YU Lu (Chinese Academy of Sciences) - Non-BCS Superconductivity in underdpoed cuprates by spin-vortex attraction

We will deal with the t-J model in 2D, with a view to application of this model to explain some of the properties of Cuprates.


Spin-Charge Gauge approach

Will use the strong coupling approach:

single occupancy constraint in the standard form, then introduce gauge particles (holons, spinons etc..)

Too many degrees of freedom: there is a gauge symmetry
First look at model with expanded dof, then fix with guage.


Slave boson approach: h boson, z_a fermion
so c_a = h*z_a is a fermion

But in 2D, there are more possibilities - can add fluxes to h and z
U(1) charge and SU(2) spin
with "statistical" compensation - so that c remains a fermion.

In other words, this is the gauge fixing: there is a huge sector of the SU(2) not allowed by U(1) compensation.

We can then start with h as a spinless fermion, then try to optimize these charge and spin fluxes (compatible with statistics) for whatever approximation scheme we use (here, Mean Field theory) Extra things to play with allow to e.g. optimize both t and J terms within the mean field.

In PG phase on 2D lattice, find charge U(1) pi-flux per plaquette works best, independent of doping up to a point.

Having optimized charge flux, look at the effects of the compatible SU(2) spin flux: basically, it attaches a spin vortex to the holon positions with opposite directions on the two sublattices. So the optimal spin flux gives short range AF.

Under doping, long range AF->short range AF via magnon formation by spinon binding.

A long wavelength treatment of the J-term in the MF gives a CP^1 NLsM with an additional term - which ultimately gives a spinon mass.

Metal Insulator crossover

What is origin of M-I crossover? The "obvious" explanation as 2D localization doesn't work - (k_F l) far from Ioffe-Regal limit.

Also, is M-I crossover with and without magnetic field the same phenomenon?

Calculation of conductivity: use Ioffe-Larkin formula R = R_h + R_s
But spinon and holon are bound by the gauge field - the contribution is decided by the slowest and not the fastest component. The holon can show metallic or insulating behavior depending on properties of the spinon: in fact, can even find that insulating behavior is compatible with a finite FS.

These effects give rise to the metal-insulator crossover - and can fit resistivity data very well, with no fitting parameters.

A new mechanism for Superconductivity

Proposed 3-step non-BCS mechanism for hole-doped underdoped cuprates.
a) Glue for SC comes from attraction between spin-vortices on different Neel sublattices - leading to incoherent holon pairs BCS-like at T_ph
b) A gauge attraction (originating from constraint) between spin and charge then induces at T_ps an RVB like pairing between spinons
c) Finally at some T_c the holes (holon-spinon bound pairs) condense, leading to superconductivity.

Some more details of this procedure:

Holon pairing: spin-vortices (which are bound to the holons) behave like a Coulomb gas. Introduce holons on left and right of MBZ, and there is a screening effect as not all vortices are paired. This gives a d-wave pairing OP, composed of two p-waves.

Nodons: Including the gauge fluctuations brings back the gauge invariance. This gives a slave-particle gauge field coupled by minimal substitution, known as the nodon.

Spinon pairing: Our physical particle excitations are the holes, not the holons; must work out how to dress the holons with spinons to get these back. Introduce a RVB-pair field, whose spectrum has a minimum at |k|=|\Delta^s| (c.f. roton minimum in liquid He due to backflow of vortices dressing the bare excitation). This eventually leads to some spinon pairing, breaking the global symmetry from SU(2) to Z_2 via the Anderson-Higgs mechanism. However for this to occur, need sufficiently large holon pair density - so this only occurs at finite doping.

In very simple terms however, the point is that the pairing of the spinons becomes easier because of the prior pairing of the holons.

Finally, one gets phase coherence - i.e. breaking of the global gauge symmetry (including the U(1) original physical one), i.e. this is finally superconductivity.

Summary

To revise: there are 4 characteristic temperatures:

Two crossover temperatures T_ph for holon pairing and T_ps for spinon-pairing
Phase transition T_c for full phase coherence
And another crossover temperature, T^* where the holon FS changes from large to small - which is the starting point of most of these calculations.

Good features of this mechanism for SC:
--Not simple BCS - also involves various crossover phenomena
--At T=0, SC appears at finite doping when doping is greater than the critical long range AF
--vortices allowed above TC, supporting Nernst signal and giving Fermi arcs, not nodes.
--Two positive branches in spinon dispersion which may give something like the hour-glass seen in neutron experiments.

