Natalia Perkins (Wisconsin) - Spin-orbital physics in frustrated vanadium oxides

It's about ten past five on Tuesday evening, Natasha kindly moved her talk to this late slot to allow some of us to go snorkling by Miramare castle this afternoon - thank you Natasha!!


Natahsa starts by saying that she has a very general title, but we will quickly go into some specific aspects, after a general introduction.

Motivations to study vanadates and other TMO's:
--understanding of reality!! many compunds, many experiments, novel phases and possible applications
--Intellectual adventure: TMO's are complex systems with number of different degrees of freedom (charge, spin, orbitals...)

Let's look at more detail about the orbital degrees of freedom:
--electrons surrounding atoms fall into different orbitals
--much degeneracy, but partially split by crystal field

For example, the octahedral ligand field in LaMnO3: spins align, t2g levels filled, eg levels have only one electron in them, giving a degeneracy of 2, or in other words, a pseudo-spin of 1/2.

In MnV2O4, MgTi2O4, find pseudo-spin 1, etc...

Question from Andriy - can you clarify what psuedo-spin means?
Answer: Psuedo-spin is simply counting the degeneracy of the orbital manifold, i.e. the number of (low-energy) ways of filling the orbitals with the required number of electrons.

As it will be important, now lets speak a bit more about the single ion properties of V^3+ (d^2, S=1)
--the 3d level splits into 2g (very high in energy, forget about them), and t2g manifold - effective orbital angular momentum L'=1. There will also be some spin-orbit coupling \lambda (S.L) splitting this t2g manifold.

So for a description of physics of Vanadates, need a multi-orbital description
.. here we see a generic Hamiltonian of multi-orbital Hubbard model ..

Then make a second-order perturbation theory in t (the materials are insulators) - giving two degrees of freedom per site: spin and pseudo-spin (orbital) - known as the Kugel-Khomskii model.

This model is still very complicated due to the presence of the two different degrees of freedom. However, if these happen at sufficiently different energy scales, then simplifications are possible.

So let's get some inspiration from the materials: normally, orbital ordering happens first - the pseudo-spin variables gain an expectation value - mapping to an anisotropic spin model is possible, which will then order at a lower temperature order.

For example: magnetic ordering on pyrochlore, ZnV2O4
--See two transitions: First structural (Ts=52K, cubic->tetragonal), then magnetic (AFM at Tn=44K)
--This gives us AFM 1D order, consistent with a set of spin chains
--This is a temperature-induced change in dimensionality
[see S.H. Lee et al. PRL 93 (2004)]

By 1D magnetic order, means you see different regimes - a temperature regime where 1D correlations are clearly seen with no magnetic order, then at a lower temperature, get full magnetic order.

Question (Paul C.) : So we need to remove this orbital degeneracy to get to the transition leading to AFM?
Answer: the structural change is very important to lead to the low-T structure of the magnetic Hamiltonian.
P: So the transitions could in principle happen at the same place?
N: Nothing prevents it, but I don't know of any examples when this happens.

Aside: there is big development in this field by group of Bela Lake, with a related Mg compound where they have very good (neutron) measurments of 1D fluctuations.


However, we can't always get rid of orbital degrees of freedom. For example, spin-order may come first (or even spin disorder - correlated paramagnet so know correlation function), leading to a pure orbital model.


Super-exchange in vanadates (AV2O4)
--Octahedra are edge sharing - VOV angles are about 90 degrees so V-O-V superexchange is weak. Small V-V distance means direct overlap of t2g orbitals. Structural distortions life orbital degeneracies... A static Potts-like orbital interaction happens if only dd\sigma overlap is taken into account. In this case, the spin exchange is AF if the bond is occupied by the `good' orbitals, FM for bonds with one `good' and one `bad' orbital, and no exchange otherwise.

Now that the generic introduction is over, we have the plan for:

Rest of talk:
--Spin-Orbital model for MnV2O4
--Spin-Orbital model for CaV2O4
--Open questions

Collaboration with:
Gia-Wei Chern (Wisconsin)
Zhihao Hao (John Hopkins)
Gia Japaridze (Tbilisi)
Shura Nersesyan (Tbilisi/ICTP)

Spin-Orbital model for MnV2O4

Two sub-lattices: A diamond lattice interweaving with B pyrochlore lattice
On A: Mn^2+, on B V^3+

Experiments tell us: Two phase transitions in all insulating AV2O4. Specifically for MnV2O4, find
--magnetic (para-ferrimangetic) at Tn=56K
--tetragonal+ collinear-noncollinear (orthogonal in xy plane) magnetic at Ts=53K
[see Garlea et al, PRL 100, 066404 (2008) ]

Suggestions that magnetic order is stabalized by some form of orbital order (generally believed to be A type - but we will see that this is not correct).

