Henri Alloul (Orsay): NMR studies of the pressure induced Mott transition to superconductivity in the two phases of Cs3C60

Henri started by asking the audience whether they are familiar with the fact that fullerenes are superconducting and also strongly correlated systems [majority knew, by the way]

He also pointed out the major players in the field in a past: P.W Anderson and B. Batlogg

Highlight of the field: new results on the superconductivity induced by pressure, Takabayashi et
al, Science 2009, A. Ganin et al., Nature Materials, Aprl 2008

Outline of the talk:
- Introduction: A3C60 and their superconductivity
- electronic corrleations and Jahn-Teller Distotions
- Expanded magnetic moments
- Conclusion


He introduced the crystal structure (cubic and fcc - bipartite) of fullerides. Electronic structure from LDA: from molecular levels to the Bloch states [transfer integrals are weak W~ 0.5 eV] t_{1u} bands are filled by introducing dopants (alkali ions)

Exp. fact: Tc depends linearly on the lattice constant, a. It was originally interpreted as a sign that BCS formula Tc~ \hbar \omega_D exp[-1/(V N_0)] works well. Thus it was concluded at a time that it is a phonon-mediated superconductor [in 1991!]

1995: the story is not so simple - different behavior of the slopes Tc vs a [lattice constant] for Na2AC60 and A3C60 - signs of the correlations.

Furthermore, for A_nC_60 it was found that they have different (from metallic to insulating) behaviors. He gave two examples: A_4 C_60 (bct structure) and N2C60 (cubic). Both show small spin gap from NMR and large charge gap from optics - Mott insulators?!

The reason: large Coulomb interaction U, two electrons on a ball costs an energy U plus there is a Jahn-Teller effect [deformation of the molecule]. Remarkable result is that for n=2 and n=4
there is a larger energy gain per electron [these are results from the molecular structure calculations by Tosatti]. Especially there is additional U_{eff} which arises due to Jahn-Teller distortion and adds to the usual U for even n (in A_nC_60) and U-U_{eff} for odd n. It gives Mott insultor for n=2 and n=4.

Nevidomskii: why it is not simply a band inslutor: Answer: the reason is that n=2 and n=4 have a different crystal structure, it cannot be explained on the level of band insulator.

The rest of the talk was about odd value of n in A_nC_60

Special case: CsC_60 (A_1C_60): - Mott insulator. The reasoning: from a high_T_c cubic phase phase - to a polymer phase at 200K - and finally dimer phase at 77 K (all from NMR). Experimental justification comes from NMR which sees 3 different nuclei sites with different electronic surrounding. Studying the intensity of NMR you realize the proportional compositions of the phases. By doing that you find 12% of sites in a spin singlet state.

Then he moved to the A_nC_60 series with n=3: here the most intriguing perspective is a search for the Mott insulator (originally studied in 90's K_3C_60 is not a Mott insulator). The idea is to take a larger alkali ionic radius (going from Li to Cs). Success by chemists: Cs_3C_60 has been recently prepared. Here you do find the AF Mott state and overall the phase diagram as a function of pressure resembles many of those which are typical for Mott insulators: AF at small pressure, AF+SC at intermediate pressure (doping) and SC at larger doping with a dome like structure. The slight complication is that there is also a structural transition in these compounds: A15 structure for Cs_3 C_60 - you see a single Cs site with non-cubic local symmetry (MIT and SC part of the phase diagram); fcc (only Mott part) structure of Cs3C60 you find two Cs sites and the ratio 1:2. Difference allows for selective NMR experiments.

In the next few slides Henri has analyzed the magnetic dynamics and crystal structure by means of NMR in Cs3C60 in the fcc phase. Special emphasis was put on the enhanced magnetic fluctuations in the paramegnatuc phase. Upon pressure you find the transitions from fcc phase to A15 phase as well the transition from AF to SC phase. Important remark: Mott insulator to metal transition is not directly related to the crystal structure transition. (here the argument is that you do have MIT also in the A15 phase)

Superconductivity: definitely singlet superconductivity most likely s-wave (there is a Hebel-Slichter peak). But more measurements have to be taken.



Summary:
- fullerides are correlated
- original due to disorder, icosaedral symmetry of the soccer ball
- supercondctors near MIT
- static charge segregation in Cs1C60
- importance of the Jahn-Teller effect: different between odd and even n
- excellent possibility to study the multiorbital Mott transitions.


