Andrey Chubukov: Collective instabilities of doped graphene

Blogged by Rafael Fernandes

 Contrasting the talk given by Oscar Vafek, who looked at many-body instabilities in double-layer graphene, Andrey will talk about instabilities in (doped) single-layer graphene.  

 Glossary: SDW = spin-density wave; SC = superconductivity; CDW = charge-density wave; FS = Fermi surface; VHS = van Hove singularity; RG = renormalization group.

 Andrey explains that when the chemical potential of graphene is changed (via electron doping, for instance), the Fermi surface changes from two Dirac points to a hexagonal FS with three saddle points, corresponding to VHS. He mentions that experiments in doped graphene find a hexagonal FS, which is somewhat similar to the one expected theoretically. However, there is no consensus that this is really the VHS of the band structure.

 Andrey proposes to study the "ideal" situation of 3/8 doping. Besides the VHS, there are three nesting vectors connecting different sides of the hexagonal FS. He mentions that such a FS is stable even when one includes next-nearest neighbor hopping. Due to the presence of VHS, the particle-particle bubble acquires an extra log, and the SC instability becomes log^2. Furthermore, due to the presence of perfect nesting, the particle-hole bubble also acquires an extra log, and density wave instabilities also becomes log^2. Other exotic instabilities may also appear, but they do not have log^2. Thus, there are several many-body instabilities, and one needs to investigate in which channels there are small attractive interactions that can take advantage of the these log instabilities. Ultimately, Andrey wants to find which of the several instabilities is the leading one.

 To address this problem, he will use parquet RG, which treats particle-particle and particle-hole channels on equal footing. In the RPA level, the pairing interaction is repulsive, but SDW and CDW channels are attractive. However, these are the bare interactions, which become renormalized at low energies. Andrey now writes all the possible interactions between the low-energy fermions near the VHS in each of the three FS patches. There are four of them, including density-density interactions, pair-hopping, exchange, etc. Doing the one-loop parquet RG, these interactions are renormalized and change as higher energy modes are integrated out. The question Andrey wants to answer is which of these four coupling constants diverges, and which order it triggers. Formally, one has a set of coupled first-order differential equations. Andrey considers both the cases of perfect nesting and non-perfect nesting - the latter via an effective parameter.

 Q: is umklapp considered? A: sure, pair-hopping is an umklapp process.

 Andrey shows the solution of the flow equations - all couplings diverge at a certain energy scale, but one of them diverges first. By writing down the SC, SDW, and CDW vertices, and plugging in the renormalized couplings, he can identify which one diverges first. The two leading instabilities are the SDW (whose bare coupling was positive - attractive) and singlet SC (whose bare coupling was negative - repulsive). Eventually (i.e. at lowest energies), SC wins and becomes the leading instability. Several technical questions pop up, from the validity of the RG procedure for log^2 instability to the fate of the flow equations away from perfect nesting.

 Now the symmetry of the SC state is discussed. There are three gaps corresponding to the three VH points, and three possible eigenvectors/eigenvalues for the linearized gap equations. Two of these eigenvalues are actually degenerate - the other one, whose eigenvector corresponds to the s-wave solution, remains always negative, i.e. it is not realized. The two degenerate solutions have d-wave symmetry: d_xy and d_{x^2-y^2}. By performing a Ginzburg-Landau expansion, Andrey finds that the solution with lowest energy is actually a combination of these two symmetries: it is a (d+id) state, which breaks time-reveral symmetry. Interestingly, functional RG gives the same result.

 Conclusion: graphene doped to the saddle-point of the band structure displays as the leading instability a time-reversal (d+id) SC state!

Maxim Dzero: Topological Kondo Insulators

Your blogger today: Andy Schofield

OK well due to a blogging malfunction - this blog is starting half-way through the talk - no it seems to be the end of the talk!

Max has defined two types of Topological Insulators: weak or strong. A large N analysis has been done.

