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Motivated by recent STM experiments, we explore the magnetic field induced Kondo effect that takes place at symmetry protected level crossings in finite Co adatom chains. We argue that the effective two-level system realized at a level crossing acts as an extended impurity coupled to the conduction electrons of the substrate by a distribution of Kondo couplings at the sites of the chain. Using auxiliary-field quantum Monte Carlo simulations, which quantitatively reproduce the field dependence of the zero-bias signal, we show that a proper Kondo resonance is present at the sites where the effective Kondo coupling dominates. Our modeling and numerical simulations provide a theoretical basis for the interpretation of the STM spectrum in terms of level crossings of the Co adatom chains. PACS numbers: 72.15.Qm,75.20.Hr,75.10.Pq,75.30.Hx The Kondo effect is one of the most extensively studied and adequately addressed many-body process occurring due to the screening of a local moment by a conduction electron cloud 1-3 . On the experimental side, recent advances in scanning tunneling microscopy (STM) open greater opportunities to realise and investigate the Kondo effect in various Kondo nanostructures 4-11 . For instance, the recent STM experiments on finite atomic spin-chain realizations of Co adatoms in the presence of an external magnetic field have revealed an interesting interplay between the Kondo problem and the physics of quantum spin chains in a field 11 . Co adatoms on a Cu 2 N/Cu(100) surface carry a spin-3/2 with a strong uniaxial hard-axis anisotropy(D) 10,12 , and applying an external magnetic field perpendicular to the surface, the Co adatom chain effectively behaves like a spin-1/2 XXZ chain in transverse field. The magnetic field induced level crossings of finite XXZ and SU(2) chains are similar so that the experiments of Ref. [11] can be discussed in the context of the SU(2) invariant version of the model:Here, t is the hopping parameter of the conduction electrons, J k the antiferromagnetic Kondo coupling between a Co adatom and the conduction electrons, J h the Heisenberg antiferromagnetic coupling, h z an external magnetic field in the z direction, L the length of the Heisenberg chain,Ŝ l spin-1/2 operators andŜ c l = 1 2 σ,σ ĉ † l,σ σ σ,σ ĉ l,σ denotes the spin of conduction electrons. Throughout the calculation we set t, µ B = 1 and g = 2.When the Kondo coupling is switched off (J k = 0), the chain undergoes a series of level crossings that lead to steps in the magnetization curve. In the geometry used in the experiment of Ref.[11], J h and the Kondo energy k = k B T k are both of the order of 0.2 meV, with two important consequences: There is a competition between the Heisenberg coupling and the Kondo effect, and one can reach the saturation field of the isolated chain.The main result of the STM experiments of Ref.[11] is to demonstrate that the differential conductance exhibits a series of anomalies as a function of the field, and that these anomalies coincide with the fields at which the isol...

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We study a class of one-dimensional antiferromagnetic quantum spin-1/2 models using DMRG. The exchange interaction in these models decreases linearly with the separation between the spins, Jij = R − |i − j| for |i − j| < R, where R is a positive integer 2. For |i − j| R, the interaction is zero. It is known that all the odd-R models have the same exact dimer ground state as the Majumdar-Ghosh (MG) model. In fact, R = 3 is the MG model. However, for an even R, the exact ground state is not known in general, except for R = 2 (the integrable nearest-neighbor Heisenberg chain) and the asymptotic limit of R in which the MG dimer state emerges as the exact ground state. Therefore, we numerically study the ground-state properties of the finite even-R = 2 models, particularly for R = 4, 6 and 8. We find that, unlike R = 2, the higher even-R models are spin-gapped, and exhibit robust dimer order of the MG type in the ground state. The spin-spin correlations decay rapidly to zero, albeit showing weak periodic revivals.

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