week ending PRL 102, 256803 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009
Kondo Effect in the Helical Edge Liquid of the Quantum Spin Hall State
Joseph Maciejko,1 Chaoxing Liu,1,2 Yuval Oreg,3 Xiao-Liang Qi,1 Congjun Wu,4 and Shou-Cheng Zhang1 1Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA 2Center for Advanced Study, Tsinghua University, Beijing, 100084, China 3Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 4Department of Physics, University of California, San Diego, California 92093, USA (Received 13 January 2009; published 26 June 2009) Following the recent observation of the quantum spin Hall (QSH) effect in HgTe quantum wells, an important issue is to understand the effect of impurities on transport in the QSH regime. Using linear response and renormalization group methods, we calculate the edge conductance of a QSH insulator as a function of temperature in the presence of a magnetic impurity. At high temperatures, Kondo and/or two- particle scattering give rise to a logarithmic temperature dependence. At low temperatures, for weak Coulomb interactions in the edge liquid, the conductance is restored to unitarity with unusual power laws characteristic of a ‘‘local helical liquid,’’ while for strong interactions, transport proceeds by weak tunneling through the impurity where only half an electron charge is transferred in each tunneling event.
DOI: 10.1103/PhysRevLett.102.256803 PACS numbers: 73.43. f, 71.55. i, 72.25.Hg, 75.30.Hx
The quantum spin Hall (QSH) insulator is a topologi- interactions K>1=4 where K is the Luttinger parameter cally nontrivial state of matter [1] that has recently been of the HL, G is restored to unitarity at T ¼ 0 due to the observed in transport experiments carried out in HgTe formation of a Kondo singlet. This is in stark contrast with quantum wells (QW) [2] following its theoretical predic- the Kondo problem in a usual spinful 1D liquid where G tion [3]. The QSH insulator has a charge excitation gap in vanishes at T ¼ 0 for all K < 1 where K is the Luttinger the bulk but supports 1D gapless edge states forming a so- parameter in the charge sector [8]. At low but finite T, G called helical liquid (HL): on each edge, a Kramers’ pair of decreases as an unusual power law G / T2ð4K 1Þ due to states with opposite spin polarization counterpropagate. correlated two-particle backscattering (2PB). The edge The QSH insulator is robust against weak single-particle liquid being helical, the decrease in G is a direct measure perturbations that preserve time-reversal symmetry (TRS) of the spin-flip rate [9]. (3) For strong Coulomb interac- such as weak potential scattering [4–6]. tions K<1=4, 2PB processes are relevant and the system This theoretical picture is consistent with experimental is insulating at T ¼ 0. At low but finite T, G is restored by observations: the longitudinal conductance G in a Hall bar 2 measurement is approximately quantized to G0 ¼ 2e =h, independent of temperature, for samples of about a micron length [2,7]. However, larger samples exhibit G
0031-9007=09=102(25)=256803(4) 256803-1 Ó 2009 The American Physical Society week ending PRL 102, 256803 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009 tunneling of excitations with fractional charge e=2, and we 1 ~ 2 GK ð2Þ ðKÞ 2 2ð1=4K 1Þ ¼ SðS þ 1Þ½ J ðTÞ ; (2) obtain GðTÞ/T . 2 1 ~ 3 k 1 e =h ð2 þ KÞ Model.—We model the impurity by a S ¼ 2 local spin K~ 1 coupled by exchange to the 1D HL with Coulomb inter- where JkðTÞ¼ JkðT=DÞ to OðJkÞ and D ¼ @v= actions. The HL having the same number of degrees of is a high-energy cutoff on the order of the bulk gap. G2 is freedom as a spinless fermion, a single nonchiral boson the correction due to 2PB, is sufficient to bosonize the HL[5]. The system is described 1 4 2 G2 ð Þ ð4KÞ a 2ðTÞ by the Hamiltonian H ¼ H0 þ HK þ H2 where H0 is the ¼ 2 ; (3) v 2 1 4 2 4 4 usualR Tomonaga-Luttinger Hamiltonian H0 ¼ 2 e =h ð2 þ KÞ D 2 1 2 dx½K þ K ð@x Þ , with K the Luttinger parameter 4K 1 where 2ðTÞ¼ 2ðT=DÞ . There are no crossed terms and v the edge state velocity. The Kondo Hamiltonian HK 1 of the form OðJk 2Þ or OðJz 2Þ for a S ¼ 2 impurity since has the form 1 the 2PB operator flips two spins but a S ¼ 2 spin can be flipped only once. We expect that such terms would be J a pffiffiffi J a k i2 ð0Þ zffiffiffiffi generated for impurities with higher spin. HK ¼ ðS : e : þH:c:Þ p Sz ð0Þ; (1) 2 These results can be complemented by a RG analysis. The RG equation for 2 follows by dimensional analysis d 2 ¼ð1 4KÞ 2 with ‘ ¼ lnðD=TÞ so that 