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.