Ce-based Kondo Insulators expected to be weak and unstable to disorder.
Mixed valence systems are most likely to be the ones where strong TI insulators would be observed. Hybridization with the conduction electrons can create an infinite spin-orbit coupling.
These are adiabatilcally connected to topological band insulators: small bandwidth and infinite spin-orbit.
Question: Could a Hopf term be useful? Possibly.
Question: Why are all topological insulators cubic and does this impact your discussion? Not sure and the questioner does not know either.
Question: Your analysis looks like 2D - aren't the supposed to be 3D creating surface states? Sure - layers will create it.
Question: Why called Dirac point? Answer: the dispersion looks like a Dirac cone.
Question: Why do we need the notion of entanglement entropy? It provides a way of distinguishing between charge density wavesand topological insulators.
Question: Can all of this physics be deduced from the Green function? No you might need a 4 particle correlation.
Question: Why do you get band narrowing from spin orbit? It is nothing to do with spin orbit. Instead it is simply the usual Kondo lattice phenomenon.
Question: Where is this in the periodic table of the topological insulators? Yes - it is A1.

Serguei Borisenko: Fermiology and Order Parameter of Fe Based Superconductors

SVBorisenko(Institut fuer Festkorper Physik, Dresden, Germany)
Fermiology and Order  Parameter of Fe based superconductors (FeSC)

–      Blogged by Saurabh Maiti

Blogger’s notation-
SV Borisenko (SVB)

Superconductor (SC)

Anti Ferro Magnetism (AFM)

Spin Density Wave (SDW)

SVB starts by advertising 1-cubed ARPES that can reach to temperatures below 1K. He explains how low energy excitations can be angle resolved by rotating the sample. ARPES can be used to probe-

(*)Fermiology— get information abt:Fermi Surface (FS), band structure, reconstruction due to (e.g magnetic) ordering

(*)Self Energy- Get information abt: V_F renormalization, scattering, coupling const. (these last two are related to imaginary and real parts of the self energy respectively)

(*)Order Parameter (mostly talks abt extracting SC order parameter)

SVB mentions this was very useful for the cuprates in all the three above mentioned aspects [although Self energy study was a little involved because the bosonic mode was likely to be of electronic origin]. For the cuprate case it was clearly established that there was no 3D and the gap was d-wave in character with clear nodes – max gaps of 20-30mev.

SVB mentions then talks abt how comparing ARPES band structure  and LDA band structure can be used to measure correlations, velocity renormalizations…

Having mentioned the success in Cuprates, SVB now moves on to FeSC-specifically the pnictides. He reminds us that the photoemission data is a product of

<f|p.A|i> A(k,e)f(e) X R(k,E)---the last term is resolution

Explains step by step how the matrix elements, polarization, resolution convolutions are taken care of and finally we get A(k,e)*f(e) and then final division by the Fermi-function gives the electronic spectral function which is used to extract info abt self energy and gap structure.

SVB then talks about LiFeAs-mentions presence of one large hole pocket and two crossed elliptical electron pockets at (\pi,\pi) in the two iron unit cell Brilliouin zone. But what happens very close to \Gamma point its more involved [different results for different polarizations---resolved by scanning across kz---he concludes that in a particular region of kz there is hole pocket… kind of like cigar shaped. Points out the 3D nature of this material.]

SVB now talks abt Co-NaFeAs shows ARPES results- combines results from different polarizations and concludes presence of two crossed elliptical electron pockets and 3D small hole pocket at \Gamma point.

SVB now moves to K-BaFeAs

Three hole pockets at \Gamma point—for optimally doped material

But !! Result!!  At corner there are 4 hole barrels [blogger’s note- star shaped] with an electron band crossing in the centre.

KFeAs          

Same as above but no central electron band crossing; So three hole pockets  at \Gamma and 4 hole barrels at the corner —

Co-BaFeAs

Two crossed elliptical electron pockets

One clear hole pocket, but two other bands possibly cross the FS. Its complicated due to hybridization of xz /yz.

