Doug Scalapino, UCSB: Is There Pairing Glue in the Hubbard Model?

Doug's presentation was his response to the recent article in Science magazine by Phil Anderson questioning the validity of a bosonic glue for the high temperature cuprate superconductors (Science 317, 1705 (2007)). Phil's contention is that the pairing is coming from the superexchange energy, J, which is in essence instantaneous in nature. As a consequence, he contends that a proper theory must be quite different from the strong coupling theories developed in the 1960s in regards to electron-phonon mediated pairing. The implication is that J is the "elephant", and that any dynamics causing bumps and wiggles in ARPES, tunneling, and optics spectra is the "mouse".

What Doug and his colleagues like Mark Jarrell have done is to use a dynamical cluster approximation to calculate the singlet pairing vertex in the Hubbard model for values of the Hubbard U ranging from 4t to 12t, where 8t is the bandwidth of the electronic states. This vertex, Gamma, can be thought of as a sum of an irreducible part, Lambda, plus induced interaction terms due to repeated scattering in the particle-hole channel. The latter can be divided into an S=0 and S=1 part. As the temperature is lowered, Gamma develops a momentum structure with a peak at q=(pi,pi), despite the fact that Lambda is structureless (the latter defines an effective U, denoted as U-bar). This behavior of Gamma is mirrored in the S=1 part of the induced interaction (the S=0 part is depressed around q=(pi,pi) instead), indicating that it is this term where the real action lies. They then write down a gap equation, and find out that the dominant eigenvalue displays the d-wave cos(kx)-cos(ky) behavior.

Now what about the dynamics? The pairing self-energy decays as a function of Matsubara frequency out to an energy scale of t, and this is mirrored by the frequency dependence of the dynamic spin susceptibility, chi. Equating the two, he finds that Gamma is equal to (3/2) (U-bar)^2 chi(q,omega), as expected for a spin fluctuation picture for the pairing. Therefore, Doug concludes that at the level of the Hubbard model, one can indeed think in terms of a pairing glue.

Next week, we will find out what Phil has to say about all of this.

Several questions were raised after Doug's talk.

Jorge Hirsch - What are the consequences? Doug - There will be structure in the ARPES and infrared conductivity that can be related to the frequency dependence of the normal and pairing self-energies.

Chandra Varma - Is this approach equivalent to RPA? Doug - Yes, except that the dynamic spin susceptibility is quantitatively different from what RPA gives.

Gabi Kotliar - What about analytic continuation, and how is this work related to my own? Doug - We have a lower Hubbard band, coherent structure near the Fermi energy, and an upper Hubbard band, as in your work.

Peter Hirschfeld - How does U-bar vary with doping? Doug - We don't know yet.

Chandra Varma - Does your pair vertex, etc., scale with the antiferromagnetic correlation length? Doug - We don't know yet.

Peter Hirschfeld: extracting new kinds of information from recent STM data

Peter Hirschfeld

"Extracting new kinds of information from recent STM data"

Peter remarked that he was going to give an extremely rare type of STM talk - one without color pictures, presented on a blackboard. Two topics were presented :

• The extraction of life-times from STM data
• Modulation of the gap in response to the BSSCO supermodulation.

The first topic concerns a new method, developed by Aldridge et al. They have found, empirically, that one can fit the local density of states
with two variables, a local scatering rate, and a local gap. The fit form looks something like





Emprically, these two variables are correlated with the gap determined from the coherence peaks in N(r,E), i.e. Delta_1(r) is found to be basically the same as the former type of gap maps.
Now however, one can pull out a scattering rate as well. All of these variables follow the approximately 30A (five unit cell) correlation length seen in previous gap maps.

Hirschfeld argued that this indicates that the local DOS is therefore sensitive to disorder effects that must be on length scales that are shorter than 30A. The local Green function is determined by



and he argues that r-r1 and r-r2 must be smaller than about 30A for consistency.


One can also extract an effective scattering rate



What is the meaning of this?

Chandra Varma asked whether this is sensible, because the scattering is low-angle scattering?

Hirschfeld pointed out that the scattering rate extracted this way is about 10 times smaller than anisotropic scattering rates extracted from ARPES measurements. From STM

Gamma(max) = 10meV (p=0.08) - underdoped
Gamma(max) = 2 meV (p=0.02) - optimally doped

Comparable figures from Arpes are 100 and 40 meV respectively. The origins for this mismatch may be

• Broadening effects of the inhomogeneity in oxygen content
• Resolution of the ARPES

Hirschfeld then turned to the second topic. In Bisco, there is a 26A, or 4.8 unit cell modulation of the structure. Slezak et al have been able to map out the phase of the modulation, defining contours of constant phase across the maps, and from this, they are able to correlate the local gap with the phase of the modulation. It is found that the gap drops by 10% as the phase goes from zero to pi. In Boguilubov-de Gennes phenomenology, this corresponds to a reduction of the coupling constant by about 30% (?). Peter mentioned similar types of conclusion in a t-J model by Zhang and Rice.