Questions: 1) blogger: is Jahn-Teller splitting is comparable to the bandwidth, Answer: there is no clear indication, calulations are done for the molecule.
2) Vojta: what is the role of possible spin frustration in the MIT Answer: not really known
Remark: coexistence region between SC and AFM is possibly an inhomogeniety effect
3) Nevidomskii: fcc phase can be a spin glass - any results for zero field. Answer: similar to the previous ones: the field just started, Remark from someone in the audience: there are data from muSR which indicates phase transition or something similar at 2K, origin is unclear.

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

Matthias Vojta: Kondo impurities in Graphene

Matthias began with a summary of Kondo physics.

Matthias turned to the quest for realistic models for magnetic impurities. eg. its not so simple to figure out the Kondo model for Fe in gold.  Not well known until very recently, because of the interplay of spin and orbital degrees of freedom. Is it spin 1/2, or two channel spin 1, three channel spin 3/2, four channel spin 2?

Alloul pointed out from the high temperature chi in the early days, it was known to be larger than S=1/2.  Matthias argued that it did not really become clear until recently - Costi et al, PRL 102, 056802 (2009). Best fit to experimental dephasing rate suggests S=3/2, three channels.

The dephasing rate grows from T^2 up to a broad plateaux, and the subtle different fits favour S=3/2 for Fe in Au.

Outline of talk:

1 Impurities in Graphene, Dirac fermions, STM expts, orbital physics of d electron impurities.

2. Review: pseudogap Kondo model.  Quantum phase transitions.  Critical field theories.

3. Pseudogap Kondo model with voltage bias. (You can tune from linear density of states to a finite density of states. Maximal electron hole asymmetry. Spectral functions.

Turned to Dirac Fermions in graphene. Two atoms per unit cell.  When you diagonalize the short-range hopping Hubbard model (U=0), you get two Dirac cones with Hamiltonian

H ~ vF (p-K).sigma_sublattice
E_k = v_F }| k - K|

Pauli matrix acts in "sublattice space".  There are two copies of this Dirac Hamiltonian. The Dirac cones are "topologically protected".  (Semenoff 1984, Haldane 1988).   The Fermi points are robust against next nearest neighbour hopping etc.

By gateing, you can tune the Fermi surface to go from a Fermi point to a Fermi surface.

Dos (E) \propto |(E-E_F)|   linear density of states.

Kondo effect in graphene: first observation

Manoharan group. STM shows the hexagonal structure, with a puckered, rippled surface. Schofield asked why there was a superstructure.  There was no obvious answer from the croud.  From dI/dV you can see the Co on the surface. Now you can see "blue dots" representing the Cobalt atoms. You can now see the dI/dV spectrum.  You see a peak on some cobalt atoms, on others you see a dip.  Can extract a width, or Kondo temperature of TK~ 15K.   These pictures corresponded to an effective gate voltage of 200mV.

The point is, there are two different locations of the Cobalt atom.  Site A corresponds to atom on top of a C atom (t-site, dip structure, pseudo-spin breaking) whereas site B (h-site) is in the middle of a hexagon (peak, pseudospin conserving).  Add a field, the structures split, proving that the peak is of magnetic origin.

What is the correct Kondo model for Co on the graphene sheet?   The symmetries are very important here. In the graphene you have band degeneracies - C3nu, C6nu - three and six fold degeneracies.

Orbital physics and spin orbit coupling crucial.  Models such as SO(4) Kondo model are possible.   Using Generalized Gradient Approximation + U, a first principles study. Spin resolved DOS for Co in center of Hexagon and above C.  Three orbitals E1 (dxz,dyz) E2 (dx2-y2,dxy), A1 (d3z^2-r^2).  h-site, spin 1/2, SOC lifts 4 fold degeneracy - SU(2) Kondo possible.  h-site, spin 1 (SOC stablizes singlet, no Kondo expected.).  t-site spin 3/2 in E1, E2, A1 that would lead to a two stage, small TK effect.
(Wehling et al, PRB 81, 115427 (2010)). DOS is strongly particle-hole asymmetric.  J~ 2eV, bandwith from t=2.8eV.

