Kondo length in bosonic lattices arXiv:1704.07485v2 [cond-mat.mes-hall] 5 Sep 2017 (1) Domenico Giuliano(1,2) , Pasquale Sodano(3,4) , and Andrea Trombettoni(5,6) Dipartimento di Fisica, Universit` della Calabria Arcavacata di Rende I-87036, Cosenza, Italy a (2) I.N.F.N., Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy (3) International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil (4) Departemento de F´ ısica Teorica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil (5) CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy (6) SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy (Dated: July 21, 2018) Motivated by the fact that the low-energy properties of the Kondo model can be effectively simulated in spin chains, we study the realization of the effect with bond impurities in ultracold bosonic lattices at half-filling. After presenting a discussion of the effective theory and of the mapping of the bosonic chain onto a lattice spin Hamiltonian, we provide estimates for the Kondo length as a function of the parameters of the bosonic model. We point out that the Kondo length can be extracted from the integrated real space correlation functions, which are experimentally accessible quantities in experiments with cold atoms. PACS numbers: 67.85.-d , 75.20.Hr , 72.15.Qm , 75.30.Kz . I. INTRODUCTION The Kondo effect has been initially studied in metals, like Cu, containing magnetic impurities, like Co atoms, where it arises from the interaction between magnetic impurities and conduction electrons, resulting in a net, low-temperature increase of the resistance1–3 . It soon assumed a prominent role in the description of strongly correlated systems and in motivating and benchmarking the development of (experimental and theoretical) tools to study them2,4 . Indeed, due to the large amount of analytical and numerical tools developed to attack it, the Kondo effect has become a paradigmatic example of a strongly interacting system and a testing ground for a number of different many-body techniques. The interest in the Kondo effect significantly revitalized when it became possible to realize it in a controlled way in a solid state system, by using quantum dots in contacts with metallic leads, in which the electrons trapped within the dot can give rise to a net nonzero total spin interacting with the spin of conduction electrons from the leads, thus mimicking the behavior of a magnetic impurity in a metallic host5–8 . An alternative realization of Kondo physics is recovered within the universal, low energy-long distance physics of a magnetic impurity coupled to a gapless antiferromagnetic chain9,10 . In fact, though low-energy excitations of a spin chain are realized as collective spin modes, the remarkable phenomenon of ”spin fractionalization”11 implies that the actual stable elementary excitation of an antiferromagnetic spin-1/2 spin chain is a spin-1/2 ”half spin wave”12,13 (dubbed spinon). Spinons have a gapless spectrum and, therefore, for what concerns screening of the impurity spin, they act exactly as itinerant electrons in metals, as the charge quantum number is completely irrelevant for Kondo physics. A noticeable advantage of working with the spin chain realization of the Kondo effect is that a series of tools developed for spin systems, including entanglement witnesses and negativity, can be used to study the Kondo physics in these systems14,15 . Another important, long-lasting reason for interest in Kondo systems lies in that the multichannel ”overscreened” version of the effect16,17 provides a remarkable realization of non-Fermi liquid behavior18 . Finally, the nontrivial properties of Kondo lattices provide a major arena in which to study many-body nonperturbative effects, related to heavy-fermion materials19,20 . A recent example of both theoretical and experimental activity on multichannel Kondo systems is provided by the topological Kondo model21–24 , based on the merging of several one-dimensional quantum wires with suitably induced and possibly controllable Majorana modes tunnel-coupled at their edges, and by recent proposals of realizing topological Kondo Hamiltonians in Y -junctions of XX and Ising chains25–27 and of Tonks-Girardeau gases28. Finally, the effects of the competition between the Kondo screening and the screening from localized Majorana modes emerging at the interface between a topological superconductor and a normal metal has been recently discussed in [29] using the techniques developed in [30]. The onset of the Kondo effect is set by the Kondo temperature TK , which emerges from the perturbative renormalization group (RG) approach as a scale at which the system crosses over towards the strongly correlated nonpertubative regime2,31 . The systematic implementation of RG techniques has clearly evidenced the scaling behavior characterizing the Kondo regime, which results in the collapse onto each other of the curves describing physical quantities in terms of the temperature T , once T is rescaled by TK 31,32 . The collapse evidences the one-parameter scaling, that is, there is only one dimensionful quantity, which is dynamically generated by the Kondo interaction and invariant under RG 2 trajectories. Thus, within scaling regime, one may trade T for another dimensionful scaling parameter such as, for instance, the system size ℓ. In this case, as a consequence of one-parameter scaling, a scale invariant quantity with the dimension of a length emerges, the Kondo screening length ξK , given by ξK = vF /kB TK , where vF is the Fermi velocity of conduction electrons and kB is the Boltzmann constant31 . Physically, ξK defines the length scale over which the impurity magnetic moment is fully screened by the spin of conduction electrons, that is, the ”size of the Kondo cloud”33 . Differently from TK , which can be directly measured from the low-T behavior of the resistance in metals, the emergence of ξK has been so far only theoretically predicted, as a consequence of the onset of the Kondo scaling31 . Thus, it would be extremely important to directly probe ξK , as an ultimate consistency check of scaling in the Kondo regime. As the emergence of the Kondo screening length is a mere consequence of the onset of Kondo scaling regime, ξK can readily be defined for Kondo effect in spin chains, as well9,15,34 . Unfortunately, despite the remarkable efforts paied in the last years to estimate ξK in various systems by using combinations of perturbative, as well as nonperturbative numerical methods10 , the Kondo length still appears quite an elusive quantity to directly detect, both in solid-state electronic systems as well as in spin chains33 . This makes it desirable to investigate alternative systems in which to get an easier experimental access to ξK . A promising route in this direction may be provided by the versatility in the control and manipulation of ultracold atoms35,36 . Indeed, in the last years several proposals of schemes in which features of the Kondo effect can be studied in these systems have been discussed. Refs.[37,38] suggest to realize the spin-boson model using two hyperfine levels of a bosonic gas37 , or trapped ions arranged in Coulomb crystals38 (notice that in general the Kondo problem may be thought of as a spin-1/2, system interacting with a fermionic bath39 ). Ref.[40] proposes to use ultracold atoms in multi-band optical lattices controlled through spatially periodic Raman pulses to investigate a class of strongly correlated physical systems related to the Kondo problem. Other schemes involve the use of ultracold fermions near a Feshbach resonance41 , or in superlattices42 . More recently, the implementation of a Fermi sea of spinless fermions43 or of two different hyperfine states of one atom species44 interacting with an impurity atom of different species confined by an isotropic potential has been proposed43 . The simulation of the SU (6) Coqblin-Schrieffer model for an ultracold fermionic gas of Yb atoms with metastable states has been discussed, while alkaline-earth fermions with two orbitals were also at the heart of the recent proposal of simulating Kondo physics through a suitable application of laser excitations45 . Despite such an intense theoretical activity, including the investigation of optical Feshbach resonances to engineer Kondo-type spin-dependent interactions in Li-Rb mixtures46 , and the remarkable progress in the manipulation of ultracold atomic systems, such as alkaline-earth gases, up to now an experimental detection of features of Kondo physics and in particular of the Kondo length in ultracold atomic systems is still lacking. In view of the observation that optical lattices provide an highly controllable setup in which it is possible to vary the parameters of the Hamiltonian and to accordingly add impurities with controllable parameters47,48 , in this paper we propose to study the Kondo length in ultracold atoms loaded on an optical lattice. Our scheme is based on the well-known mapping between the lattice Bose-Hubbard (BH) Hamiltonian and the XXZ spin-1/2 Hamiltonian49 , as well as on the Jordan-Wigner (JW) representation for the spin 1/2 operators, which allows for a further mapping onto a Luttinger liquid model50–52 . Kondo effect in Heisenberg spin-1/2 antiferromagnetic spin chains has been extensively studied53–55 , though mostly for side-coupled impurities (i.e., at the edge of the chain). For instance, in Ref.