Of course, perhaps if once one correlate this modulation of the gap with the structure, one can gain insight into the pairing mechanism of high Tc. Is this possible?

Hirschfeld pointed out that the supermodulation corresponds to a "tipping" of the octahedra, and that in the region where the gap is largest, the apical oxygens are furthest apart. This may cause the t' - next nearest neighbor hopping to be larger, giving rise to a large Delta.

Philip Phillips, UIUC: " Exact Integration of the High Energy Scale in Doped Mott Insulators"

Philip points out a problem with the naive procedures used to integrate out high energy degrees of freeedom.
As an example, in eliminating the upper Hubbard band in order to
derive the low energy spectra of the large-U Hubbard model,one might miss the enhanced spectral weight for the addition spectrum of holes in the lower Hubbard band. (Comment: a fact which is missed in simple Hartree Fock theory.)

In his talk, Philips sets out to preserve the "2X" sum rule (i.e. the holes' spectral weight is -twice- their doping concentration, as hole doping pulls states down from the upper Hubbard band).

Philips' method of choice is an introduction of addtional fields, a fermion field which counts the high energy doublons (sites with two electrons), and a constraint field -phi- which projects the enlarged Hilbert space back to the original electrons states.
The result is a formally quadratic Lagrangian, with matrix (and space-time dependent) coupling parameters which is a starting point for a saddle point expansion. Its saddle point includes the t-J model terms and is argued to be a better description of the Hubbard model's low energy spectral weight. Predictions were made about a second dispersing peak which may be observed in the ARPES data.
Time limitations have restricted questions to a minimum, but Patrick Lee commented that
although the effective Lagrangian is formally correct, its fluctuations are large in the large U/t limit.

I argued that the traditional renormalization procedures (Brillouin-Wigner perturbative expansion, Real space Contractor Renormalization (CORE)) are somehat simpler and that the t-J model correctly captures the low energy spectrum. However in a later discussion with Philip and Patrick, Philip's approach was understood as an attempt to simplify the calculation of
quasiparticle renormalization and the intermediate energy scale excitations.

Thursday 9th Aug, Patio Discussion

Thursday 9th Aug. 10.30am-1.00am Patio

Short talks and Discussion: provisional schedule

Philip Phillips, UIUC

"Exact Integration of the High Energy Scale in Doped Mott Insulators"

Boris Shklovskii, UMN

``A simple model of superconductor-insulator transition in Coulomb disorder"

Douglas Scalapino UCSB

``Is there pairing glue in the Hubbard model?"

Mike Norman ANL

"What is the Fermi arc?"

Peter Hirschfeld UF

"Extracting new kinds of information from recent STM data"

Ribhu Kaul: Lattice Deconfined Quantum Criticality

In a beautiful and inspiring talk, Ribhu Kaul described his research on DQC ( a pun on QCD) - deconfined quantum criticality. He began with a review of the various ideas about spin liquids - mentioning two ideas -

• the algebraic spin liquid that occurs in RVB, involving Dirac fermions in a gauge field.
• DQC - the unusual fixed point involving deconfined spinons that is conjectured to lie between the valence bond solid and the Neel state in certain two dimensional antiferromagnets.

The physics of the DQC is thought to be described by the non-compact CP1 gauge theory - in which the fields are spinors interacting with a U(1) gauge field. The critical physics of this model is distinct from that of the O(3) sigma model.

The big question however - is how should one disorder the Heisenberg model? Frustration
is difficult to treat using numerical methods. There is a sign problem for Monte Carlo approaches and direct diagonalization can not reach lattices with more than about 40 spins. He described the ring exchange approach of Anders Sandvik, the so-called "J-Q" model, in which a nearest neigbor Heisenberg model has an additional term of the form

Q(S_iS_j - 1/4)(S_k.S_l-1/4)

where the spins are arranged around the plaquet. This model can be treated using an overcomplete RVB basis, and there are no sign problems.

Kaul described his new work with Roger Melko, to be found at http://arxiv.org/abs/0707.2961
where, by using an S_z basis, they have been able to show that the model has a kind of Marshall sign property, where all off-diagonal matrix elements in the Hamiltonian are negative. They can treat this model using Monte Carlo methods at finite temperatures.

They are able to see many interesting things in their simulation. They can measure the spin spin correlation function, and find that it has an anomalous dimension

eta = 0.35,

to be compared with eta - 0.038 for the O(3) non-linear sigma model.

