Quantum dot in a two-dimensional topological insulator: The two-channel Kondo fixed point The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Law, K. T. et al. “Quantum dot in a two-dimensional topological insulator: The two-channel Kondo fixed point.” Physical Review B 81.4 (2010): 041305. © 2010 American Physical Society. As Published http://dx.doi.org/10.1103/PhysRevB.81.041305 Publisher American Physical Society Version Final published version Accessed Thu May 26 19:14:38 EDT 2016 Citable Link http://hdl.handle.net/1721.1/56577 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms RAPID COMMUNICATIONS PHYSICAL REVIEW B 81, 041305共R兲 共2010兲 Quantum dot in a two-dimensional topological insulator: The two-channel Kondo fixed point K. T. Law,1,2 C. Y. Seng,3 Patrick A. Lee,2 and T. K. Ng3 1Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China 2 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Department of Physics, Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China 共Received 14 December 2009; published 14 January 2010兲 A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter K ⬍ 1, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which K ⬍ 1 / 2 is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a K dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated. DOI: 10.1103/PhysRevB.81.041305 PACS number共s兲: 73.63.Kv, 71.10.Pm, 72.15.Qm, 72.25.Hg The two-channel Kondo model has been under intense theoretical study1 after the seminal work of Noziere and Blandin,2 in which they pointed out that a non-Fermi liquid fixed point exists for the two-channel Kondo model with noninteracting electrons. Recently, the authors of Ref. 3 demonstrated experimentally that a quantum dot coupled to an infinite reservoir with noninteracting electrons and a finite reservoir with interacting electrons may display two-channel Kondo effect. The key idea is that Coulomb blockade suppresses the exchange of electrons between the finite reservoir and the infinite reservoir.4 Consequently, the two reservoirs couple to the quantum dot independently. Indeed, using electron-electron repulsion to suppress interreservoir tunnelings was suggested earlier by Fabrizio and Gogolin, in the context of a quantum dot coupled to two Luttinger liquid leads.5–7 They show that the two-channel Kondo fixed point can be reached when the repulsive interactions of the electrons in the leads are strong enough, i.e., with Luttinger liquid parameter K ⬍ 1 / 2. Recently, a class of materials called topological insulators were first theoretically proposed and then experimentally fabricated.8–12 Two-dimensional topological insulators have gapless helical edge states despite the presence of a bulk gap.8 The edge states are called helical because the directions of the spins and the momentum of the electrons are tied together.13 The one-channel Kondo effect of helical edge states was first studied in Ref. 14 as a truncated model of Kondo effect in Luttinger liquids. A more detailed recent study is carried out in Ref. 15. In this work, we study a quantum dot coupled to two helical edge states as depicted in Fig. 1共a兲. If the electrons on the edge states are noninteracting, it is well known that the system can be described by a one-channel Kondo Hamiltonian.16,17 However, we show that a weak repulsive interaction, with K ⬍ 1 is enough to drive the system to the two-channel Kondo fixed point. This is in sharp contrast to the case of Luttinger liquid leads in which K ⬍ 1 / 2 is