**Final-day (Feb 1 ^{st} 2019) workshop on NRG**

To close the School, the last day of activities (02/01/2019, from 9:00 to 13:10, at room 5RA) will be in the format of a workshop with lectures presented by invited speakers who are specialists in Kondo physics. They will join Prof. Žitko in delivering lectures describing specific aspects of their research in strongly correlated systems:

- 9:00 to 9:20: Opening remarks
- 9:20 to 10:10: Prof. George Martins, Physics Institute, UFU, Uberlândia, MG
- 10:10 to 10:45: Prof. Ginetom Diniz, Physics Institute, UFJ, Jataí, GO
- 10:45 to 11:15: Coffee break
- 11:15 to 12:00: Prof. Luis Gregório Dias, Physics Institute, USP, São Paulo, SP
- 12:00 to 13:00: Prof. Rok Žitko, Jozef Stefan Institute, Ljubljana, Slovenia
- 13:00 to 13:10: Closing remarks

**Titles and abstracts:**

**The Kondo cloud: a(nother) numerical investigation**

*George B. Martins*

In this work, using the finite-U Slave Bosons Mean-Field Approximation to solve the single impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ξ_{K} through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ξ_{K}^{L}), while the second relies on the analysis of spin correlations (ξ_{K}^{Σ}). Our calculations show that there is exact quantitative agreement, in the way ξ_{K} depends on U=Γ, between ξ_{K}^{Σ} and the results obtained through the heuristic expression ξ_{K} / v_{F}=T_{K}, while there is very close quantitative agreement between ξ_{K}^{Σ} and ξ_{K}^{L}. The use of the slave bosons technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the Density Matrix Renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous Numerical Renormalization Group calculations [Phys. Rev. B 84, 115120 (2011)].

**Kondo Effect in Graphene Nanoribbons with Magnetic Impurities**

*Ginetom S. Diniz*

In this work we have performed a detailed analysis of the Kondo effect of a magnetic impurity in graphene nanoribbons with zigzag edge termination. We have considered an adatom coupled to the graphene nanoribbon via a hybridization amplitude in two different configurations: hollow or top site. In addition, the adatom is also weakly coupled to a metallic STM tip by a hybridization function that provides a Kondo screening of its magnetic moment. We have described the entire system by the well-known Anderson-like Hamiltonian whose low-temperature physics is accessed by employing the numerical renormalization group approach, which allows us to obtain the thermodynamic properties used to compute the Kondo temperature of the system. In our numerical calculation, two screening regimes were observed: local singlet and Kondo singlet, which strongly depends on the impurity adatom location and on the coupling strength of the carbon sites of the graphene nanoribbon with the adatom impurity.

We acknowledge financial support received from FAPEG, FAPEMIG, FAPERJ, CAPES and CNPq.

**Application of Wilson’s NRG to Majorana-Kondo systems**

*Luis G. Dias*

The realization of Majorana bound states (MBSs) in condensed matter systems has been a very active research topic, with possible applications to topological quantum computation. A promising platform for the realization of MBSs are semiconductor quantum wires with induced superconductivity. In practice, the measurement of MBSs in these systems is usually done by looking for zero-bias signals in transport experiments across the wire and some interesting and fundamental questions arise. For instance, how to differentiate the zero-bias signal from other non-topological effects (such as the Kondo effect) and whether or not MBSs would survive in the presence of interactions. We advocate that one way to test these questions is to couple the wire to interacting quantum dots.

In this talk, I will show how Wilson’s numerical renormalization group (NRG) can effectively help us to address these questions by modeling a interacting quantum dot (QD) coupled to a topological quantum wire as an Anderson-like quantum impurity model. I will discuss the subtle symmetries of the model and how we can use Wilson’s NRG to investigate its low-temperature transport properties. We find evidence of a strong interplay between Majorana and Kondo physics. Our results show a strong renormalization of the Kondo temperature by the QD-Majorana interaction, which can drive the system from a Kondo-dominated to a Majorana-dominated ground state. By quenching the Kondo effect with Zeeman fields, we found a persistent 0.5e^{2}/h conductance coming from the Majorana mode leaking into the QD. These properties can be used for the experimental distinction of Majorana and Kondo modes in topological quantum wires.

**Revival of the physics of sub-gap states in superconductors: from Shiba to Majorana states**

*Rok Žitko*

Recent advances in the fabrication of nanometer scale hybrid semiconductor-superconductor devices as well as in the scanning tunneling spectroscopy of adsorbate covered surfaces of superconductors have made possible very detailed experimental studies of the old problem of paramagnetic impurities in a superconducting host using local probes with very high energy and spatial resolution. At the same time, improved theoretical tools have been devised to reliably and accurately calculate the excitation spectra of the corresponding quantum impurity problems with gapped continuum electrons. These developments have enabled very stringent tests between experiment and theory. The results demonstrate the importance to describe the magnetic impurities as quantum objects with non-trivial internal dynamics due to the coupling to their environment.

I will discuss the physics of bound states induced by the exchange interaction between the magnetic impurities and the Bogoliubov quasiparticles in superconductors. They are observable as spectroscopically sharp resonances in the tunneling spectra, located well inside the superconducting gap for sufficiently strong exchange coupling. They provide a new way to study the effects of strong correlations. Particularly interesting behavior is found for cases of high-spin impurities in the presence of magnetic anisotropy, and when nearby impurities are coupled through exchange interaction, as in double-quantum-dot in nanowires. I will also briefly comment on the non-Fermi-liquid features found in the two-channel Kondo model with superconducting channels.