One of the questions raised by Coleman, was whether these techniques can see the two correlation lengths expected in the deconfined quantum criticality scenario? Kaul reminded us that in this scenario, there are two correlation lengths = a "short" one that describes the size of the critical magnetic region, surrounded by a larger one, defining the length scale on which the spinons are deconfined. Beyond this length scale, one either develops valence bond, or neel order. The upper length scale is currently too large to be measured independently, but they certainly see the regime with power-law valence bond order.

Ribhu Kaul raised two key questions

• Can one find another lattice model that displays the same anomalous dimensions?
• Can one confirm this anomalous dimension by a lattice simulation of the CP(1) model?

Tesanovic mentioned that work on the non-compact CP(2) model seems to have ended with a first order phase transition. When questioned by Sachdev, Tesanovic cited this work - of Prokofiev (et al?) - but it seems that this may not apply anyway, becuase it was an x-y model with U(1) rather than SU(2) symmetry.

This blogger came away from this great talk with a very optimistic sense that genuine progress is bein made on the topic of deconfined criticality in spin systems. My questions: When will we have evidence that the same physics can occur with both spins and charges? Can we find a controlled expansion, eg a 1/N expansion, epsilon expansion, in which the observed anomalous exponent can be obtained approximately?

Subir Sachdev: "Theory of the Nernst Effect near the superfluid-insulator transition"

Sachdev presented general approach to transport in quantum critical systems based on the (broadly speaking) Ginzburg-Landau-type field theories. An example of a 2D bosonic superfluid-insulator transition at integer fillings, with its 2+1 “relativistic” symmetry, was worked out in some detail. Elegant connections and parallels were made to various CFTs of highly supersymmetric field-theory models where the universal numbers, critical exponents and critical scaling functions entering quantum transport frequently can all be computed explicitly.

Sachdev stressed that the well known misfortunes of condensed matter physics, with its paltry supply of symmetry, limit our ability to compute quantum critical transport to the hydrodynamic (as opposed to collisionless) regime. Still, he showed how general hydrodynamic arguments and conservation laws in 2D can be effectively used to infer various transport coefficients from the knowledge of only one and how that particular one, say electrical conductivity, can be computed from the 2+1 relativistic quantum critical field theory. He then discussed the effects of perturbations taking one away from the ideal relativistic (particle-hole) symmetry, like the chemical potential, as well as finite magnetic field and impurity disorder. He formulated the connection between his theory and the phenomenology of cuprates and discussed how the calculated Nernst coefficient appears to fit the experimentally observed trends. He also briefly discussed how the theory can accommodate Dirac-type fermions which do not demand a finite Fermi surface. For those interested in the overall philosophy and inner workings of Sachdev’s theory the best resource is http://www.arxiv.org/abs/0706.3215 . Those interested in the Nernst effect in cuprates might also enjoy www.princeton.edu/~npo/VortexNernst/Nernst.html .

The presentation was punctuated and followed by a spirited debate: Patrick Lee noted that Sachdev’s Drude-like expression for conductivity implied significant temperature dependence and stated that this does not seem to be the case for the experimentally observed Drude part of optical conductivity in cuprates. Balatsky and Varma both inquired about the values of various parameters and their connections to measurable physical parameters of cuprates. In response, Sachdev noted the importance of the “speed of light” in the theory as a dimensionful parameter that plays a crucial role in phenomenology. Coleman wanted to know more about supersymmetric theories and to what extent was the loss of supersymmetry injurious to our ability to calculate everything analytically. Sachdev explained how loss of supersymmetry makes it difficult and often impossible to compute general \omega/T scaling functions; one ends up limited to the hydrodynamic regime where the conservation laws can be utilized to evaluate transport coefficients. Scalapino asked about pair correlations.

Cigdem Capan: Superconductivity and Quantum Criticality in CeIrIn_5

Cigdem Capan began with a reprise of the key physics of Kondo lattice symmetries, describing how the physics of heavy electron physics is driven by a competition between the screening of the local moments and the RKKY interaction between them.





She raised three general key questions:

• How does the single ion Kondo effect relate to the Kondo lattice effect?
• Is the phase diagram organized by the singular Quantum Critical Points?
• Does the presence of a QCP favor superconductivity?

Capan introduced the 115 heavy electron materials, already discussed in blogs last week. She introduced the Pagliuso phase diagram that links the Co, Rh and Iridium versions of these tetragonal systems. She mentioned CeCoIn5, for which a variety of measurements indicate dx^2-y^2 pairing. Her main focus is on CeIrIn5. This is a HF superconductor, thought to have a line of gap nodes, but the symmetry is currently unknown.

This was followed by a brief summary of the physics of CeRhIn5 - for which I will refer you to the previous talk by Joe Thompson.