needed.5 This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a K dependent boundary term. We study the ground state, thermaldynamic and transport properties near this two-channel fixed point. More specifically, we show that the impurity entropy at zero temperature is ln冑2K, the impurity specific-heat 1098-0121/2010/81共4兲/041305共4兲 Cimp ⬀ T2/K−2 when 2 / 3 ⬍ K ⬍ 1, and Cimp ⬀ T when K ⬍ 2 / 3, and the linear conductance across the two helical edges through the quantum dot has nontrivial temperature dependence as a result of the renormalization group flow. The schematic diagram of our model is depicted in Fig. 1共a兲. Two helical edge states of an two-dimensional 共2D兲 topological insulator are brought close to each other at a tunneling junction. A quantum dot is placed at the middle of the junction and coupled to the two edge states through electron tunnelings. The system is described by an Anderson Hamiltonian HA = H0 + Hi + Hd + Ht where H0 = 兺 − ivF i=1,2 Hi = 冕 兺 兺 i=1,2 ␣=R,L + g2 冕 † † 关iR↑ 共x兲xiR↑共x兲 − iL↓ 共x兲xiL↓共x兲兴dx, g4 2 冕 i†␣共x兲i␣共x兲i†␣共x兲i␣共x兲dx † † iR↑ 共x兲iR↑共x兲iL↓ 共x兲iL↓共x兲dx Hd = 兺 ⑀d† d + Ud†↑d↑d†↓d↓ , 1a) S1 S2 1b) 1 Edge 1 t t + −−2 Edge 2 1/2 < < 1 1 −−2 2 −−2 < 1/2 0 D1 D2 FIG. 1. 共a兲 A quantum dot coupled to two helical edges of a 2D topological insulator through electron tunneling. S and D denote the source and drain, respectively. 共b兲The schematic picture of the linear conductance G from S1 to D2 versus temperature T. The solid line and the dashed line depict the 1 / 2 ⬍ K ⬍ 1 and K ⬍ 1 / 2 cases, respectively. 041305-1 ©2010 The American Physical Society RAPID COMMUNICATIONS PHYSICAL REVIEW B 81, 041305共R兲 共2010兲 LAW et al. Ht = t共d†↑iR↑ + d†↓iL↓兲 + h.c. 兺 i=1,2 K= H0 and Hi denote the kinetic energy and the interaction of the helical edge states, respectively. From H0, we see that the right/left moving electrons are spin up/down. Hd is the Hamiltonian of the quantum dot where ⑀ is the single particle energy level at the dot and U is the on-site interaction energy. Ht describes the coupling between the dot and the edges and we have tuned to the channel symmetric point such that t is independent of the edge index. In the presence of spin-orbit coupling, spin 共↑ and ↓兲 denotes pseudospin of the edge modes. In the large U limit, one obtains the Kondo Hamiltonian H = H0 + HK through perturbation theory,18 where HK = 兺 J1Sជ · i=1,2 冉 i†␣ · 冊 冉 冊 ␣ ␣ · i + 兺 J2Sជ · i†␣ · · j . 2 2 i⫽j 共1兲 Initially, J1 = J2 ⬎ 0 and without interaction, Eq. 共1兲 reduces to a single channel Kondo problem.16,17 Our goal is to show that J2 normalizes to zero at low temperatures when g2 ⬎ 0. Because of helicity, one of the R/L and the spin ↑↓ indexes are redundant, we keep only one of them starting from Eq. 共1兲. It is important to note that, in general, there are extra † 共0兲iL共0兲 + h.c. caused backscattering terms of the form iR by the quantum dot in the Hamiltonian H. The backscattering terms cut each edge into two separate parts at low temperatures if K ⬍ 1 and render the geometry in Fig. 1共a兲 unstable. However, in our case, single particle back-scattering terms are forbidden because of time-reversal symmetry. Twoparticle back scatterings preserve time-reversal symmetry but they are irrelevant so long as K ⬍ 1 / 4.13 Thus, our discussion below is valid for K ⬎ 1 / 4. By Abelian bosonization,19,20 the electron operators can be written as: iR/L = 冑21a e⫾i共冑4iR/L共x兲+kFx兲. We define bosonic fields i共x兲 = iL共x兲 + iR共x兲 and i共x兲 = iL共x兲 − iR共x兲. The