Also find in this material a trigonal distortion and modulation of electronic density, which may be very important. Will also find at low-T a tetragonal distortion - but dealing with both will be non-fun, so lets assume that the trigonal distortion is the largest one.

The trigonal distortion gives a splitting \Delta between the a1g and the eg1/eg2 doublet, which after careful examination will lead to an appropriate super-exchange Hamiltonian, with various projectors whose form comes directly from the structure of the trigonal distortion.

At temperatures higher than the structural transitional, can reduce to an effective orbital Hamiltonian, known as the 120 degrees orbital model (see e.g. Khomskii and Mostovoy; Nussinov and Fradkin; etc...) Note that this is related to the eg-orbital model in manganites with a cubic-lattice.

This model on the cubic lattice is highly frustrated (due to various rotations), however in our case this frustration is not so bad... we'll see why in a moment.

Classical Orbital Order:
Coupling constant is positive - AFM order in pseudospins - but frustrated lattice so can't satisfy them all.
Find a discrete degeneracy of the ground state (total degeneracy=6, all of which are collinear), and can write staggered orbital order parameters, which are the difference in orbital occupation on bonds of the same type.

Let's look at one of these possible GS. Then find that all three order parameters are the same (2/3), which is a different state than that previously proposed (which had 2 out of 3 orbitals occupied, rather than all occupied by 2/3).

Of course, orbital ordering will give some modulation of the electron density - basically from the staggering of the trigonal axes. We then see some nice pictures of this.

What are the effects of orbital-ordering on the magnetic state?
Note that the magnetic frustration is relieved by:
--interaction with magnetic A-sites
--an anisotropic spin exchange in the presence of orbital ordering
--a single-ion anisotropy in the presence of orbital ordering due to spin orbit coupling

Competition between orbital exchange and spin-orbit coupling: both are of the same order or magnitude -> need to go to second order in \lambda -> gives staggered spin anisotropy, which is consistent with the proposed magnetic state found from experiment for MnV2O4.

No time for CaV2O4 - ... blogger sees slides flashing before his eyes...
... but one point: here, the one-dimensionality comes from the beginning, due to the structure - zig-zag spin-chains. Find a toy model with symmetry U(1)xZ2xZ2 (first U(1)xZ2 is from a reduction of SU(2) spin (with S=1), second Z2 is from orbital Ising-like chain)... shame there was no time for this as it sounds very interesting. Another few words: Can de-couple spin/orbitals by mean-field -- see G.-W Chern, N.Perkins, G.Japarize, arXiv:1007.3472. Find two phase transitions, both Ising-like - but in another limit there is only one Gaussian transition - but this is work in progress.

Open Questions:
--Reduction of dimensionality due to orbital anisotropy...
--Orbital excitations: both experiment and theory...
--Entanglement of orbital and spin degrees...
--New models...


Questions from the audience

Ordering in Haldane-like phase? Well, true magnetic ordering is always 3D.

A discussion has now broken out between people in the front row, relating to mutli-ferroics in these materials, which the blogger is not really able to hear clearly.

No more questions? I think we are all tired. Good evening all.

Sang-Wook Cheong (Rutgers): Multiferroic vortices

Sang starts by introducing the materials with multiple directional orders which are most known in the community by now: (Anti)ferromagnetics and (anti)ferroelectrics: TbMnO3, DyMnO3, TbMn2O5 and DyMn2O5.

The most recent example is Eu(Y)MnO3: magnetization is along c and ferroelectric moment P is along a direction, both orders compete, and the balance between these two phases can be tuned by the external time-dependent magnetic (or electric) field. Piers: the scale of the application of the magnetic field is hours, why so? Reply: it depends on the scale of coupling between ferroelectric and magnetic orders and is determined by the behavior of the domain walls which can be different for the magnetic and ferroelectric orders. Usually the timescale is shorter for the magnetic order: Canfield: for Delta M/H shown on the viewgraph: How large it is in percents to the susceptibility values. Reply: few percents.

Canfield is also amused by the millimeter paper which Cheong uses as a background for his sample shown on the photograph, Cheong refers to the Bell Lab resources back in 80-ies [general appreciation of the good old times].

Next Sang shows the picture from the conference (seemingly our conference as Piers is also there) with a banana which he argues has something to do with multiferroics. The analogy is P vs E hysteresis curve which to Cheong looks like a banana. He further remarks that multiferroics are usually good insulators and they NEED to be a good insulators otherwise they are bad ferroelectrics. On the next slide Cheong shows the switchable photodiode device based on the multiferroic BiFeO3 (published in Science) with a potential to be used in the applied science. The most interesting thing is that you find a finite current at zero voltage which, as Cheong explains, is due to ferroelectric component.