Question (Piers): Can you write a Jastrow like wavefunction for your condensate?
Answer: No, not yet, and also not clear how much of this could possibly be seen in numerics.

There were some other questions, but I didn't quite catch them.

Massimo Capone: Signatures of strongly corelated superconductivity in expanded fulleride Cs3C60

Massimio gives us the outline: pairing in the alkali doped fullerides is most likely phonons but with pecularities which grows with lattice spacing. There is a Mott transition close to the first order superconducting transition.
We stand now with apparently two distinct classes of superconductivity:

• Low Tc: phonon mediated, normal metal above Tc
• High Tc: electronic mechanism and a strange metallic state

But are these two distinct divisions. After all the heavy fermions are presumably electronic but with low Tc (my comment: of course all the energy scales are low there). Claim that the fullerides may show aspects of both. So where do the alkali metal doped fullerides belong?

Start with the parent: C60 - a molecular crystal, a band insulator with three fold degenerate LUMO.

Now doped: AnC60: the alkali donate their s-electrons. Highest Tc is with n=3. But why a superconductor. At first sight these are ordinary BCS superconductors. There is a carbon isotope effect, a regular specific heat jump and Tc increases with density of states as the lattice spacing is changed. Piers asks for clarification about the relation between lattice spacing and the density of states: naively the increasing in spacing lowers the wavefunction overlap and so increases the density of states - is that right? Yes: but perhaps the correlations are effected too - a key point for the talk.
However some interesting features emerge on expanding the lattice with NH3, or mixing Cs and Rb as the metals. In both cases the DOS (from Knight shift) increases but Tc goes down. Moreover some related compounds become Mott insulators - suggestive of enhanced correlations. K4C60 is a spin-gapped insulator.

Now enter the A15 structured (bcc) Cs3C60 which is non-superconducting at ambient pressure and yet has very large lattice spacing. It superconducts from 4-7 kbar with Tc(max)=38K reminiscent of the phase diagram of the cuprates and further pressure gets you to an AFM phase with TN=40K S=1/2. There seems to be some coexistence of superconductivity and AFM. Andrey asks about the coexistence region. Henri answers: this is a materials issue and that a pure system has a first order transition with no coexistence. Contrasting with the fcc structure Cs3C60 has exactly the same Tc dome as a function of pressure but the scale of the AFM is an order of magnitude lower. Thus magnetism is not setting the scale of the superconductivity. Massimo claims that Mott is the key to superconductivity - but Piers plays devils advocate and says its separated from the metal by a first order transition so why should it have anything to do with the superconductivity? Claim that the talk will answer this...
Now to theory (which predates these recent experiments...) . A three band Hubbard model (with band width of order 0.5eV and U ~ 1 or 1.5eV) so correlations playing a role. The add a Hund's rule:
-JH(2S^2_i + (1./2)L_i^2) - (5/6)J_H(n_i-3)^2
and then add electron-phonon: a t_{1u} electrons coupled to the H_+{1g} which contains some Jahn-Teller physics.
The claim is that this coupling mirrors the Hund's rule coupling but with an opposite sign - they play a negative role. Unlike the usual case where slow phonons overscreen the local Coulomb interaction, here retardation plays less of a role since it fights with the Hund's rule rather than Coulomb. Taking realistic numbers the e-ph coupling exceeds the Hund's rule so the phonons favour a small S state inverting the usual Hund's rule.
Now we try and extract some physics first from the small U/W limit: Perturbation theory gives an s-wave superconductivity - adding U then there is a competition and if U> J_eff there is no superconductivity.
Now think of U>>W. Three electrons become stuck on the site (bucky ball) but J is free to act and so it forms a low spin S=1/2 state (consistent with later expeiments). But still no superconductivity. So to look for superconductivity Massimo and colleagues to DMFT. Its a good first step since the interactions are all assumed to be local to the Buckyballs. Now we are being reminded of the DMFT approach to the Mott transition showing the upper and lower Hubbard bands and a quasiparticle band in between which has Z->0 and m^* diverging. The solution (on a bipartite Bethe lattice) of the DMFT equations gives a phase diagram with metal, s-wave superconductor and then AFM as U/W is increased (ie like increased pressure). The plot seems to show coexistence of AFM and superconductivity and Ken questions it. Massimo clarifies that two solutions exist at that point but it does not mean they coexist. Erio says it is a bad figure.
Question: Andriy asks how come s-wave exists in a large U limit? Erio answers that this is a multi-orbital system which allows the electrons to avoid each other while still pairing in the s-wave channel. In more detail: they find that U is effectively renormalized downwards with the quasiparticle Z as the Mott transition approaches for the quasiparticle, but J is unrenormalized and so eventually wins to and drives superconductivity. But how come J is unrenormalized. Mott freezes out the charge fluctuations but the spin and orbital degrees if freedom are still fluctuating: J_eff ~ Z^2 L_s^2 J ie the vertex correction cancels the Z since L+s ~1/Z. YuLu asks is there a Ward identity protecting it? Piers asks what symmetry is there left which is conserved and would spin-orbit coupling prevent this? An open question...
Now into the final part of the talk and Massimo introduces a simple model with just two bands and an inverted J which they can also treat in DMFT. Some insight comes from the impurity model which has a critical point sick with groundstate entropy. On the lattice superconductivity quenches this and sharpens the quasiparticle peak and gives a contribution to the condensation energy (like, it is claimed, in the underdoped cuprates). So a prediction is that the superfluid contribution to the optical conductivity is larger than the Drude weight of the metal. They also compute the specific heat jump, uniform susceptibility and photo-emission spectra (with a pseudogap). So the normal state should not be so normal and there are testable predictions.
See Rev Mod Phys 81, 943 (2009).