The talk then proceeded to discuss CeCoIn5, where there is a field tuned QCP at the upper critical field, yet to be identified. Here, as one lowers the field towards Hc2=5T,

• The linear specific heat grows and appears to diverge at Hc2 (Bianchi et al)
  
• The A coefficient of the resistivity (rho(T) = rho(0) + AT^2), A diverges as one approaches Hc2 (Paglione et al)
  

It turns out that the QCP is not pinned to the top of the sc phase diagram. First, the position of the QCP is suppressed much faster than Hc2 under pressure. One can carry out an analysis of the Hall coefficient RH(T) which has an interesting field dependence. Cigdem claimed that the Hall data can be collapsed onto a single scaling curve, by scaling the field with respect to the field at which the Hall constant is a minimum. This procedure suggests that the QCP appears to be located at Hc =4.1T.

There is also recent work with tin (Sn) and Cadmium(Cd) doping. Tin doping fails to separate Recent Cd doping by Pham et al that indicates a link with antiferromagnetism. At the "top" of the Superconducting phase diagram, there is an additional phase - which may have links with antiferromagnetism and the FFLO incommensurate superconductor.

Returning to CeIrIn5 - here there is indication that some characteristic temperature scale in the specific heat drops with magnetic field, suggesting a QCP at 27T. The specific heat at high temperatures shows a transition, which appears to extrapolate to this same point. The talk focusses on this putative phase diagram. Here, more careful measurements suggest that at
the system narrowly misses a QCP at 27T, but makes a close "flyby", leading to a rapid evolution in the magnetization, or "meta-magnetic transition".

Sachdev suggested that this transition might be a very weak antiferromagnetic phase transition.

Capan showed dHvA data and argued that there is no significant change in the alpha orbits in the passage past the "metamagnetic transition". There is no significant change in the effective mass m* through the transition.

Varma pointed out that the observed masses were far too small to account for the huge linear specific heat (1000mJ/mol/K2) in this system. The amplitudes are however, anomalously depressed near the MMT - Capan would like to understand the origin of this anomally.

Key questions raised by Cigdem Capan:

• Is there a simple phenomenology to understand the scaling of the Hall effect in CeCoIn5?
• How do we undertand the electron hole assymetry in CeConIn5 (sn vs cd doping)?
• What is the damping mechanism for the dHva oscilations at the MMT transition in CeIrIn5?
• What is the origin of the upturn in the resitivity near the MMT transition in CeIrIn5?
• Why is there no QCP in CeIrIn5?

There is an interesting dichotomy between the larger Fermi surface of CeIrIn5, and the much higher mass of the system. Are the f-electrons more localized - as suggested by the large mass - or are they more delocalized - as suggested by the large Fermi surface. Is the Co or the Ir closer to the antiferromagnetic instability?

Maxim Dzero: Symplectic spins and Pu 115

Time reversal and the symplectic spin of the electron: application to Pu 115 superconductors

Collaborators: R. Flint, P. Coleman

(1) The discovery of the talk

- Superconductivity in the fluctuating valence compounds Pu 115 may arise from two-body interference between two-Kondo screening channels.

Results and contact with experiment:

- The superconducting critical temperature reaches its maximum when the energy levels of excited valence configurations are almost degenerate. This is the case of PuGaIn5.
- One can probably explain the fact that Curie Weiss behavior in these compounds ends at the critical superconducting transition temperature.
- It is predicted that the symmetry of the order parameter is determined by the product of the Wannier factors in the interfering conduction channels. For example, kz2(kx2-ky2).

Assumptions

- That it is the virtual valence fluctuations of the magnetic Pu configuration that create two conduction channels of different symmetry.
- That the two-channel Kondo lattice model is an appropriate description.
- That the mean-field theory corresponding to the large N limit of the symplectic Sp(N) representation of the SU(2) spins is an accurate description.


(2) Questions it raises

Subir: Superconductivity does not occur in this formalism in the single-impurity limit. What symmetry garantees that the V2 and D2 coefficients in the Hubbard-Stratonovich transformation are identical?
Answer: Particle-hole symmetry.

Does the Sp(N) large N solution correspond to the SU(N) large N solution?
Answer: Yes at mean-field but not for the fluctuations.

Why should we consider that the Sp(N) representation is better?
Answer: Because it gets rid of the "dipole" degrees of freedom of the SU(N) representation that do not transform like spin under time-reversal and charge conjugation symmetry. Sp(N) preserves that fundamental property of the physical spins.

Can the antiferromagnetic phase be described in this formalism?
Answer: One probably needs to use bosons.

(3) Questions left open

How big are the fluctuations, at least at the gaussian level?

Should we expect that all two-channel Kondo systems should have superconducting ground states?


Reference

Experimental Talks and Discussion: Week 2.

Tuesday 7th July. 10.30am-1.00pm, Flug Auditorium

Experimental Talks, Short Theory presentations and Discussion

Cigdem Capan, LSU. (50 mins)

``Superconductivity and quantum criticality in CeCoIn5 and CeIrIn5"

Chandra Varma, UCR (10 mins)

``Brief update on experimental evidence for a Phase transition entering the pseudo-gap region of the cuprates".

