Approaching multichannel Kondo physics using correlated bosons: Quantum phases and how to realize them
AApproaching multichannel Kondo physics using correlated bosons:Quantum phases and how to realize them
Siddhartha Lal, ∗ Sarang Gopalakrishnan, † and Paul M. Goldbart ‡ Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
We discuss how multichannel Kondo physics can arise in the setting of a localized level coupledto several bosonic Tomonaga-Luttinger liquid leads. We propose one physical realization involvingultracold bosonic atoms coupled to an atomic quantum dot, and a second, based on superconduct-ing nanowires coupled to a Cooper-pair box. The corresponding zero-temperature phase diagramis determined via an interplay between Kondo-type phenomena arising from the dot and the conse-quences of direct inter-lead hopping, which can suppress the Kondo effect. We demonstrate that themultichannel Kondo state is stable over a wide range of parameters. We establish the existence oftwo nontrivial phase transitions, involving a competition between Kondo screening at the dot andstrong correlations either within or between the leads (which respectively promote local number-and phase-pinning). These transitions coalesce at a self-dual multicritical point.
PACS numbers: 72.15.Qm, 71.10.Pm, 73.23.-b
Introduction.
Magnetic impurities can drastically al-ter the low-temperature properties of metals, leading toanomalous temperature dependence in, e.g., the heat ca-pacity, resistance, and magnetoresistance. These prop-erties, collectively known as the Kondo effect [1], areexhibited when the impurity interacts antiferromagneti-cally with the conduction electrons of the metal, and aredue to the dynamic screening, as T →
0, of the spins ofthe individual impurities by a cloud of electrons. Thelow-energy scattering properties of each screened impu-rity are those of a bound but spin-polarizable impurity-cloud singlet , whose effects on the electron gas can bedescribed using Fermi-liquid theory [2]. For a ferromag-netic impurity-cloud interaction, on the other hand, thespins of the impurity and its polarization cloud align inone of two degenerate triplet ground states. A quantumphase transition separates the ferro- and antiferromag-netic (i.e., Kondo) cases.A striking generalization of this “1-channel” Kondo ef-fect is the multichannel version [3], in which each im-purity couples separately to conduction electrons thatpropagate in N ( >
1) channels (e.g., distinguished by or-bital angular momentum). When
N > s (with s beingthe spin of the impurity), the conduction electrons over-screen the impurity, and the low-energy behavior can nolonger be described by Fermi-liquid theory; instead, thethermodynamic properties follow anomalous power-lawsgoverned by a quantum critical point [4]. In metallic sys-tems, observing the N -channel Kondo effect has provendemanding [5]. Although advances in nanoscience haveenabled the exploration of Kondo physics in the morecontrolled setting of 2DEG-based quantum dots [6] con-nected to leads (which realize the channels), even herethe engineering of multichannel Kondo phenomena re-mains a challenge. This is primarily due to the dif-ficulty in preventing interchannel hybridization, whichgives rise at low energies to a single composite channel that screens the impurity via the 1-channel Kondo effect.For instance, signatures of the two-channel Kondo effecthave recently been observed in a quantum-dot-based set-ting [7], but required fine tuning to prevent hybridization.In view of these challenges, it is desirable to explorethe N -channel Kondo effect and its onset in a more read-ily controllable setting: this can be achieved, e.g., usingleads having tunable interparticle correlations. For the1-channel Kondo effect the possibility of such control wasdemonstrated in Ref. [8] in the context of a quantum dotcoupled to a Tomonaga-Luttinger liquid (TLL) lead [9].The strength of the repulsive interactions in the lead (asencoded in the TLL parameter K ) was found to tunethe position of the Kondo-to-ferromagnetic phase tran-sition. Motivated in part by this result, in this paperwe explore the case of N TLL leads coupled to a quan-tum dot; this case is expected to exhibit richer physicsarising from the possibility of not only intra-lead butalso inter-lead correlations. We focus on the case ofbosonic leads because for them a wide range of K val-ues can be experimentally accessed with relative ease:e.g., ultracold bosons with short-range repulsions have K >
K < N -channel Kondo effect and interactions inthe leads; these interactions either suppress lead-dot tun-neling or generate inter-lead phase-locking (which wouldshort-circuit the dot). Our analysis yield a phase diagramcontaining four distinct phases (see Fig. 3, below) and ex-hibiting a pair of unusual phase boundaries, which meet a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t at a self-dual multicritical point. As discussed below, animportant advantage of our proposed experimental real-ization of the N -channel Kondo effect is that it is robustagainst interlead hybridization. We discuss the condi-tions for observing the N -channel Kondo effect using coldatoms, and suggest that the unusual phase boundariesmay be accessed experimentally using dipolar bosons. Kondo and resonant-level models.
