Patio Discussion on Shubnikov de Haas oscilations in the cuprates

On Wednesday morning, the group met for a very animated and very exciting Patio discussion about the implications of the Shubnikov de Haas oscillations recently observed by the Taillefer group (Doiron-Leyraud et al.) at high fields in YBa2Cu3O6.5 and more recently in the double layer YBa2Cu4O8 compound (Bangura et al arXiv:0707.4601). There is a lot of excitement about these measurements, which may be linked to the mysterious Fermi arcs seen in underdoped cuprates using ARPES spectroscopy.

The discussion was hosted by Andre Marie Tremblay.

Andre summarized the key observations. In underdoped YBCO with a nominal hole doping of

p=0.1





The effective mass of the carriers, obtained by fitting the temperature dependence of the oscillations (See c) is

m* ~ 2-3 m_e

Measurements were made at above 50T on YBCO. Paradoxically, even though the SdA oscillations suggest small hole pockets. Andre Marie discussed how band theory can not account for these small hole pockets.

Here are some of the key issues that came up in the discussion

1. If the measurement is made on hole pockets, then why is the Hall constant negative (corresponding to electrons)?
2. Are we sure that 60T - or even higher 80T measurements are really in the "normal state".
3. The huge size of the pseudogap, the observations of the Nernst effect all suggest that the flux flow regime of the underdoped cuprates may extend far further than these fields.
4. Do the oscillations represent conventional Schrodinger Landau level oscillations (possibly damped by pair fluctuations) - or could this be some kind of Landau level quantization of quasiparticles - even Boguilubov quasiparticles around a nodal point?
   

Five speakers then gave brief presentations.


Subir Sachdev discussed the effects of holes moving in an antiferromagnetic, or possibly a quantum critical spin background. In a antiferromagnet, the unit cell is doubled, and work carried out long ago by Schraiman and Siggia, supported by numerous subsequent work leads to the prediction of two hole pockets, so now

n = 0.075/2

and the hole density of 0.075 is closer to the nominal p=0.1 - there is a smaller discrepancy with Luttinger's theorem.




Subir asked: is it possible to get hole pockets without broken symmetry. He argues (see previous blog) that if there is topological order, with gauge excitations, one can have holon pockets - which are spinless - invisible to ARPES but which still give dHva and SdH oscillations.

The audience cruelly asked Subir if this scenario predicts a negative Hall constant. Subir admitted that it probably would not.

Subir also discussed the possibility that a superconducting-insulator transition might be able to give a negative Hall constant, but there was not enough time to pursue this point.

Michael Norman gave a brief review of earlier attempts to carry out quantum oscillation measurements on cuprates at high fields. He pointed out that three other measurements gave
oscillations in broad agreement with the Taillefer measurements - if less reliably.



He also mentioned Zlatko Tesanovic's work on dHvA in the mixed state - the main point here is that the average gap around a quasiparticle orbit is zero in a d-wave superconductor. Providing that the gap is smaller than the cyclotron frequency, one can have Landau Levels.

Such oscillations have been seen in conventional superconductors too - such as V3Si and NbSe3.

Norman also discussed the Hall constant, which changes sign as a function of doping in the cuprates in the flux phase. Larkin et al made a theory of this, correlating it with dT_c/dmu,
but the sign was wrong. He reported that whereas old measurements had found negative Hall constants over a narrow range of the phase diagram - Taillefer now finds it extends over a large region at high fields, also in the 248 material.

Mike raised the following questions

• Is there a conflict with Photo-emission?
• Are the pockets electrons or holes? He noted that this can be determined by looking at the relative phase of the rho_xx and rho_xy quantum oscillations, or M and rho_xy - and work is now underway in this direction?

Mike discussed whether the observed pockets might somehow be associated with the "banana-shaped" surfaces of constant quasiparticle energy that are known to exist around the nodes of the d-wave superconductor. He pointed out that since the ratio of v_Delta/v_F ~ 20, these long, thin regions would have to extend out to the zone edge to get the right areas.

He ended by asking

• Is it a field induced effect?
• Is it a field induced state?
• Is it oscillations around a nodal particle-hole ordered state like a d density wave?

Phil Anderson then gave a brief presentation. Phil pointed out that we really don't know Hc2 for these systems. He argued that we know that T^*, the pseudo-gap temperature scale sets the scale of the gap anti-nodes - this is a pairing energy at energies of order J/2 he said, which is several 100 meV. This is much larger than the observed fields, so he said, crossing Hc2 is simply out of the question, even at 80T.