Peter Hirschfeld asked where the spin is localized.  Henri Alloul suggested that the GGA+U might not have enough correlations to locate the spin. The blogger thinks these methods are probably good enough to get the spin form factor.

So what happens for the Kondo effect in a non-magnetic host.  If the DOS vanishes at the Fermi level, there is no Kondo screening at small J_K.  (Fradkin and Withoff - though not referenced).  Two possibilities

Hard gap - first order transition upon varying J_K at T=0.

Pseudogap - continuos transition upon variation of J_K. Non-trivial finite T behavior arising from quantum critical point.  DOS ~ epsilon^r.   r >0 gives phase transition.  (d-wave, graphene r=1).

Pseudogap Kondo model - 

Two axes - Kondo axis J.  Particle-hole asymmetry V.

For small r < r* = 0.3748, get simple Fradkin-Withoff behavior. Jc ~ r.  Also an ASC, asymmetric strong coupling fixed point.  beta (j) = rj - j^2.

For r* < r < 1/2  a new fixed point appears at finite Vc and Jc.

r=1 is upper critical dimension. r=0 is lower critical dimension. Hyperscaling is obeyed for r<1.

Chubukov asks can you do an expansion in epsilon = r-1?  Matthias says yes - but to do it requires the Gaussian theory at r>1.  The answer is a level crossing between a doublet of single impurity and a singlet of a screened impurity.  Simple model with doublet of energy epsilon-0 hybridized to a singlet via an Anderson screening - it is a non-interacting pseudo-gap Anderson model.   Can now do an epsilon expansion.  Vojta Fritz PRB 70, 094502 (2004)

r<1 have a finite hybridization fixed point (Wilson Fisher fixed point).  r>1 have gaussian fixed point. Critical fixed point is maximally p-h asymmetric near r=1. Hybridization becomes irrelevant above r=1, relevant below r=1.

Pseudogap Kondo model with voltage bias Sofar, only neutral graphene.  Next, mu>0. Now the moment will ultimately be screened at low T.  But if the chemical potential is of order the TK, there will be critical physics.  Chemical potential provides a fan of NFL physics.  J=Jc, then predict TK = kappa * mu.

RG now done with chemical potential effect on flow equations.   The leading effect is that one drives the impurity to the  screened, or unscreened phase.  (mu <0 epsilon = -infinity screened; mu > 0, epsilon = + infinity).  Depends on sign of mu.

Ultimate results - TK as a function of gate voltage.  (Vojta, Fritz, Bulla EPL (2010)).

Conclusions

• Magnetic impurities in graphene. Kondo criticality possible.

• Critical theory is not of Landau Ginzburg Wilson. but intrinsically fermionic

• TK(mu) extreme asymmetry between electron and hole doping, not only near criticality resulting from structure of critical fixed point.

• Systematic measuremets of Co impurities as function of gate voltage required. 

Kenji Ishida: NMR in Pncitides Talk 4 August 2010 11:30 am

Main message: AF fluctuations with quantum critical character induce superconductivity in Fe-based systems.

Focus in talk was primarily on BaFe_2(As_1-xP_x)_2 (P-doped 122). This system is interesting and unusual because a) it shows transport properties suggestive of quantum critical behavior near optimal doping at P=0.33; b) the SC dome is created by “doping” with P, which is however isovalent to As; c) the superconducting state properties are strongly suggestive of line nodes. Here are resistivity measurements from Matsuda group quite reminiscent of cuprates showing relatively large range linear fit to resistivity at x=0.33, and deviations away from this doping.

P-doping affords the possibility for NMR to do very clean experiment, since ^31P nucleus is spin-1/2 & eliminates quadrupole contribution to signal. At first glance the symmetric position of the As or P above Fe plane might make you think it would not be sensitive to AF correlations, but Ishida showed that this is incorrect when one takes into account full structure of hyperfine interaction tensor. Below, for example, is shown the expected response in the presence of certain assumed type of order on the Fe atoms (red).

Ishida said by changing angle of field, and noting that the experiment is sensitive to the fluctuations transverse to the field direction, they can verifiy that the order and low-frequency fluctuations are indeed of the stripe type shown in 1^st panel above.