[54], the Kondo impurity is coupled to a single site of a gapless XXZ spin chain, while in Ref.[9] a magnetic impurity is coupled at the end of a J1 − J2 spin-1/2 chain. At variance, in trapped ultracold atomic systems, it is usually difficult to create an impurity at the edge of the system. Accordingly, in this paper we propose to study the Kondo length at an extended (at least two links) impurity realized in the bulk of a cold atom system on a 1d optical lattice. In particular, we assume the lattice to be at half-odd filling, so to avoid the onset of a gapped phase that takes place at integer filling in the limit of a strong repulsive interaction between the particles. Since the real space correlation functions are quantities that one can measure in a real cold atom experiment, we address the issue of how to extract the Kondo length from the zeroes of the integrated real space density-density correlators. Finally, we provide estimates for ξK and show that, for typical values of the system parameters, it takes values within the reach of experimental detectability (∼ tens of lattice sites). Besides the possible technical advances, we argue that, at variance with what happens at a magnetic impurity in a conducting metallic host, where one measures TK and infers the existence of ξK from the applicability of oneparameter scaling to the Kondo regime, in an ultracold atom setup one can extract from density-density correlation functions the Kondo screening length, that is in principle easier to measure, so that, to access ξK , one has not to rely on verifying the one parameter scaling, which is what tipically makes ξK quite hard to detect. The paper is organized as follows: • In section II we provide the effective description of a system of ultacold atoms on a 1d optical lattice as a spin-1/2 spin chain. In particular, we show how to model impurities in the lattice corresponding to bond impurities in the spin chain; • In section III we derive the scaling equations for the Kondo running couplings and use them to estimate the 3 corresponding Kondo length; • In section IV we discuss how to numerically extract the Kondo length from the integrated real space densitydensity correlations and compare the results with the ones obtained in section III; • In section V we summarize and discuss our results. Mathematical details of the derivation and reviews of known results in the literature are provided in the various appendices. II. EFFECTIVE MODEL HAMILTONIAN Based on the spin-1/2 XXZ spin-chain Hamiltonian description of (homogeneous, as well as inhomogeneous) interacting bosonic ultracold atoms at half-filling in a deep optical lattice, in this section we propose to model impurities in the spin chain by locally modifying the strength of the link parameters of the optical lattice, eventually resorting to a model describing two XXZ “half-spin chains”, interacting with each other via a local impurity. When the impurity is realized as a spin-1/2 local spin, such a system corresponds to a possible realization of the (two channel) Kondo effect in spin chains9,54 . Therefore, our mapping leads to the conclusion that spin chain Kondo effect may possibly realized and detected within bosonic cold atoms loaded onto a one-dimensional optical lattice. To resort to the spin-chain description of interacting ultracold atoms, we consider the large on-site interaction energy U -limit of a system of interacting ultracold bosons on a deep one-dimensional lattice. This is described by the extended BH Hamiltonian56–58 ℓ−1 HBH = − tj;j+1 (b† bj+1 j + b† bj ) j+1 j=−ℓ U + 2 ℓ j=−ℓ ℓ−1 nj (nj − 1) + V j=−ℓ ℓ nj nj+1 − µ nj . (1) j=−ℓ In Eq.(1), bj , b† are respectively the annihilation and the creation operator of a single boson at site j (with j = j −ℓ, · · · , ℓ) and, accordingly, they satisfy the commutator algebra [bj , b†′ ] = δj,j ′ , all the other commutators being j equal to 0. As usual, we set nj = b† bj . Moreover, tj;j+1 is the hopping amplitude for bosons between nearest j neighboring sites j and j + 1, U is the interaction energy between particles on the same site, V is the interaction energy between particles on nearest-neighboring sites. Typically, for alkali metal atomes one has V ≪ U while, for dipolar gases59 on a lattice, V may be of the same order as U 60,61 . Throughout all the paper we take U > 0 and V ≥ 0. To outline the mapping onto a spin chain, we start by assuming that tj;j+1 is uniform across the chain and equal to t. Then, we discuss how to realize an impurity in the chain by means of a pertinent modulation of the tj;j+1 ’s in real space. In performing the calculations, we will be assuming open boundary conditions on the 2ℓ + 1-site chain and we will set the average number of particles per site by fixing the filling f = NT where NT is the total number of N particles on the lattice and N = 2ℓ + 1 is the number of sites. In the large-U limit, one may set up a mapping between the BH Hamiltonian in Eq.(1) and a pertinent spin-model Hamiltonian HS , with HS either describing an integer,61,62 , or an half-odd spin chain63 , depending on the value of f . An integer-spin effective Hamiltonian is recovered, at large U , for f = n (with n = 1, 2, . . .), corresponding to µ = µ0 (n) = n(U + 2V ) − U/2 and U ≫ t60 , which allowed for recovering the phase diagram of the BH model in this limit by relating on the analysis of the phase diagram of spin-1 chains within the standard bosonization approach62,64 . In particular, the occurrence of Mott and Haldane gapped insulating phases for ultracold atoms on a lattice has been predicted and discussed61,65,66 . Here, we rather focus onto the mapping of the BH Hamiltonian onto an effective spin-1/2 spin-chain Hamiltonian. This is recovered at U/t ≫ 1 and half-odd filling f = n + 1/2 (with n = 0, 1, 2, . . .), corresponding to setting the chemical potential so that µ = (U + 2V )(n + 1 ) − U . In this regime, the effective low-energy spin-1/2 Hamiltonian 2 2 for the system is given by63 ℓ−1 Hspin−1/2 = −J ℓ−1 + − + − Sj Sj+1 + Sj+1 Sj + J∆ j=−ℓ z z Sj Sj+1 , j=−ℓ (2) 4 a with the spin-1/2 operators Sj defined as + Sj = − Sj = 1 n+ 1 2 1 n+ 1 2 P 1 b† P 1 2 j 2 , P 1 bj P 1 2 2 , z 1 1 Sj = P 2 [b† bj − f ]P 2 j . (3) 1 and P1/2 being the projector onto the subspace of the Hilbert space F 2 , spanned by the states ⊗N n + j=1 ˜ The parameters J and ∆ are given by J = J 1 − with σ = 1 ±2. ˜ ∆ = t2 (2n2 +6n+4) ˜ JU V ˜ J − − 4t2 (n+1)2 ˜ JU and ρ = U(n+1) ˜ 2J − U(n+1) ˜ 2J 2 ˜ 2J U ρ and ∆ = ˜ ∆ ˜ 1− 2J ρ U 1 2 + σ j, ˜ , with J = 2t n + 1 2 , + n + 2. In the regime leading to the effective Hamiltonian in Eq.(2), the large value of U/t does not lead to a Mott insulating phase, as it happens for a generic value of f . Indeed, the degeneracy between the states |n and |n + 1 at each site allows for restoring superfluidity, similarly to what happens in the phase model describing one-dimensional arrays of Josephson junctions at the chargedegenerate point67 . z Notice that the spin-1/2 Hamiltonian in Eq.(2) has to be supplemented with the condition that j Sj = 0, implying that physically acceptable states are only the eigenstates of j nj belonging to the eigenvalue NT : this corresponds to singling out of the Hilbert space only the zero magnetization sector. As discussed in detail in Ref.[63], Hspin−1/2 provides an excellent effective description of the low-energy dynamics of the BH model at half-odd filling. Although the mapping is done in the large-U limit, in Ref.[63] it is shown that it is in remarkable agreement with DMRG results also for U/J as low as ∼ 3 − 5 and for low values of NT such as NT ∼ 30. ℓ Additional on-site energies ǫi can be accounted for by adding a term j=−ℓ ǫj nj to the right-hand side of Eq.(1). ℓ z Accordingly, Hspin−1/2 in Eq.(2) has to be modified by adding the term j=−ℓ ǫj Sj . As soon as the potential energy scale is smaller than U , we expect the mapping to be still valid (we recall that with a trapping parabolic potential typically ǫj = Ωj 2 with Ω ≡ mω 2 λ2 /8, m being the atom mass, ω the confining frequency and λ/2 the lattice spacing68 ). Yet, we stress that recent progresses in the realizations of potentials with hard walls69,70 make the optical lattice realization of chains with open boundary conditions to lie within the reach of present technology. Another point to be addressed is what happens slightly away from half-filling, that is, for f = n + 1/2 + ε, with ε ≪ 1. In this case, one again recovers the effective Hamiltonian in Eq.(2), but now with the constraint on physically z acceptable states given by (1/N ) j Sj = ε. Since keeping within a finite magnetization sector is equivalent to having a nonzero applied magnetic field71 , one has then to add to the right hand side of Eq.(2) a term of the form z H j Sj , where H ∝ ε: again, we expect that the mapping is valid as soon as that the magnetic energy is smaller than the interaction energy scale U , and, of course, that the system spectrum remains gapless72 . To modify the Hamiltonian in Eq.(2) by adding bond impurities to the effective spin chain, we now create a link defect in the BH Hamiltonian in Eq.