Next we move to the most recent trends in the multiferroics:
1) use non-zero d-electrons
2) employ interaction among d-electroncs - drives antiferromagnetic order
3) look for collective phenomena - should give colossal magnetoelectric effect
4) and what everyone likes - Mott-type charge gap

This directs us to the Doped Mott Ferroelectrics (blogger again suspects that this new keyword in multiferroics goes back to Cheong), new aspect of the field of multiferroics.

Next he moves to the main topic of the talk - multiferroic vortices.

the system in question is h-YMnO3 with a hexagonal (h) crystal structure, Tc~900K T_N=90K - ferroelectric insulator. There is additional transition at 1325 K (is that right!?) which is slightly below the melting point which again raises the comment by Canfield whether this transition is well separated from the melting point. Reply: well separated

Mn3+ ions are in S=2 state and the magnetic structure above and below T_N looks fairly complicated. An observation by second harmonic generation (SHG) is that various domain walls are formed with purely ferrolectric (FEL ~P) component, mixed FEL & AFM (~PI)and AFM (~I) [blogger thinks that I refers to the magnetic moment and/or interaction]. Origin of the ferroelectricity in YMnO3 (Van Aken et al., Nature 2004) is a trimerization of Mn ions. This is supported by the fact that there are also two separate transitions: first structural and then ferroelectric one. The consequence of this is a very subtle effect on the magnetic properties as there are 6 possible domain walls expected. On the next slide Sang provides its visualization by means of the high-resolution TEM, AFM and conductive (c)-AFM techniques. Combining these techniques Cheong argues that you may identify 6 domain wall structures. The visualization of these domains on the next slide looks fairly complicated. Cheong claims that this is easy to follow, blogger doubts that but continues with a fear of messing up things.

Basically Cheong suggests to look on the domain boundary as a topological defect. Canfield points out that if this were topological defects you could heat the sample and then cool it down again and by doing this you would see the annihilation and creation of vortices at different places on the surface. Cheong agrees but says they do not have a picture yet to show.

Then Cheong shows the I(V) characteristics of this system. Negative voltage is more conducting than a positive one (range is from +10 to -10 eV). The explanation is given in terms of a non-ohmic conductance where various effects can contribute: transport along the domain boundaries, Schottky barrier, and non-linear effects.

Next slides are concerned with the control of the domain boundaries by an external electric field. Showing maps of c-AFM at various fields there is a poling effect at Ec ~ 45kV/cm, however above this field you still find vortices, up to 70 kV/cm [blogger guesses that it is in favor of 6 domain boundaries to be the topological defect]. A word of caution is that these pictures are obtained for different samples thus nothing can be said on whether the vortex is moving in a field or not.

Again the SHG picture comes and Canfield raises an issue whether all the curves on the cartoon agree with the actual data shown in b/w maps. Somehow he is convinced by the reply while the blogger simply has nothing to say as some very experimental words has been said.

Then Cheong speculates that the topological defect has to have a phase of 2pi for the magnetism, and 6pi for the ferroelectric order. Next slide says something about the interpretation power of the TEM pictures where Cheong sees (a) vortex-antivortex pairs and (b) face of Marilyn Monro.

Some speculations are made on whether or not these vortex-ativortex pairs are good or bad for the technical applications. In liquid crystals they are not, in multiferroics these not yet clear as the applications are not yet well-spread.



Questions:

1) blogger: origin of asymmetry between I(V) curves in YMnO3? Answer: it is basically the non-linear effect and many features may affect the non-ohmic behavior
2) Piers: the model of the vortex you offer: is it equivalent to the 6-vortex model, then you may expect BKT transition in thin films ? Answer: Yes, indeed it might be very interesting
3) Canfield pointed out that the vortices can be pinned by the dislocations

Meigan Aronson (Stony Brook and Brookhaven NL) Geometric Frustration and Quantum Criticality in Heavy Fermion Compounds

Meigan began with a short introduction about the formation of quantum critical points.  She emphasized how in  wide variety of compounds, heavy fermions, organics, arsenides, quantum critical points play an organizing role. We've come to know that if we are such a QCP, that we will see unusual critical phenomena - C/T ~ log(T), chi ~ T^(1 + lambda), chi'' ~ f(E/T).   Yet if we are away from the QCP - Fermi liquid, C/T~ gamma_0, chi~ chi_0, rho~ A T^2, m*/m ~ 1/T_F.   This reveals that one has an underlying electronic state with Landau quasiparticles.

In some, not all, she said, as we come towards the QCP, we often see that the effective mass m*/m diverges - a phenomenon that is associated with the break-down of the Fermi liquid at the QCP.