2 is relevant where S ¼ Sx iSy and Sz are the spin operators for the d‘ impurity localized at x ¼ 0. a is the lattice constant of the for K<1=4 and irrelevant for K>1=4. The renormalized 4K 1 coupling is 2ðTÞ¼ 2ðT=DÞ , and second order re- underlying 2D lattice and corresponds to the size of the 2 impurity (we assume that the impurity occupies a single normalized perturbation theory G2ðTÞ/ 2ðTÞ simply lattice site). , the penetration length of the helical edge reproduces the Kubo formula result (3). Perturbation the- 0 1=ð1 4KÞ states into the bulk, acts as a short-distance cutoff for the ory fails for T & T2 where T2 /ð 2Þ is a scale for 0 1D continuum theory in the same way that the magnetic the crossover from weak to strong 2PB with 2 the bare length, the penetration length of the chiral edge states, acts 2PB amplitude. The 1-loop RG equations [5,15] for the as a short-distance cutoff for the chiral Luttinger liquid Kondo couplings Jk, Jz read theory of the quantum Hall edge [10]. In addition to Kondo dJk 1 dJz 2 scattering, 2PB is allowed by TRS [5,6]. In HgTe QW, the ¼ð KÞJk þ JkJz; ¼ Jk: (4) wave vector where the edge dispersion enters the bulk is d‘ d‘ usually much smaller than =2a such that the uniform 2PB The family of RG trajectories is indexed by a single scaling 4 2 2 ~ 2 ~ (umklapp) term requiring kF ¼ =a can be ignored. invariant c ¼ð JkÞ ð JzÞ where Jz Jz þ 1 K, 4 The impurity can however provide a kF momentum trans- which is fixed by the couplings at energy scale D.In fer, and we must generally also consider a local impurity- contrast to the spinful case [8], the absence of spin-flip 2 ffiffiffiffi 2a p ¼ 2 2 : cos4 ð0Þ forward scattering in the HL preserves the stability of the induced 2PB term [11] H2 2 : where 2 is the 2PB amplitude. ferromagnetic fixed line, as is the case in the usual Kondo Weak coupling regime.—We first consider the weak cou- problem. The renormalized spin-flip amplitude JkðTÞ is given in terms of the bare parameters J0, J0 by pling regime where Jk;Jz; 2 are small parameters. We first k z calculate G to second order in the bare couplings J and 2 k J0 using the Kubo formula. This result is then extended to J ðTÞ¼ k ; (5) k sinh 0 ln include all leading logarithmic terms in the perturbation ½ Jk ðT=TKÞ expansion by means of a weak coupling renormalization group (RG) analysis of the scale-dependent couplings such that GK is obtained to all orders in perturbation theory in the leading-log approximation by substituting JkðTÞ and 2ðTÞ where the scale is set by the tempera- ¼½ð~0 0Þ2 1 1=2 ture T. Eq. (5) in Eq. (2). In Eq. (5), Jz =Jk is an The Jz term can be removed from the Hamiltonian anisotropy parameter [16] and TK is the Kondo tempera- by a unitary transformation [12] of the form U ¼ ture, pffiffiffiffi i Sz ð0Þ e with ¼ Jza=vK , which changesffiffiffi the 1 arcsinh p exp : i2 : T ¼ D 0 : scaling dimension of the vertex operator e . The K J ~ y k transformed Hamiltonian is H UHU ¼ H0 þ H2 þ pffiffiffi 0 Jka i2 ð0Þ In the isotropic case ¼ , one recovers the usual form ðS : e : þH:c:Þ, where ¼ 1 J =2K 0 2 z 1= J 0 0 k 1 TK ¼ De . In the limit K Jk, Jz , Coulomb with ¼ a= v thepffiffiffi density of states of the HL. The scaling i2 ~ 2 interactions dominate over Kondo physics, and we obtain : e : K K 0 dimension of is . J ~ k 1=ð1 KÞ Using H, we calculate GðTÞ¼ GKðTÞþ G2ðTÞ to TK ’ Dð1 KÞ , a power-law form similar to the result second order in Jk and 2 using the Kubo formula, where for spinful Luttinger liquids [17] . From the exponent, we GK is the correction due to Kondo scattering [13], see that TK corresponds to the scale of the mass gap opened 256803-2 week ending PRL 102, 256803 (2009) PHYSICAL REVIEW LETTERS 26 JUNE 2009 in a spinless Luttinger liquid by a nonmagnetic impurity of 0 strength Jk and the corresponding crossover is that of weak to strong single-particle backscattering. max & In the high-temperature limit fT2 ;TKg T D, both the Kondo and 2PB processes contribute logarithmi- cally to the suppression of G, GðTÞ þ lnðD=TÞ 0 0 where , are functions of the bare couplings K, Jk, Jz , 0 and 2. Strong coupling regime.