dual fields satisfy the commutation relations 关i共x兲 , j共x⬘兲兴 = −2i ␦ij sgn共x − x⬘兲, where sgn共x兲 = 0 when x = 0. The symmetric and antisymmetric combinations of i and i are denoted as: s/a = 冑12 共1 ⫾ 2兲 and s/a = 冑12 共1 ⫾ 2兲. In terms of s/a and s/a, the above Hamiltonians can be written as: H0 + Hi v⬘ 1 = 2F 兰 K 共xs兲2 + K共xs兲2 + K1 共xa兲2 + K共xa兲2dx and HK = − 冑 2 z J Szxs共0兲 1 + Jxy 冑 冑 1 关S−e−i 2s共0兲 + S+ei 2s共0兲兴cos关冑2a共0兲兴 a + 2Jz2 z S sin关冑2a共0兲兴sin关冑2a共0兲兴 a + Jxy 冑 冑 2 关S−e−i 2s共0兲 + S+ei 2s共0兲兴cos关冑2a共0兲兴. a Here, the Luttinger liquid parameter is defined as 冑 1 + g4/2vF − g2/2vF 1 + g4/2vF + g2/2vF and vF⬘ = vF冑共1 + g4 / 2vF兲2 − 共g2 / 2vF兲2.20 The bosonic form of HK is spin anisotropic, we denote the two components of Ji as Jz/xy i , respectively. This is reasonable for our problem, because spin-orbit coupling breaks SU共2兲 symmetry of the electrons on the edges. The spin-coupling terms acquire different scaling dimensions when K ⫽ 1 as we see below. With the bosonic Hamiltonian H = H0 + Hi + HK, we may calculate the scaling dimensions of the operators in HK. At the vicinity of the fixed point where Jzi = Jxy i = 0, the scaling terms are 1, and K, respectively. On dimensions of the Jz1, Jxy 1 the other hand, the scaling dimensions of the Jz2 and Jxy 2 terms are 21 共K + 1 / K兲 ⬎ 1. Thus, when K ⬍ 1, the J2 terms decrease but the J1 terms grow when temperature is lowered. In order to study the physics at the strong coupling regime when J1 is of order 1, we use the Emery-Kivelson method by z applying a unitary transformation U = ei冑2Ks共0兲S to H.21 After the unitary transformation and the rescaling of the bosonic fields, we have H̃ = U†HU = H0 + Szxs共0兲 + ⫻ 冋冑 2Jz2 z S sin a 再 册 2 a共0兲 sin关冑2Ka共0兲兴 + 共S− + S+兲 K 冋冑 Jxy Jxy 1 cos关冑2Ka共0兲兴 + 2 cos a a 2 a共0兲 K 册冎 , 共2兲 where = 冑2KvF⬘ − 冑 2K Jz1. At the vicinity of the point z xy = Jxy 1 = J2 = J2 = 0, we may calculate the scaling dimensions of xy the Ji terms as before. The Jxy 1 and J2 terms become more relevant because of the elimination of the e⫾is共0兲 factors by K the transformation. The Jxy 1 term has scaling dimension 2 . It xy is relevant when K ⬍ 1. The J2 term has scaling dimension 1 2K and it is relevant when 1 / 2 ⬍ K ⬍ 1. A quantum dot coupled to two Luttinger liquid leads is described by a Hamiltonian HL which is very similar to H̃ of Eq. 共2兲. However, there is one crucial difference between HL and H̃. In H̃, the two terms attached to the spin operators S+ + S− are cos共冑2Ka兲 and cos共冑2 / Ka兲, respectively. In HL, the corresponding terms attached to S− + S+ are cos共兲 and cos共兲, where and are charge and spin bosonic fields, respectively, which are independent from each other.5 In the present problem, a and a are dual fields of each other, pinning the value of the a field increases the fluctuation of the a field. We will see below that this is responsible for the fact that the two-channel Kondo fixed pointed is stable for K ⬍ 1. When K ⬍ 1, the Jxy 1 term is the most relevant term. At low temperatures, one expect the effective Hamiltonian of H̃ is described by the fixed point Hamiltonian: 041305-2 RAPID COMMUNICATIONS PHYSICAL REVIEW B 81, 041305共R兲 共2010兲 QUANTUM DOT IN A TWO-DIMENSIONAL TOPOLOGICAL… TABLE I. Scaling dimensions of the operators 共Op.兲 at different fixed points 共FPs兲. 共J2 = 0 for all fixed points, only coefficients of corresponding operators shown.兲 Op.