One picture that has emerged, there is a transition from a small to a large Fermi surface.   g the tuning parmeter. When g< g_c, one has a small Fermi surface, but when g > g_c, a large Fermi surface, in which the localized spin is absorbed into the large Fermi surface.

Meigan discussed how it now appears, that the localization delocalization transition may sometimes separate from the magnetic ordering QCP, leaving open the possibility of an intermediate phase that she called a "spin liquid".

The sequence at T=0 would then be

   metallic AFM ---------metallic "spin liquid" ----large fermi surface Fermi liquid.

This then motivates a search for new materials in which geometric frustration is important.

Meigan reviewed the Kondo effect.  T>>TK, localized moments decoupled from conduction electrons - f-electrons excluded from the small Fermi surface.  T<

1/T ---> chi_0
spectral function T< TK develops a Kondo resonance. eg inverse photoemission shows formation of Kondo resonance in CeNi2

Once one goes to the Kondo lattice, there is, she said, the potential for magnetic order, with a TN ~ J'~ J^2. This requires that TN> TK.  But if TN < JK, get a large FS Fermi liquid.  From this (Doniach) perspective, we expect a QCP separating the small FS AFM from the large FS Fermi liquid.

At this point, Meigan introduced the concept of the Shastry Sutherland Kondo lattice.  This model has the Hamiltonian

H =  J * sum over nearest neigbor bonds + J' * sum over diagonal bonds.

Intrinsic QCP at J'/J ~ 0.7, and for J' bigger than this point, a "spin liquid" develops. (Dimer fluid).

Q: (Schofield) why did you call it a liquid, rather than a solid?
A: What I meant is that you have local dimers, with no magnetic order.

Meigan then introduced the combined global phase diagram that combines the two ideas, frustration and Kondo effect.

Q: (Raphael Fernandes)  you showed a phase diagram where TK goes to zero - yet here it has a finite value at the quantum critical point.
A: The scale that you are refering to, is really T*, the delocalization scale - its relation to the Kondo temperature is complicated.  Here the quantity we are using is the single ion Kondo temperature.

Q: (Andrey Chubukov) can the Fermi liquid just go into a Fermi liquid with no major change in FS.
A: It could go either way. Localizing transition or spin density wave. (Blogger liberally interpreted Meigan's answer).

So that inbetween the small FS spin liquid and the large FS Fermi liquid, there should be a phase transition.

Some discussion with Paul Canfield about what it means to have a small or large Fermi surface.

Blogger: Piers Coleman

Blogger mentions work of Sachdev Senthil and Vojta - if you have an odd no of spins per unit cell - will donate a non-integral amount of electrons per spin, and there is definitively a difference between small and large FS.

Megian intrdouces the Shastry Sutherland mateiral SrCu2(BO3)2. This is a spin gap material with gap appearing that is seen in thermodynamics.  The hallmark of  a SS lattice is the presence of lattices in the

magnetization. Once the triplet becomes the ground-state, the possibility of AFM emerges - as you increase the field, a larger and larger no of the dimers will enter the AFM. You don't just see a smooth increase in magnetization - dimer triplets crystallize into SDW with long range AF order - these are incompressible leading to plateaux in the magnetization M(H).  The SDW co-exists with gaples, intinerant "superfluid' = triplets and singlets, and this leads to a rounding of the plateaux.




R2T2X Compounds

So Meigan's system takes us to the next level - the inclusion of conduction electrons.  She introduced the R2T2X compounds. These 221 compounds contain a frustrated rare earth layer separated by a transition metal TX layer.  These systems are interesting because there is the possibilty of Kondo physics, for a wide variety of compounds.   She introduced a huge list of Ce 221 compounds. Many are AFM ordered. Others are valence fluctuating.

She presented pictures of three single crystal compounds, Ce2Ge2Mg, Yb2Pt2Pb and Ce2Pt2Pb.
The first is magnetic, the second close to a QCP (but magnetic) and the third is a heavy fermion metal?
Ce221Mg  and Yb2Pt2Pb 221Pb are highly anisotropic, with moment lying in the plane of the  SS lattice.

She showed Schottky anomalies in the specific heat.  The Ce221 Mg and Yb221 show magnetic ordering peaks, but in the Ce221Pb, there is a smooth peak suggesting no magnetic order. It looks as if the Yb221 has an ordering state in which some percentage of the Yb states are in dimers.  Ce221 Mg - S= 0.49R at Tc suggesting that 93% of triplets - but in Yb 221, 77% triplets (0.4R) while Ce 221Pb 0.29R - 56% triplets.

Yb moments R log(2) = 0.68R
Yb triplets    1/2 R log(3) = 0.55 R

Canfield - I assume that further data will solidify the argument that the J'/J axis plays an important role.
Aronson - I agree with you - up to this point, I have not definitively shown that it is important.

