Building complex Kondo impurities by manipulating entangled spin chains

The creation of molecule-like structures in which magnetic atoms interact controllably is full of potential for the study of complex or strongly correlated systems. Here, we create spin chains in which a strongly correlated Kondo state emerges from magnetic coupling of transition-metal atoms. We build chains up to ten atoms in length by placing Fe and Mn atoms on a Cu 2 N surface with a scanning tunneling microscope. The atoms couple antiferromagnetically via super-exchange interaction through the nitrogen atom network of the surface. The emergent Kondo resonance is spatially distributed along the chain. Its strength can be controlled by mixing atoms of different transition metal elements and manipulating their spatial distribution. We show that the Kondo screening of the full chain by the electrons of the non-magnetic substrate depends on the inter-atomic entanglement of the spins in the chain, demonstrating the prerequisites to build and probe spatially extended strongly correlated nanostructures.

Conduction electrons scattering off a magnetic impurity can screen the localized magnetic moment by forming a non-magnetic many-particle singlet state. This is known as the Kondo effect 1 . On a metallic surface, Kondo screening of a magnetic moment manifests itself as a prominent zero-bias resonance in scanning tunneling microscope (STM) differential conductance measurements 2 . Ample experimental evidence exists for Kondo resonances of single spins on metals, either atomic or molecular 2-6 . However, an entirely different world is revealed when several spins interact on metallic hosts [7][8][9][10][11] . An extensive body of theoretical work explores the rich range of new states of matter that can be created from the competition of Kondo screening and inter-spin interactions [12][13][14][15][16][17] . Such quantum many-body phenomena exhibit extraordinary complexity but offer great potential if the emergent quantum states can be controlled 18,19 . The Kondo effect in spin-coupled chains, has been proposed as a means of transmitting quantum coherent information in solid-state environments 20 . This concept requires ways to extend 21,22 and manipulate 3,23,24 the Kondo effect on the atomic level. To date, in most scenarios considered experimentally, each spin is individually Kondo-screened and the interaction among spins drives the pre-existing Kondo effect into a new phase of matter 8,22,25 . These Kondo chains and lattices are keystones in understanding heavy-fermion compounds 26,27 , entanglement in condensed matter 28 or in transmitting information quantum coherently through chains of spins 20,29,30 .
However, the intriguing possibility to create emergent Kondo states in nanostructure systems built from atoms that, individually, are not Kondo screened is unexplored.
Using the atom manipulation capabilities of the STM, we assemble one-dimensional chains of atoms composed of Fe and Mn atoms on a monolayer film of copper nitride, Cu 2 N, grown on Cu (100). We use a low-temperature STM to assemble chains of up to ten atoms in length, Fig. 1a,c.
We find that chains with one Fe and an odd number of Mn atoms exhibit a collective Kondo state. The Kondo state in the composite system appears only if the spin-spin interactions within the chain are engineered to entangle all atomic spins with each other and form a doubly degenerate ground state that enables electrons from the host metal to flip all spins in the chain by a single electron scattering event. Atomically precise construction allows exact determination of the chain composition and tuning of coupling strengths. This in turn determines the degree of inter-atomic entanglement which influences the probability of spin-flip scattering and the resulting efficiency of the Kondo screening. Thus the control we have over atomic entanglement in the atom chain allows us to create and tune the emergence of a many-body electronic state.
The magnetic moments of Fe and Mn atoms on the Cu binding site of Cu 2 N are well-described as spins with magnitude 2 (Fe) and 5/2 (Mn) 7,31 . Due to their easy axis magnetic anisotropy, neither Fe nor Mn forms a Kondo state individually at the temperature of our experiment, T = 0.5 K 4,32 . Spectra of the differential conductance, dI/dV(V), recorded with the tip positioned over each atom show spin excitations at finite energy but no Kondo resonance at zero bias (Fig. 1b).
Surprisingly, the two-atom chain, one Mn and one Fe atom, shows a drastically altered conductance spectrum, Fig. 1b. Spin excitations appear at ±16 mV and ±24 mV bias and a clear resonance is visible at zero bias. The spectra show spin excitations compatible only with an antiferromagnetic coupling of the atomic magnetic moments as in Mn dimers 7 or Fe dimers 33 . The zero-bias resonance on MnFe shows the behaviour expected for a Kondo resonance: the peak can be fit by a Frota function with a width of    = 1.47±0.05 meV (see Supplementary   Fig. S4 for fit details) corresponding to a Kondo temperature of T K =17 ± 1 K, (Fig. 1b). It quickly broadens with increasing temperature and splits in magnetic field confirming its magnetic many-body nature ( Supplementary Fig. S1).
To test whether this zero-bias resonance is a robust effect related to the chain's spin ground state rather than properties of the constituent atoms we assembled chains of the type Mn x Fe with x = 1, …, 10 ( Fig. 2a,b). We find that the presence of Kondo  permitting the formation of a collective Kondo resonance (Fig. 2e).