FeSe-

Very tiny two elliptical electron pocket

And very tiny hole pocket at \Gamma point of xz/yz character

Rb-FeSe

Rb expels iron and causes vacancy ordering but will not discuss this.

Want to discuss the metallic behavior of  RbFe2Se2

Tiny hole pocket and two large elliptical electron pockets

In this case the disordered vacancies give metallic regions.

Two electron pockets at corner and 3D ELECRON pocket at gamma point (does not cross for all Kz values)

MAIN MESSAGE is that the standard picture of 2 circular hole pockets and two circular electron pockets is never experimentally realized.

Also points out that conventional mapping of FS topology to phase diagram is not realized experimentally…[insert picture].

SVB now discusses probing order parameter.

Start with FeSe (T_c=8K)—remind yourself that it has tiny electron pockets and tiny hole pocket. Then moves on to K-BaFeAs

Message- the gaps are strongly orbital dependent and kz dependent.

Points out that---

-xy band is irrelevant (gaps are small and some times give large FS, sometimes small, etc)

-xz/yz are the important bands

-Wherever the xy content is present the gap has a minima---

-Absence of xy character—large gap

SVB returns to his favorite material  ‘LiFe As’ to discuss its order parameter—The important results is the anisotropy of the SC gap which was claimed to be isotropic before. [blogger’s note: The oscillations are conts+cos4(\theta) type as expected from the A1g symmetry]

Electron pockets—also anisotropic.

The tiny hole pockets; cannot probe for anisotropy. [but gap is large]

[blogger’s note—the pocket is of xz/yz character; the outer FS is of xy character and the gap structure anisotropic and weaker]

SVB believes that it is likely to be s++ gap. He tried the fitting with usual form factors for the s+- wave gap structure like cosKx + cosKy and cosKx*cosKy. The only consistency he finds is with the gap structure predicted by Kontani et al. which describes the in-phase oscillations of the electron pockets. And since this required phonon mechanism in the theory, it might well be that electron-phonon mechanism is likely in this material.

Nevertheless, SVB points out that it can also be explained by S+- picture (S. Maiti et al. PRB 85, 014511 (2012)).

SVB thus invites the theorists to investigate this material more thoroughly because the self energy is known, band structure is known orbital characters are known, gaps are known; and this should be enough to find the right theory for this material.

SVB stresses on the fact that \Gamma point has cigar like features of the hole/electron pocket and suggests it  be taken into account instead of simply considering cylindrical FS. Simply put 3D is very likely important for SC. Life becomes hard but needs to be dealt with.---

Thanks collaborators and restates conclusions—

Q. Did you observe gaps induced by SDW gap in your?

A. Yes. It is clearly seen. But involves a lot of work.

Q. Are there data below Tc which show change electronic spectrum above and below Tc

A. Yes. No problem. Can be provided.

Q. Can your Spectroscopy give insight into the local  quantum chemistry of these materials?

A. Ok. Periodic potential gives good qualitative picture.

Local picture needs special attention and seems to be theorist dependent that is why I am not using that picture. Band structure picture seems to have some universal results that most people are getting. If I get a more or less universal local picture I don’t mind using it.

Q. 11 sample  FS looks like 122 (FeAs based); why? Even though the latter is heavily electron doped.

A. They are not really same. FeSe has a HOLE like  also the pockets are very Tiny. Doping with electrons makes electron pockets bigger.

Q. Is the phase diagram applicable to all pnictides.? What abt LiFeAs? No AFM… where does it fall?

A. The phase diagram is really for 122. 111 is different.

Q. Laser v/s your method; comparison?

A. Prof. Shin has an excellent technique. We have the same resolution as there’s.

Disadvantage of Laser is cannot scan kz and matrix elements can play crucial role in those measurements.

Q. In Li-111 what are the bands/orbitals responsible for SC?

A. xz and yz character of the inner pocket, I think is most relevant.

Q. Is there any criticality associated with change in FS topology?

A. It is interesting question—the main problem is that we don’t have data for continuous doping to probe that feature.

Time Reversal Symmetry Breaking & Charge Ordering in Pseudogap Phase of HTSC

Time Reversal Symmetry Breaking & Charge Ordering in Pseudogap Phase of HTSC
Aharon Khapitulnik, Stanford University

Good morning everybody, the second talk of the morning is by Aharon on time reversal symmetry breaking and charge ordering in the pseudo gap phase of HTSC.