Nevidomskyy - can you clarify the entropy.
Aronson - integration of C/T from T=0

Chubukov - what is the peak?
Aronson - would be the triplet-singlet excitation in an insulating SS magnet.

Magnetization of Ce2Ge2Mg - you cansee that lowering into the magnetic sate, one sees steps in the magnetization curve. Peaks in dB/dM that occur at M/Msaturation = 1/3, 4/9, 8/11 - exactly as in SrCu2(BO3)2. Perhaps most striking is the field scale - 14 tesla enough to get to 85% of saturation - but in Sr system, 33T gives 1/4 saturation magnetization, suggesting a much large interaction in the Sr case.

Also note that the steps are sharp in Ce2Ge2Mg, compared with the Sr221 system.

Takagi - question - the Strontium is Heisenberg - but this one is Ising like.
Aronson - Sr is also Ising out of plane - but ours are Ising like in the plane.
Takagi - but then you can't use a simple singlet-triplet picture.

Similar DB/dM for Yb2Pt2Pb- much more small index states, much more like Sr221, but broader and more hysteretic - much broader - so closer to melting in this case than in Ce221Mg.

Ce221Pb SDW completely absent - itinerant dimer triplets only.

Yb221 field-temperature phase diagram.  Actually, done using Hall sensor magnetometer - but you can see that the magnetization is highly structured.  The sharp peaks are the magnetization plateau. The highest temperature one is the analog of the Neel temperature - suppressed to zero at B~ 1T.Saw a lot of hysteresis beneath the neel dome - corresponding to strongly 1st order transitions. The upper transition does not show hysteresis, suggesting that it is second order.

TN/TN(B=0) vs B-BN/TN lines lie on top of one-another. Can place Ce2Pt2Pnb  and the other materials onto a single phase diagram.

Now turn to Ce2Pt2Pb.  Powelaw like divergence of magnetic susceptibility.
Heat capacity of Ce221Pb, has C/T ~ T^1.4.

C/T = large gamma_0 * T + C_o T^alpha

Ended talk presenting many materials on phase diagram.

Electronic structure very modified in Yb221 and Ce 221 - large resisitivity and collapsed anisotropy.   Still missing, evidence for Fermi liquid and spin liquid phases. Large to small Fermi surface transitions/crossovers.

Q how did you grow the crystals
A magnesium flux.

Q how close are these systems to perfect SS systems?
A one reason that they are not idealized - there are two SS bonds on one plane.  May also be important interplane interactions.  We haven't considered them yet?
Q what about dipolar interactions
A interesting question - typically, we don't think that the dipolar interactions are so important in HF compounds because they are metallic - screens out. But there might be short range interactions and these could contain dipole components?

Paul Canfield - possibility that there is a lot of disorder cerium. Do you single crystal x-ray refinements on the lanthanum or cerium.
Aronson - we don't have on La, but in Ce we do, and don't see signs of disorder in the X-ray refinement.

Silke Paschen.  Frustration diagram. I think you should allow the possibility of large FS magnetic order - consider extending the T* part of your diagram into the magnetically ordered phase.
Meigan - since we don't know anything about where the FS is small or large, will have to report back on it.
Paschen - maybe safer to put large FS on magnetic side.

Chubukov - you said there is anomalous T dependence of Sheat - can be magnetic, or electronic. Can you change magnetic field go away from critical point and measure whether T1/4 changes into T^2.
Aronson - yes - but we haven't yet.

Rebecca Flint (Rutgers U.): How Spin become pairs: composite pairing and magnetism in the 115 heavy fermion superconductors

articles Rebecca reports on:

Flint, Dzero and Coleman Nature Physics 4, 643 (2008)

Flint and Coleman arXiv (2009)

Outline of her talk:

* Why are 115 SC special?

* How do spins form pairs? two different pairing mechanisms:

magnetic pairing and composite pairing

* tool symplectic-N

* illustration: two-channel Kondo Heisenberg model

* experimental consequences

1.) Introduction: Why are 115 SC special?

Rebecca explained that a general theme in correlated systems is the distinction/competition between localized and itinerant electrons: localization vs. itineracy. For example, in the cuprates this competition is induced by doping and in the heavy fermions this competition is inherent in the interaction between local spin and the conduction electrons. This dichotomy leads to rich phase diagrams with magnetism, superconductivity etc.

She then continued by introducing the 115 family of unconventional superconductors. For example, CeMIn5 exhibits various phases as a function of doping: AFM, SC with many quantum critical points. The critical temperature for superconductivity can be raised by considering other 115 compounds like PuCoGa5.

2.) How do spins form pairs?

Next, the conventional picture for the formation of heavy Cooper pairs was explained.