(1) by making use of the fact that optical lattices provide a highly controllable setup in which it is possible to vary the parameters of the Hamiltonian as well as to add impurities with tunable parameters47,48 . This allows for creating a link defect in an optical lattice by either pertinently modulating the lattice, so that the energy barriers among its wells vary inhomogeneously across the chain, or by inserting one, or more, extra laser beams, centered on the minima of the lattice potential. In this latter case, one makes the atoms feel a total potential given by Vext = Vopt + Vlaser , where the optical potential is given by Vopt = V0 sin2 (kx), with k = 2π/λ and λ = λ0 / sin (θ/2), λ0 being the wavelength of the lasers and θ the angle between the laser beams forming the main lattice47 (notice that the lattice spacing is d = λ/2). For counterpropagating laser beams having the same direction, θ = π and d = λ0 /2, while d can be enhanced by making the beams intersect at an angle θ = π. Vlaser is the additional potential due to extra (blue-detuned) lasers: with one additional laser, centered at or close to an energy maximum of 2 2 Vopt , say at x ≡ x0;1 among the minima x0 = 0 and x1 = d, the potential takes the form Vlaser ≈ V1 e−(x−x0;1 ) /σ . When the width σ is much smaller than the lattice spacing, the hopping rate between the sites j = 0 and j = 1 is reduced and no on-site energy term appears, as shown in panels a) and b) of Fig.1. Notice that we use a notation such that the j-th minimum corresponds to the minimum xj = jd in the continuum space. When x0;1 is equidistant from the lattice minima x0 and x1 , corresponding to x0;1 = λ/4 = d/2 and σ < d, then only the hopping t0,1 is practically altered (see Fig.1a) ). When x0;1 is displaced from d/2 one has an asymmetry and also a nearest neighboring link [e.g., t−1;0 in Fig.1 b)] may be altered (an additional on-site energy ǫ0 is also present). With d ∼ 2 − 3µm, one should have σ 2µm, in order to basically alter only one link. Notice that barrier of few µm can be rather straightforwardly implemented28,73 and recently a barrier of ∼ 2µm has been realized in a Fermi gas74 . 5 As discussed in the following, this is the prototypical realization of a weak-link impurity in an otherwise homogeneous spin chain75,76 . In general, reducing the hopping rate between links close to each other may either lead to an effective weak link impurity, or to a spin-1/2 effective magnetic impurity, depending on whether the number of lattice sites between the reduced-hopping-amplitude links is even, or odd (see appendix A for a detailed discussion of this point). To ”double” the construction displayed in panels a) and b) of Fig.1 to the one we sketch in panels c) and d) of Fig.1, we consider 2 2 2 2 a potential of the form Vlaser ≈ V1 e−(x−x0;1 ) /σ + V2 e−(x−x−1;0 ) /σ with x−1;0 lying between sites j = −1 and j = 0: assuming again σ d, when V1 = V2 and x0;1 = −x−1;0 = d/2 then only two links are altered, and in an equal way [the hoppings t−1;0 and t0;1 in Fig.1 c)], otherwise one has two different hoppings [again t−1;0 and t0;1 in Fig.1 d) ]. When σ is comparable with d, apart from the variation of the hopping rates, on-site energy terms enter the Hamiltonian in Eq.(1), giving rise to local magnetic fields in the spin Hamiltonian in Eq.(2). Though this latter kind of “site defects” might readily be accounted for within the spin-1/2 XXZ framework, for simplicity we will not consider them in the following, and will only retain link defects, due to inhomogeneities in the boson hopping amplitudes between nearest neighboring sites and in the interaction energy V . Correspondingly, the hopping amplitude tj;j+1 in Eq.(1) takes a dependence on the site j also far form the region in which the potential Vlaser is centered. In the following, we consider inhomogeneous distributions of link parameters symmetric about the center of the chain (that is, about j = 0). Moreover, for the sake of simplicity, we discuss a situation in which two (symmetrically placed) inhomogeneities enclose a central region, whose link parameters may, or may not, be equal to the ones of the rest of the chain. We believe that, though experimentally challenging, this setup would correspond to the a situation in which the experimental detection of the Kondo length is cleaner. In fact, we note that all the experimental required ingredients are already available, as our setup requires two lasers with σ d (ideally, σ ≪ d) and centered with similar precision. As we discuss in detail in Appendix A, an “extended central region” as such can either be mapped onto an effective weak link, between two otherwise homogeneous “half-chains”, or onto an effective isolated spin-1/2 impurity, weakly connected to the two half-chains. In particular, in this latter case, the Kondo effect may arise, yielding remarkable nonperturbative effects and, eventually, “sewing together” the two half chains, even for a repulsive bulk interaction53,54 . Denoting by G the region singled out by weakening one or more links, in order to build an effective description of G, we assume that the mapping onto a spin-1/2 XXZ-chain works equally well with the central region, and employ a systematic Shrieffer-Wolff (SW) summation, in order to trade the actual dynamics of G for an effective boundary Hamiltonian, that describes the effective degrees of freedom of the central region interacting with the half chains. One is then led to consider the Hamiltonian in Eq.(1), with link-dependent hopping rates tj;j+1 . To illustrate how the mapping works, we focus onto the case of M = 2 altered links, corresponding to two bluedetuned lasers, and briefly comment on the more general case. To resort to the Kondo-like Hamiltonian for a spin-1/2 impurity embedded within a spin-1/2 XXZ-chain, we define the hopping rate to be equal to t throughout the whole chain but between j = −1 and j = 0, where we assume it to be equal to tL , and between j = 0 and j = 1, where we set it equal to tR , corresponding to panels c) and d) of Fig.1. On going through the SW transformation, one therefore gets the effective spin-1/2 Hamiltonian Hs = Hbulk + HK , with Hbulk = HL + HR and HL = −J −2 + − + − Sj Sj+1 + Sj+1 Sj + J∆ j=−ℓ z z Sj Sj+1 j=−ℓ ℓ−1 ℓ−1 HR = −J −2 + − + − Sj Sj+1 + Sj+1 Sj + J∆ z z Sj Sj+1 . (4) j=1 j=1 The ”Kondo-like” term is instead given by ′ ′ ′ ′ − + + + − − − + z z z z HK = −JL (S−1 S0 + S−1 S0 ) − JR (S0 S1 + S0 S1 ) + JzL S−1 S0 + JzR S0 S1 , ′ ′ ′ (5) where Jα = tα f and Jzα ≈ V − 3Jα2 /4U (with α = L, R). Our choice for HK corresponds to the simplest case in which G contains an even number of links – or, which is the same, an odd number of sites, as schematically depicted in Fig.2b). We see that the isolated site works as an isolated spin-1/2 impurity SG , interacting with the two half chains via the boundary interaction Hamiltonian (1) HB ≡ HK . The other possibility, which we show in Fig.2a), corresponds to the case in which an odd number of links is altered and G contains an even number of sites. In particular, in Fig.2a) we have only one altered hopping coefficient. This latter case is basically equivalent to a simple weak link between the R- and the L- half chain, which is expected to realize the spin-chain version of Kane-Fisher physics of impurities in an interacting one-dimensional electronic system77 . In Appendix A, we review the effective low-energy description for a region G containing an in principle arbitrary number of sites. In particular, we conclude that either the number of sites within G is odd, and 6 therefore the resulting boundary Hamiltonian takes the form of HK in Eq.(5), or it is even, eventually leading to a weak link Hamiltonian75,76 . Even though this latter case is certainly an interesting subject of investigation, we are mostly interested in the realization of effective magnetic impurities. Therefore, henceforth we will be using Hs as the main reference Hamiltonian, to discuss the emergence of Kondo physics in our system. a) c) 4 4 V /V ext 0 V /V ext 0 2 2 0 −4 b) x/d 0 0 −4 4 d) x/d 0 4 0 4 4 4 V /V ext 0 V /V ext 0 2 0 −4 2 x/d 0 4 0 −4 x/d FIG. 1: External potential Vext (in units of V0 ) as a function of x (in units of d) for different values of V1 /V0 and V2 /V0 (with σ = 0.2d). In panels a) and b), we consider V2 = 0 so that only two hopping parameters are altered: panel a) corresponds to x1 /d = 0.5 and panel b) to x0;1 /d = 0.25 (in both cases V1 /V0 = 2). In panels c) and d), we have x0;1 /d = 0.5 and x−1;0 /d = −0.5 in both cases, but V1 /V0 = V2 /V0 = 2 for panel c) and V1 /V0 = 2, V2 /V0 = 3 for panel d). III. RENORMALIZATION GROUP FLOW OF THE IMPURITY HAMILTONIAN PARAMETERS In this section, we employ the renormalization group (RG) approach to recover the low-energy long-wavelength physics of a Kondo impurity in an otherwise homogeneous chain. From the RG equations we derive the formula for the invariant length which we eventually identify with ξK . In general, there are two standard ways of realizing the impurity in a spin chain, which we sketch in Fig.2. Specifically, we see that the impurity can be realized as an island containing either an even or odd number of spins. The former case is equivalent to a weak link in an otherwise homogeneous chain, originally discussed in Refs.[77,78] for electronic systems, and reviewed in detail in Ref.[79] in the specific context of spin chains. In this case, which we briefly review in Appendix C, when ∆ > 0 in Eqs.