Subir Sachdev, Harvard (30 mins)

``Theory of the Nernst Effect near the superfluid-insulator transition"

Ribhu Kaul, Harvard (30 mins)
``Lattice deconfined quantum criticality: The search for exotic physics in spin models"

Maxim Dzero, Rutgers/Columbia (30 mins)

``Time reversal and the symplectic spin of the electron: application to Pu 115 superconductors"

Nick Curro: Droplets of Magnetism in Cadmium Doped CeCoIn5

(Delayed Blog posting from Week 1, Thursday, 2nd Aug).



Nick Curro described the results of a new series of NMR measurements that his group (see Urbano et al.) has carried out on Cadmium doped CeCoIn5, Ce(Co_1-x Cd_x)In5 . CeCoIn5 is a heavy fermion superconductor. One of the recent excitements has been the discovery by Pham et al, that the addition of Cadmium induces antiferromagnetism within the superconductor. There is a lot of similarity between the phase diagram of the Cd doped CeCoIn5 and the phase diagram of CeRhIn5. (See Joe Thompson blog, below). Cd is like "negative pressure" in the CeRhIn5 phase diagram. Thus the Cadmium helps to experimentally unify the physics of the 115 materials.

Nick described CeCoIn5 as a Kondo lattice material, in which mobile electrons move through a lattice of localized moments, interacting with the moments via an antiferromagnetic super-exchange J. He showed the Doniach phase diagram, and compared it with the phase diagram of this Cadmium doped material. Cadmium doping is loosely equivalent to "hole doping", and by reducing the size of the conduction sea, the system is driven to the left on the Doniach diagram. (See figure).

NMR measurements are carried out on the Indium sites of this 115 material - there are two indium sites - one of high symmetry, lying in the Ce plane (I) and another of low symmetry, out-of-plane. When NMR is carried out, the NMR line at the I site splits into three peaks, corresponding to three different environments (A, B, C). The A environment is the predominant "bulk" environment, and the 1/T1 signal shows the higher Neel temperature and the lower superconducting temperature. Two interesting features here:

• The normal state above TN does not display a Korringa relaxation rate, but a slower T^1/4 temperature dependence.
• There is no observed effect on the T^1/4 region, due to Cadmium doping.

But the SC Tc is Cadmium dependent, and at high dopings, completely disappears.

The B and C sites are exposed to progressively higher local Weiss fields, and indicate a non-uniform antiferromagnetic environment. Curro says these results show that the Cd is inducing antiferromagnetic droplets, and the C and B sites may correspond to the nearest and next-nearest neighbors, respectively.

In conclusion, the effect of the Cadmium seems two fold:

1. It changes the uniform bulk environment by reducing the hole density and uniformlysuppressing the superconducting Tc
2. It induces droplets of antiferromagnetism which percolate to produce long-range order.

Joe Thompson: "Proximity of Superconductivity and Magnetism in the 115s"

Joe described his work with T. Park and other collaborators on CeRhIn5,
a member of the 115-family of heavy fermion materials that are layered
derivatives of the cubic CeIn3. The attached figure shows the
magnetic/superconducting phase diagram as a function of pressure,
magnetic field, and temperature.

At a sufficiently high magnetic field (above 9T), increasing pressure
induces a quantum phase transition (at P1) from an antiferromagnetic metal
phase to a non-magnetic metal phase. There is some indication that
the transition is second order. Yet, dHvA measurements of Onuki's group
find a jump in the Fermi surface, with effective mass showing a tendency
of divergence at P1. In the low-pressure AF phase, the f-electrons are
localized since the measured Fermi surface is similar to that seen
in the f-less reference material LaRhIn5. In the higher-pressure non-magnetic
phase, on the other hand, the same f-electrons are itinerant because
the measured Fermi surface is similar to that of the bandstructure
calculations in which these f-electrons are assumed mobile.

Joe went on to describe the pressure-induced transitions inside the
superconducting part of the phase diagram. For finite fields (which
are smaller than 9T), there is evidence for the second-order nature
of the transition from a co-existing AF+SC phase (which appears to
be homogeneous) to a pure SC phase. At H=0, the residual specific
heat coefficient was found to undergo a rapid decrease as the pressure
is increased through a threshold value (P2, smaller than P1),
as did the Fermi velocity fitted from the finite-T Hc2.

Joe suggested that the f-electrons remain localized in the SC+AF
phase. He did so based on the aforementioned dHvA observation
of localized f-electrons in the high field AF state, along with
the observation that the ordered moment at temperatures just above
Tc of the AF+SC phase is large.

There was discussion about how strong an evidence the above entail
for the localized nature of the f-electrons inside the AF+SC phase.
There was also discussion on the extent to which the change of \gamma
and v_F across P2 should be associated with the transition in magnetism,
or is instead a reflection of a distinction in superconductivity between
the co-existing AF+SC phase and the pure SC one.