Usually one considers the following sequence of mechanisms:

1. The local moments are quenched by means of the Kondo effect and the heavy electron liquid forms.

2. The Cooper pairs of heavy quasiparticles form; the glue is provided by the residual spin fluctuation, which is known as "magnetic pairing"

Rebecca explained that this two-stage pairing mechanism is reflected in the T-dependence of the magnetic susceptibility of various compounds. For example, in UPt3 there is a crossover from an Curie behavior at high temperatures reflecting the local spins to a Pauli behavior at low temperatures characteristic for a (heavy) Fermi liquid. In addition, the quenched entropy during this crossover is usually large, ie. on the order of R Log 2 so that the residual entropy attributed to the (quenched) local moments is usually small. The important aspect of this two-stage mechanism is that the energies associated with the quenching of local moments and the formation of Cooper pairs are well separated.

However, this seems to be different in CeCoIn5. There is in fact no Pauli susceptibility observed at lowest T and residual entropy is rather large. As a consequence, the nature of the Cooper pairs might be different in this 115 compounds. Rebecca explained that there is alternative scenario for Cooper pair formation, ie., "composite pairing" (going back to works of Abrahams, Balatsky Scalapino Schrieffer 1995). Here, the Cooper pair is not form by a pair of heavy electrons as for "magnetic pairing" but instead by a pair of (light) conduction electrons PLUS a spin flip of the local moment. In fomula the order parameter would read schematically as c+c+S-. Rebecca explained that whereas magnetic pairing is favoured by spin-fluctuations, the composite pairing is favoured by two channel Kondo physics.

Next, Rebecca motivated why a two-channel Kondo model might be important for the 115 compounds. The Ce3+ atoms are situated in a tetragonal cage of In atoms. This leads to several crystal field doublet Gamma6, Gamma7- and Gamma7+ as a function of decreasing energy.

The Gamma7+ doublet has the lowest energy and provides the effective local spin 1/2. There is a direct coupling of the conduction electrons to this effective spin providing a first screening channel. In addition, there is another orthogonal channel mediated by the other two crystal field doublets. (Did I get this correctly?)

There was a question, if it is known that Gamma7+ is indeed the lowest doublet. The answer was yes due to neutron scattering data.

Afterwards, Rebecca explained the important features of the two-channel Kondo effect (Noziere Blandin 1980). The important aspect for Rebeccas work is that this model exhibits a quantum critical point as a function of the ratio of the two couplings J1/J2 of the two channels. In the context of the two-channel Kondo lattice model, the strong fluctuations close to this critical point provide a strong glue for composite pairing leading to a dome of superconductivity as a function of J1/J2.

3.) tool symplectic-N

The analytical formalism that was used is the so-called symplected large N approach. The problem of Kondo lattice models in general is the absence of any small parameter that could be used as an expansion parameter. So one is forced to introduce an artifical one by for example enhancing the Hilbert space by generalizing the local spin 1/2 to something else like N/2 and use 1/N as a small parameter. However, Rebecca explained carefully that this procedure is not unique and that it is important to keep the essential physics intact upon this generalization. In particular, one should ensure that the following symmetries are not violated: (i) spin rotation symmetry, (ii) time reversal symmetry and (iii) charge neutrality (the spins are neutral objects). A simple SU(N) generalization of the local spin SU(2) in fact misses the discrete symmetries. A generalization that keeps these symmetries intact is the symplectic large N approach. Rebecca presented many formulae in this context, see here paper.

She then continued with a discussion how the Kondo lattice can be treated within the symplectic large N limit. A Hubbard-Stratonovich decoupling leads to two different pairs in the Hamiltonian: a hybridization term between the local spins and the conduction electrons and a pairing term that is important for superconductivity. She then argued that this large N treatment does not give a superconductiviting ground state if there is only a single Kondo screening channel present in agreement with known results. In the present of two channel, however, there is the possibility to obtain SC. Whereas the Kondo effect is an intra-channel effect, the SC arises from an effective inter-channel interaction. The spin provide an effective coupling that can be understood in terms of an resonant Andreev scattering finally leading to the composite pairing.

Rebecca continued with explaining that the nature/quantum numbers of the two channels arising from the crystal field splittings naturally explains the d-wave SC order parameter within the composite pairing mechanism.

A question was raised whether a self-consistent solution for composite pairing is already obtained for infinitesimal small coupling, ie., N->\infty, or if there is a threshold behavior as a function of coupling. This issue was further discussed at the end of the talk.

Rebecca then presented a phase diagram of the two-channel Kondo lattice. as a function of J2/J1 there are three phases: two heavy fermion phases HFL1 and HFL2 than sandwich a composite paired SC phase close to the critical point.