Aharon starts with an intro on Time reversal symmetry, T = i \sigma^y K (K is complex conjugation)

Magneto-optics and T, Kerr effect.

By axial symmetry, the index of refraction for left and right light is related by:

\epsilon_{r,l} = 1 + 4 pi i \sigma_{r,l} \omega^-1

To measure the Faraday effect, they use a Sagnac loop with 2 quarter waveplates to select the circular polarization.

The new feature is the use of interferometry to detect broken rotational symmetry.

Pseudogap Phase in HTSC:

Theory 1. T* represents a cross-over into a state with pre-formed pairs and a d-wave symmetry.

Theory 2. T* marks a true phase transition into a phase with broken symmetry that ends at a QCP, usually within the SC dome.

Recent neutron scattering, Kerr effect and ARPES all show a broken symmetry phase at T*, and the possibilities include a structural phase, electronic nematic, smectic, stripe phase, DDW, etc.

Kerr effect measurement data on YBCO (detwinned):

1. First they cool in high field 5T, and then measure Kerr while warming up at ZF.
2. Kerr effect is seen above T_c, and disappears completely at T_s.
3. The response below T_c is due to trapped vortices, and cooling in ZF eliminates the vortices.
4. Note that the Kerr effect seen below T_c, within the SC phase as well!

Additional experiments on YBCO also signatures of TRSB, including neutron scattering measurements to detect signatures of the Varma loop current state; and resonant ultrasound spectroscopy that measure compression and shear modes, and they see a transition at 280 K and 200 K, which correspond to neutron scattering and Kerr respectively.

Resonant soft X-ray scattering shows 2D charge fluctuations with incommensurate periodicity, and high energy X-ray shows a co-exiting CDW and SC phase in YBCO. This indicates that there is a structural-type transition at a higher temperature with evidence of a short-range CDW state, and then TRSB at a lower temperature T*.

TRSB Data in LSCO:

There is a 1st order transition from LTO to LTT phase measured using birefringence, and when Aharon measures the Kerr effect, he finds a large signal at the LTT phase transition, which peaks at the spin-orbit temperature, and then levels off below T_c, and remains broken in the SC state!

Moreover, there is no training effect for the Kerr, meaning it does not change sign even when we flip the B field. One possibility is that there is TRSB  even at high temperatures, but Aharon will discuss another possibility later.

The zero-field Nernst effect also shows TRSB at the charge-ordering temp, T_co = 54 K, and the Nernst effect is seen also in the SC state, i.e. TRSB in SC state! Ong's group tried to change the sign by heating up to 290 K and then cooling in an opposite B field, but the sign of the Nernst signal remains the same!

ARPES data in Bi2201 also shows evidence of TRSB at T*. The energy position of k_F is the measure at which the pseudogap becomes nonzero below T*, and this agrees with the T-dependent data from Kerr effect.

Aharon has also measured the Kerr effect in HgBa2CuO4, and TRSB is also seen in this system.

Aharon predicts that below T*, a weak charge ordering will be found!

Key Observations:
1. The Kerr effect occurs at the charge-order transition T_co.

2. Possibility of TRSB is being broken at higher temperatures, due to a pre-existing magnetic phase that changes its coupling to the off-diagonal conductivity at T*, and changes due to cystal symmetry changing at T*.

3. Another possibility is that HTSC are magnetoelectric below T*, and below T_co, symmetry is further lowered to acquire AHE effect, and this allows for a finite Nernst effect and Kerr signal.

Free energy, F_{ME} = \alpha_{ij} E_i H_j

4. Comments on Varma Loop-Current state with net moment = 0 in unit cell.

Type 1 Varma state: In-plane loops that breaks T and I(Inversion), but not TI.