Joe went on to describe the effect of Cd-doping in Co-115 and Ir-115.
A co-existing AF+SC region occurs in the Cd-doped Co-115, but is absent
in the Cd-doped Ir-115.

The main questions that Joe raised are:

* What is the nature of the 4f electrons when they participate
simultaneously in magnetism and superconductivity?

* If we accept the notion that the f-electrons are localized while
participating in the superconductivity, what implications does it
have for the non-Fermi liquid behavior in the normal state?

* In particular, does it suggest some form of Kondo breakdown?

Herb Mook: Search for Magnetism in YBCO Superconductors

H. Mook, ORNL

“Search for Magnetism in YBCO Superconductors”

H. Mook described his work with P. Bourges and their collaborators at the Laboratoire Léon Brillouin (LLB), Saclay, France. The doping corresponded to O(6.6). The Oxygen chain ordering was striking and corresponded to the Ortho-II state (for details, see the reference below). The large crystal weighing 25g was previously studied for the detection of d-density wave: Mook et al. Phys. Rev. B 66, 144513 (2002). Mook reiterated that the previous experiment did exhibit evidence of d-density wave although the signal was weak. The present experiment was designed to detect circulating currents proposed by C. M. Varma in the same sample.

What makes this effort puzzling is that in an unpublished work done at NIST no evidence for circulating currents was found, while the same sample examined in LLB showed strong evidence of circulating currents in agreement with the experiment of B. Fauqué et al. Phys. Rev. Lett. 96, 197001 (2006). To quote Mook, “If you can see it in France, why can’t you see it here?” The joke apart, there may be some serious reasons for this. The beam at LLB was better, with a higher flux and therefore higher intensity. More importantly, the flipping ratio, crucially important for polarized neutron scattering, was much larger, 70 as opposed to 23.3 in the NIST experiment. According to Mook these factors contributed significantly to the success of the measurement in LLB.

Indeed, the rise of the spin flip intensity at (101) at 200K was very sharp and continued through without the slightest hint of the superconducting transition. However, no such scattering was found at (200). The size of the moment is not yet known but the scattering cross section is 1 mb, probably corresponding to a few hundredths of a Bohr magneton. The real difficulty is that the observed direction of the moments is at an angle of 45 degrees, not along the c-direction.

P. A. Lee suggested that it would be useful to repeat the previous measurements at
(1/2,1/2) at LLB. I completely agree.

The bigger issue involves the elusive signatures of magnetic order in the pseudogap phase of high temperature superconductors. I find it particularly puzzling that the same sample shows two signatures of orbital magnetic order: one that breaks translational symmetry of the lattice and the other that does not. However, it is heartening that competing groups are now collaborating to unravel some of the mysteries of the pseudogap. This is real progress.

Laura Greene: Andreev reflection in Ce 115's

Laura presented point-contact spectroscopy results on CeCoIn5/Au contact,
namely the data on normalized tunneling conductance for the wide range of
temperatures. Data shows an asymmetry which appears below certain temperature,T*, most probably associated with a formation of a heavy Fermi liquid. As CeCoIn5 becomes superconducting, the small (~13%) Andreev signal is
observed. Laura presented fits to the data based on the the Blonder-Tinkham-Klapwijk (BTK) model. To account for an asymmetry, the "Kondo-like" resonance peak is included in the density of states. The Andreev reflection data for various tunneling directions convincingly shows the d-wave symmetry of the order parameter. To account for an amplitude of Andreev signal it is assumed that there are two contributions coming from the light gapless Fermi surface and heavy Fermi surface which is gapped. Questions from the audience (C. Capan, A. Balatsky, A. Millis, Q. Si) were mostly concerned with the origin of an asymmetry in tunneling conductance and whether one might think of this asymmetry as a result of the formation of a heavy quasiparticles. Laura's BIG QUESTIONS are:

1. Why does BTK model works so well?

2. What would be theoretical justification for "two-fluid" picture of gapless
and gapped Fermi surfaces?

3. Does the peak in the density of states exist?

Jim Allen: Quasi-1D "Purple Bronze" Evidence for a new state of matter.

J. W. Allen, U. Michigan

"Evidence, arguments and challenges for showing a new quantum state of
matter in the normal phase of quasi-1D Li_0.9Mo_6O_17"


Jim Allen gave an extensive review of the highly anisotropic physics of Blue Bronze. This material is a quasi-1D conductor that develops superconductivity at 1.9 K. This system has a resistance anisotropy

rho|| : rho _|_ (1): rho _|_(2) = 1:10:25

Down to about 26K, rho|| follows a T^0.4 variation and rho _|_ follows a T^1.15 variation.
Below this temperature, both resistivities show an upturn. The upturn was initially thought to be the result of a spin or charge density wave, but susceptibilility, optics, photo-emission and X-ray diffraction seem to rule this out.