(4), the impurity corresponds to an irrelevant perturbation, which implies an RG flow of the system towards the fixed point corresponding to two disconnected chains, while for ∆ < 0 the weak link Hamiltonian becomes a relevant perturbation. Though this implies the emergence of an ”healing length ” for the weak link as an RG invariant length scale, with a corresponding flow towards a fixed point corresponding to the two chains joined into an effectively homogeneous single chain, there is no screening of a dynamical spinful impurity by the surrounding spin degrees of freedom and, accordingly, no screening cloud is detected in this case79 . At variance, a dynamical effective impurity screening takes place in the case of an effective spin-1/2 impurity34 . In this latter case, at any ∆ such that −1 < ∆ < 1, the perturbative RG approach shows that the disconnected-chain weakly coupled fixed point is ultimately unstable. In fact, the RG trajectories flow towards a strongly coupled fixed point, which we identify with the spin chain two channel Kondo fixed point, corresponding to healing the chain but, 7 a) b) FIG. 2: Sketch of two different kinds of central regions in an otherwise uniform spin chain, respectively realizing an effective weak-link impurity (a)), and an effective spin-1/2 impurity (b)). at variance with what happens at a weak link for 0 < ∆ ≤ 1, this time with the chain healing taking place through an effective Kondo-screening of the magnetic impurity53 . A region containing an odd number of sites typically has a twofold degenerate groundstate and, therefore, is mapped onto an effective spin-1/2 impurity SG . The corresponding impurity Hamiltonian in Eq.(A2) takes the form of the Kondo spin-chain interaction Hamiltonian for a central impurity in an otherwise uniform spin chain9 . To employ the bosonization formalism of appendix C to recover the RG flow of the impurity coupling strength, we resort to Eq.(B12), corresponding to the bosonized spin Kondo Hamiltonian HK given by ′ HK = α=L,R + −Jα [S0 e i − √2 Φα (0) − + S0 e i √ Φα (0) 2 ′ z 1 ∂Θα (0) ] + Jzα S0 √ 2π ∂x . (6) The RG equations describing the flow of the impurity coupling strength can be derived by means of standard techniques for Kondo effect in spin chains54 and, in particular, by considering the fusion rules between the various operators entering HK in Eq.(6). In doing so, in principle additional, weak link-like, operators describing direct tunneling i √ [Φ (0)−ΦR (0)] between the two chains can be generated, such as, for instance, a term ∝ e 2 L , with scaling dimension 1 hA = g . However, one may safely neglect a term as such, since, for g < 1, it corresponds to an additional irrelevant boundary operator that has no effects on the RG flow of the running couplings appearing in HK . For g ≥ 1 it becomes ′ marginal, or relevant, but still subleading, compared to the terms ∝ Jα , as we discuss in the following and, therefore, it can again be neglected for the purpose of working out the RG flow of the boundary couplings. This observation effectively enables us to neglect operators mixing the L and the R couplings with each other and, accordingly, to factorize the RG equations for the running couplings with respect to the index α . More in detail, we define the dimensionless variables Gα (ℓ) and Gz,α (ℓ) as Gα (ℓ) = ℓ ℓ0 1 1− 2g ′ ′ Jα J and Gz,α (ℓ) = Jzα J , (7) (see Appendix B for a discussion on the estimate of the reference length ℓ0 ) with α = L, R. The RG equations for the running couplings are given by dGα (ℓ) = hg Gα (ℓ) + Gα (ℓ)Gz,α (ℓ) d ln( ℓℓ ) 0 dGzα (ℓ) = G2 (ℓ) , α d ln( ℓℓ0 ) (8) with hg = 1 − 1/(2g). For the reasons discussed above, the RG equations in Eq.(8) for the L - and the R -coupling strengths are decoupled from each other. In fact, they are formally identical to the corresponding equations obtained for a single link impurity placed at the end of the chain (“Kondo side impurity”)9 . At variance with this latter case, as argued by Affleck and Eggert53, in our specific case of a ”Kondo central impurity” the scenario for what concerns the possible Kondo-like fixed points is much richer, according to whether GL (ℓ0 ) = GR (ℓ0 ) (“asymmetric case”), or GL (ℓ0 ) = GR (ℓ0 ) (“symmetric case”), as we discuss below. 1 To integrate Eqs.(8), we define the reduced variables Xα (ℓ) ≡ Gα (ℓ) and Xz,α (ℓ) = Gz,α (ℓ) + 1 − 2g for α = L, R (since the equations for the two values of α are formally equal to each other, from now on we will understand the index α). As a result, one gets dX(ℓ) = X(ℓ)Xz (ℓ); d ln( ℓℓ ) 0 dXz (ℓ) = X 2 (ℓ). d ln( ℓℓ0 ) (9) Equations (9) coincide with the RG equations obtained for the Kosterlitz-Thouless phase transition80 . To solve them, we note that the quantity 2 κ = Xz (ℓ) − X 2 (ℓ) , (10) 8 is invariant along the RG trajectories. In terms of ′the microscopic parameters of the BH Hamiltonan one gets ′ κ = κ(ℓ0 ) = (V /J − 3J 2 /(4U J) + 1 − 1/(2g))2 − (J /J)2 . To avoid the onset of Mott-insulating phases, we have to assume that the interaction is such that g > 1/2. This implies hg > 0 and Xz (ℓ0 ) > 0: thus, we assume X(ℓ0 ), Xz (ℓ0 ) > 0. This means that the RG trajectories always lie within the first quarter of the (X, Xz )-parameter plane and, in particular, that the running couplings always grow along the trajectories. Using the constant of motion in Eq.(10), Eqs.(9) can be easily integrated. As a result, one may estimate the RG invariant length scale ℓ∗ defined by the condition that, at the scale ℓ ∼ ℓ∗ , the perturbative calculation breaks down (which leads us to eventually identify ℓ∗ with ξK ). As this is signaled by the onset of a divergence in the running parameter X(ℓ)27 , one may find the explicit formulas for ℓ∗ , depending on the sign of κ, as detailed below: • κ = 0. In this case, as the symmetry at ℓ = ℓ0 between K and Xz is preserved along the RG trajectories, it is enough to provide the explicit solution for Xz (ℓ)(= X(ℓ)), which is given by Xz (ℓ) = Xz (ℓ0 ) 1 − Xz (ℓ0 ) ln( ℓℓ ) 0 . (11) From Eq.(11), one obtains 1 Xz (ℓ0 ) ℓ∗ ∼ ℓ0 exp , (12) which is the familiar result one recovers for the ”standard” Kondo effect in metals34 . • κ < 0. In this case, the explicit solution of Eqs.(9) is given by Xz (ℓ) = X(ℓ) = √ −κ tan atan 2 −κ + Xz (ℓ) , √ Xz (ℓ0 ) √ + −κ ln −κ ℓ ℓ0 (13) which yields   ℓ∗ ∼ ℓ0 exp  √ π − 2 atan( Xz (ℓ0 ) ) |κ| 2 |κ| • κ > 0. In this case one obtains  √  √  [Xz (ℓ0 ) − κ] Xz (ℓ) = − κ   [X (ℓ ) − √κ] z 0 X(ℓ) = ℓ ℓ0 ℓ ℓ0 √ 2 κ √ 2 κ    . (14)  √  + [Xz (ℓ0 ) + κ]  − [Xz (ℓ0 ) + 2 −κ + Xz (ℓ) . √  κ]  (15) As a result, we obtain ℓ∗ ∼ ℓ0 √ Xz (ℓ0 ) + κ √ Xz (ℓ0 ) − κ 1 √ 2 κ . (16) To provide some realistic estimates of ℓ∗ , in Fig.3 we plot ℓ∗ /ℓ0 as a function of the repulsive interaction potential V , keeping fixed all the other system parameters (see the caption for the numerical values of the various parameters). ′ The two plots we show correspond to different values of J . We see that, as expected, at any value of V /J, ℓ∗ decreases ′ on increasing J . We observe that with realistically small values of V /J, say between 0 and 0.5, one has a value of the Kondo length order of 20 sites (for J ′ /J = 0.2) and 5 sites (for J ′ /J = 0.6), that should detectable from experimental data. Also, we note a remarkable decrease of ℓ∗ with V /J and, in particular, a finite ℓ∗ even at extremely small values of ′ V , which correspond to negative values of Jz and, thus, to an apparently ferromagnetic Kondo coupling between the impurity and the chain. In fact, in order for the Kondo coupling to be antiferromagnetic, and, thus, to correspond ′ ′ to a relevant boundary perturbation, one has to either have both J and Jz positive, or the former one positive, the 9 a) b) 6.2 30 l /l0 * 25 20 l /l0 * 4.7 15 10 5.7 5.2 4.2 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 V/J V/J FIG. 3: ℓ∗ /ℓ0 as a function of V /J for 0 ≤ V /J ≤ 0.5. The other parameters are chosen so that U/J = 4 and J ′ /J = 0.2 (panel a)), and J ′ /J = 0.6 (panel b)). As discussed in Appendix B, ℓ0 is of order of the lattice spacing d. latter negative. In our case, the RG equations in Eqs.(9), show how the β-function for the running coupling X(= G) 1 is proportional to Xz G, rather than to Gz G. Thus, what matters here is the fact that Xz − Gz = 1 − 2g > 0, which ′ ′ makes Xz (ℓ0 ) positive even though Gz (ℓ0 ) is negative. As a result, even when both J and Jz are negative as it may happen, for instance, if one starts from a BH model with V ∼ 0, one may still recover a Kondo-like RG flow and find a finite ℓ∗ , as evidenced by the plots in Fig.3. Being an invariant quantity along the RG trajectories, here ℓ∗ plays the same role as ξK in the ordinary Kondo effect, that is, once the RG trajectories for the running strengths are constructed by using the system size ℓ as driving variable, all the curves are expected to collapse onto each other, provided that, at each curve, ℓ is rescaled by the corresponding ℓ∗ 2,34,81,82 . In fact, in the specific type of system we are focusing onto, that is, an ensemble of cold atoms loaded on a pertinently engineered optical lattice, it may be difficult to vary ℓ by, in addition, keeping the filling constant (not to affect the parameters of the effective Luttinger liquid model Hamiltonian describing the system). Yet, one may resort to a fully complementary approach in which, as we highlight in the following, the length ℓ, as well as the filling f , are kept fixed and, taking advantage of the scaling properties of the Kondo RG flow, one probes the scaling properties by varying ℓ∗ . Indeed, from our Eqs.