Question time:
Andrei C.: Is Z a constant or should it really be frequency dependent - do you see this? Answer the complexity of the three-band model means Massimo can not reliably be confident in any statements about this, but he does not think it is relevant.
Henri: What is the pseudogap in this model? Answer: it is J in the model and is more or less constant with U/W. Henri so would you not expect it to go to zero in the metallic side? Yes
Zlatko: This is really just an electron-phonon superconductor? Answer: yes.
Piers: what about the spin entropy - how much goes into the condensation energy? If it is a lot, as 3/2 to 1/2 to 0 suggests, then this should be very different from an electron-phonon superconductor. Answer: I think it is log(2). Piers - so this is massively different from an electron-phonon superconductor so the answer to Zlatko's question should be no!
Q: (Sorry I missed this in typing the above...)

Peter HIRSCHFELD: Accidental order parameter nodes in Fe-pnictide superconductors

OUTLINE:
(I) FeAs experiments in SC state
(II) spin fluctuation theory of FeAs superconductors
(III) phenomenology: qp transport in 122 systems

Started with the overview of DFT-calculated band structure.
LOFP(1111): Lebegue 2007 (Tc=6K)
LOFA : Singh & Du 2008 (Tc=26K)



(I) FeAs experiments in SC state
Phase diagram shows diversity from one compound to another: e.g. Ce-1111 has no AF-SC coexistence, whereas La-1111 does.
Inconsistency in early measurements: NMR T1~T^3, whereas penetration depth and ARPES showed full-gapped superconductor (not unlike the early days of the cuprates).

Possible symmetries of OP, classified by symmetry representations:
A1g (s-wave)
A2g (g-wave): xy(x^2-y^2)
B1g (d-wave): x^2-y^2
...
Nodes or no nodes?

*PENETRATION DEPTH
Assuming line nodes,
\lambda(w) ~ T^2 (dirty limit) or ~T (clean limit)
Experimental data all over the place, e.g.:
*Sm-1111 (SOFFA): exp-l dependence, i.e. full gap!
*La-1111: ~T^2 dependence (Ames group)

*THERMAL CONDUCTIVITY
LaFePO: \kappa/T -> const as T->0 (nodes)
K-doped Ba-122: \kappa/T ~ T

(II) Spin-fluctuation theory of FeAs superconductors
History: Berk & Schrieffer (1961)
RPA: SC pairing proportional to spin susceptibility
Implications: Peak in spin fluctuations at (pi,pi) is taken advantage of by d-wave order parameter, even with repulsive interactions.

Graser (2009): pairing functions display gap nodes
Also: Kuroki '08, '09; Ikeda '09,'10

What is origin of gap anisotropy? [Maier et al PRB'09]

1. Orbital character on Fermi sheets
2. scattering between beta-1 and beta-2 sheets
3. intraband Colulomb repulsion

Kuroki et al. discovered a hole-like \gamma-pocket at (pi,pi), which grows upon hole-doping. It is d-xy in character and turns out important for the symmetry of the OP. This pocket helps overcome frustration by intraband Coulomb repulsion and beta-beta pocket scattering.

Large inter-orbital pairing stabilizes s+- state on xz, yz portions of Fermi surface.
This extended s-wave state may still develop (accidental) nodes: depends on the details of the model.