We consider the gen-eral anisotropic Kondo Hamiltonian H K = H leads + H int ,where H leads = (cid:80) k v F k c † kσ c kσ describes a free-electrongas (or a Fermi liquid) and H int describes the couplingto the impurity spin, which is located at r = 0: H int = J ⊥ (cid:104) S + c †↓ (0) c ↑ (0) + h . c . (cid:105) + J z S z (cid:88) σ σc † σ (0) c σ (0) , where c σ (0) annihilates a conduction electron of spin σ atthe impurity location, the Pauli matrices S act on the im-purity spin state, and J z and J ⊥ , respectively, the ampli-tudes for the lead-dot Ising and spin-flip processes. Theantiferromagnetic case, J z >
0, leads to Kondo screen-ing; for J z < | J ⊥ | < | J z | , the impurity couplesferromagnetically to the conduction electrons.We shall primarily be concerned with a model thatis equivalent to the Kondo model, viz., the interactingresonance-level model (iRLM). This model consists of alocalized level d (i.e., a quantum dot) at the Fermi energyhybridized with N channels of spinless noninteractingconduction electrons c j k , together with a short-rangedrepulsion between dot and lead electrons: H iRLM = H leads + H onsite + H int . Here, the leads are describedby H leads = (cid:80) k (cid:80) Nn =1 (cid:15) ( k ) c † n k c n k , the on-dot potentialby H onsite = B d † d , and the dot-lead couplings by H n int = V (cid:0) d † c n (0)+h . c . (cid:1) + U (cid:0) d † d − (cid:1)(cid:0) c † n (0) c n (0) − (cid:1) (1)where V ∼ J ⊥ and U ∼ J z + const. The canonicaltransformation that maps the Kondo model on to theiRLM consists of identifying the spin density waves ofthe Kondo model with the particle density waves of theiRLM. The analogy is most evident as U → ∞ : the pres-ence of an electron on the dot ensures the absence ofelectrons near the dot, and vice versa, i.e., an anticor-related state resembling the Kondo singlet. The Kondoeffect manifests itself in the iRLM via an enhancement,as T →
0, of number fluctuations on the dot.We now turn to the iRLM with TLL leads, governed bythe Hamiltonian H leads = (cid:80) k E ( k ) b † k b k , in which { b k } are bosonic fields that describe free collective phononmodes that disperse linearly [9]: E ( k ) ∼ | k | . As discussedin Ref. [8], this version of the iRLM can also be mappedon to the standard Kondo model, but the location ofthe ferro- to antiferromagnetic transition depends on K .The boundary between the ferro- and anti-ferromagneticphases occurs at U = (cid:126) v s ( (cid:112) / K − v s is thespeed of sound in the leads.In the N -channel generalization of the iRLM, eachchannel couples independently to the impurity via H n int . By inverting the iRLM-Kondo mapping, one arrives atthe N -channel Kondo Hamiltonian studied in Refs. [3, 4],in which each of N channels couples independently tothe spin. However, the interchannel particle transferterms (i.e., hybridization) H nn (cid:48) Γ ∼ c † nα (0) σ αβ c n (cid:48) β (0),which destabilize the N -channel Kondo effect, do notarise in the iRLM. This is because the particle-number(or “charge”) sector of the leads in the iRLM—the modelof physical relevance here—maps on to the spin sector ofthe equivalent Kondo model. In contrast, the particle sector of the leads in the equivalent Kondo model hasno physical significance in the iRLM and, accordingly,cannot couple to the dot. Realizations of the iRLM.