4.) Illustration: two channel Kondo-Heisenberg model

Now the presented discussion was extended to include also a Heisenberg term, ie., a direct spin interaction between the local spins. Such a term favours magnetic pairing so that a competition/coexistence of magnetic and composite pairing arises, which Rebecca also called "tandem pairing". The result of this extended model were compared to the experimental phase diagram. As a function of chemical doping, Rh-Ir-Co, one observes two SC phases (Sarrao, Thompson (2007)) These were interpreted as SC phases where different pairing mechanisms, magnetic or composite, prevail.

5.) Experimental consequences

Interestingly, the tandem condensate is electrostatically active. As the composite pairing redistributes f-electron charges, the condensate acquires an effective quadrupole moment. This moment will couple to tetragonal strain c/a that, as a consequenc, will correlate with the critical temperature Tc. Such relations are known, see Bauer PRL 93 147005 (2004). Furthermore, the quadrupolar moment could be directly probed with the help of nuclear quadrupolar resonance at the In position. Such a resonance signal close to the SC transition temperature is predicted by the theory.

Summary of the talk:

* spins quench as they pair in 115 and this must be incorparated in the condensate itself

* composite and magnetic pairing in tandem to drive SC

* composite pairing redistributes charge that is

observable as a sharp NQR shift at Tc

or as a sharp shift in the f-electron valence at Tc

Open issues:

* How does disorder affect the different mechanisms

* quantum criticality

* tandem pairing extended to other families of SCs

There were a couple of questions and discussion after the talk:

Q: Are there transitions between different FL phases possible at higher temperatures?

A: You have to discriminate between different symmetries. At high T, possible crossover between FL phases.

Q: Do 1/N corrections increase or decrease the SC

A: not considered, probably decrease it. Appearance of a pseudogap?

Q: Did you consider the magnetically ordered phases within this formalism?

A: not possible in large N

Q: Is this scenario relevant for other materials with high spin entropy at low temperatures?

A:

Then there was again a discussion about the stability of the composite paired SC phase in the large N limit, questions were raised by Eremin, Chubukov and Tesanovich.

Piers Coleman: Questions on the theory of iron-based superconductors.

This post is linked as a comment to the various excellent talks [1, 2, 3, 4] on the theory of iron-based superconductors presented at the ICTP2010 PDSCES conference. Almost all of the proposed theories are set in momentum space. Many of these theories are based on the s+- gap scenario as proposed by Mazin et al and Kuroki et al. This gap structure  can take unique advantage of the nested electron-hole pockets to develop a robust paired state with gap-nodes positioned between the pockets.  Nevertheless questions remain - the state appears to be far more robust to disorder than current theories can account for. Furthermore, there are clear links between the pairing and the local structure that are not naturally accounted for by the momentum-space based spin-fluctuation theories.

My key question:


what is the relation of iron-based superconductivity  to the local structure and chemistry of the material?  

One might argue that since these systems are of intermediate coupling strength, and that consequently, momentum space is the right venue for theories of  these systems. However, there are three reasons to question such answers:

• The observation by Lee et al (JPSJ, 77, 083704 (2008), that the superconducting transition temperature is sensitive to the local tetrahedral environment surrounding the iron atoms, peaking quite sharply at the point where the structure attains perfect tetragonal structure. To my knowledge, this remarkable observation can not be accounted for by any existing theory. Some have suggested it is connected with the one-body band-structure - but I know of no mechanism by which the sharp selection of the tetrahedral angle is produced by band theory.  In a spin-fluctuation picture, magnetic interactions would also be sensitive to these bond angles, but as yet, there is also no convincing accounting of the tetrahedral maximum.  This leads open the possibility of a many body explanation - could there be a multiparticle (eg pair, or composite pair) bound-state that is symmetry-selected favored by the tetrahedral environment? 

• From a purely pragmatic point of view, if we are to generalize these systems, with for example the goal of raising the transition temperature,  then we need to understand the relationship between the pairing and the local chemistry. 

• Since these systems lie between strong coupling and weak coupling, their physics should be equally accessible from a strong coupling perspective, and one might learn a lot from the exercise of starting from a strong coupling perspective, as suggested by Si and Abrahams.   There is perhaps here, a good analogy with intermediate valence f-electron systems, which can be treated starting with a finite U Anderson model, or as the large Kondo coupling limit of a Kondo model.  Hund's coupling will play an essential role in such starting points.

Andy Mackenzie (U. St. Andrews): Thermodynamics of Sr3Ru2O7

OUTLINE:
- Sr3Ru2O7 and the phase diagram
- thermodynamic probes
- qu. critical entropy pileup
- a novel phase with nematic transport properties
- broader perspectives: relevance to surface spectroscopy, electronic nematics, heavy fermions...