Type II Loop current state: Breaks T and C(Chirality), and gives rise to AHE.

Mixed State: Breaks T and Chirality (AHE).

Aharon suggests a Possible Scenario:

At higher T*, there is a Varma loop state, and at T_co a Type II state occurs, giving rise to a mixed state below T_co and thus gives rise to a finite Kerr effect.

Another Scenario: Start with Type I at T*, and at T_co there is canting of the moments that gives rise to a Kerr effect.

Consequence of the magnetoelectric state: This means that there should be no sign of Kerr effect alignment with perpendicular B field, which is what is seen and also in the Nernst effect measurement. Furthermore, there should be the same sign on opposite side of sample, which is also seen in Kerr effect.

Qn: Pierls transition can be measured using derivative of resistance, d \rho/ dT. Was this measured?

Ans:  In Bi2201, yes this was measured and found at the same T* as Kerr effect.

Qn: Is there a possibiltity of a QCP within SC dome?

Ans: Yes, there is a Kerr effect below the SC dome and also evidence from resonant ultrasound sepctroscopy, indicating that this should terminate at a QCP.

Qn: What is the wavelength used in the Kerr effect, and do they have data for different wavelengths?

Ans: Wavelength is 1.55 microns, and they have a new system at 830 nm. But they do not have wavelength-dependent measurements.

Qn: Does the community have a consensus that fluctutations due to TRSB and current loops are the cause of linear-T resistivity?

Ans; No consensus in community on this point yet.

We thank the speaker for a very interesting talk!

Satoru Nakatsuji (ISSP Tokyo)

Satoru Nakatsuji (ISSP Tokyo)
"Unconventional quantum criticality, anomalous metal with strong valence/orbital fluctuations"
Blogged by Maxim Dzero

    One of the greatest challenges to Landau's Fermi liquid theory - the standard theory of metals - is presented by complex materials with strong electronic correlations. In these materials, non-Fermi liquid transport and thermodynamic properties are often explained by the presence of a continuous quantum phase transition which happens at a quantum critical point (QCP). QCP can be revealed by applying pressure, magnetic field, or changing the chemical composition. In the first part of his inspiring talk, Prof. Nakatsuji has discussed manifestation of quantum criticality in the f-orbital material beta-YbAlB4. Starting with the review of the data on the magnetization and resistivity measurements, he showed that beta-YbAlB4 displays an anomalous (non-Fermi liquid) behavior at low field and temperatures. Increase in magnetic field fully recovers conventional (metallic) Fermi-liquid properties. Prof. Nakatsuji discussed how the data analysis unambiguously points to the presence of the QCP at very small (of the order of 1G) magnetic field. This results show that beta-YbAlB4 displays quantum criticality without an apparent QCP suggesting the non-Fermi liquid metal is a phase in itself rather than is caused by system’s proximity to QCP.
    To get further insight into the physics of that system, the results of the hydrostatic and chemical pressure studies have been presented. As it turns out, pressure induces the phase transition in magnetically ordered state, while the quantum criticality remains intact. Therefore, the results of the pressure studies strongly suggest that the microscopic mechanisms, which govern quantum criticality in this material, are different from those responsible for the onset of magnetic order. In addition, measurements of the Seebeck and Zommerfeld coefficients show that quantum criticality anomalous transport and thermodynamic properties are likely governed by the quasi-particle excitations from the same heavy-Fermi surface. To explain the anomalous features in the Hall constant strongly suggest that one needs to invoke the excitations from another Fermi surface. In addition, there is an indication that there may be a line of nodes in the hybridization gap[. These results triggered the questions from the audience as to what extend the existence of the “quantum critical” Fermi surface can be reconciled with the absence of the hybridization for the specific values of momenta in the Brillouin zone. Another intriguing question is whether it would be possible to identify the Fermi surface, which develops the leading superconducting instability.
    In the second part of the talk, Prof. Nakatsuji has discussed the manifestation of an interplay between conduction and orbital degrees of freedom which under certain circumstances may lead to a phenomenon of the orbital (or quadrupolar) Kondo effect. The candidate materials, which may display this physics, are PrTr2Al20 with Tr=V, Ti. Prof. Nakatsuji has presented a set of thermodynamic and transport data consistent with the idea of quadrupolar ordering in PrTi2Al20. On the contrary, PrV2Al20 shows the dominant role of the Kondo screening processes.
    I think, the presented results mark novel and important developments in the studies of f-orbital materials.
     