(12,14,16), one sees that in all cases of interest, the relation between ℓ∗ and the microscopic parameters characterizing the impurity Hamiltonian is known. As a result, one can in principle arbitrarily tune ℓ∗ at fixed ℓ by varying the tunable system parameter. As we show in the following, this provides an alterative way for probing scaling behavior, more suitable to an optical lattice hosting a cold atom condensate. In order to express the integrated RG flow equations for the running parameters as a function of ℓ and ℓ∗ , it is sufficient to integrate the differential equations in Eqs.(9) from ℓ∗ up to ℓ. As a result, one obtains the following equations: • For κ = 0: X(ℓ) = Xz (ℓ) = • For κ < 0: Xz (ℓ) = X(ℓ) = • For κ > 0: √ −κ tan Xz (ℓ0 ) − ln( ℓℓ ) ∗ π √ − −κ ln 2 ; ℓ∗ ℓ 2 −κ + Xz (ℓ) ;  √  Xz (ℓ) = κ  X(ℓ) =  √ ℓ∗ 2 κ +  ℓ √ ℓ∗ 2 κ − 1 ℓ 2 −κ + Xz (ℓ) . (17) (18) (19) From Eqs.(17,18,19), one therefore concludes that, once expressed in terms of ℓ/ℓ∗ , the integrated RG flow for the running coupling strengths only depends on the parameter κ. Curves corresponding to the same values of κ just collapse onto each other, independently of the values of all the other parameters. 10 We pause here for an important comment. As discussed in9 , in the spin chain realization of the Kondo model, one exactly retrieves the equation of the conventional Kondo effect at g = 1/2 only after adding a frustrating secondneighbor interaction, thus resorting to the so-called J1 − J2 model Hamiltonian. In principle, the same would happen for the XXX-spin chain with nearest-neighbor interaction only, except that, strictly speaking, the correspondence is exactly realized only in the limit of an infinitely long chains. In the case of finite chains, the presence of a marginally irrelevant Umklapp operator may induce finite-size violations from Kondo scaling which, as stated above, disappear in the thermodynamic limit. Yet, as this point is mostly of interest because it may affect the precision of numerical calculations, we do not address it here and refer to Ref.[9] for a detailed discussion of this specific topic. Another important point ′to stress is that, strictly speaking, we have so far neglected the possible effects of the ′ ′ ′ ′ ′ ′ ′ asymmetry (JL = JR and Jz,L = Jz,R ), versus symmetry (JL = JR and Jz,L = Jz,R ) in the bare couplings. In fact, the nature of the stable Kondo fixed point reached by the system in the large scale limit deeply depends on whether or not the bare couplings between the impurity and the chains are symmetric, or not. Nevertheless, as we argue in the following, one sees that, while the nature of the Kondo fixed point may be quite different in the two cases (two-channel versus one-channel spin-Kondo fixed point), one can still expect to be able to detect the onset of the Kondo regime and to probe the corresponding Kondo length by looking at the density-density correlations in real space, though the correlations themselves behave differently in the two cases. We discuss at length about this latter point in the next section. Here, we rather discuss about the nature of the Kondo fixed point in the two different situations, starting with the case of symmetric couplings between the impurity and the chains. ′ ′ ′ ′ When JL = JR and JzL = JzR , since, to leading order in the running couplings, there is no mixing between the L - and the R coupling strengths, the L − R symmetry is not expected to be broken all the way down to the strongly coupled fixed point which, consequently, we identify with the two-channel spin-chain Kondo fixed point, in which the impurity is healed and the two chains have effectively joined into a single uniform chain. Due to the L − R symmetry, one can readily show that all the allowed boundary operators at the strongly coupled fixed point are irrelevant53,54 , leading to the conclusion that the two-channel spin-Kondo fixed point is stable, in this case. Concerning the effects of the asymmetry, on comparing the scale dimensions of the various impurity boundary operators, one expects them to be particularly relevant if the asymmetry is realized in the transverse Kondo coupling ′ ′ strengths, that is, if one has JL ≫ JR . We assume that this is the case which, moving to the dimensionless couplings, implies GL (ℓ0 ) ≫ GR (ℓ0 ). Due to the monotonicity of the integrated RG curves, we expect that this inequality keeps preserved along the integrated flow, that is, GL (ℓ) ≫ GR (ℓ) at any scale ℓ ≥ ℓ0 . In analogy with the standard procedure used with multichannel Kondo effect with non-equivalent channels, one defines ℓ∗ as the scale at which the larger running coupling GL (ℓ) diverges, which is the signal of the onset of the nonperturbative regime. Due to the coupling asymmetry, we then expect GR (ℓ∗ ) ≪ 1, that is, at the scale ℓ ∼ ℓ∗ , the system may be regarded as a semi-infinite chain at the left-hand side, undergoing Kondo effect with an isolated magnetic impurity, weakly interacting with a second semi-infinite chain, at the right-hand side. To infer the effects of the residual coupling, one may assume that, at ℓ ∼ ℓ∗ , the impurity is “re-absorbed” in the left-hand chain53,54 , so that this scenario will consist of the left-hand chain, with one additional site, connected with a link of strength ∼ GR (ℓ∗ ) to the endpoint of the right-hand chain. Within the bosonization approach, the weak link Hamiltonian is given by77 Asym VB ∼ −GR (ℓ∗ )e i √ [ΦL (0)−ΦR (0)] 2 + h.c. . (20) Asym 1 VB has scaling dimension g . Depending on whether g > 1, or g < 1, it can therefore be either relevant, or irrelevant (or marginal if g = 1). When relevant, it drives the system towards a fixed point in which the weak link is healed. When irrelevant, the fixed point corresponds to the two disconnected chains. In either case, the residual flow takes place after the onset of Kondo screening. We therefore conclude that Kondo screening takes place in the left-hand chain only and, accordingly, one expects to be able to probe ℓ∗ by just looking at the real space density-density correlations in that chain only. From the above discussion we therefore conclude that Kondo effect is actually realized at a chain with an effective spin-1/2 impurity whether or not the impurity couplings to the chains are symmetric, or not, though the fixed point the system is driven to along the RG trajectories can be different in the two cases. IV. DENSITY-DENSITY CORRELATIONS AND MEASUREMENT OF THE KONDO LENGTH In analogy to the screening length ξK in the standard Kondo effect83,84 , in the spin chain realization of the effect, the screening length ℓ∗ is identified with the typical size of a cluster of spins fully screening the moment of the isolated magnetic impurity, either lying at one side of the impurity itself (in the one-channel version of the effect-side impurity at the end of a single spin chain), or surrounding the impurity on both sides (two channel version of the effect-impurity embedded within an otherwise uniform chain). So far, ℓ∗ showed itself as quite an elusive quantity to experimentally detect, both in electronic Kondo effect, as well as in spin Kondo effect33 . In this section, we propose to probe ℓ∗ in the effective spin-1/2 XXZ chain describing the 11 BH model, by measuring the integrated real-space density-density correlation functions. Real-space density-density correlations in atomic condensates on an optical lattice can be measured with a good level of accuracy (see e.g. Refs.[35,85].) Given the mapping between the BH- and the spin-1/2 XXZ spin Hamiltonian, real-space densitydensity correlation functions are related via Eq.(3) to the correlation functions of the z-component of the effective spin operators in the XXZ-Hamiltonian (local spin-spin susceptibility), which eventually enables us to analytically compute the correlation function within spin-1/2 XXZ spin chain Hamiltonian framework. The idea of inferring informations on the Kondo length by looking at the scaling properties of the real-space local spin susceptibility was put forward in Ref.[86]. In the specific context of lattice model Hamiltonians, the integrated real-space correlations have been proposed as a tool to extract ξK in a quantum dot, regarded as a local Anderson model, interacting with itinerant lattice spinful fermions81 . Specifically, letting SG denote the spin of the isolated spin-1/2 impurity and Sj the spin operator in the site j, assuming that the impurity is located at one of the endpoints of the chain and that the whole model, including the term describing the interaction between SG and the spins of the chain, is spin-rotational invariant, one may introduce the integrated real-space correlation function Σ(x), defined as81 x Σ(x) = 1 + y=1 SG · Sy SG · SG . (21) The basic idea is that the first zero of Σ(x) one encounters in moving from the location of the impurity, identifies the portion of the whole chains containing the spins that fully screen SG . Once one has found the solution of the equation Σ(x = x∗ ) = 0 , one therefore naturally identifies x∗ with ℓ∗ . It is important to stress that this idea equally applies whether one is considering the spin impurity at just one side of the chain (one-channel spin chain Kondo), or embedded within the chain (two-channel spin chain Kondo). Thus, while in the following we mostly consider the two-channel case, we readily infer that our discussion applies also to the one-channel case. To adapt the approach of Ref.