Rafael Fernandes

Interplay between superconductivity, magnetism and nematic/orbital order in the iron pnictides

Blogged by Andy Schofield and Kedar Damle 

Started with in introduction of spin-fluctuations and other degrees of freedom. We have heard much about cuprates and the Mott transision We will be looking at pnictides where we are dealing with band physics. But the bands are complex yet it simplifies to some q2d electron and hole pockets ( a theorist’s view): Hole pockets in the centre of the zone (circular) and elliptical electron pockets at the X points:

Question: What about experiements? Good agreement with LDA in terms of shape though not always the mass, but see ARPES data later.

1. Unconventional Superconductivity:

A review of BCS superconductivity with phonons: The Cooper instability set as an exercise! The problem of computing Tc from first principles is difficult because of the lack of ability to predict the Coulomb pseudopotential. Rafael now moves to the blackboard. Want compute the partition function and he introduces the action for the electron phonon interaction: in the path integral formalism with coherent states. Then we get the Nambu representation of the problem where the electrons are a spinorial like combination of particle and hole. Constructing the perturbation theory gives a Dyson equation for the self-energy - and an Ansatz for developing superconductivity has a self-energy with off-diagonal parts representing the energy gap, W. It has a new dispersion E and a quasiparticle renormalization Z. BCS gets obtained in the limit that Z=1, E=dispersion and W is the gap.
Question: What is the physics of this limit? Answer: it is the limit of weak coupling.
Question: Is BCS limit applicable away from weak coupling - like in cuprates? Answer: no - but see later.
Question: Could you better define weak-coupling in terms of the parameters you introduced? Yes - I can introduce lambda which is like g^2 x density of states. There is another factor (m/M) which is small - ratio of the electron mass to ionic mass - so we can neglect vertex corrections.

So - are the iron based superconductors electron-phonon coupled? One calculation claimed maximum from e-ph is 0.8K in the pnictides - so that looks unlikely. Now let us look at the symmetries of the possible superconducting state: Crystal Lattice (X) Rotations (R) Time-reversal (T) and U(1) gauge. We look at the Cooper pair wavefunction in systems with inversion symmetry: singlet must be s- or d-wave while the triplet could be p-wve or f-wave etc. We will restrict to tetragonal singlet states with no T-breaking. Looking in 2D we have some simple possibilities:
s-wave: A_1g: no nodes - or x^2+y^2 extended s-wave with line nodes that are not fixed to symmetry lines.
g-wave: A_2g:  state unchanged by 90 degree rotation, but nodes along every 45, 90, etc degree - 4 lines of nodes
d_x^2-y^2: B_1g: state unchanged by 180 degree rotation - nodes at 90, 180 - 2 line nodes
d_xy: B_2g: line nodes
Electron-electron repulsion tends to favour nodal superconductors and we will consider pairing fluctuations by spin-fluctuations (Berk and Schrieffer, PRL 1966). We revisit these calculations and change to spin-fluctuations from bosons. The electrons now couple to a vector field and the new inverse greens function of the boson is the spin-propagator discussed in Andy Schofield's quantum criticality tutorial.
Question: what about vertex corrections? I will come back to this.
The spin-density means that the sigma_z involving the density goes to the identity which changes a sign to make the interaction effectively look repulsive not attractive.
Vertex corrections: now instead of being a small term (m/M) the spin fluctuations are also from the electrons so we cannot use Migdal's theorem. 
Question: I worked in Migdal's lab - and the theorem does not work if you use bare variables? AVC answers: there is a critical value below which this approach still works. 
Question: are we here in 2D where the boson coupling might be a relevant operator? Answer it is always relevant and we going to develop a theory here.
When visualizing the interaction it is repulsive at short range but changes sign and oscillates etc - and the diagonals it is purely repulsive. So put the nodes along the diagonal lines.
More on this in the next lecture...

Good evening blog watchers---we're back for the second installment
of Rafael Fernandes's double-header on superconductivity, magnetism,
and nematic/orbital order in the iron pnictides.


In the last lecture, Rafael gave a sketch of Eliashberg theory of phonon-mediated superconductivity (conventional)
and introduced the symmetry analysis of the SC order parameter,
focusing on singlet superconductivity with no time reversal
symmetry breaking in 2 dimensions (see description of previous lecture)
and then asked:
 
Why would the gap have nodes when the gap helps gain energy?
 
And his answer was: repulsive interactions.
 
To see how that works, he took the example of interactions mediated by
spin fluctuations and repeated Eliashberg theory, this
time for an interaction which couples the electron spin density to antiferromagnetic spin fluctuations, and showed
how the final gap equation gets an additional minus sign.

This minus sign comes from
the fact that the antiferromagnetic spin fluctuations mediate
a bonafide repulsive interaction.

Now in the second lecture, he's promised to start with this repulsive
interaction and show us how it leads naturally to a gap function with nodes when we look for solutions to the gap equation. So here we go with the
ball by ball commentary, a la cricinfo, of Rafael's second innings...

To begin, Rafael notes that the effective interaction in real space is proportional to \chi(r).
The gap would like to vanish across diagonals to avoid unfavourable
part of \chi(r), and that is the origin of the diagonal nodes in the gap.
 
For the detailed analysis, it is better to go back to momentum space and think
in terms of different parts of the Fermi surface connected by Q (Q is \pi,0
or 0,\pi). 

Simplification:  We can assume
V_{k-k'} = \chi(Q) \delta_{k-k',Q}
since the  real interaction has a strong peak at Q.