The relation between entanglement and strength of the Kondo resonance is substantiated by
investigating MnFe chains where the ratio between the inter-atomic exchange coupling strength, J, and the magnetic anisotropy energy, D, can be adjusted by varying the separation between the atoms (Fig. 3a). We record the amplitude of the Kondo resonance in the dI/dV spectra of each chain (Fig. 3c)  Conductance spectra acquired along the Mn x Fe chains, dI/dV(V,d), show that the Kondo resonance is delocalized (Fig. 4a). However, the entanglement among the atoms in the chain does not lead to a structureless magnetic object as could be expected for a macrospin. Contrarily, the resonance increases along the chain and is strongest at the Mn atom furthest from the Fe atom. This is surprising because the Fe atom is required for the emergence of the Kondo resonance and emphasizes that, even for the longest chain (Mn 9 Fe), it must be caused by collective screening of entangled spin states of the entire chain.
Remarkably, the maximum amplitude of the Kondo resonance increases approximately linearly with chain length. For Mn 9 Fe it reaches up to 80% of the amplitude of the spin-excitationinduced conductance steps (Fig. 4c). This behaviour is consistent with an increased efficiency of electron-spin scattering between the two degenerate ground states by substrate electrons. The increased number of scattering sites provided by longer chains increases the interaction between the spin chain and the electron bath. In addition, the Fe atom's magnetic anisotropy becomes effectively diluted with increasing number of Mn atoms allowing for a more efficient Kondo screening. Therefore length, geometry and composition of the chain are tunable parameters that permit us to vary the degree of entanglement of the atoms in the chain and as a consequence, the All atom chains were created by vertical atom manipulation 7 . Prior to manipulation Fe and Mn atoms were identified using their characteristic conductance spectra 31 . Differential conductance spectra, dI/dV(V), record the differential conductance of the tunnel junction as a function of sample bias detected by Lock-In detection of the tunnelling current caused by adding a modulation voltage of 72 V rms amplitude at 691 Hz to the bias.
Hamiltonian to describe the spin states of the Mn x Fe. Conductance spectra are calculated by considering electron-spin scattering with tunnelling electrons (Supplementary Figure S2). The spin states of the MnFe chains represented by their density matrices, , are completely determined by the ratio J/|D| (Fig. 2d). We quantify the degree of entanglement for the dimers (Fig.  2fg) by entanglement entropy 38 , I(). For this analysis we add an infinitesimal magnetic field to the spin Hamiltonian and set temperature to zero to ensure that is a pure state. I() is obtained by calculating the von Neumann mutual information of the Fe and Mn spin subsystems, in the basis that minimizes I() (see Supplementary Fig. S5). We find that this basis coincides with the coordinate system determined by the magnetic anisotropy axis of the Fe atom. S is the von Neumann entropy and  Fe and  Mn are the reduced density matrices of the Fe and Mn spin obtained by partial trace over . Since the spins considered here are larger than ½ we normalised I() to the entanglement entropy of the maximally entangled state, so that I() ranges from 0 for chains where Fe and Mn are not correlated or only classically cor-