Our first realization involvesultracold atoms, and extends the ideas of Ref. [13]. Astar-shaped pattern of N one-dimensional leads meet-ing at a point can be constructed by passing a laserbeam through a phase mask or spatial light modulator(SLM) [15]. Such a device is a sheet of glass of spatiallyvarying thickness, which distorts flat wavefronts, givingrise to a prescribed intensity pattern at a “screen” somefixed distance away. Algorithms for the construction ofappropriate phase patterns are discussed in Refs. [15, 16]. x yz A B CDSLM leaddot
FIG. 1:
Candidate cold-atom setup.
The spatial lightmodulator (SLM) distorts the wavefronts of laser A so as tocreate a star-shaped pattern at the “screen.” Atoms are con-fined to the screen using two lasers (B) propagating at a rel-ative angle θ (cid:28) π so as to create an optical lattice of spacing L = λ/ [2 sin( θ/ b at the “dot” (dark blue region). As for the dot itself, it can be realized as follows [13].Suppose that the atoms discussed in the preceding para-graph are in a hyperfine state a . A tightly-confining trapfor a different hyperfine state b is now created at the in-tersection of the leads. An atom in state a can make aRaman transition to state b , and vice versa; while in b ,it is confined at the “dot.” The Raman transition thuscreates a lead-dot hopping amplitude. This setup real-izes an iRLM having the following couplings: K is de-termined by the scattering length g aa for atomic state a , U by the a ↔ b scattering length g ab , and V by the am-plitude (i.e., effective Rabi frequency) ∆ of the Ramantransition. Double occupancy of the dot is prevented bya large repulsive interaction g bb between atoms on thedot (with g bb (cid:29) g aa , g ab ). All interactions are tunablevia a Feshbach resonance. The direct interlead hoppingamplitude is governed by the intensity of the laser thattraps a -state atoms at the intersection of the leads.One can realize a similar model in a mesoscopic set-ting by using superconducting nanowires as the leads,together with a Cooper-pair box [14]—i.e., a supercon-ducting island that holds at most one Cooper pair—asthe dot. The normal modes of the leads are plasmonexcitations, and the Hamiltonian for the box-lead sys-tem has the same form as that for the cold atom system,provided the leads are connected to the dot via Joseph-son couplings (which determine V ). The coupling U isdetermined by the lead-dot Coulomb repulsion. Analysis of the model.
In addition to processes thatinvolve the dot, we account for those in which bosonshop directly between leads. It is useful to writethe Hamiltonian for the uncoupled leads as H leads = (cid:80) Nn =1 v s π (cid:82) L dx [ K ( ∂ x θ n ) + K − ( ∂ x φ n ) ], where the den-sity fluctuation modes of the TLLs are given by theoperator ρ n ( x ) ∼ ∂ x φ n ( x ) /π and the canonically con-jugate momenta by ∂ x θ i ( x ). Direct hopping processesbetween the leads can be described using the bosonannhilation/creation operators ψ n ( x ) ∼ e iθ n ( x ) | x =0 atthe end-points of the semi-infinite TLLs: H tunn = t (cid:80) n,n (cid:48) ( e i ( θ n (0) − θ n (cid:48) (0)) + h . c . ) . At low frequencies, theground state of the complete system consists of either N uncorrelated wires (i.e., the disconnected fixed point,DFP, t = 0), or N maximally correlated wires (i.e.,the connected fixed point, CFP, t → ∞ ). The CFPmanifests itself via the mutual pinning of the phasefields at the junction: θ n ( x, t ) = θ n (cid:48) ( x, t ) | x =0 for allpairs ( n, n (cid:48) ) [17]. Additionally, current conservation de-mands that (cid:80) Nn =1 ∂ t θ n (0 , t ) = 0. In the language ofthe renormalization group (RG), the hopping t flowsto 0 (i.e., DFP) for K <
K > λ gives dλ/dl = { − (2( N − K/N ) } λ . Therefore, the CFP isstable for K > N/ (2( N − N/ [2( N − < K <
1; for K in thisinterval there must therefore be a quantum phase tran-sition at some t ∗ (cid:54) = 0, separating the ground states of N uncorrelated and N maximally correlated wires. Thistransition is analogous to that exhibited by a quantumBrownian particle on an ( N − N -wire phase diagram shownin Fig. 2a is that the CFP is stable against weak backscat-tering close to the junction, even for K < N − K eff = ( N − K > J J z K K t = 0 t = ∞λ = 0 a. b. = 1 N N − 1) M FIG. 2: (a) Phase diagram for N leads with interlead hopping t and TLL parameter K . The phase boundary (thick line)separates the disconnected (i.e., DFP, t = 0) and maximallyconnected (i.e., CFP, t → ∞ ) fixed points, and has a self-dualpoint ( M ). (b) Kosterlitz-Thouless flow for lead-dot couplingsat fixed ( K, t ). The left separatrix (thick line) demarcates theboundary between the ferromagnetic and Kondo phases.
