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Kondo Impurities Coupled to a Helical Luttinger Liquid: RKKY-Kondo Physics Revisited

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Kondo Impurities Coupled to a Helical Luttinger Liquid: RKKY-Kondo Physics Revisited PHYSICAL REVIEW LETTERS 120, 147201 (2018) Kondo Impurities Coupled to a Helical Luttinger Liquid: RKKY-Kondo Physics Revisited Oleg M. Yevtushenko1 and Vladimir I. Yudson2,3 1Ludwig Maximilians University, Arnold Sommerfeld Center and Center for Nano-Science, Munich DE-80333, Germany 2Laboratory for Condensed Matter Physics, National Research University Higher School of Economics, Moscow 101000, Russia 3Russian Quantum Center, Skolkovo, Moscow Region 143025, Russia (Received 2 September 2017; published 2 April 2018) We show that the paradigmatic Ruderman-Kittel-Kasuya-Yosida (RKKY) description of two local magnetic moments coupled to propagating electrons breaks down in helical Luttinger liquids when the electron interaction is stronger than some critical value. In this novel regime, the Kondo effect overwhelms the RKKY interaction over all macroscopic interimpurity distances. This phenomenon is a direct consequence of the helicity (realized, for instance, at edges of a time-reversal invariant topological insulator) and does not take place in usual (nonhelical) Luttinger liquids. DOI: 10.1103/PhysRevLett.120.147201 The seminal problem of the indirect exchange interaction Helicity means the lock-in relation between electron spin [Ruderman-Kittel-Kasuya-Yosida (RKKY)] between two and momentum: helical electrons propagating in opposite spatially localized magnetic moments, i.e., Kondo impu- directions have opposite spins. This protects the helical rities (KIs), weakly coupled to propagating electrons has a transport against effects of spinless impurities. HLL can well-known solution [1]. The paradigmatic approach can be appear at edges of time-reversal invariant 2D topological reformulated in the contemporary language as follows: one insulators [15–19] and in purely 1D interacting systems integrates out fermionic degrees of freedom and reduces [20,21]. The Kondo effect [22–26] and RKKY [27–32] in the resulting nonlocal Lagrangian to the effective spin the topological insulators have been intensively studied for Hamiltonian. The second step is usually justified by a scale the past several years. This increasing interest is partly separation; the spin dynamics is slower than the electron related to the hypothesis that Kondo-RKKY effects can be one if the electron-spin coupling is weak. RKKY induces responsible for deviations of the helical conductance from perceptible interimpurity correlations if an interimpurity its ideal value; see Refs. [33–37] and discussions therein. distance R is smaller then the thermal length and the electron At a simple phenomenological level, one can find “the coherence length. This RKKY theory is the obvious winner of the RKKY-Kondo competition” by comparing T E simplification since it neglects another fundamental phe- K with the characteristic energy of RKKY, RKKY. The nomenon, namely, the Kondo effect [2]. If the temperature latter has the meaning of the energy gap, which opens after is below the Kondo temperature, T<TK, the antiferromag- the RKKY correlations lift a degeneracy in the energy netic Kondo coupling drives the single KI to the strong of the uncorrelated KIs. In the absence of Coulomb coupling limit where the electrons screen KI. Hence, the ð0Þ ð0Þ 2 d interactions, TK ∝ expð−1=ρ0JÞ and E ∝ J =R , Kondo screening is an antagonist of the RKKY interaction. RKKY where ρ0 is the density of states of the electrons at the The RKKY-Kondo interplay has attracted large attention Fermi surface, J is the Kondo coupling constant, and d is for several decades [3–7] and remains a hot topic of the space dimension. If ρ0J ≪ 1, there is a broad range of research because of its importance for new systems, such Eð0Þ ≫ Tð0Þ T as graphene [8,9], strongly correlated quantum wires, and macroscopic distances where RKKY K , . In this carbon nanotubes, which are described by the Luttinger case, RKKY is expected to overwhelm the Kondo liquid model [10,11]. The latter are especially interesting screening. because the Kondo effect can be enhanced by the inter- The situation drastically changes in HLL with the strong XXZ actions [12–14]. The common wisdom is that the RKKY interaction. Let us concentrate on the Kondo coupling J J ≪ 1=ρ physics dominates in a broad macroscopic range of R in with small constants ⊥, z 0; see the formal J three- and low-dimensional systems. definition in Eqs. (5) and (6) and temporarily neglect z. In this Letter, we will demonstrate that, surprisingly, the The electron repulsion is reflected by the Luttinger param- K ≤ 1 K 1 paradigmatic RKKY approach breaks down in strongly eter of HLL: [38]; ¼ corresponds to non- E T correlated helical systems—helical Luttinger liquids interacting fermions. Both RKKY [see Eq. (13)] and K (HLLs). We will show that the reason for this unexpected (see Ref. [22]) are modified by the interaction finding is the nontrivial and unusually increased RKKY- E ∼ D ρ J 2 ξ=R 2K−1; 1=2 <K≤ 1; Kondo competition. RKKY ð 0 ⊥Þ ð Þ ð1Þ 0031-9007=18=120(14)=147201(6) 147201-1 © 2018 American Physical Society PHYSICAL REVIEW LETTERS 120, 147201 (2018) ( ð0Þ TK ; 0 < 1 − K ≪ 1; explain how RKKY stops Kondo renormalizations and TK ∝ ð2Þ why the paradigmatic theory of RKKY is valid in the range 1 ð0Þ D ρ J 1−K ≫ T ; 1 − K ≫ ρ J : ð 0 ⊥Þ K 0 ⊥ 1=2 < K˜ and fails at K<˜ 1=2. By exploiting the extreme situation close to the decoupling limit [24], we will ξ D Here ( ) is the spatial (energy) UV cutoff, i.e., the lattice demonstrate that the strong effective interaction makes spacing (bandwidth). A naive formal extension of Eq. (1) the RKKY-induced spin correlations irrelevant. We thus K<1=2 to the regime would lead to a paradoxical result: could conclude that the physics is fully dominated by the E RKKY seems to grow without bound with the increase of Kondo effect at K<˜ 1=2. the interimpurity distance. The results presented below The model.—We use functional integrals in the ultimately refute any possibility of such an effect. Matsubara formulation with the imaginary time τ. The Based on the above explained phenomenological argu- bosonized Lagrangian density of HLL [18,22,24,35,40] is E R ∼ ξ >T ments, we expect that, if RKKYð Þ K, there exists a broad range of macroscopic distances where RKKY 2 2 L ¼½ð∂τϕÞ þðu∂xϕÞ =ð2πuKÞ; ð4Þ dominates over the Kondo effect. In the opposite case, HLL E R ∼ ξ <T RKKYð Þ K, the Kondo physics dominates every- here u is the velocity of bosonic excitations. The electron- where. The border between these two phases is defined KI interaction is described by Lagrangians of the forward E ∼ T by the condition RKKY K. We will show that it or backward scattering, corresponds to the critical value of the effective interaction X parameter L iJ a = πuK δ x − x Sz∂ ϕ; fs ¼ z fs ð Þ ð jÞ j τ ð5Þ j¼1;2 K˜ K 1 − ρ J =2K 2; K˜ 1=2; ¼ ð 0 z Þ crit ¼ ð3Þ X −2ia ϕ L J = 2πξ δ x − x Sþe bs : : : bs ¼ ⊥ ð Þ ð jÞ½ j þ c c ð6Þ see the phase diagram in Fig. 1 [39]. The paradigmatic j¼1;2 K>˜ 1=2 RKKY theory is valid only at and fails at μ K<˜ 1=2. Namely, the spin subsystem cannot be described Here xj are impurity positions with R ¼jx1 − x2j, Sj are by an effective (RKKY-like) Hamiltonian at K<˜ 1=2. fields describing KI spin degrees of freedom, and we have a These statements, which are proven below at a more formal introduced auxiliary dimensionless constants fs;bs, which level, are our main result. are explained below. Equations (4)–(6) describe the low The rest of this Letter is organized as follows: First, energy physics; i.e., all fields are smooth on the scale of ξ. we will rederive the RKKY Hamiltonian by integrating In particular, 2kF oscillations (kF is the Fermi momentum) L SÆe∓2ikFxj → out HLL degrees of freedom and discuss the difference are eliminated from bs by the spin rotation j Æ between helical and usual (not helical but spinful) cases. Sj . We note in passing that, unlike previously studied We will combine the microscopic diagrammatic approach examples [41–43], features of a single particle density of with one-loop renormalization group (RG) arguments to states and the precise level of the chemical potential are unimportant for the RKKY-Kondo physics, which we 1 explore. This is the peculiarity of the interacting 1D systems described by the bosonization approach [40]. We emphasize that the helicity of our model implies that it has only U(1) spin symmetry, but no SU(2) symmetry. 0.5 Decoupling Limit Kondo Phase z We restrict ourselves to the case of spin-1=2 KIs and choose J 0 ρ a parametrization for S fields in terms of Grassmann fields corresponding to Dirac fermions [44,45], 0 þ ¯ z RKKY Phase Sj ¼ðdj þ djÞc¯j;Sj ¼ c¯jcj − 1=2: ð7Þ Each Grassmann field has the usual dynamical Lagrangian 0.25 0.5 0.75 1 L ψ¯ ∂ ψ ψ c ;d K f ¼ j τ j, j ¼f j jg [46,47]. a 1 In the initial formulation, one chooses fs;bs ¼ ; FIG. 1. Phase diagram of the two Kondo impurities coupled however, the gauge transformation of c fermions, cj → to the HLL. Green and orange regions demonstrate the RKKY cj exp½iλϕðxjÞ, which is equivalent to the Emery-Kivelson and the Kondo phases, respectively. Axes show values of the rotation [24,48], allows one to represent the theory in two Luttinger parameter 1=4 ≤ K ≤ 1 and the dimensionless cou- extreme forms, pling constant jρ0Jzj < 1; see Eqs.
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