\FPs Jz1共␦兲 Jxy 1 Jz2 Jxy 2 J1 = 0 = Jxy 1 =0 = 0, Jxy 1 →⬁ 1 K 1 2 共K + 1 / K兲 1 2 共K + 1 / K兲 1 K/2 1 2 共K + 1 / K兲 1 / 2K 1 + 1 / 2K N/A 1 / K + 1 / 2K 1/K 2Jxy H̃ef f = H0 + 1 Sx cos关冑2Ka共0兲兴. a ˜ ˜ ˜ ˜ 具− 兩eHef f Sz共0兲e−Hef f Sz共0兲兩− 典 = 具− 兩eH−W†共0兲e−H−W共0兲兩− 典 = 具− 兩W†共兲W共0兲兩− 典H˜− 共3兲 To show that this fixed point is indeed stable, we need to calculate the scaling dimensions of the x 冑 2 operators Jxy ␦Szxs共0兲, and 2 S cos关 K a共0兲兴, Jz2Sz sin关冑 2K a共0兲兴sin关冑2Ka共0兲兴 at the vicinity of the fixed point. ␦ denotes the deviation from the EmeryKivelson line. The results are given in the third column in Table I. We see that all these operators are irrelevant if K ⬍ 1. Which confirms that Eq. 共3兲 is the low-temperature effective Hamiltonian. Below we give some details of the calculation. To calculate the scaling dimension of 2 x x 冑 S cos关 共0兲兴, we note that S commutes with H̃ , so Jxy ef f 2 K a that we may set Sx to ⫾1 / 2. As a result, one needs to calculate the scaling dimension of cos关冑 2K a共0兲兴 with the Hamiltonian Jxy H̃⫾ = H0 ⫾ 1 cos关冑2Ka共0兲兴. a The calculations of the scaling dimension of the Szxs共0兲 term is similar to what have been done before in Ref. 23. Since s is governed by a free Hamiltonian, we have 具Sz共兲xs共兲Sz共0兲xs共0兲典 = 具Sz共兲Sz共0兲典具xs共兲xs共0兲典 In order to calculate 具Sz共兲Sz共0兲典H˜ef f , we note that Sz is the sum of the raising and lowering operators of the eigenstates of Sx, 兩 ⫾ 典. Thus, we have 共4兲 Equation 共4兲 has the form of the Hamiltonian of a spinless Luttinger liquid wire with an impurity backscattering term at point x = 0. One may interpret the cos关冑2Ka共0兲兴 as an impurity back scattering term with Luttinger liquid parameter K⬘ = K / 2. At low temperatures, the “back scattering term” is relevant for K⬘ ⬍ 1. Thus, the cos关冑2Ka共0兲兴 term cuts the Luttinger liquid wire into two separate pieces at x = 0.22 As a result, the point x = 0 can be regarded as a point at the boundary when Jxy 1 → ⬁. The effective Hamiltonian for half of the wire is H0⬘ v = 2F 兰L0 兵关xa共x兲兴2 + 关xa共x兲兴2其dx. Define chiral fields L and R by the relations a共x兲 = L共x兲 − R共x兲 and ˜ a共x兲 = L共x兲 + R共x兲 and apply the conditions R共x兲 = R共x兲 ˜ R共−x兲, we have H⬘ = vF兰L 关x ˜ R共x兲兴2dx and L共x兲 = − 0 −L ˜ 共−x兲+ ˜ 共x兲兴冑2/K −i关 ia共x兲冑2/K R and e =e R . Define free bosonic ˜ L共x兲 − ˜ R共x兲 fields which are defined from −L to L, ˜共x兲 = i 共x兲 ˜ 共x兲 = ˜ L共x兲 + ˜ R共x兲, and we have e a 冑2/K ˜ ˜ ˜ ˜ = e−i共共−x兲−共−x兲+共x兲−共x兲兲冑/2K. At x = 0, eia共0,t兲冑2/K ˜ 共0,t兲−˜共0,t兲兲冑2/K −i共 =e . As a result, the scaling dimension of eia共0,t兲冑2/K at the boundary can be easily calculated. It is 1 / K instead of 1 / 2K as is the case in the bulk. This well x 冑 2 known result20 is crucial because the Jxy 2 S cos关 K a共0兲兴 term is now irrelevant when K ⬍ 1. 共5兲 where W is a unitary transformation which satisfies ˜ ˜ W†e−H−W = e−H+. Actually, we need to find a unitary operator W which transforms a to a + 冑 2K . Indeed, W = e−i冑/2K关a共L兲−a共−L兲兴, where 2L is the length of the system. In order to calculate 具−兩W†共兲W共0兲兩−典H˜−, we come back to the spinless Luttinger liquid Hamiltonian interpretation of H̃−. Suppose we have a spinless Luttinger liquid wire with an impurity at x = 0 and the left and right ends locate at −L and L, respectively. At the strong coupling fixed point, the wire is cut into two separate parts by the impurity. However, the left half and the right half are described by free Hamiltonians H0L and H0R, respectively. As a result, we have 具W†共兲W共0兲典H˜− ⬇ 具ei 冑/2K关a共−L,兲−共−L,0兲兴 ⫻具ei 典H0L 冑/2K关a共L,兲−共L,0兲兴 典H0R ⬀ 1 1/K . 