Anderson then went on to talk about the Fermi arcs. Here's his argument, as close as I could capture, verbatim

"Near the nodes, there's still a gap (at these fields) and its fairly hefty. The Fermi arcs form by the electrons buming against the gap in the anti-nodes, giving rise to Andreev scattering. In Andreev scattering a pair of electrons go on, a hole goes back in the opposite direction. The arcs form because electrons are bumping against the gap. Its Andreev reflection. "

Phil mentioned an old paper of his, which he said that Marcel Franz and Zlatko Tesanovic (in the audience) had effectively destroyed- but he felt it is still relevant. He said that when an electron turns into a hole through Andreev scattering - it leads to a kind of Zwitterbewegung. He drew a picture of electrons scattering back into holes inside a square well.

Phil ended by saying that he worries that all the current proposals don't take into account
Andreev scattering, and Andreev scattering is not normal scattering. The advantage of such a scenario, he said, is that the sign of the Hall effect would not be important.


As you can imagine a lot of discussion followed. Here's a brief summary

Chandra Varma:" We would like to know Hc2"

Phil Anderson: "Its some kind of Vortex matter state."

Chandra Varma: "These are just murky words that don't amount to anything very much."

Zlatko Tesanovic: "There is very strong sign of some kind of vortex matter state"

Chandra Varma: "Zlatko has solved this problem at large field ,he should talk by himself."

David Pines: "I'm quite taken with Phils notion that one has to understand how the quasiparticle goes (Andreev reflects) around the Fermi surface. The gaps are so big that they are not going to be affected by the magnetic fields."




Next up came Assa Auerbach. Assa presented a low energy Hamiltonian called the plaquet fermion model, which he argued has features that can account for the observed phenomena. The plaquet fermion model is a Hamiltonian containing mobile 2e bosons and mobile fermions that scatter via Andreev scattering. The dispersion and existence of these objects he argued, can be determined from finite size diagonalization of the Hubbard model on small plaquets.

The blogger is not sure he understood the full gist of this theory, but he said that the f-fermions in his theory have fermion arcs, and when the bosons condense, this produces a standard BCS dispersion

E(k) ~ Sqrt[(epsilon(k)-\mu)^2 + (d_k b)^2]

where epsilon(k) is the dispersion of the electrons moving on a small pocket.

Chandra Varma argued that this is not consistent with ARPES, where one has never seen two peaks in the spectrum. Mike Norman agreed with him.

Muramtsu argued that the finite plaquet diagonalizations were, in effect, holes moving in an antiferromagnetic background - in other words - he felt that Auerbach's pockets were really hole pockets in an AFM. Assa Auerbach disagreed strongly.



After this, by popular request, Zlatko Tesanovic stood up to discuss his ideas, developed with Marcel Franz, on the theory of quantum oscillations in superconductors. Zlatko has since posted a set of notes on these ideas, which you can obtain here. Zlatko began by remarking that the main issue divided into whether the underlying order was

particle-particle (pairing or pair fluctuations)

particle-hole (density waves, circulating currents..)

In this latter category would like graphene, d-density waves, Chandra Varma's theory of circulating currents (see blog by Y. B. Kim).

Conventional normal state de Haas van Alphen, he said, is caused by the Landau quantization of the Fermi sea. According to conventional wisdom, there could be no dHvA oscillations in a superconductor, because the gap mean there were literally, no states that could undergo any sort of Landau Quantization. ("No density of states, no oscillation").

However, both experiment, and detailed theory, paint a different picture. If one lowers the field at low temperatures, he said there were three regions

I - Above Hc2 - normal dHva oscillations

II- Below Hc2, where Delta is smaller than the cyclotron frequency - dHvA oscillations with the same frequency, but damped by the pairing

II - Region III, where Delta is larger than the cyclotron frequency -here - dHvA dies.

He made a remark that this was an example of KTN^2 - a kind of topological transition.
I did not understand.

Zlatko then turned to the situation in d-wave superconductors. He referred to the ideas of Gorkov and Schrieffer, and later Anderson - who proposed that in a d-wave superconductor, some sort of Landau quantization would occur around the nodes of the 2D d-wave sc.

Unfortunately, this idea turns out to be wrong at the lowest energy scales, because the phase of the superconductor has to be taken into account, and when this is done so- the gauge field associated with it cancels the effect of the field for those quasiparticles that wind around the vortices in the mixed state. There are he said, unfortunately, no Dirac Landau levels
in a d-wave superconductor.

He sketched the energy levels of the superconductor, showing how the Schrodinger Landau levels ultimately die as one reduces the energy down towards the node. Here's the point he said. In a Dirac Landau Level, the Hamiltonian looks like




However, in a field, the field is replaced by p-> p - e A, so that this becomes ( a minimal gauge coupling)


The change of sign in the coefficient of A for particles and holes is significant. This is different to the situation in graphene, where the vector potential couples to both diagonal elements with the same sign.


Zlatko's discussion was interrupted by lunch, and resumed on Thursday morning.

Zlatko's online notes on electrons in the mixed state.