 

Wednesday, August 15th
Tom Timusk, "The normal state of URu2Si2: spectroscopic evidence for an anomalous Fermi liquid"
Blogged by Andrey Chubukov 

Tom starts  with the summary of 1988 view of URu2Si2 (the time of first ICTP workshop on correlated electrons)

He discusses  normal state properties of URuSi2: mass enhancement, scattering rate, Drude peak.

 Scattering rate has  a peak  at 20 meV.  There is a coherent Drude peak at 20K,  it becomes incoherent avove 20K.   The conclusion was that at low T/\omega the system is a Fermi liquid  with m^* =25m

In 1988 the community believed that hidden order below 18K is  spin-density-wave (SDW).  Superconductivity (SC) emerges  below 1K. Tom presented arguments for SDW which sounded reasonable back in 1988 but later were found to be in disagreement with the measurements.

20 years have passed (but ICTP workshop is still running!)

He cites STM data by S. Davis group, which show that  at 18.6K (right above  transition into hidden order phase)  the system shows  normal metallic dispersion with m^* =3m. At 5.9K  the  gap develops due to hybridization.   This gap was not detected in earlier opt conductivity data because of  too much noise .

He next describes new method – refined thermal reflectance, which allows one to  change T without moving the sample. He divides all the spectra by reflectance at 25K and argues that accuracy increases substantially (noise is reduced by 8). All  T-independent features in conductivity are lost, all T-dependent features stay.

Tom shows data for \sigma (w) down from 75K.  At T decreases, Drude peak forms.  No change in the total spectral weight  at 15 meV ->  total Drude weight is independent on T (m^* remains the same). Conclusion – heavy mass is already present at 75K.  

Second result – 1/tau  is smaller that \omega up to 30K .  The implication is that a coherent Fermi liquid forms at 30K. He shows plot of \pho (w,T)  [=Re 1/\sigma(w,T)].

 \pho (w, T) follows A (w^2 + b (pi^2 T^2))

Discussion on the value of b follows.  In a Fermi liquid, b should be =4.  He finds,  from high-frequency part, \rho (w,T) = A T^2, A = 0.3 \muOm* cm/K^2  . He next looks at DC resistivity, takes  derivative with respect to T , gets the same A.

Shows Matsuda data in a wider range of T d^2 \pho/dT^2  is almost a const up to 100K (hence T^2 form works).

Discusses the difference  between single fermion lifetime  (m \Sigma (w,T) \propto (w^2 + \pi^2 T^2) and optical lifetime  (1/tau \propto w^2 + b \pi^2 T^2) with b=4. He cites many examples of T^2 resistivity with various A

Examples of b=4?  There are none

UPt3    b<1

Nd0.9TiO3  b =1.1   Ce0.95Ca0.05TiO3   b=1.7

Tom discusses theory proposal from two Russian gringo about resonance impurity scattering.  If Im \Sigma has no T^2 term  then optical conductivity has w^2 + b \pi^2 T^2 form with  b=1

He argues that resonant impurities  are un-hybridized uranium f-electrons.

Last part – hidden order phase

Tom shows data   fitted  by Dynes formula for a dirty SC.  He finds one gap  along ab axis and two gaps along c axis (3.1 meV and 2.7 and 1.8  meV) .  Roughly the same gaps have been found from the resistivity fit and from neutrons.

Questions:  (i) about w/T range and about direct vs indirect gap, (ii) is  the two-component electronic model valid in the hidden order phase,(iii) is b=1 vs b=4 related to vertex corrections?

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.