of topologically protected spin-1=2 edge states.This seems to correspond to the magnetic phase found byAlexandrov and Coleman [26], which for a uniform Kondocoupling JK is characterized by Kondo-unscreened spins atthe end of the Heisenberg chain. However, the emergenceof these edge states again corresponds to a very differentmechanism than the one provided by the mean-field theory.Interestingly, within the bosonization framework, we areable to obtain a Z2 topological invariant [see Eq. (53)] interms of the magnetization at an end of the chain.Our work opens the possibility to explore the

physics of broken Z2 Z2 hidden symmetry and theexistence of a nonvanishing string order parameter

Ostring limjlmjhSl eiP

lj

APPENDIX: DYNAMICALLY GENERATEDINTERACTIONS AND DERIVATION OF THERENORMALIZATION-GROUP EQUATIONS

Here, we derive an effective action for the system and therenormalization-group equations [Eqs. (34)(36)]. The ideais to show that umklapp processes, which mimic a repulsive

interaction among electrons in the half filled conductionband, arise at order OJ2K and open a gap in the chargesector of the model. To that end, we expand the generatingfunctional of the system (i.e., the partition function) up tosecond order in the coupling constants G followingRefs. [73,74],

ZZ0

1 X

Ga2

Zd2rhOi0

1

2

X;

GGa4

Z Zjrr0j>a

d2rd2r0hOrOr0i0 ; A1

where indices and run on 2c, 3, and K. Here, is the scaling dimension of the operator O definedin Eqs. (31)(33) (3 2c 2, K 32), Z0 R Q

if1c;gDi; ieS0i;i is the generating functionof the free theory, and the mean values h i0 alsocorrespond to that theory. This formalism is standard inthe analysis of 1D quantum systems, and has been appliedin several previous works (see, for example, Ref. [75],where the method is explained in detail).The third term of the rhs in Eq. (A1) takes the same form

as the second one if we assume that for r r0 the productof two operators fulfills the following operator productexpansion property [73,74] :

OrOr0 X

COrr02

r r0more irrelevant operators; A2

where O includes all the operators generated fromeach OPE.Let us focus on the OPE between two OK operators,

which is the most relevant perturbation in the RG sense,and is precisely the contribution that leads to the umklappprocesses we are trying to describe. To simplify thediscussion, here we return to the representation of thebosonic fields in terms of left and right movers Lz andRz [see Eq. (8)], with z vF ix and z vF ix.We now assume to be sufficiently far away from theboundaries. In these conditions, the boundary conditions(9) and (10) and the commutation relation (11) can beeffectively neglected, and the fields Lz and Rzbecome independent (i.e., they do not mix). This allowsus to focus on only the processes involving the left-movingfield Lz (for right-moving fields, we just need to changeL R and z z). The basic OPE we need is

eiLz ei0Lz0 2

L

0=2

ei0Lz0

1

z z00 iz0Lz0z z001

; A3

which is obtained by normal ordering the rhs expressionand then developing for z0 near z. Through repeated use ofthis expression, we obtain the desired OPE, which reads

OKz; zOKz0; z0 3

4

O3z z0

3

4

O2cz z0

34

Oz z0

1

4

Oz z0 ; A4

where we define the operatorsO 14 zRzL [whichalso appear in the z component of the product of the right andleft smooth-varying spin currents, in the first two lines inEq. (3.8) of Ref. [45]]. Note that these terms break the SU(2)invariance of the model. This is a well-known feature of theAbelian bosonization prescription, which is not explicitlySU(2)-invariant formalism [36,37]. This means that one hasto keep track of all contributions to recover the SU(2)invariance and, vice versa, neglecting irrelevant operators

[as we do to obtain the action in Eq. (26)] might result inapparent inconsistencies in the formalism. In our case, thisproblem has no consequences for our purposes because theoperators O renormalize the couplings of the marginalcontributions, which we, in any case, neglect in relation tothe relevant contribution N1 N2. Therefore, we will notconsider these operators.On the other hand, the first line in Eq. (A4) is physically

more interesting, as the operators O3 and O2c were alreadypresent in the action (28) corresponding to the Hubbardmodel. If we insert (A4) into (A1), change variables asr r r0 and R r r0=2, and integrate over r imposinga cutoff of order a, we obtain an expression that renorm-alizes the first-order contribution. We identify the effectivecoupling for operators O2c and O3 as

G02c G02c 3

8G2K; A5

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and the same for G03. Therefore, we show that the interchainKondo coupling generates an effective Hubbard repulsionUeff 3a=8JKm22=vF in the conduction chain. Theequation above can be physically interpreted as umklappprocesses (generated by integrating out fast spin fluctua-tions in the Heisenberg chain at second order in theinterchain Kondo coupling) which mimic the effect of aninteraction in the conduction band.To determine the actual dependence of the charge gapc

with respect to the parameters of the model, we need toderive the RG flow equations. This is achieved followingsimilar steps as in the previous paragraphs. The main ideais that the theory defined with a microscopic cutoff ashould remain invariant under a scaling transformationa a1 dl, where dl is a dimensionless infinitesimal.Therefore, the couplings Ga in Eq. (A1) must bechanged in such a way that they preserve the generatingfunctional, i.e., Za Za1 dl. The method is stan-dard and we refer the reader to Ref. [73] for details. Therenormalization group flow equations can be written interms of the coefficients C as

dGdl

2 G X

CGG; A6

where the coefficients C2cKK C3KK 3=4 areextracted from Eq. (A4). The remaining coefficients areobtained by the OPEs between the corresponding oper-ators. Following the lines of Ref. [39], we straightforwardlyobtain

C32c 3 12 ; C2c3 3 1 ;

CK2cK 18 ; CK3K 12 : A7

Inserting these values into Eq. (A6), we obtainEqs. (34)(36) in the main text.

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