Here are Ishida’s spin-lattice relaxation rate (T_1) data for various P dopings (Nakai et al., to appear in PRL). Ishida noted the large upturn in the (T_1T)^(-1) signal in the underdoped (magnetic) materials just above optimal doping. These fluctuations appear to grow until, near optimal doping, they reach a maximum (at this point T_N

And T_c are nearly equal— Ishida said whether they cross is not directly clear from NMR, and there is no neutron work because crystals are too small).

Data were fit to Curie-Weiss term b/(T+theta) plus constant offset. Variation of Curie-Weiss theta with doping shows theta goes through zero at/near optimal doping, where effective mass of one of the orbits from dHvA also appears to diverge. Ishida: this is consistent with claimed QCP at or near optimal doping. Alloul: there is no evidence for quantum cricitcal *behavior* from NMR alone; fact that theta-->0 at x=0.33 merely indicates T_N going to zero. Chubukov: is there low-T data showing QCP or only high-T extrapolated? Ishida: only the latter.

Ishida: qualitative behavior of Co-doped system is very similar, again with theta crossing zero around optimal doping. Some differences with K-doped material could be understood by changes in band structure, DOS at Fermi level which decreases as one e-dopes. Ishida reminded us that this emphasizes one great advantage of P-doping on the As site: unlike cuprates, or Co,K-doped 122 systems, changing “dopant” P does not change the DOS and one can examine changes in spin dynamics independently. Evidence: relative x independence of Knight shift.

Fernandes asked if there was evidence for re-entrant behavior of the superconductivity near the magnetic transition as in Co-doped systems. Ishida said he was not aware of any.

While the focus of this talk was not on SC state, Ishida pointed out that near optimal doping T-dependence was close to T^3 with low-T linear term (consistent with line nodes and some dirt). Benfatto pointed out that the overdoped data do not show the expected decrease of (T_1T)^(-1) below Tc at all, and Ishida said this was not understood.

Comparison with La-1111 (As NMR, powdered samples). Ishida pointed out that 1/T_1T changes by 2 orders of magnitude with F doping over a range x=0.05-0.15 where Tc changes by less than a factor of 2. Data show that while a similar, if weaker, increase in spin fluctuations occurs just above T_N in the magnetic phase, when one reaches the superconducting F doping concentrations (recall T_N has a first order drop to zero with doping in this system) these fluctuations are gone and (T_1T)^-1 is flat with T. His group is trying hard to understand these differences with 122 systems. Ishida claimed other 1111 systems, including those with higher Tc, show similar behavior when one subtracts rare earth magnetism (Ce,Sm, …).

Hirschfeld asked “devil’s advocate question” : if higher Tc superconducting family 1111 shows weaker spin fluctuations above the transition at optimal doping, is this not evidence that spin fluctuations are NOT responsible for superconductivity, or at least that another mechanism may be in play? Ishida agreed this was an important open question that they are investigating.

Lara BENFATTO (La Sapienza) -- Superconducting properties of pnictides within a low-energy multiband approach

Lara started by comparing the iron-based high Tc superconductors to the older cuprates. There are similarities like the close relation between superconductivity and magnetism and a potential role of spin fluctuations in the superconductivity mechanism. However, there are crucial differences as well, and she stressed the multiband nature of superconductivity in pnictides, in contrast to the effective single band description of cuprates. Furthermore, cuprates are near half-filling of this single band while iron-pnictides involve nearly filled or nearly empty bands and there is significant particle-hole asymmetry.

There are three main aspects of superconductivity in iron-pnictides: they are multiband superconductors; the interband pairing interaction related to nesting among electron and hole pockets on the Fermi surface (FS) is the dominant mechanism of superconductivity; and there is strong particle-hole asymmetry in the problem. Lara proceeded to explain that these different features are studied within a general multiband Eliashberg-style formalism, in which a fermion self-energy is computed in presence of a coupling to a bosonic mode at energy \omega_0. Among other quantities, this allows one to compute a quasiparticle renormalization factor, Z(\omega \to 0), and extract the interaction-renormalized effective mass m^* from m^*= Zm_b \sim (1+\lambda)m_b, where m_b is the band mass and \lambda is the dimensionless coupling to the boson mediating superconductivity. The results can be compared to the available experimental information, including the ARPES and specific heat measurements from which the effective mass and other dynamical information can be extracted. She pointed out that the general model is still too and perhaps unnecessarily complicated and the further simplifications included ignoring electron-phonon coupling and intraband repulsion which are too weak and irrelevant under RG, respectively, and retaining only the repulsive interband interaction.