Lonzarich's insight (as well as Belitz, Kirkpatrick): Take a low-Tc ferromagnet and presurize it. What will happen? The result is a bit unexpected: a 2nd order transition will bifurcate to a tri-critical point with two lines of second-order transitions coming out from it and finishing in quantum critical endpoints at T=0. Two tuning parameters - pressure and magnetic field.

Take H=8 T. Find \rho~T^2 to the left and right of this critical point, but directly above it, one finds \rho linear in T, indicating appearance of a new phase (to be discussed in what follows).
S. Grigera, Science ...

Low temperature magnetization of Sr-237. Experimentally - use Faraday force magnetometer with customized design. Picks up a mix of torque and magnetization signals, which one can separate from each other by doing a clever subtraction of two signals.

de Haas van Alphen effect:
J. Mercure et al: PRL 09, PRB '10
dHvA osicallations are seen below and above the metamagnetic transition, with similar frequencies, i.e. the carrier density does not change much through these transitions.

Magnetocaloric effect and specific heat:
A.W. Rost, Science 325, 1360 (2009). Plot of Cv/T coefficient and entropy dS/T both show divergence as Bc=8T is approached from above and below. However right around 8T, dS/T shows a wiggle. This would indicate a diverging mass of the Fermi liquid, however no direct observation of strongly B-dependent mass in dHvA. A puzzle!

[Animated discussion of this puzzle with P. Canfield, P. Coleman, A. Schofield, S.-W. Cheong...]
Remark: the fact that C~\gamma T in the FL would mean that also entropy S~\gamma T. And indeed, on approach to Bc, both entropy and specific heat are on top of each other. Conclusion - Sr327 is a solid Fermi liquid away from Bc=8T.

Specific heat: C/T ~ log(T/T0) below T~10K - a work by S. Nakatsuji and Y. Maeno.

• Sr-214 shows a classic FL behaviour with C/T = \gamma + b*T^2.
• Sr-327 at B=7.9T (at Bc): Above 15K, Sr-327 looks like a FL with gamma = 75 mJ/(K^2.mol). However at low-T it shows log-divergence.

* Quantum critical entropy pileup
The speaker shows on the board his interpretation of the data:
Assume that DOS has a narrow peak at E=Ep just below the Fermi level. In finite temperature so that Ep < T < E_Fermi, this would explain the excess entropy is dS =0.1Rln2 (as measured between the B=0 specific heat data and the Fermi liquid fit C/T = gamma + bT^2). However to explain how this excess entropy is seen at low temperature, the logical assumption is that this narrow peak in the DOS must become narrower and narrower as temperature goes to zero, so that Ep < T is always satisfied. This is obviously a phenomenological observation, and we don't know how to justify it microscopically. [Note: it is pointed out from the audience, that Rln2 is the spin entropy (for S=1/2). Generally, there are also electron charge degrees of freedom, so that total entropy would be Rln4. Unless of course one has strong enough Hubbard U that would suppress charge fluctuations.]
Whatever that feature is in the DOS, it is NOT just a rigid band feature through which one simply Zeeman shifts. See Iwaya, PRL 2007; J. Farrel, PRB .

* The novel nematic phase
Observed in a narrow region of fields around the Bc=7.9T below T~1K (showing a curved "roof" feature).
Tentatively explained by a weak Pomeranchuk distortion in the d-symmetry (l=2) channel. See theory by Ho and Schofield who analysed the region of stability of such a Pomeranchuk (nematic) phase.

At Bc, C/T keeps rising as T->0. However away from Bc, C/T saturates to a Fermi liquid value.
Use the experimental reconstruction of S(T) to try to delineate which theory is right!

E.g. (C/T)exp ~ [H-Hc]^{-1}
whereas Hertz-Millis prediction would be [H-Hc]^{-1/3}.
Something else we might be missing? S. Hartnoll, PRB 76, 144502 (2007).



CONCLUSIONS:

Questions:
Q. Could you comment on spin anisotropy? What is the dependence on the field direction?
A. Everything shown was for field along c-axis. We are able, in principle, to measure the angle-dependent specific heat, but so far not possible due to design of the experimental setup.

Q. A. Chubukov: Belitz and Kirkpatrick claim that C ~ T^3 log T in their theory. Do you see any deviation from C~T^3 at higher temperatures?
A: You can indeed see C/T deviating from T^2, but the details are unclear. Postpone till private discussion.

Q. S.-W. Cheong.
1) Are all 3 transitions true phase transitions or crossovers.
2) you may have a phase coexistence?
A: 2) Indeed, coexistence, with domains may be possible.
1) only one is a true transition, the other two are crossovers.

Q. R. Fernandez: Can you see nematic phase by measuring anisotropy in resistivity?
A. It's more complicated, since the domains, if they exist, would change their orientation as the field is rotated.