The locking of the phase field in the first wire to thatof the composite TLL precludes any chemical-potentialdrop across the junction, although only a fraction of theincoming current enters any individual lead of the com-posite TLL [17]. Said another way, the local inertia ofthe phase fields strongly suppresses backscattering eventsinvolving high-momentum phase fluctuations. This phe-nomenon is dual to the enhanced inertia in the numberfields at the endpoints of the wires at the DFP for
K < N leads via H n int [see Eq. (1)], and develop the phase dia-gram via an RG analysis around the CFP and DFP. Nearthe CFP, we find the following scaling equations: dJ z dl = J ⊥ , dJ ⊥ dl = (cid:18) − N − KN + J z (cid:19) J ⊥ , (2)to second order in all couplings, where the couplingshave been scaled by the high-frequency cutoff ω c = v s /ξ (where ξ is the healing length [13]). In addition, wefind that dλ/dl = { − (2( N − K/N ) } λ + ( J /ω c ). Byshifting J z to J (cid:48) z = J z + 1 − N − K/N , Eqs. (2) as-sume the well-known Kosterlitz-Thouless form [20] (seeFig. 2b), with a Kondo temperature scale given by T K ∼ ω c e − /J (cid:48) z . Thus, J z is found to be RG-marginal at firstorder but RG-relevant at second order, and independentof K and N . On the other hand, J ⊥ has the same scal-ing dimension as λ , and is thus dependent on K and N .For K > [ N/ N − J ⊥ is RG-irrelevant(from its scaling dimension), it can turn relevant, dueto the growth of J z . The scaling equation for J ⊥ ad-mits a nontrivial fixed point at ˜ J z ≡ [2( N − K/N ] − J z > ˜ J z , all flows lead to the N -channel Kondo fixedpoint; for J z < ˜ J z , flows lead to zero dot-to-lead hopping.If K < [ N/ N − dJ z dl = J ⊥ , dJ ⊥ dl = (cid:18) − K + J z (cid:19) J ⊥ . (3)As with λ , the flow for t acquires a positive contributionof order J ⊥ /ω c from the dot-mediated hopping (i.e., thedot promotes interlead hopping, as one might expect).By shifting J z to J (cid:48) z = J z + 1 − (1 /K ), Eq. (3) assumesthe Kosterlitz-Thouless form. Similarly, for K < J ⊥ has a nontrivial fixed point at J ∗ z = (1 − K ) /K . For J z > J ∗ z , all flows lead to the N -channel Kondo fixed point; for J z < J ∗ z , flows lead tozero dot-to-lead hopping. K KK t J J J M z z z I II IIIIVIII III IV K J z III IIIIV
A B C FIG. 3: Phase diagram of the dot-lead system. The discontin-uous critical surface separates the Kondo and ferromagneticphases. The curved vertical ribbon (orange) is the phaseboundary between the DFP and CFP phases (see Fig. 2).The discontinuity of the critical surface shrinks to zero at themulticritical point M . The diagram has four phases— I: com-pletely decoupled wires; II: wires coupled only through dot;III: wires connected both directly and through dot; IV: wiresconnected but decoupled from dot. The transitions betweenthem (shown in insets A, B, and C) are described in the text. Bringing together the flows of ( t, J ⊥ , J z ) yields thethree-dimensional phase diagram shown in Fig. 3. Thetuning of t and/or K allows one to access two non-trivial transitions between phases that have opposingcharacters in both their Kondo coupling to the dot andtheir direct interlead hopping. One is a transition be-tween phase II (in which N -channel Kondo physics dom-inates TDOS suppression) and phase IV (in which Kondophysics is suppressed by local phase pinning); see Fig. 3B.The other is a transition between phase I (in whichKondo physics is dominated by TDOS suppression) andphase III (in which Kondo screening overcomes localphase pinning); see Fig. 3C. In addition, we find a mul-ticritical point (see point M in Fig. 3A) at intermedi-ate coupling in t ; this occurs when ˜ J z = J ∗ z [so that J J Jt t t
