Hastatic Order in URu2Si2 by Rebecca Flint

Will discuss her theory of hastatic order breaking double time reversal symmetry as related to URu2Si2.

In heavy fermion containing Ce, U etc get the competition between local and itinerant built into the atom itself. 

Describes Kondo's theory of a single impurity in a metal with nice PPT images. 

On the lattice: Local moments surrounded by conduction electrons. Moments screened by conduction electrons. 

In momentum space this is describes as a band hybridization between a flat band and a dispersive conduction electron band. No symmetries are broken and there are no phase transitions. 

Now turn to the hidden order in URu2Si2. A heavy fermion system with a phase transition but 27 years after the discovery no definitive information about the nature of the magnetic order. 

question from Andy Schofield: Can you really have a Kondo effect for an Ising spin? Answer it is not truly an Ising spin there are other levels to worry about: OK

Impressive list of many theories offered to describe this. Emphasize that many of the theories have been ruled out and several more are on the way out. 

Summary of major points to explain: Large entropy change at phase transition, small or absent magnetic ordered moment, proximity to antiferromagnetism. 

New insights from STM: an effective measurement of the band structure. Hybridization develops at the hidden ordering transition. BCS development of the hybridization gap. This gap has also been seen in optical spectroscopy. 

The experimental claim is that the Hybridization is the order parameter and it develops in a mean field fashion. Timusk points out that the data is not good enough to distinguish the critical exponent. There is agreement on this. 

Then describe the Kyoto experiments revealing four fold symmetry breaking at the phase transition. 

The absence of a transverse f-electron response indicates this is conduction electrons scattering off the hidden order. This leads to constraints on the spin dependent scattering t-matrix. 

Now return to the giant Ising anisotropy. This is a generic feature of the non-Kramers ion. 

Describe deHaas van Alpen data that show the carriers really are ideally Ising. The consequence is a Kramers index (-1)^2J. The Kramers index in this case must be -1. 

This leads to the prediction that the hybridization operator must break double time reversal symmetry like a spinor. 

Spinorial order parameter is also indicated by the proximity of the large moment antiferromagnetic phase. The nature of the  multi phase diagram indicates that the order parameter of the hidden ordered phase and the AFM phase are relates by a simple rotation. This indicates that if the AFM order breaks time reversal symmetry then the hidden order must also.

Now discuss spinorial hybridization. In fluctuations from a non-Kramers doublet the fluctuations will also be to a doublet. Since the excited state is a doublet the hybridization operator must itself have spinor character. 

hastacit means spear like and is associated with a rotation of the AFM phase. Now write a Landau Free energy that is to describe the behavior and phase diagram of the spinor field. This can produce the hidden order and by rotation the AFM order. 

The theory predicts that the gap to longitudinal spin fluctuations should vanish at the first order phase transition between the HO and AFM phases. 

The Landau theory also is able to describe the non-linear susceptibility and it's anisotropy. This is another aspect of URu2Si2 that had remained for many years since the discovery of the effect by Ramirez. 

Now back to the microscopic theory. The Non-Kramers Gamma 5 doublet is relevant here. This is a magnetic non-Kramers doublet. It has a quadropolar moment and so is not exactly an  Ising spin. This may address the previous comment about how one can have a Kondo effect for an Ising spin. 

Now to the board! Write down two hybridization terms; Valence fluctuations terms. Now user Slave boson mean field theory. A Scwinger Boson representing the excited doublet. This is simultaneously a slave and a Schwinger Boson. 

Develop the hybridization  term which has gamma6 and gamma7 components. Develop a hidden order ansatz. The interchannel hybridization is staggered whereas the intra channel hybridization is uniform. 

With the mean field hamiltonian can apply all the usual machinery and extract the results. 

One is that there must be a magnetic moment that must be in the basal plane. The magnitude is about 0.01 muB and is an upper bound on what can be seen. There is no large f-electron moment. 

The xy anisotropy should appear below TN. 

Now address comparison to experimental results: There is generally consistency: 

no large moments

hybridization gap is the order parameter

Ising quasi-particles

Broken tetragonal symmetry 

inelastic neutron scattering is a pseudo goldstone mode. 

predict longitudinal spin fluctuations that should vanish across the phase transition

resonant nematicity

Things to be done: How to generate superconductivity from the hastatic state. 

Are there other examples of hastatic order. 

In the hidden ordered phase: Kapitulnik says there is no Kerr effect in the hidden ordered phase but there is in the superconducting phase. 

Fernandez: What would be the elastic data across the hidden ordered phase transition. Answer this should be looked at in new better samples.

Flint states that if the mixed valency is 20% then the transverse moment in the hidden ordered phase should be 0.01 muB. 