[81] to our specific case, first of all, since our impurity is located at the center of the chain, one has to modify the definition of Σ(x) so to sum over j running from −x to x. In addition, in our case both the bulk spin-spin interaction, as well as the effective Kondo interaction with the impurity, are not isotropic in the spin space. This requires modifying the definition of Σ(x), in analogy to what is done in Ref.[81] in the case in which an applied magnetic field breaks the spin rotational invariance. Thus, to probe ℓ∗ we use the integrated z-component of the spin correlation function, Σz (x), defined as x Σz (x) = 1 + y=−x z z z z SG Sy − SG Sy z z z SG SG − SG 2 . (22) In general, estimating ℓ∗ from Σz (x) would require exactly computing the spin-spin correlation functions by means of a numerical technique, such as it is done in Ref.[81] – nevertheless one in general expects that the estimate of ℓ∗ obtained using perturbative RG differs by a factor order of 1 from the one obtained by nonperturbative, numerical means. For the purpose of showing the consistency between the estimate of ℓ∗ from the spin-spin correlation functions and the results from the perturbative analysis of Sec.III, one therefore expects it to be sufficient to resort to a perturbative (in ′ Jz , J ′ ) calculation of Σz (x), eventually improved by substituting the bare coupling strengths with the running ones, computed at an appropriate scale34 . To leading order in the impurity couplings, we obtain ′ ∞ ′ 0 ∞ z z SG Sy = −Jz,R z z SG Sy = −Jz,L 0 dτ Gz,z (y, 1; τ |ℓ) , (y > 0) dτ Gz,z (y, 1; τ |ℓ) , (y < 0) , (23) with the finite-τ correlation function Gz,z (x, x′ ; τ |ℓ) defined in Eq.(C2). To incorporate scale effects in the result of Eq.(23), we therefore replace the bare impurity coupling strengths with the running ones we derived in Sec.III, computed at an appropriate length scale, which we identify with the size x of the spin cluster effectively contributing to impurity screening. Therefore, referring to the dimensionless running coupling Xz (λ) defined in Eqs.(8), we obtain ′ Σz (x) = 1 − 8Jz (x)ℓ πu x y=1 ∞ dw Gz,z y, 1; 0 = 1 − 8ϕ(∆) Xz (x) + 1 −1 ℓ 2g πuw ℓ ℓ x y=1 ∞ 0 dw Gz,z y, 1; πuw ℓ ℓ , (24) 12 a) Σ z (x) b) 1.2 1.2 Σ z (x) 0.9 0.6 0.9 0.6 0.3 0.0 1 0.3 10 15 x 0.0 1 20 25 30 10 15 20 25 30 35 x FIG. 4: a): Plot of Σz (x) vs. x for U/J = 4, V = 0 (corresponding to ∆ = −0.1875) and J ′ /J = 0.2. From the plot one infers ℓ∗ ∼ 26, which is in good agreement with the value obtained from the plot in Fig.3a). b): Same as before, but with V /J = 2.1875 (corresponding to ∆ = 0.2) and J ′ /J = 0.1. As expected, the lower value of J ′ yields a larger ℓ∗ ∼ 32. with ϕ(∆) given by ϕ(∆) = arcos π2 1− ∆ 2 ∆ 2 2 . (25) Remarkably, ϕ(∆) → 1 as ∆ → 0. In Fig.4, we show Σz (x) vs. x (only the positive part of the graph) for two paradigmatic situations: in Fig.4a) we consider the absence of nearest-neighbor ”bare” density-density interaction (V = 0). In Fig.4b) we consider a rather large, presently not straightforward to be implemented in experiments, value of V (V /J ∼ 2.2) to show the results for the Kondo length with a positive value of the XXZ anisotropy parameter. We see that there is not an important dependence of the Kondo length upon V , since the main parameter affecting ℓ∗ is actually given by J ′ /J. From the analysis of Ref.[63], one sees that, even at V = 0, a nonzero attractive density-density interaction between nearest-neighboring sites of the chain is actually induced by higher order (in t/U ) virtual processes, which implies that, for V = 0, g keeps slightly higher than 1. At variance, for finite V , g can be either larger, or smaller than 1, as it is the case in the plot in Fig.4b). In both cases we see the effect of ”Friedel-like” oscillations in the density-density correlation, which eventually conspire to set Σz (x) to 0 at a scale x ∼ ℓ∗ (see the caption of the figures for more details on the numerical value of the various parameters). In general, Eq.(24) has to be regarded within the context of the general scaling theory for Σz (x)34 . In our specific case, at variance with what happens in the ”standard” Kondo problem of itinerant electrons in a metal magnetically interacting with an isolated impurity34 , the boundary action in Eq.(B12) contains terms that are relevant as the length scale grows. In general, in this case a closed-form scaling formula for physical quantities cannot be inferred from the perturbative results, due to the proliferation of additional terms generated at higher orders in perturbation theory87 . Nevertheless, here one can still recover a pertinently adapted scaling equation, as only dimensionless contributions B to SG effectively contribute Σz (x) to any order in perturbation theory. The point is that, as we are considering a boundary operator in a bosonized theory in which the fields ΦL,R (x, τ ) obey Neumann boundary conditions at the z z boundary, the fields ΘL.R (0, τ ) appearing in the bosonized formula for S1,L , S1,R in Eqs.(B8) are pinned at a constant for any τ . As a result, the corresponding contribution to the boundary interaction reduces to the one in Eq.(B12), ′ which is purely dimensionless and, therefore, marginal. As for what concerns the contribution ∝ JL,R , it is traded for ′ ′ a marginal one once one uses as running couplings the rescaled variables XL and XR , rather than JL , JR . Now, from z Eqs.(B8) we see that the bosonization formula for Sj contains a term that has dimension d1 = 1 and a term with dimension d2 = (2g)−1 . Taking into account the dynamics of the degrees of freedom of the chains comprised over a segment of length x, we therefore may make the scaling ansatz for Σz in the form Σz [x, ℓ, Xz , X] = ω0 ˜ x x , Xz , X + ℓ1−g ω1 , Xz , X ˜ ℓ ℓ , (26) with ω0 , ω1 scaling functions. Now, we note that, due to the existence of the RG invariant κ, which relates to each other the running parameters Xz and X along the RG trajectories (Eq.(10) in the perturbative regime), we may trade 13 ω0,1 ˜ x ℓ , Xz , X for two functions ω0,1 of only x ℓ and X. As a final result, Eq.(26) becomes Σz [x, ℓ, Xz , X] = ω0 x x , Xz (x) + ℓ1−g ω1 , Xz (x) ℓ ℓ . (27) Equation (27) provides the leading perturbative approximation at weak boundary coupling, as it can be easily checked from the explicit formula in Equation (C2). Eq.(27) illustrates how the function we explicitly use in our calculation can be regarded as just an approximation to the exact scaling function for Σz (x). A more refined analytical treatment B might in principle be done by considering higher-order contributions in perturbation theory in SG . Alternatively, one might resort to a fully numerical approach, similar to the one used in Ref.[81]. Yet, due to the absence of an intermediate-coupling phase transition in the Kondo effect2 , in our opinion resorting to a more sophisticated approach would improve the quantitative relation between the microscopic ”bare” system parameters and the ones in the effective low-energy long-wavelength model Hamiltonian, without affecting the main qualitative conclusion about the Kondo screening length and its effects. For this reason, here we prefer to rely on the perturbative RG approach extended to the correlation functions which, as we show before, already provides reliable and consistent results on the effects of the emergence of ℓ∗ on the physical quantities. The obtained estimate of ξK , although perturbative, provides, via the RG relation kB TK = vF /ξK , an estimate of the Kondo temperature. When the measurements are done at finite temperature, of course thermal effects affect the estimate of ξK : we anyway expect that if the temperature is much smaller than TK , then such effects are negligeable. Considering that TK has been estimated of order of tens of nK 28 , and that TK may be increased by increasing vF , which may be up to hundreds nK, and by increasing J ′ /J, we therefore expect that with temperatures smaller than the bandwidth one can safely extract ξK . One should anyway find a compromise since by increasing J ′ /J the Kondo length decreases (and the Kondo effect itself disappears). A systematic study of thermal effects on the estimate of ξK is certainly an important subject of future work. V. CONCLUSIONS In this paper we have studied the measurement of the Kondo screening length in systems of ultracold atoms in deep optical lattices. Our motivation relies primarily on the fact that the detection of the Kondo screening length from experimentally measurable quantities in general appears to be quite a challenging task. For this reason, we proposed to perform the measurement in cold atom setups, whose parameters can be, in principle, tuned in a controllable way to desired values. Specifically, after reviewing the mapping between the BH model at half-filling with inhomogenous hopping amplitudes onto a spin chain Hamiltonian with Kondo-like magnetic impurities, we have proposed to extract the Kondo length from a suitable quantity obtained by integrating the