Even this simplified version of the model is still a challenge. Two main questions, important for understanding of experiments, were addressed: How many bands are necessary to reproduce experimental data? Does one really need the full Eliashberg formalism or are the retardation effects relatively unimportant and the BCS theory will suffice? At the end of the talk, it turned out that the answers to these questions are “four” and “yes.” Thus, the minimal model needed all four bands and the full Eliashberg calculation was necessary to reproduce different superconducting gap amplitudes observed in experiments like ARPES. The anisotropic orbital character of the interband interactions also had to be included.

Some important results of the work were presented (the full account can be found in L. Benfatto et al, arXiv:0909.3735). One example is that BCS model is not sufficient since it produces the wrong hierarchy of gap sizes. Second, the theory gives a good agreement with m^* extracted from experiments, and, in particular, \lambda \sim 1, which indicates a reasonably strong degree of coupling.

Next, it turns out there are three different gaps whose magnitude can be fitted rather well to the experimental observations. Interestingly, while these magnitudes cannot easily arise within a BCS theory, once we adopt their T = 0 values from the full Eliashberg approach, a reasonable account of quasiparticle thermodynamics does in fact follows from the two-band BCS treatment. Finally, the kinks in the ARPES dispersion are also reproduced with a similar \lambda \sim 1. Lara also mentioned an alternative approach (arXiv:1001.1074) which gives somewhat larger \lambda.

The rest of the talk dealt with the dHvA experiments and the issue of renormalization of the size of electron and hole pockets. Such renormalization arises naturally within a multiband Eliashberg approach. Lara made an insightful observation that interband interactions lead to shrinking of FS pockets and that this is just what is observed, when the experimental dHvA FS cross-sections are compared to those derived from LDA (band-structure) calculations. She also discussed the experimental results using the optical sum rule to estimate effective masses of carriers. This is a more complex exercise in multiband systems and many in the audience asked questions and made comments concerning just how should optical sum rule be interpreted (Alloul), pointing the fact that not all pockets change in the same way (Hirschfeld), debating whether or not Luttinger theorem holds (it does, Chubukov), what is the shift in the chemical potential, and other assorted issues (Nevidomskyy, Burch), etc.

Blogged by Zlatko Tesanovic.

Ilya EREMIN (Ruhr-Uni. Bochum): Selection of magnetic order and magnetic excitations in the metallic SDW state of ferropnictides

OUTLINE:
(I) Introduction
(II) Peculiarities of the SDW state in the itinerant scenario
(III) Spin excitations

I. Introduction
Cu-oxides vs. Fe-pnictides - similarities in the (T-doping) phase diagram. Proximity to AFM phase is important in both cases. However, unlike CuO2, all regions of FeAs phase diagram are metallic.

Two somewhat contradictory observations:

• Metallic transport in Ba-122 (N. Kurita, PRB'09): below T_Neel, the resistivity is metallic [Remark from the audience: for samples prepared in Sn-flux, the resistivity is known to go up, not down, below T_Neel.]
• well-defined Fe-moments, with spin waves inside the ordered phase.
  

I will present the itinerant scenario, leaving aside the alternative scenario (local moments).


II. Peculiarities of the SDW state: itinerant scenario

Nesting properties:
\eps_holes(k) = - \eps_el(k+Q),
with Q=(pi,pi) an AFM ordering wave-vector (in a 2-Fe unit cell convention).
Nesting provides a boost for SDW, with susceptibility \chi(Q,w) diverging as log(w/Ef).

Below T_Neel: \sqrt{2} x \sqrt{2} order with Q=(pi,pi).

Unfolding the bands to an extended BZ corresponding to 1-Fe unit cell. In this unfolded picture, there are 2 nesting wave-vectors: Q1=(0,pi) and Q2=(pi,0). Two vector order parameters with Delta_1 and Delta_2. Two sublattice order parameters: (Delta_1 + Delta_2) and (Delta_1 - Delta_2). How do we know which of the two orders is selected?

Simplest model: 1 hole and 2 electron pockets [in the unfolded 1-Fe BZ]
I. Eremin and A. Chubukov, PRB 81, 024511 (2010)

Only 1 equation, from which \Delta_1 and \Delta_2 SDW components cannot be both determined: only |\Delta_1|^2 + |\Delta_2|^2 is fixed. The absolute values and the angle can vary!
Result: very degenerate ground state: O(6) degeneracy, 5 Goldstone modes
(more degenerate than the purely magnetic J1-J2 model).