I II III
FIG. 4: RG flows in the t - J ⊥ plane for N/ N − < K < p N/ N − J ⊥ (cid:28) | J z − J ∗ z | , | J z − ˜ J z | . I. J z 0, hopping via thedot is irrelevant for t < t ∗ and relevant otherwise. However,sufficiently large J ⊥ can drive t past t ∗ toward the regimein which both t and J ⊥ grow at low energies. III. J z > J ∗ z :hopping via the dot is relevant on both sides. Figures I andIII would remain identical for p N/ N − < K < 1, whileII would become inverted about the dashed line. K = (cid:112) N/ N − N = 2, the phase boundary in Fig. 3a becomes a marginalline at K = 1 (i.e., the Tonks-Girardeau gas [12]), andthe point M becomes a multicritical line at ˜ J z = J ∗ z = 0.The four phases are characterized by the following baretwo-lead transmission coefficients across the junction: G = 0 (i.e., minimal) at the DFP and G = 4 K/N (i.e.,maximal) at the CFP. These bare coefficients acquirepower-law corrections of order t T ν (DFP) and λ T µ (CFP), arising from direct interlead scattering [18]; here, T represents an energy scale (e.g., temperature) that cutsoff the RG flows, and ( ν, µ ) are exponents determinedby the leading irrelevant perturbations around the cor-responding fixed point. In experiments with cold atoms,these power-law contributions should be detectable viareal-time lead dynamics [17].Number fluctuations on the dot can be accessed via anon-destructive measurement scheme such as, e.g., thatsuggested in Ref. [21] for the Bose-Hubbard model: insuch a scheme the dot would be located in the waistof a high-finesse optical cavity that has a resonance fre-quency near an optical transition of the hyperfine state b (but not of a ). A fixed-number state on the dot (i.e.,the unscreened spin) would merely shift the cavity’s res-onance; by contrast, a fluctuating-number state (i.e., theKondo state) would lead to a double-peak structure inthe transmission spectrum of the cavity, with peaks cor-responding to an empty dot and to an occupied dot. Ac-knowledgments . We thank E. Demler and M. Pasienskifor stimulating discussions. This work was supported byDOE Award No. DE-FG02-07ER46453 (S.L., P.M.G.),and NSF DMR-0605813 (S.L.) and DMR 09-06780 (S.G.) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] J. Kondo, Prog. Theor. Phys. , 37 (1964).[2] P. Nozi`eres, J. Low. Temp. Phys. , 31 (1974).[3] P. Nozi`eres and A. Blandin, J. Phys. (Paris) , 193(1980).[4] I. Affleck, Nucl. Phys. B , 517 (1990); I. Affleck andA. W. W. Ludwig, Nucl. Phys. B , 849 (1991).[5] D.L. Cox and A. Zawadowski, Adv. Phys. , 599 (1998).[6] D. Goldhaber-Gordon et al., Nature , 156 (1998).[7] R.M. Potok et al., Nature , 167 (2007).[8] A. Furusaki and K. A. Matveev, Phys. Rev. Lett. ,226404 (2002).[9] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2004).[10] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[11] C. Kollath, J.S. Meyer, and T. Giamarchi, Phys. Rev.Lett. , 130403 (2008); R. Citro et al., Phys. Rev. A , 051602(R) (2007).[12] B. Paredes et al., Nature (London) , 277 (2004).[13] A. Recati et al., Phys. Rev. Lett. , 040404 (2005).[14] V. Bouchiat et al., Phys. Scr. T76 165-170 (1998).[15] D. McGloin et al., Optics Express , 158 (2003).[16] M. Pasienski and B. DeMarco, Optics Express , 2176(2008).[17] A. Tokuno, M. Oshikawa, and E. Demler, Phys. Rev.Lett. , 140402 (2008).[18] C.L. Kane and M.P.A. Fisher, Phys. Rev. B , 15233(1992).[19] H. Yi and C.L. Kane, Phys. Rev. B , 5579 (1998).[20] J.M. Kosterlitz, J. Phys. C , 1046 (1974).[21] I.B. Mekhov, C. Maschler and H. Ritsch, Nature Phys.3