real space density-density correlation functions. The corresponding estimates we recover for the Kondo length are eventually found to assume values definitely within the reach of present experiments (∼ tens of lattice sites for typical values of the system parameters). We showed that the Kondo length does not significantly depend on nearest-neighbor interaction V , and it mainly depends on the impurity link J ′ . Concerning the Kondo length, a comment is in order for quantum-optics oriented readers: in a typical measurement of the Kondo effect at a magnetic impurity in a conducting metallic host, one has access to the Kondo temperature TK , by just looking at the scale at which the resistance (or the conductance, in experiments in quantum dots) bends upwards, on lowering T . The very existence of the screening length ξK is just inferred from the emergence of TK and from the applicability of one-parameter scaling to the Kondo regime, which yields ξK = vF /kB TK . However the latter relation stems from the validity of the RG approach. Thus, ultimately probing directly ξK in solid-state samples would correspond to verifying the scaling in the Kondo limit, which is what makes it hard to actually perform the measurement. At variance, as we comment for solid-state oriented readers, in the ultracold gases systems we investigate here, one can certainly study dynamics (e.g., tilting the system) but a stationary flow of atoms cannot be (so far) established, so that the measure of TK may be an hard task to achieve. Rather surprisingly, as our results highlight, it is the Kondo length which can be more easily directly detected in ultracold gases and our corresponding estimates (order of tens of lattice sites) appear to be rather encouraging in this direction. Several interesting issues deserve in our opinion further work: as first, it would be desirable to compare the perturbative results we obtain in this paper with numerical, nonperturbative findings in the Bose-Hubbard chain, to determine the corresponding correction to the value of ℓ∗ . It would be also important to understand the corrections to the inferred value of ξK coming from finite temperature effects, that should be anyway negligeable for T (much) smaller that TK . Even more importantly, we mostly assumed that it is possible to alter the hopping parameters in a finite region without affecting the others. This led us to infer, for instance, the existence of the ensuing even-odd effect 14 – however, having two lasers with σ ≪ d is a condition that may be straightforwardly implementable. In this case, one has to deal with generic space-dependent hopping amplitudes tj;j+1 . It would therefore be of interest to address, very likely within a fully numerical approach, the fate of the even-odd effect in the presence of a small modulation in space of the outer hopping terms. In particular, a theoretically interesting issue would be the competition between an extended nonlocal central region and the occurrence of magnetic and/or nonmagnetic impurities in the chain. Another point to be addressed is that an on-site nonuniform potential may in principle be present (event though its effect may be reduced by hard wall confining potentials) and an interesting task is to determine the interplay between the Kondo length and the length scale of such an additional potential. In conclusion, we believe that our results show that the possible realization of the setup proposed in this paper could pave the way to the study of magnetic impurities and, in perspective, to the experimental implementation of ultracold realizations of Kondo lattices and detection of the Kondo length, providing, at the same time, a chance for studying several interesting many-body problems in a controllable way. We thank L. Dell’Anna, A. Nersesyan, A. Papa, and D. Rossini for valuable discussions. A.T. acknolweldges support from talian Ministry of Education and Research (MIUR) Progetto Premiale 2012 ABNANOTECH - ATOM-BASED NANOTECHNOLOGY. Appendix A: Effective weak link- and Kondo-Hamiltonians for a spin-1/2 XXZ spin chain In this appendix we review the description of a region G, singled out by weakening two links in a XXZ spin chain, in terms of an effective low-energy Hamiltonian HG . In particular, we show how, depending on whether the number of sites containined within G is odd, or even, either HG coincides with the Kondo Hamiltonian HK in Eq.(5), or it describes a weak link between two ”half-chains”75,76 . In general, Kondo effect in spin-1/2 chains has been studied for an isolated magnetic impurity (the “Kondo spin”), which may either lie at the end of the chain (boundary impurity), or at its middle (embedded impurity)53,54 . In the former case, the impurity can be realized by “weakening” one link of the chain, in the latter case, instead, it can be realized by weakening two links in the body of the chain. Following the discussion in Sec.III of the main text, here we mostly focus on the latter case. In general, in a spin chain, impurities may be realized as extended objects, as well, that is, as regions containing two, or more, sites. Whether the Kondo physics is realized, or not, does actually depend on whether the level spectrum of the isolated impurity takes, or not, a degenerate ground state. A doubly degenerate ground state is certainly realized in an extended region with an odd number of sites, without explicit breaking of “spin inversion” symmetry (that is, in the absence of local “magnetic fields”). For instance, let us consider a central region realized by three sites (j = −1, 0, 1), lying between the weak links. Let the central region Hamiltonian be given by + − + − Middle z z z z H3J = −J S−1 S0 + S0 S1 + h.c. + J z S−1 S0 + S0 S1 , (A1) and let the central region be connected to the left-hand chain (which, as in the main text, we denote by the label L ), and to the right-hand chain (denoted by the label R ) with the coupling Hamiltonian ′ ′ ′ ′ − + − + z z z z HCoupling = − JL S1,L S−1 + JR S1,R S1 + h.c. + Jz,L S1,L S−1 + Jz,R S1,R S1 . (A2) Middle A simple algebraic calculation shows that the ground state of H3J is doubly degenerate and consists of the spin-1/2 doublet given by √ θ θ 1 1 | 2 = √ {sin( )[↑↑↓ + | ↓↑↑ ] + 2 cos( )| ↑↓↑ } , (A3) 2 2 2 2 and |− 1 2 2 √ θ θ 1 = √ {sin( )[↓↓↑ + | ↑↓↓ ] + 2 cos( )| ↓↑↓ } 2 2 2 with Jz cos(θ) = √ 2 2J + Jz 1 2 whose energy is given by E2 = −Jz − as √ 2J , cos(θ) = √ 2 2J + Jz , , (A4) (A5) 2 Jz + 2J 2 . Defining an effective spin-1/2 operator for the central region, SG , + SG ≡ | 1 2 2 2 1 1 z − | , SG ≡ 2 2 b|b b=±1 1 2 2 2 1 b | , 2 (A6) 15 Middle allows to rewrite H3J + HCoupling as ′ ′ ′ ′ ′ ′ + + − − − + 3J z z z VB = −{[JL sin(θ)S1,L + JR S1,R ]SG + [JL S1,L + JR S1,R ]SG } + cos(θ)[Jz,L S1,L + Jz,R S1,R ]SG . (A7) Thus, we see that we got back to the spin-1/2 spin-chain Kondo Hamiltonian, with a renormalization of the boundary couplings, according to √ ′ ′ Jz,L(R) Jz 2JL(R) J ′ ′ ′ ′ , Jz,L(R) −→ Jz,L(R) cos(θ) = √ 2 JL(R) −→ JL(R) sin(θ) = √ 2 . (A8) 2J + Jz 2J + Jz A local magnetic field h may break the ground state degeneracy, thus leading, in principle, to the breakdown of the Kondo effect. However, in analogy to what happens in a Kondo dot in the presence of an external magnetic field7,8,88 , Kondo physics should survive, at least as long as h ≪ EK , with EK (∼ kB TK ) being the typical energy scale associated to the onset of Kondo physics. At variance, when the central region is made by an even number of sites, the groundstate is not degenerate anymore. As a consequence, the central region should be regarded as a weak link between two chains. For instance, we may consider the case in which the central region is made by two sites. Using for the various parameters the same symbols we used above, performing a SW resummation, we obtain the effective weak link boundary Hamiltonian + − + − z z 2J VB = −λ⊥ SL,1 SR,1 + SR,1 SL,1 − λz SL,1 SR,1 , (A9) with ′ ′ (J )2 λ⊥ ∼ J + 2Jz (J )2 , λz ∼ z 2J . (A10) Appendix B: bosonization approach to impurities in the XXZ spin chain In this section we review the bosonization approach to the XXZ spin chain as it was originally developed in Refs.[53,54]. As a starting point, we consider a single, homogeneous spin-1/2 XXZ spin chain, with ℓ sites, obeying open boundary conditions at its endpoints, described by the model Hamiltonian HXXZ , given by ℓ−1 ℓ−1 HXXZ = −J + − + − Sj Sj+1 + Sj+1 Sj + J z z z Sj Sj+1 . (B1) j=1 j=1 The low-energy, long-wavelength dynamics of such a chain is described53 in terms of a spinless, real bosonic field Φ(x, τ ) and of its dual field Θ(x, τ ). The imaginary time action for Φ is given by SE [Φ] = g 4π ℓ β dx dτ 0 0 1 u ∂Φ ∂τ 2 +u ∂Φ ∂x 2 , (B2) where the constants g, u are given by g=  1 − ( ∆ )2 2  π π  , u = vf  2 arccos( ∆ ) 2(π − arccos( ∆ )) 2 2 , (B3) with vf = 2dJ, d being the lattice step, and ∆ = J z /J. The fields Φ and Θ are related to each other by the relations ∂Φ(x,τ ) 1 1 = u ∂Θ(x,τ ) , and ∂Θ(x,τ ) = u ∂Φ(x,τ ) . A careful bosonization procedure shows that, in addition to the free ∂x ∂x ∂x ∂x Hamiltonian in Eq.