Nesting is not perfect. Pockets are elliptic, introducing a positive term (mx-my)^2, resulting in the term ~C|\Delta_1|^2 |\Delta_2|^2. This term comes from charge-charge interactions (and won't be there in the purely magnetic J1-J2 model).
As a result of this term, the ground state is either \Delta_1 = 0 or \Delta_2 = 0.
To summarize:
- no need to appeal to qu. fluctuations
- charge fluctuations are crucial to determine the order parameter.

NOTE:
One of the electronic pockets decouples from the problem, so that even for arbitrarily strong interaction, there will always remain an ungapped electron pocket at the Fermi level, resulting in a metallic state even inside the SDW phase!

Inclusion of the 4th (hole) pocket: the picture remains basically unchanged, with the same type of the magnetic order.
[Proviso: However, for U>Ucr, both \Delta_1 and \Delta_2 may become non-zero, resulting in a stripe order that will be distorted.]

Comparison with ARPES:
one electron pocket does indeed survive at the Fermi level (Dresden group ARPES: V. Zabolotnyy et al, Nature'2009)

III. AFM excitations
J. Knolle, I.Eremin, A.Chubukov, R. Moessner, PRB 81, 140506 (2010).

Now that we determined the ground state, let us consider excitations.
Compute transverse spin susceptibility -> spin waves.

If nesting is complete -> continuum is gapped, Goldstone modes only near (0,pi) and (pi,0). No Landau damping as long as the energy is below the size of the SDW gap.

If ellipticity is included -> finite continuum, with present Landau damping. Well-defined spin waves around Q1=(pi,0), but only diffuse paramagnons around (0,pi).
Results in anisotropic spin wave velocity along x- and y- directions, even though the underlying interactions are isotropic.

CONCLUSIONS:

• ellipticity of the electron pockets and the e-e interaction at (pi,pi) stabilize the metallic AFM state with (0,pi) or (pi,0) order
• well-defined spin excitations near Q, with anisotropic spin velocities in x- and y-direction
  
• AFM state is always metallic with more FS crossings than in the normal state. In the folded BZ, (0,0) and (pi,pi) are not equivalent.
  
• no correspondence to the J1-J2 model in strong coupling limit

QUESTIONS:

Q: A. Chubukov: What if there is coupling to the lattice?
A: Depending on the scenario, either SDW is of electronic origin, and structural transition follows. OR, the structural transition leads to spin nematic phase ( see Fernandez, arXiv:0911.3084).

Q: M. Vojta: Can Dirac points develop in the pnictides? Can you comment on this?
A: In Vishvanath's picture, the Dirac point can develop due to q-dependent interaction. However in our picture, accidental Dirac points can appear.
Z. Tesanovic's comment: unlike Vishvanath et al, there is also a possibility of a topologically protected Dirac point (only arises in pure xz,yz (2 band) models).

Q. Yu Lu: 1) what is the size of the moment?
2) above the magnetic transition, how do you explain the linear-T behaviour of
susceptibility (R. Klingeler et al, )?
A: 1) the moment of order 0.6 bohr-magneton, and depends on how much of the Fermi surface is gapped.
2) non-analytic, linea-T, corrections to susceptibility can arise due to proximity to SDW. See Korshunov, I. Eremin PRL 102 (2009).

Q: Rafael Fernandez: Do you need orbital physics to explain the spin excitation dispersion, or is it enough to only consider spin physics?
A: no comment. Perhaps.

Zlatko Tesanovic: What is the theory of the Fe-pnictides?

Outline of the talk:

1) Fe-pnictides: semimetals turned superconductors
2) pairing states
3) minimal model
4) multiband magnetism and superconductivity

Zlatko started his talk by providing some background information on the Fe-pnictides: The Fe-pnictides were discovered by the group of H. Hosono in 2008 with a Tc of 26K. Currently, the highest Tc in the Fe-pnictides is approximately 57K. The pnictides, whose name comes from Greek meaning "choking, suffocating", are made of elements from group V of the periodic table. The 1111-materials exhibit a larger Tc than the 122-materials, though the later are easier to fabricate. The Fe-pnictides are layered, quasi-2D materials.