—We now investigate the low min temperature regime T fT2 ;TKg. The topological na- ture of the QSH edge state as a ‘‘holographic liquid’’ living on the boundary of a 2D system [5] results in a drastic change of the low-energy effective theory in the vicinity of FIG. 2 (color online). Strong coupling regime: the Kondo the strong coupling fixed point as compared to that of a singlet effectively removes one site from the system. usual 1D quantum wire. As suggested by the perturbative (a) Luttinger liquid with K < 1: a punctured 1D lattice is RG analysis, the nature of the T ¼ 0 fixed point depends disconnected, (b) QSH edge liquid with K>1=4: the edge liquid on whether K is greater or lesser than 1=4. follows the boundary of the deformed 2D lattice. (c) Half-charge For K>1=4, 2PB is irrelevant and G2 flows to zero. tunneling for K<1=4 by flips of the Ising order parameter m. On the other hand, for antiferromagnetic Jz the Kondo strong coupling fixed point J , J !þ1is reached at T ¼ k z e S=ð2jhI ijÞ obtained from a measurement of the shot 0, with formation of a local Kramers singlet and complete B noise S in the backscattering current hI i to be e ¼ 2e screening of the impurity spin by the HL. As a result, the B [11,19]. formation of the Kondo singlet effectively removes the For K<1=4, the j 2j!1fixed point is reached at T ¼ impurity site from the underlying 2D lattice [Fig. 2(b)]. 0 and the system is insulating. The field ðx ¼ 0; Þ is In a strictly 1D spinful liquid, this has the effect of cutting pinned at the minima of the cosine potential H2 located at the system into two disconnected semi-infinite 1D liquids pffiffiffiffi pffiffiffiffi ð2n þ 1Þ =4 for 2 > 0 and 2n =4 for 2 < 0, [Fig. 2(a)] and transport is blocked at T ¼ 0 for all K < 1 with n 2 Z. G is restored at finite T by instanton processes [8]. In contrast, due to its topological nature, the QSH edge corresponding to tunnelingffiffiffiffi between nearby minima sepa-ffiffiffiffi state simply follows the new shape of the edge, and we p p rated by ¼ =2. From the relation je ¼ e@t = expect the unitarity limit G ¼ G0 to be restored at T ¼ 0. where je is the electric current, the charge pumped by a For finite T T , the effective low-energy Hamiltonian e e K single instanton [Fig. 2(c)]is Qinst ¼ pffiffiffi ¼ . This contains the leading irrelevant operators in the vicinity of 2 the fixed point. In the case of spinful conduction electrons, fractionalized tunneling current can be understood as the the lowest-dimensional operator causing a reduction of G Goldstone-Wilczek current [20–22] for 1D Dirac fermions L 5 is single-particle backscattering. However, the helical with a mass term ¼ g ðm1 þ i m2Þ pwhereffiffiffiffi g cos2 property of the QSH edge states forbids such a term, and 2, andpffiffiffiffi the mass order parameters m1 ¼ , m2 ¼ sin2 it is natural to conjecture that the leading irrelevant opera- change sign during an instanton process. The tor must be the 2PB operator with scaling dimension 4K. order is Ising-like because the 2PB term explicitly breaks 1 v We thus expect a correction to G at low temperatures T theR spin Uð Þ symmetry of the helical liquid H0 ¼ 2 2 1 2 Z 1 4 2ð4K 1Þ dxðK z þ Þ down to 2, where z ¼ þ and TK for K> = of the form G / ðT=TKÞ . K In particular, in the noninteracting case K ¼ 1, we pre- ¼ þ þ are the spin and charge densities, and dict a T6 dependence in marked contrast to both the usual are the chiral densities for the two members of the Kramers Fermi liquid [18] and spinful 1D liquid [8] behaviors. pair. This dependence characteristic of a ‘‘local helical Fermi Fractionalization of the tunneling current is confirmed liquid’’ can be understood from a simple phase space by a saddle-point evaluation of the low-energy effective argument. The Pauli principle requires the 2PB operator action Seff for large 2 in the dilute instanton gas approxi- to be defined through point-splitting [5] with the short- mation, which yields a Coulomb gas model that can be distance cutoff , which translates into a derivative cou- mapped exactly to the boundary sine-Gordon theory c y 0 c y c 0 c 2 c y c y c c pling Rð Þ Rð Þ Lð Þ Lð Þ! R@x R L@x L in X Z ffiffiffiffi K 2 p the limit of small . In the absence of derivatives, the Seff½ ¼ j!njj ði!nÞj þ t d cos ð Þ; 2 1 0 i! 4-fermion term contributes T to the inverse lifetime k . n The derivatives correspond to four powers of momenta close to the Fermi points in the scattering rate k;k0!p;p0 / where t is the instanton fugacity. The RG equation for t 0 2 0 2 dt 1 1 2 ðk k Þ ðp p Þ , which translates into an additional fac- follows as d‘ ¼ð 4KÞt and GðTÞ/tðTÞ is a power law 4 2ð1=4K 1Þ 1 4 tor of T . Furthermore, since for T TK, the suppression G /ðT=T2 Þ for T T2 , K< = . 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B 46, 10812 conductance with decreasing temperature in HgTe samples (1992). with only a backgate. Because of lower Fermi velocities [13] This result holds for @v=L T 256803-4