(B2), an additional Sine-Gordon, Umklapp interaction arises, given by SG HL = −GU ℓ √ dx cos[2 2Θ(x)] . (B4) 0 SG Since the scaling dimension of HL is hU = 4g, it will be always irrelevant within the window of values of g we are SG considering here, that is, 1/2 < g. In fact, HL becomes marginally irrelevant at the “Heisenberg point”, g = 1/2, 9 which deserves special attention , though we do not consider it here. Within the continuous bosonic field framework, 16 the open boundary conditions of the chain are accounted for by imposing Neumann-like boundary conditions on the field Φ(x, τ ) at both boundaries76,89–91 , that is ∂Φ(ℓ, τ ) ∂Φ(0, τ ) = =0 . ∂x ∂x (B5) Equation (B5) implies the following mode expansions for Φ(x, τ ) and Θ(x, τ )   2 iπuτ α(n) πnx − πn uτ  Φ(x, τ ) = e ℓ P +i cos q−  g ℓ n ℓ n=0    πx α(n) πnx − πn uτ  e ℓ , P+ sin Θ(x, τ ) = 2g θ +   ℓ n ℓ (B6) n=0 with the normal modes satisfying the algebra [q, P ] = i , [α(n), α(n′ )] = nδn+n′ ,0 . (B7) The bosonization procedure allows for expressing the spin operators in terms of the Φ- and Θ-fields. The result is92 + Sj −→ c(−1)j e i √ Φ(xj ,τ ) 2 + be √ 2Θ(xj ,τ ) i √ Φ(xj ,τ )+i 2 √ 1 ∂Θ(xj , τ ) z Sj −→ √ + a(−1)j sin[ 2Θ(xj , τ )] ∂x 2π . (B8) The numerical parameters a, b, c in Eq.(B8) depend only on the anisotropy parameter ∆ = Jz /J 92–96 . While their actual values is not essential to the RG analysis in Sec.III, it becomes important when computing the real-space correlation functions of the chain within the bosonization approach, in which case one may refer to the extensive literature on the subject, as we do in Sec.IV. To employ the bosonization approach to study an impurity created between the L and the R chain, we start by doubling the construction outlined above, so to separately bosonize the two chains with open boundary conditions 2J (which is appropriate in the limit of a weak interaction strength for either HK in Eq.(5), or VB in Eq.(A9)). Therefore, on introducing two pairs of conjugate bosonic fields ΦL , ΘL and ΦR , ΘR to describe the two chains, the corresponding Euclidean action is given by SE [ΦL , ΦR ] = g 4π ℓ β dx dτ 0 0 X=L,R 1 u ∂ΦX ∂τ 2 ∂ΦX ∂x +u 2 , (B9) supplemented with the boundary conditions ∂ΦL (ℓ, τ ) ∂ΦR (x, 0) ∂ΦR (ℓ, τ ) ∂ΦL (x, 0) = =0 , = =0 . ∂x ∂x ∂x ∂x (B10) Taking into account the bosonization recipe for the spin-1/2 operators, Eqs.(B8), one obtains that, in the case in which G contains an even number of sites (and is, therefore, described by the prototypical impurity Hamiltonian 2J VB ), the effective weak link impurity between the two chains is described by the Euclidean action B SG = −λ⊥ β 0 dτ {e i √ [ΦL (τ )−ΦR (τ )] 2 +e i − √2 [ΦL (τ )−ΦR (τ )] }− λz 2π 2 β dτ 0 ∂ΘL (τ ) ∂ΘR (τ ) ∂x ∂x , (B11) with ΦL,R (τ ) ≡ ΦL,R (0, τ ), and ΘL,R (τ ) ≡ ΘL,R (0, τ ). Similarly, in the case in which G contains an odd number of sites, in bosonic coordinates, the prototypical Kondo Hamiltonian HK yields to the Euclidean action given by B SG = − β 0 ′ dτ {[JL e i √ ΦL (τ ) 2 ′ + JR e i √ ΦR (τ ) 2 1 − ]SG + h.c.} + √ 2π β dτ 0 ′ Jz,L ′ ∂ΘL (τ ) ∂ΘR (τ ) z SG + Jz,R ∂x ∂x . (B12) Equation (B12) provides the starting point to perform the RG analysis for the Kondo impurity of Section III. To illustrate in detail the application of the RG approach to link impurities in spin chains, in the following part of this appendix we employ it to study the weak link boundary action in Eq.(B11). Following the standard RG recipe, to 17 describe how the relative weight of the impurity interaction depends on the reference cutoff scale of the system, we have to recover the corresponding RG scaling equations for the running coupling strengths associated to λz and to λ⊥ . This is readily done by resorting to the Abelian bosonization approach to spin chains applied to the boundary action in Eq.(B11)53 . From Eq.(B11) one readily recovers the scaling dimensions of the various terms from standard Luttinger liquid techniques, once one has assumed the mode expansions in Eqs.(B6) for the fields ΦL (x, τ ), ΘL (x, τ ), 1 as well as ΦR (x, τ ), ΘR (x, τ )53,54 . Specifically, one finds that the term ∝ λ⊥ has scaling dimension h⊥ = g , while the term ∝ λz has scaling dimension h = 2. As we use the chain length ℓ as scaling parameter of the system, to keep in touch with the standard RG approach, we define the dimensionless running coupling strengths L⊥ (ℓ) = ℓ ℓ0 1 1− g λ⊥ J −1 λz and L (ℓ) = ℓℓ0 J , with ℓ0 being a reference length scale (see below for the discussion on the estimate of ℓ0 ). To leading order in the coupling strengths, we obtain the perturbative RG equations for the running parameters given by77 dL⊥ (ℓ) d ln ℓ ℓ0 dL (ℓ) d ln ℓ ℓ0 = 1− 1 L⊥ (ℓ) g = −L (ℓ) . (B13) Equations (B13) encode the main result concerning the dynamics of a weak link in an otherwise uniform XXZ chain53,76,79 . Leaving aside the trivial case g = 1, corresponding to effectively noninteracting JW fermions, which do not induce any universal (i.e., independent of the bare values of the system parameters) flow towards a conformal fixed point, we see that the behavior of the running strengths on increasing ℓ is drastically different, according to 2J whether g < 1 (∆ > 0), or g > 1 (∆ < 0). In the former case, both h⊥ and h are > 1, which implies that VB is an irrelevant perturbation to the disconnected fixed point. The impurity interaction strengths flow to zero in the low-energy, long-wavelength limit, that is, under RG trajectories, the system flows back towards the fixed point corresponding to two disconnected chains. At variance, when g > 1, L⊥ (ℓ) grows along the RG trajectories and the system flows towards a ”strongly coupled” fixed point, which corresponds to the healed chain, in which the weak link has been healed within an effectively uniform chain obtained by merging the two side chains with each other77 . The healing takes place at a scale ℓ ∼ ℓHeal , with75,76 ℓHeal ∼ ℓ0 1 L(ℓ0 ) g g−1 . (B14) As we see from Eq.(B14), defining ℓHeal requires introducing a nonuniversal, reference length scale ℓ0 . ℓ0 is (the plasmon velocity times) the reciprocal of the high-energy cutoff D0 of our system. To estimate D0 , we may simply require that we cutoff all the processes at energies at which the approximations we employed in Appendix B to get the effective boundary Hamiltonians break down. This means that D0 must be of the order of the energy difference δE between the groundstate(s) and the first excited state of the central region Hamiltonian. From the discussion of Appendix B, we see that δE ∼ J, which, since we normalized all the running couplings to J, implies ℓ0 ∼ d, d being the lattice step of the microscopic lattice Hamiltonian describing our spin system. To conclude, it is important to stress that, though an RG invariant length scale ℓHeal emerges already at a weak link between two chains with ∆ < 0, there is no screening cloud associated to this specific problem. Indeed, in the case of a weak link impurity, the healing of the chain is merely a consequence of repeated scattering off the Friedel oscillations due to backscattering at the weak link97–99 , which conspire to fully heal the impurity at a scale ℓHeal . At variance, when there is an active spin-1/2 impurity, the density oscillations are no longer simply determined by the scattering by Friedel oscillations, but there is also the emergence of the Kondo screening cloud induced in the system79 . Appendix C: Bosonization results for the correlation functions between spin operators at finite imaginary time In this appendix we provide the generalization of the equal-time spin-spin correlation functions on an open chain, derived in Ref.[92], to the case in which the spin operators are computed at different imaginary times τ, τ ′ . As discussed in the main text, such a generalization is a necessary step in order to compute the contributions to the spin B correlations due to the impurity interaction in SG . The starting point is provided by the finite-τ bosonic operators over a homogeneous, finite-size chain of length ℓ, which we provide in Eqs.(B6) of the main text. Inserting those 18 formulas for Φ(x, τ ) and Θ(x, τ ) in the bosonic formulas in Eqs.(B8) and computing the imaginary-time ordered − + z z correlation functions G+,− (x, x′ ; τ |ℓ) = Tτ Sx (τ )Sx′ (0) and Gz,z (x, x′ ; τ |ℓ) = Tτ Sx (τ )Sx′ (0) , one obtains: G+− (x, x′ ; τ |ℓ) = x−x′ 2 c (−1) +b2 πx 2ℓ sin π ℓ +bc sgn(x − x′ ) × (−1)x 1 4g πx 2ℓ sin π ℓ 2ℓ sin π 1 4g −g 1 4g πx 2ℓ sin π ℓ πx′ ℓ −g 1 4g −g πx′ ℓ 2ℓ sin π 2ℓ sin π − (−1)x 1 4g πx′ ℓ 2ℓ sin π ′ π 2ℓ sinh [uτ + i(x − x′ )] π 2ℓ π 2ℓ sinh [uτ + i(x − x′ )] π 2ℓ 1 4g πx′ ℓ 1 − 2g −2g 2ℓ π sinh [uτ + i(x − x′ )] π 2ℓ −g 2ℓ πx sin π ℓ 1 − 2g π 2ℓ sinh [uτ + i(x + x′ )] π 2ℓ π 2ℓ sinh [uτ + i(x + x′ )] π 2ℓ 1 − 2g 1 − 2g 1 − 2g +2g 2ℓ π sinh [uτ + i(x + x′ )] π 2ℓ , 1 − 2g (C1) as well as  ′ 1 − cosh πuτ cos π(x−x ) ℓ ℓ g  Gzz (x, x′ ; τ |ℓ) = − 2 ′ ′ 4ℓ 1 + cos2 π(x−x ) − 2 cos π(x−x ) cosh πuτ + sinh2 ℓ ℓ ℓ   π(x+x′ ) 1 − cosh πuτ cos ℓ ℓ  + ′ ′ 2 π(x+x ) − 2 cos π(x+x ) cosh πuτ + sinh2 πuτ 1 + cos ℓ ℓ ℓ ℓ + −g ′ 2ℓ a2 πx (−1)x−x sin 2 π ℓ sinh sinh π 2ℓ [uτ π 2ℓ [uτ −2g + i(x − x′ )] + i(x + x′ )] 2ℓ sin π sinh − sinh πx′ ℓ −g π 2ℓ [uτ π 2ℓ [uτ πuτ ℓ   × + i(x − x′ )] + i(x + x′ )] 2g −g ′ 2ℓ aig πx′ × (−1)x sin 2ℓ π ℓ π π π π coth (uτ + i(x + x′ )) − coth (uτ − i(x + x′ )) − coth (uτ + i(x − x′ )) + coth (uτ − i(x − x′ )) 2ℓ 2ℓ 2ℓ 2ℓ − −g aig 2ℓ πx × (−1)x sin 2ℓ π ℓ π π π π (uτ + i(x + x′ )) − coth (uτ − i(x + x′ )) + coth (uτ + i(x − x′ )) − coth (uτ − i(x − x′ )) coth 2ℓ 2ℓ 2ℓ 2ℓ − . 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