Double dot Kondo


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Kondo effect and Numerical Renormalization-Group


    Remarkable achievements in the development of nano-scale devices in the last decade provide unique opportunities for research in nanoelectronic systems. Recent technological developments in both sample materials and experimental techniques have made it possible to reach control over the dynamics of individual electrons in nanodevices such as single-electron transistors (SETs).

Particular attention has been devoted to the effect of strong electron-electron interactions (also referred to as "strong correlation effects") in the transport properties of nanodevices. Electrons can now be confined and manipulated in a controllable way in semiconductor quantum dots, scanning tunneling microscopy set-ups and molecular junctions, allowing for a myriad of single-particle and many-body effects to be probed in detail.
    Prominent among these is the Kondo effect, arising from the screening of a local magnetic moment (such as a single electron spin) by the surrounding electrons in a continuum, forming a many-body bound state. The essential physics of the Kondo effect in equilibrium is captured by quantum impurity models describing a magnetic impurity coupled to Fermi reservoirs, such as the Kondo model or, more generally, the Anderson model. The formulation of the latter includes charge fluctuations, thus allowing for the description of equilibrium transport properties through the impurity. One of the most accurate schemes for obtaining the low-energy excitation spectra in these and other quantum impurity models is given by Kenneth Wilson's Numerical Renormalization Group (NRG) method.

    The non-perturbative nature of this method allows the calculation of physical properties (such as spectral functions and magnetization curves) at arbitrarily low temperatures and excitation energies, precisely in the region where the Kondo effect is fully developed. In this sense, the NRG method constitutes a very powerful tool to explore different effects in transport properties of strongly correlated systems. In particular, there has been a large interest on the use of NRG for equilibrium transport calculations in quantum dot systems.
My intention is to expand the scope of such applications, by considering different scenarios where the method could be applied.


Kondo effect in QDs and NRG for beginners:

I suggest reading Chapters 2, 3 and 4 of Michael Sindel's Ph.D. Thesis (Munich, 2004)

A nice review on the Kondo effect in quantum dots:

Kowenhoven, Glazman Review - Physics World

Essential NRG references:

  1. Wilson, RMP 47 773 (1975) - Wilson's original paper on the method, with a detailed and rigorous discussion of its application to the Kondo model.

  2. Krishna-Murthy, Wilkins & Wilson PRB 21 1003 (1980) - The basic paper on the NRG calculations for the Anderson Hamiltonian.

  3. Bulla, Costi, Pruschke RMP 80 395 (2008) - A recent and comprehensive review on the method.

Other relevant references on NRG:

  1. Wanda Oliveira & Luiz Oliveira PRB 49 11986 (1994) - Describes the useful "z-trick" in the discretization procedure. Very useful for the computation of thermodynamical properties and spectral densities.

  2. Chen & Jayaprakash PRB 52 14 436 (1995) - A very nice reference on non-flat conduction bands. Explains the "trick" to get the Lanczos algotithm to converge. Without it, the algorithm is quite unstable.

  3. Gonzales-Buxton & Ingersent PRB 57 14 254 (1998) - A comprehensive paper by Kevin Ingersent's group. Very nice description of the Lanczos algorithm for the discretization of a arbitraty conduction-band

  4. Hofstetter's DM-NRG method PRL 85 1508 (2000) - Explains the basics of a method for a more accurate calculation of the Spectral Function for systems with broken spin degeneracy.

  5. Other relevant papers coming soon!