Dirk K. Morr: Defects, Density of States and Differential Conductors in Heavy Fermion Materials

Dirk's work is done with Jeremy Figgins.

Dirk begins with the puzzle of the resistance minimum in metals (de Haas et al., Physica 1, 1115 [1934]) and in "pure" gold sees that minimum tuned with magnetic impurities. We get the history...Kondo, Wilson, large-N...: Then Dirk introduces the spectroscopic signature in dI/dV (V. Madhavan et al. Science 280, 567 [1998]) where the Fano lineshape is identified in tunnelling as a signature of the Kondo resonance developing. So much for the single impurity.
Now we start thinking about the Kondo lattice of magnetic atoms with a conduction fluid. Dirk shows us the bad actors of quantum criticality (Au doped CeCu6; and magnetic field tuned YbRh2Si2). Question: about the interplay of doping and quantum criticality and how close you can actually get to the QCP with a discrete parameter in CeCu6-xAu? Stefan Wirth comments about how it can be combined with pressure to get to the QCP. The signatures of the non-Fermi liquid physics that emerge in these QCPs: Resistance T^n where n \neq 2 and T log T specific heat. (Note for the students from Andrey: technically n<=1 is really required for a non-Fermi liquid). Dirk now is moving on to the notion that impurities can be a useful probe of unconventional/puzzling systems. Cuprates provide a case-in-point: eg impurities in the superconducting state and the induced resonant states and quasiparticle interference experiments can probe the d-wave superconducting state. What about measurements of STM tunnelling on the heavy fermion systems. There are (at least) three groups working on this: Seamus Davis', Ali Yazdani's and the Dresden group with Stefan Wirth whose data are being discussed at this meeting. Ken asks about the symmetry breaking implicit in the introduction of a surface: Dirk "parks" the question. So, Dirk's key questions are

• Can defects provide insight into the heavy fermion systems and

• Do defects discriminate between electronic and magnetic correlations

The idea is that Friedel oscillations may provide that insight.
There are two possible defects: removing a magnetic atom (Kondo hole), or replacing a magnetic atom by a non-magnetic one. How should this be described? There has been previous work (Schlottmann, Freytag, Vojta and others). Dirk's approach begins with a Hamiltonian: Its the usual Kondo-Heisenberg Hamiltonian characterized by J (Kondo), I (Heisenberg) and an additional U0 (potential term on the impurity sites). His approach is SU(N), representing the spins as fermions which will be treated in mean-field which can have a local character. Rather than explain the actual calculations we get the physical content of the mean fields:

1. There is a hybridizing field (which is the Kondo physics) mixing s electrons and the spin fermions
   
2. The magnetic bond variable (this is the Heisenberg physics which gives the fermions representing the spin become itinerent)
3. A local constrain to force nf=1 forbidding valence fluctuations

Piers asks a nasty question: how can you have hopping of the f fermions when nf=1. Dirk gives the usual answer that it is a constraint on average. Piers says that is not acceptable. Piers: You should say we have a gauge model description whereby the "f" fermion is really a composite object which when included the gauge fluctuation and the hybridization means that it is representing not simply the moment but the Kondo-ized gauge field composite.

[An interesting blackboard discussion emerges after the talk (PC and DKM): the point that PC is making is that if you start from a Hamiltonian then the f in the Vc^{\dagger}f hybridization term is not really the original fermion which represents the spin rigorously localized, but is a composite object which involves a gauge fixed choice (Anderson and Appelbaum). This is significant because tunnelling a physical electron into that f object is, in reality, tunnelling into a complex many-body state containing the physics of "co-tunnelling".]

What does an STM experiment measure in HF materials? Dirk's work (Figgins and Morr, PRL 104, 187202 [2010]) (and other people...) stresses that there are two possible paths that the tunnelling electron could take: into the magnetic ion or into the conduction band and these two routes lead to the asymmetry and the Fano shape of dI/dV and the ratio of the two processes tf/tc radically changes the shape. tf/tc=0 has one shape while tf/tc=0.08 is enough to completely reflect the symmetry of the dI/dV curve. Similarly inverting the band structure also inverts the shape of the line, so bandstructure matter. Matthias points out that there is another possible asymmetry coming from moving away from the particle-hole symmetric Anderson model which everyone uses. Andriy asks how the calculation is done: since with tf/tc=0 does this not mean that the Fano shape should be Lorentzian? Answer is that there is asymmetry in the original bandstructure (unhybridized) - but then Dirk sketches it on the board and it looks pretty symmetric at low energy scales. Blogger is not convinced here but is too busy typing to ask the obvious question...[But now I can discuss it. The answer Dirk should (in my opinion) have given is that tf=0 does not mean there is no Kondo effect - it just that you tunnel only into a c-electron. The bybridizing mean field is still there so there is a Kondo resonance which is asymmetric with respect to the chemical potential].]
Dirk then tries to explain this with pictures of how the shape changes with the various parameters in the theory for the case of Cobalt on Gold (111). Andrey asks about the small scales that appear in the plots (meV) when the bare scales are (1eV) and how they come about? Dirk says it is TK. Markus asks for further clarification of the definition of the tf and how the process happens (like how the hole left behind affects things). Piers says you cannot do SU(N) for S=3/2 cobalt since S=3/2 is a symmetric representation of SU(2) and SU(N) large N does antisymmetric representations. Then into a discussion of spin-orbit as to whether J=4 saves you, but PC says orbital physics is quenched...an impasse. Then Dirk compares his theory with some of the experiments.
Finally we move to some numerical studies of Kondo impurities in small cluster numerical studies (arXiv:1001.3875) to look for the perturbations in the electronic correlations. He looks at how the hybridizaton and bond variables get distorted in his numerics. The role of the conduction Fermi surface and the gives significant changes (with large anisotropies) to the shape of the oscillations in real space. Dirk now needs to relate the oscillations and distortions seen in his mean fields relates to things you measure in experiment. Under pressure from the chair to wind up we get a fast tour through adding non- magnetic impurities. And in closing an array of Kondo holes is looked at which drives a first order phase transition as the holes start to interfer. Open questions: out of time.
Henri: a comment - these ideas have been explored in the cuprates by Henri for 18 years - not with STM but with NMR. Why not use NMR to do this in heavy fermions? Response: you need a magnetic impurity, Henri no you don't think of Zn in the cuprates.
What is the physical reason for tf << tc? Correlations suppress tunnelling into the f-electron state. You can also have co-tunnelling where in effect a spin hops (as studied by Piers Coleman and colleagues).
Peter Hirschfeld: Do you really see such a large factor of 10 in the anisotropy in the real physical systems? DKM: It may be a consequence of an idealized band-structures.
Stefan: In reality replacing a magnetic ion with a non-magnetic one - it matters greatly which atom you mess around with and you get very subtle changes in the vicinity of quantum criticality.

Determined Blogger: Andy Schofield