Jim described that ARPES appears to suggest this system is more likely to be a Luttinger Liquid, and in the spectra, they can see a holon peak and a spinon edge. The photo-emission spectra show a power-law A(E) ~ (E-E_F)^alpha, where alpha is temperature dependent .

This temperature dependent exponent can, it seems be understood in terms of a two band Luttinger Liquid, undergoing a cross-over from a two band fixed point to a one-band fixed point.

There are several questions raised by this system

• Can the large t_perp (of order 100K) be reconciled with the quasi-one dimensional behavior seen at lower temperatures?
• Band theory predicts two quasi-one dimensional bands, yet only one is observed in ARPES (figure above). Is this because a gap is starting to develop in the second band - and if so - can this be modelled theoretically?
• Does the superconductivity restore the 3 dimensionality to the electron fluid?
  

Experimental Talks and Discussion.

Today's experimental talks will be individually blogged by five members of the audience. Please feel free to comment on any of the postings.

J. W. Allen, U. Michigan

"Evidence, arguments and challenges for showing a new quantum state of
matter in the normal phase of quasi-1D Li_0.9Mo_6O_17"

J. Thompson, LANL

"Proximity of Superconductivity and Magnetism in the 115s"

L. Greene, UIUC

"Andreev reflection in heavy-fermion superconductors and order parameter symmetry
in CeCoIn5"

H. Mook, ORNL

"Search for magnetic order in the YBCO superconductors"

N. Curro, LANL

"Antiferromagnetic droplets in the 115 superconductors"

Open Discussion and Short Presentations

Tuesday, 31st July.

Open Discussion and short presentations.
10.30am-1.pm Bethe Meeting Place.

The group met to discuss the structure of the workshop. Five speakers
gave their perspective on some of the open questions in this field.

1. Andre Marie Tremblay, Sherbrooke.

"Antiferromagnetism vs d-wave superconductivity: Insights
from the organics"

2. Gabi Kotliar, Rutgers.

"Superconductivity near the Mott transition."


3. Qimiao Si, Rice.

"Quantum Criticality and Superconductivity in Heavy Fermions"

4. Doug Scalapino, UCSB.

"Some issues motivated by the cuprate problem".

5. Subir Sachdev, Harvard.

" Fractionalization on the route from Neel order to d-wave superconductivity"


Summary of Discussions

1. Andrey Marie Tremblay: "Antiferromagnetism vs d-wave superconductivity: Insights
from the organics"

Andrey emphasized that an ultimate test of our understanding of unconventional superconductivity, is to see how successful the theory is when applied to a diverse set of componds. The Organic superconductors display many aspects in common with the 115
and the high temperature superconductors - proximity to antiferromagnetism, frustration and Mott physics.

Andrey discussed the K- (ET)2 X layered organics. The physics of these systems is believed to be described by a 2D Hubbard model on a triangular lattice, with hopping t and t'. You can think of them like the cuprates, but with only one t' cross-link per square plaquet. Unlike the cuprates, by changing the anion X that separates the layers, you can tune t'/t

U ~ 400 meV
t ~ 30 meV
t'/t - (0.6 - 1.1 variation).

Andrey presented the results of a cluster dynamical mean field theory calculation on this model. Without pairing it has a 1st order Mott Transition line. For larger U/t, the system develops antiferromagnetism, or spin liquid. For smaller U/t, it enters a d-wave sc phase and then a metal.

Questions: At large frustration, does the d-wave sc become unstable to new symmetries - e.g p-wave symmetry?

Gabi Kotliar questioned the similarity with the cuprates. Here, the superfluid stiffness is a maximum near the Mott transition, whereas it goes to zero at the Mott transition in the cuprates. Others questioned whether this is due to the difference between doping and U/t tuning.

Also - here the temperature dependence of the superfluid stiffness has an anomalous T^3/2 variation.

2. Gabi Kotliar, Rutgers.

"Superconductivity near the Mott transition."

Gabi discussed the phase diagram of the t-J model of high temperature superconductors, contrasting the predictions of slave boson theory, with that of cluster dynamical field theory
(CDMFT). CDMFT predicts significant anisotropies in k-space that are absent from a slave boson theory. In particular

• The rate at which the quasiparticle renormalization constant Z goes to zero in the approach to the Mott transition, is much faster in the antinodal regions. (Measured in the superconducting phase).
• The v_Delta - the component of the qp velocity coming from momentum dependence of the gap, decreases with the doping (linearly? ), whereas v_F remains doping independent.

This last point has some important consequences. In particular, the T coefficient of the
superfluid spin stiffness, "a" in

rho = rho_0 - a T

becomes doping independent, rather than proportional to doping squared, as in RVB theory.
A similar feature is seen in the omega coefficient of chi''(omega) in Raman spectroscopy, which is predicted to be doping independent.