Silke Paschen (Vienna): Exploring heavy fermion quantum criticality in the extreme 3D limit

Silke Paschen (TU, Vienna)
Exploring heavy fermion quantum criticality in the extreme 3D limit
Blogged by Andy Schofield

The point will be to show how experiments Ce₃Pd₂₀Si₆ (abreviated as CPS) reveal a new three dimensional (i.e. cubic) quantum critical system. This is not a superconducting system.



Introduction: The rising volume of the Fermi sea is illustrated with a cartoon from Piers Coleman's News and Views paper in Nature Materials, 11 185 (2012). The first heavy fermion system was CeAl3 [PRL, 35, 1779 (1975)] which has a mass enhancement of 1600 of a free electron. The Kadowaki-Woods ratio [SSC, 58 507 (1986)] unified the Fermi liquid response of many materials in the heavy fermion class. However subsequently materials of similar structure and composition started to show non-Fermi liquid response: eg Y doped U₂Pd₃ with C/T ~ log T. It then appeared that many of the bad-actors were close to a magnetic phase transition thereby anchoring non-Fermi liquid behaviour to quantum critical points formed by doping or other control parameters: CeCu_6-xAu_x (doping), YbRh₂Si₂ (magnetic field tuned), CeIn_3-xSn_x (doping PRL 2006), CeRhIn5 (Nature 2006).

Routes to magnetism in HF materials: Two views - either a spin-density wave instability of the heavy fermi liquid, or the breakdown of the Kondo effect and formation of order from local moments. The model for the latter case (particularly for CeCu_6-xAu_x) relied on two dimensionality.
Question: Is the SDW transition Stoner like? Yes
How could we distinguish between these two views. Coleman proposed Hall effect measurements for that to see the large heavy fermion fermi surface to magnetically ordered state with small fermi volume at the kondo breakdown scenario. YRS (YbRh₂Si₂) saw some support for this in the Hall effect, and with time this has improved.

The cross-over in the Hall sharpens in a fashion linear with T to infinitely sharp at T=0 (which will be tested with Silke's new adiabtic demag fridge so watch this space!). Cleanliness was once an issue (since the blogger proposed an alternative scenario which predicted an effect scaling with scattering) but the observed effect did not change with sample quality.
A more complete list of possible scenarios was then presented:

• Kondo breakdown
• A Lifshitz/Topogical transition
• Valence transition
• Quantum tricritical point
• Weak-field breakdown of Boltzmann transport

Now we have a new cubic material: Ce₃Pd₂₀Si₆ which has a Curie-Weiss susceptibility at high temperature, then seems to show a transition at about T_Q ~ 0.55K (quadrupolar ordering??) and then further Neel ordering at 0.25K as seen in specific heat and susceptibility. Crystal fields are analysed - there are 2 Ce sites with different crystal fields. The phase transitions are split apart in B field T_N goes down and T_Q goes up.

The non-Fermi liquid properties are as follows: Between the 2 phase transitions we see C/T = - Log T and the electrical resistivity is linear in T in a certain range. The Hall effect is then studied as a function of field which suggests two distinct regimes in terms of the slopes. The Neel ordering is now clearly seen in magnetotransport, but there is an additional line which is not associated with the magnetic ordering except at T=0 but goes off in a different direction with field and crosses the T_Q point with no effect - strikingly reminiscent of YRS with the sharpening feature as T to some power. This is in contrast to that seen at the Neel transition feature.
Question (from AJS): Are there a range of disordered systems? No - very hard to make.
Silke then made reference to a theoretical phase diagram (T=0) of the range of phases present (small vol and large vol fermi surfaces with magnetic order - either local or SDW - and paramagnetism). A picture due to Qimiao Si. Silke then placed YRS on that phase diagram.
Question (from PC): Is there an evidence for a line between multiple phases or could it be a tetracritical point? The Ir and Co doping suggest that it is a line but still basically open.
Silke claimed CPS is a transition between small and large volumes magnetic phase transitions so is a different point on the phase diagram with dimensionality being the "tuning parameter" that changes where the material is placed on the phase diagram (Custers et al Nat. Mat 2012). Matsuda's group in Kyoto work on MBE materials grown might be able to test this notion out. CeRhIn5 work shortly to be published is also suggestive of two phases.

So in summary: Seen physics in CPS very remeniscent of YRS but with the possibility that 3D materials are placed in a different point on the phase diagram.

Question (Collin Broholm): Did you consider the detailed shape of the phase boundaries under field so moving at fixed T could skim the top of the phase boundary? We avoided that part of the phase diagram.
Question (Nakatsuji):  What are the P phases on the phase diagram - are they seen? Paramagnetic regions - and we have looked in the Ge doped YRS where we do seem to see that as a non-Fermi liquid phase. Is it seen in transport only? Well T scale is very low and resistivity is the only probe we can currently use down there.