共6兲 Since xs共0兲 has a scaling dimension of 1, we conclude that the operator Szxs共0兲 is irrelevant with scaling dimension 1 / 2K + 1. One may calculate the scaling dimension of the z J2Sz sin关冑 2K a共0兲兴sin关冑2Ka共0兲兴 term in a similar way. 1 This term has scaling dimension 2K + K1 . It is irrelevant when K ⬍ 1. We have shown above that the two-channel fixed point is stable. At the vicinity of this fixed point, the system is described by H̃ef f in Eq. 共3兲. It is interesting to note that H̃ef f can be regarded as a generalization of the fixed point Hamiltonian of the two-channel and four-channel Kondo nonFermi liquid fixed-point Hamiltonians. The two-channel and four-channel cases are described by H̃ef f with K = 1 and K = 3 / 2 respectively.26 One of the most remarkable features of the multichannel Kondo effect is the existence of fractionally degenerate ground state. For example, the two-channel and four-channel Kondo models have residual entropy of 21 ln 2 and 21 ln 3, respectively.24,25 This motivates us to study the residual entropy of H̃ef f for K ⬍ 1. In order to calculate the residual entropy, we first calcu˜ late the partition function Z = Tr兵exp−Hef f 其. Since Sx commutes with the H̃ef f , the states can be labeled as 兩⫾ 21 , a典. We ˜ ˜ may write Z = Z+ + Z− = Tr兵exp−H+其 + Tr兵exp−H−其. Remember that the integral of H0 run from −⬁ to +⬁ if L → ⬁ is taken. However, we may define bosonic fields such that the positive axis is folded to the negative axis.27 After folding, the effect 041305-3 RAPID COMMUNICATIONS PHYSICAL REVIEW B 81, 041305共R兲 共2010兲 LAW et al. of the cosine term in H̃⫾ is to introduce a Dirichlet boundary condition. The actual value of a共0 , 兲, whether it is pinned at 共2n + 1兲 / 冑2K or 2n / 冑2K, does not affect the partition function.28 Thus, we have Z+ = Z− and Z = 2Z+. The impurity entropy of H̃+ has been calculated in Ref. 27. It is ln g = ln冑K / 2, where g is the “ground-state degeneracy” discussed in Ref. 25. Together with the contribution from Sx, the total impurity entropy is S = ln冑2K. 共7兲 Substituting K = 1 and K = 3 / 2 into Eq. 共7兲 reproduces the impurity entropy results, calculated by the Bethe ansatz and boundary conformal field theory, for the two-channel and four-channel Kondo effects with noninteracting electrons, respectively.24,25 We show above that the residual impurity entropy is determined by the boundary term. On the other hand, the thermodynamic properties are determined by the leading irrelevant operators 共LIO兲 near the fixed point. Since the irrelevant operators have correlation functions of the form 具具O共兲O共0兲典典 = 关 sin共TT兲 兴2⌬, the second-order correction of the free energy ␦F共兲共T兲 = −2兰/2d关 sin共TT兲 兴2⌬ where 0 is the 0 cut-off time.23 Jxy Near the strong coupling fixed point, the 2a cos关冑 2K a共0兲兴 term with scaling dimension 1 / K is the LIO if 1 / 2 ⬍ K ⬍ 1. We have ␦F ⬀ T2/K−1 + 0共T2兲, where the T2/K−1 term is universal and independent of the cut-off time 0 and the 0共T2兲 term has a 0 dependent coefficient. As a result, the impurity specific heat Cimp ⬀ T2/K−2 if 2 / 3 ⬍ K ⬍ 1. However, when K ⬍ 2 / 3, the free energy is dominated by the 0共T2兲 term and Cimp ⬀ T. L. Cox and A. Zawadowski, Adv. Phys. 47, 599 共1998兲 and references therein. 2 P. Noziere and A. Blandin, J. Phys. 41, 193 共1980兲. 3 R. M. Potok et al., Nature 共London兲 446, 167 共2007兲. 4 Y. Oreg and D. Goldhaber-Gordon, Phys. Rev. Lett. 90, 136602 共2003兲. 5 M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51, 17827 共1995兲. 6 E. H. Kim, arXiv:cond-mat/0106575 共unpublished兲. 