Gabi's questions:

• Are these features observed experimentally?
• What is the origin of this doping dependence?
• Can we understand the solutions to the CDMFT in a simple language, perhaps analytically?

3. Qimiao Si, Rice.

"Quantum Criticality and Superconductivity in Heavy Fermions"

Qimiao started his presentation with the remark

" Most of the really interesting questions about heavy fermion superconductivity have not been deeply explored.

What are these questions? Qimiao began his talk with a summary of the key properties of heavy electron quantum critical points, taking as examples, CeCu_6-x Au_x (doping tuned), CePd_2Si_2 (pressure tuned) and YbRh_2Si_2 (field tuned).

He later turned to discuss the case of the 115 materials, showing a phase diagram that was quite similar to that presented by Andre-Marie Tremblay. 115 materials have the chemical formula

CeX In_5

where X= Co, Rh, Ir. They are layered heavy electron systems that display antiferromagnetism and superconductivity. Application of pressure to CeRhIn5 leads to a transition from antiferromagnetism, to a region of co-existent superconductivity, then into a purely superconducting phase. However, if you apply a magnetic field to remove the sc, there is a single QCP between the antiferromagnet and paramagnet, where the Fermi surface volume appears to "jump". This jump is associated with the delocalization of f-electrons.

Qimiao asked:

• Can superconductivity co-exist with a state that appears to undergo "fermi surface fluctuations" ?
• How should one characterize superconductivity that forms a dome above a second order QCP, particularly one where the Fermi surface jumps?

4. Doug Scalapino, UCSB.

"Some issues motivated by the cuprate problem".

Doug Scalapino asked the question:

What is the (mother) phase that underlies superconductivity?

As an example of the importance of this question, he discussed two different models of how superconductivity emerges from an antiferromagnetic Mott insulator:

Antiferromagnetically mediated pairing, in which the "mother state" of the superconductor is
a nearly antiferromagnetic metal. In this scenario, the omega dependence of the gap function
Delta(k,omega) should reflect the underlying spectral function of the spin fluctuations.

RVB model of superconductivity, in which the "mother state" of the superconductor is a spin liquid - a Mott insulator without Neel order. In this case, one might expect a disconnect between chi(q,omega) and the frequency dependence of the gap function.

But beyond this, Doug listed all sorts of possible "mother states" of high temperature superconductors:

Stripes
d-symmetry CDWs
d-symmetry SDWs
2-leg ladders
Orbitally ordered states

"Now you may not agree of the starting point, or the mother state, but the important point is that there are a lot of different materials, and room enough for everyone!" (paraphrased).

Doug also mentioned the challenge of the s-wave superconductor Barium Potassium Bismuthate (Ba_1-xKx BiO_3) where T_c ~ 40K. He asked whether the underlying mechanism here is solely phonons, or does charge disproportionation

2Ba(4+) <----> Ba (3+) + Ba(5+)

play an important role?

There was a lot of discussion.

Andy Millis raised the issue of the Taillefer group measurements, that reveal a small Fermi surface in the high field state of underdoped high temperature superconductors. He asked - should we regard this as a "mother state" of the high temperature superconductor, or is it just a competing state?

Thomas Vojta felt that if the transition between competing states was second order, then they would still influence one another.

Shankar asked whether we should consider Grandmother states?!

Sudip Chakravarty pointed out that in simple systems like Pb, there was no ambiguity about the mother state - it was a Ferm liquid with e-phonon interactions, but already once one gets to V3Si, the underlying state might already be a CDW.

There was a philosophical discussion about why it was, when physicists manage to link
Arpes data with spin fluctuation data, it is no widely accepted..... Doug said something about Physicists being quite artful at fitting selected data....




5. Subir Sachdev, Harvard.

" Fractionalization on the route from Neel order to d-wave superconductivity"

Subir posed his questions at the beginning of his talk. They are:

• Is the low doping limit of cuprate superconductors a BCS state (+ some other unconventional order), or is it exotic or "fractionalized"?
• If there is an exotic state at low doping - can this state be obtained by doping a Neel state (and not a non-existent spin liquid?)

Subir then discussed some work he has recently done with Senthil, Levin and others,
in which they discuss the effect of doping the deconfined quantum critical point that is thought
to separate a Neel state from a valence bond solid. The key point about this, is that the quasiparticles of the state that emerges are "fractional", in the sense that they carry a non-trivial gauge charge. These particles, they believe, are possibly the origin of the pockets seen in the Taillefer experiment. Subir also described how, when they pair, they form a conformally invariant fluid, in which the superfluid stiffness has a T-dependence that is independent of doping - rather like the results of the Kotliar work

rho_s(x,T) = constant - R T

where R is universal.