What a lovely talk!

Piers Coleman: The Unsolved problem of heavy fermion superconductivity

Piers Coleman (CMT, Rutgers U.)
The unsolved problem of heavy fermion superconductivity

Blogged by Mike Norman

Piers apologizes for giving a talk since he was an organizer, but was covering for Gil Lonzarich, who unfortunately could not make it to the meeting.

Piers points out the amazing similarity of the phase diagrams for many classes of superconductors, with superconductivity often found near the boundary of magnetism.  For heavy fermions, this was identified by Gil in CeIn3 in a famous paper in 1998.  Piers next turns to NpPd5Al2, a heavy fermion superconductor with a Tc of 4.5K, which has a divergent Curie susceptibility as one approaches Tc from above, indicating that the f spins are involved in superconductivity.

He now introduces f electron physics.  Ce has an f1 configuration, with the f electron having a J=5/2.  This level will be split by crystal fields to a series of Kramers doublets.  As one lowers the temperature, the single spin of the f1 configuration will be screened by the conduction electrons, forming Kondo singlets.  Below some characteristic temperature TK, the susceptibility will flatten and the specific heat coefficient C/T will also saturate.  For a lattice of spins, once the Kondo singlets start to communicate from site to site, one forms a Kondo lattice, where the resistivity drops, forming a heavy Fermi liquid.

Now, there are two things the spins can do on a lattice.  They can form Kondo singlets, or they can interact between sites giving rise to local f magnetism mediated by the conduction electrons (RKKY interaction).  Doniach proposed in 1976 that at some intermediate coupling, the two effects have the same value, implying a quantum critical point between magnetism and a Fermi liquid.

Now, in classic superconductors, magnetic impurities are pair breaking. But in 1975, a Bell labs group discovered superconductivity in UBe13, but because of the above, felt that it had to be an extrinsic effect due to U filaments.  Later in 1983, Ott realized it was indeed a heavy fermion superconductor.  But this was preceded by Steglich's discovery of superconductivity in CeCu2Si2 in 1979.  As Frank pointed out, because TK is so low, the ratio of Tc to TK is large, implying "high temperature" superconductivity.  Moreover, it was later realized that this material is near the boundary of magnetism.  Returning to Ott, UBe13 was even more bizarre than CeCu2Si2, since Fermi liquid behavior never sets in before one hits Tc.

Then the LANL group discovered superconductivity in UPt3.  This material exhibits Fermi liquid formation well above Tc.  This material was the first where antiferromagnetic spin fluctuations were identified.  In 1986, three theoretical groups showed that such fluctuations would give rise to d-wave pairing.  Interestingly, UPt3 is now thought to be a p-wave (blogger takes this opportunity to say probably f-wave) superconductor.

Since then, a large number of superconductors have been discovered, some near the boundary of ferrromagnetism (like UGe2), some near antiferromagnetism (CeIn3), some where inversion symmetry is broken (which would give rise to mixed parity superconductivity), etc.

Piers now returns on NpAl2Pd5, which was discovered somewhat by accident by Aoki in 2007.  In this system, the ratio of Tc to TK is of order 1, so the susceptibility is still diverging when Tc is hit.  There is no question the spins are involved in the pairing.  At Hc2, the magnetization jumps by 0.2 mu_B, indicated the "release" of the f spins when superconductivity is destroyed.  So, how can neutral magnetic moments form a charged superconducting condensate?

Piers now turns to his theory of composite pairing.  The heavy electron is a product of an f spin and a conduction electron (Kondo singlet).  Cooper pairs involve pairs of electrons.  So, Piers proposes that heavy Cooper pairs are a product of two conduction electrons and an f spin flip.

Q:  Why do the two electrons have the same spin?
PC: Because to get an S=0 Cooper pair, I have to combine a triplet combination of the two conduction electrons with the spin flip of the f electron to get a net S=0.

Now, one way to realize this is by going back to the two channel Kondo model.  This model is hard to solve, but one can make progress by taking the large N limit (where N is the degeneracy of the f orbitals).  Such a model has a critical point as a function of couplings where one has a divergent tendency to form composite pairs.

One consequence of composite pairs is that the f valence and f charge distribution will change at Tc, leading to a shift in the NQR frequency, which has been recently seen.  Piers also points out that the spin resonance in CeCoIn5 has been seen to split into two by a field (Collin Broholm), implying a doublet, not a triplet.  Is this an S=1/2 excitation instead of S=1?

After the talk, there were many questions, but as I was the chairman, I was unable to blog them.