(III) 3D superconductivity in 122 systems
Ni-doped 122: penetration depth measured by Prozorov's group (Ames): Martin et al. 2010

Co-doped Ba122: \kappa/T as T->0 by Tanatar et al (Ames, Sherbrooke), PRB (2009)
undoped material: no nodes
doped: either finite value of \kappa/T (i.e. nodes), or roughly linear in T (deep minima)

Co-Ba122: Reid et al, 2010: depending on the direction J||a or J||b, \kappa/T either shows a finite value (nodes) or zero (isotropic gap) - how is it possible??

Hirschfeld's group calculations (Mishra et al, 2009): nodes near k_z=pi, so that \kappa/T is finite as T->0.
Playing with possible node structures: lambda(T) behaves as non-universal power-law, e.g. T^1.3, T^2.16 etc, and qualitatively describes the anisotropy seen by Reid et al. (2010).

CONCLUSIONS:

• the symmetry of the OP is always A1g (extended s-wave), and generically is not required to have nodes
• nodes may appear accidentally, as a consequence of orbital anisotropy on the Fermi surface and intra-band repulsion
• various experiments (penetration depth, thermal conductivity) can be explained, qualitatively, by playing with the gap structure.

QUESTIONS:
Q: H. Alloul
Asking whether any ARPES experiments were able to observe the \gamma pocket at (pi,pi).
A: not aware of a clear exp-tal signature of all 3 hole pockets.

Q: A. Chubukov
Witihin RPA approach, does one always get a gap with nodes, if one neglects the extra hole pocket at (pi,pi)?
A: In this case, we always find a strongly anisotropic order parameter, with nodes or near nodes.

Q: Y. Grin
1) Are As-p states near Fermi level important?
2) In 1111- and 122- compounds, is the orbital contribution on the Fermi surface the same?
A: 1) the effective interactions between Fe-orbitals are mediated by As atoms, so they contribute indirectly
2) in kz=0 cut, the 1111 and 122 have identical orbital composition; at kz=pi, one expects them to differ, but detailed calculations haven't been carried out yet.

Q: P. Coleman:
Is the effect of repulsive \mu* component included in the calculations? Can it be that pure s-wave is favoured alongside s+-?
A: (with comment by Chubukov): The bare repulsive intra-band U is likely to suppress s-wave pairings. But in the spirit of RG (see previous talk by Z. Tesanovic), a situation can occur where the renormalized pair-hopping value is stronger than the intraband repulsion, making SC possible.

SEBASTIAN - Quantum oscillations

Started with a brief introduction to Quantum Oscillations.

1) Oscillation Frequency -> Area of the Fermi surface

2) Can determine effective mass from temperature dependence

3) Map the Fermi Surface

What would the ground state of High Tc be if we kill superconductivity?

Checks:

A)Periodicity in 1/B strongly suggests Landau Quantization

B) Temperature Dependent Amplitude (Fermi-Dirac statistics obeyed)

Find extremely small pockets 40x smaller than band structure calculations.

Higher fields allow better resolution.

Warped cylinder leads to beating pattern. Fit data to 2 cylinders -> one normal + 1 warped.

Between YBCO and Tl2201 find 40x change in area of Fermi surface.

How to reconcile non-Fermi liquid at zero field with Quantum Oscillations

In Vortex liquid state when Q.O. seen.

Andre pushes us to stop asking questions at 9:36am... well done!

From contact-less measurements can see higher harmonics of original frequencies. See 3 fundamental frequencies.

Electron pockets suggested by negative Hall or Hole pockets at the nodes. However size of pockets doesnt match doping.

Multiple pockets suggests large warping of on pocket... putting them at different locations.... holes in one place electrons in another...

Angular dependence also agrees with warping.

Warping suggests pockets at different regions since have different c-axis hopping parameters.

Implicit translational symmetry breaking leads to different pockets. But no long range order has been observed.

Zeeman splitting of the Fermi surface? Look at this with rotating magnetic field since orbital and spin have different angular dependence. Zeeman splitting should therefore lead to flip in phase of the oscillations, not seen.

Keimer et al claim to see inellastic bragg peaks in field. but not in the same doping.

How to reconcile with Arpes?

Holes + electrons. Large part is hole part seen in Arpes. Area doesn't vary with doping but mass seems to diverge at lower doping.

Looks like MIT transition seen in resistivity goes to zero at doping where mass diverges.