7 V. R. Chandra et al., Phys. Rev. B 75, 045435 共2007兲. 8 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 共2005兲. 9 B. A. Bernevig et al., Science 314, 1757 共2006兲. 10 L. Fu et al., Phys. Rev. Lett. 98, 106803 共2007兲. 11 M. Konig et al., Science 318, 766 共2007兲. 12 D. Hsieh et al., Science 323, 919 共2009兲. 13 C. Wu et al., Phys. Rev. Lett. 96, 106401 共2006兲. 14 A. Schiller and K. Ingersent, Phys. Rev. B 51, 4676 共1995兲. 15 J. Maciejko et al., Phys. Rev. Lett. 102, 256803 共2009兲. 16 L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452 共1988兲. 17 T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 共1988兲. 1 D. In the above calculations, we have assumed that the two helical edges have equal chemical potential. If a small bias is applied across the two edges, there should be a current flowing from one edge to the other. At temperatures much higher than the Kondo temperature, the linear conductance across z 2 K+1/K−2 2 the edges can be written as G ⬀ 关2共Jxy by 2 兲 + 共J2兲 兴T pertubation to second order. At low temperatures, the conductance can be obtained by substituting the normalized value of J2 into the high temperature expression of G. As we show above, J2 decreases near the fixed point with z Jxy 1 = J1 = J2 = 0 and J1 grows. If 1 / 2 ⬍ K ⬍ 1, however, when the J1 terms are of order 1, the Jxy 2 term becomes relevant and grows. When the temperature decreases further, the strong coupling fixed point with Jxy 1 → ⬁ is reached. Near this fixed z point, Jxy 2 and J2 are irrelevant and flow to zero. As a result, the conductance acquires a nontrivial temperature dependence. On the other hand, if K ⬍ 1 / 2, the J2 terms are always irrelevant and decrease to zero. Consequently, the conductance decreases monotonically as the temperature lowers. The temperature dependence of the linear conductance at different fixed points are shown in the Fig. 1共b兲. We show above that a quantum dot coupled to two helical edge states of 2D topological insulators is described by a Kondo Hamiltonian. Weak repulsive interaction with K ⬍ 1 on the edges, drives the system to a two-channel fixed point at low temperatures. At the fixed point, the residual impurity entropy is ln冑2K. The impurity specific heat and the conductance across the two edges through the quantum dot are also studied. We thank Y. Avishai, C. H. Chung, D. Feldman and especially C. L. Kane for inspiring discussion and useful suggestions. KTL is supported by IAS-HKUST. P.A.L. acknowledge the support of NSF Grant No. DMR0804040 and the hospitality of IAS-HKUST. Hewson, The Kondo Problem to Heavy Fermions 共Cambridge University Press, Cambridge, 1993兲. 19 A. O. Gogolin et al., Bosonization and Strongly Correlated Systems 共Cambridge University Press, Cambridge, 1998兲. 20 T. Giamarchi, Quantum Physics in One Dimension 共Oxford University Press, Oxford, 2004兲. 21 V. J. Emery and S. Kivelson, Phys. Rev. B 46, 10812 共1992兲. 22 C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 共1992兲. 23 M. Fabrizio and A. O. Gogolin, Phys. Rev. B 50, 17732 共1994兲. 24 A. M. Tsvelick, J. Phys. C 18, 159 共1985兲. 25 I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 共1991兲. 26 In the Refs. 21 and 23, the sine-Gordon Hamiltonian are written in terms of chiral bosonic fields. The bosonic fields in Eq. 共3兲 are nonchiral. 27 P. Fendley et al., J. Stat. Phys. 79, 799 共1995兲. 28 H. Saleur, Topological Aspects of Low Dimensional Systems, edited by A. Comtet, et al. 共Les Houches 69, Springer, 1999兲. 18 C. 041305-4

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