Zlatko then discussed similarities and differences between the cuprate superconductors and the Fe-pnictides:
Similarities:
1) both class of materials have d-electrons (Cu vs. Fe)
2) both materials are layered and quasi-2D
3) the phase diagrams of both materials exhibit superconductivity and antiferromagnetism in
close proximity

Differences:
Fe2+ has an electronic 3d6 configuration, while Cu2+ has a 3d9 configuration. Therefore, the electronic structure of the CuO2 layers is described by a single hole in a filled 3d orbital, and a one band model might be sufficient to describe the physics of these materials. In contrast, in FeAs one has a large and even number of electrons in the 3d orbital, implying that a multiband model is necessary for their description.

Zlatko then reminded us that in the cuprate superconductors, the Mott insulating state of the undoped parent compounds evolves into a superconducting state upon doping. In the undoped cuprate compounds, the effective Coulomb interaction is much larger than the electronic hopping, resulting in a Mott insulator and a Neel AFM. The greatest challenge in the cuprate superconductors is a microscopic understanding of the pseudo-gap region in the underdoped compounds.

Zlatko then presented a schematic phase diagram (ZT, Nature 4, 408 (2008)) to explain how a correlated superconductors can evolve into a Mott insulator. At weak interactions, the superconducting state is destroyed by thermal fluctuations, while at large interactions, it is destroyed by quantum fluctuations. There are many different theoretical proposals to describe this transition.

Question by P. Coleman: is there a convergence of theories?
Answer: proposed theories seem to converge towards gauge theories for the description of the underdoped cuprates.

Zlatko then presented a schematic band structure of the Fe-pnictides to show that in these materials, the bands are either almost full or empty leading to a semi-metal and implying that these materials are far away from the Mott limit of one electron/hole per site. As a result, all regions of the FeAs phase diagram are (bad) metals, in contrast to the cuprate superconductors. Zlatko therefore argued that the appropriate starting point for the description of the Fe-pnictides is an itinerant picture. This is also supported by ARPES and dHvA experiments that seem to observe coherent propagating quasi-particles.

Zltako argued that a minimal model for the FeAs layers should be an effective 2D model that includes all 5 d-orbitals. While the As bands are below the Fermi level, they contribute to the minimal model, and one therefore should start with a minimal model that include all 5 Fe orbitals and 3 As orbitals. This gives rise to a much more complicated band structure than in the cuprate superconductors, with the hybridization between the orbitals as well as the renormalization of the band parameters being crucial. Zlatko argued that there is no Hund's rule coupling in the Fe-pnictides.

Question: is this assumption not in conflict with LDA calculations.
Answer by Zlatko: I want to give a happy talk, and therefore will not comment on these calculations.

Zlatko next discussed nesting properties and valley-density-wave (VDW) states in the pnictides. He argued that valley-density-wave states arise due to nesting enhancement of electron-hole excitations, where the latter give rise to moderate interaction strengths. Here, a VDW state refers to an itinerant multiband CDW, SDW or orbital-order-wave (ODW) state. Zlatko then presented a "bare-bone" model for the Hamiltonian that includes the electronic band structure as well as intra- and inter-band interactions. Zlatko showed that by using a particle-hole transformation for one of the electronic bands, one arrives at the negative-U Hubbard model.

This model can be solved on the mean-field level by using the Hartree-Fock approximation, where self-consistency is crucial in obtaining the correct BCS ground state of the model. Zlatko then described how the Cooper instability is obtained by summing up an infinite series of ladder diagrams. Zlatko mentioned that the relevant effective interactions should be obtained from an RG analysis. By reversing the particle-hole transformation, one then arrives at an SDW, CDW or ODW state in the Fe-pnictides.

Zlatko then turned to the question of real superconductivity in the Fe-pnictides. He pointed out that there is strong mixing of odd and even d-orbitals around the Fermi surface, and that the effective interactions at the Fermi surface need to be divided into flavor conserving and mixing vertices. This gives rise to interband superconductivity. Of great importance in the emergence of the superconducting state is that the effective interaction in the particle-particle channel is sufficiently large. An RG analysis has shown that due to the proximity of an SDW state, the pairing interaction is enhanced and thus stabilizes the superconducting state.

Blogged by Dirk Morr.