So we take two Deltas, \Delta_1 and Delta_2, on the two pockets
that are connected by Q, and get a two by two system of coupled
BCS equations for \Delta_1 and \Delta_2. Further, we can linearize
the gap equations (near T_c) and get two solutions:
One solution has an eigenvector with same sign for \Delta_1 and \Delta_2
The other has a relative minus sign. 

The equal sign solution requires
*attractive* interactions---this is the s++ state of Kontani and Onari,
which would need *orbital* fluctuations to mediate attraction.
The solution s+-, with opposite relative sign, is the one picked
by repulsive interactions, since the repulsive interaction looks effectively
attractive for this eigenvector.

Question: What is the cutoff to use in the momentum sum in the BCS eqn?
Ans: Bands are very shallow, so size of the pockets is the appropriate cutoff

Question: What about Coulomb repulsion (local)?
Ans:  Can show that this state can survive with little reduction in T_c

With this sketch of the mechanism for superconductivity, Rafael switches
focus to magnetism in the normal state:

The ordering is along two equivalent wavevectors Q_1 = \pi,0
and Q_2=0,\pi.
 
First question to address is: How do we describe the magnetically ordered state?
With an itinerant description or a localized picture of local moments ordering?

To explain this, Rafael now digresses to an elementary discussion
of antiferromagnetism in the Mott insulating state of the Hubbard
model, and explains how virtual charge fluctuations give antiferromagnetic
exchange interactions between the local moments of the Mott insulator.
This is contrasted with the prototypical example of an itinerant magnet, a Stoner
ferromagnet: 
Polarizing a free Fermi gas costs kinetic energy.
But a polarized free fermi gas pays less Coulomb repulsion energy
when we turn on interactions. At some critical value of the
Coulomb repulsion, this leads to ferromagnetism. This critical
value of repulsion is inversely proportional to the DOS at the Fermi
energy (Stoner criterion). Generalizing to ordering at some
other wavevector, say Q, the Stoner criterion becomes:
When U_Q \chi_0(Q) >1, itinerant SDW state is favoured.
Further, \chi_0(Q), the non-interacting susceptibility, is enhanced
if there is nesting with nesting wavevector Q, so it
is easier to have SDW instabilities when there is nesting.

These are the two limiting ways of thinking about magnetism.
Even in simple magnets like Ni and Fe, *neither* picture
applies literally, and we have to just make do with whichever picture
is closer to reality.

After this digression, Rafael makes his main point:
The pnictides are metallic, so it makes sense to choose the itinerant
picture to describe magnetism, even if they have significant correlations.

Question: When there is nesting, doesn't a gap open and destroy the Fermi surface?
Ans: That only happens if there is perfect nesting. Here we are not
thinking of perfect nesting.
In the pnictides, Q_1 and Q_2 are actually approximate nesting wavevectors
that connect the hole and electron parts of the Fermi surface. This nesting
gradually goes away on doping. And that is when the magnetism
goes away too. So the itinerant picture seems to hang together all right.
Of course, since the nesting is imperfect, there is still a threshold interaction
without which we would not have the SDW.

With this background, Rafael moves on to a more detailed
theory for the itinerant magnetism, basically doing a Hertz-Millis
theory for two simultaneous SDW instabilities (along Q_1 and along
Q_2). The final punch-line (details went by too quickly for a
ball-by-ball account) is

F_mag = \frac{a}{2} (\vec{m}_1^2 +\vec{m}_2^2) +\frac{u}{4}
(\vec{m}_1^2 +\vec{m}_2^2)^2 - \frac{g}{4}(\vec{m}_1^2-\vec{m}_2^2)^2

From the details that went by too quickly, 
Rafael concludes:  g>0

This g>0 picks stripes along one or the other direction from the plethora
of states that minimize the g=0 free energy. And these are exactly
the states seen in experiment. This gives qualitative agreement.
But does it work quantitatively?
The answer seems to be yes(!)
[the details again went by a bit too quickly for this blogger]

Q: What happen when g gets very large?
Ans: First order transition. But g is never that large in
the pnictides, so we don't need to worry about this possibility.

With magnetism in the bag, Rafael now moves on to discuss nematicity
in the normal state (already mentioned in earlier talks this week).

 Basic observation: The structural transition in the pnictides closely follows
the SDW transition, being slightly above the SDW transition (at slightly
higher temperatures). 

In the orthorhombic phase (below the structural
transition), experiments see strong anisotropies that cannot be attributed
to lattice distortions alone: A very tiny uniaxial pressure (of order
mega, not giga pascals) immediately leads to a resistivity anisotropy.
Now, the maximum resistivity anisotropy is at finite doping and
almost 100% in magnitude, while the maximum orthorhombic distortion
is at zero doping and tiny by comparison. So there must be
something electronic about the symmetry breaking of tetragonal
symmetry, and the lattice follows in the wake of the electrons.
This is what people mean when they talk of electronic nematicity.

Question: Are these measurements (of resistivity anisotropy) done above or below the SDW transition?
Ans: Both above and below SDW transition. Of course, the anisotropy
is largest below the SDW transition, but it is very big even above the
SDW transition (in comparison with the orthorhombic distortion).

Take-II on nematic order, starting from F_mag:

Ordering at Q_1 and Q_2 are both equally favoured.
To pick one of them, system has to break O(3) spin symmetry
and break the Z_2 symmetry (the symmetry that says Q_1 and Q_2
are equivalent).

Are both these symmetries necessarily broken simultaneously?
No!
In fact, the discrete symmetry breaking is less susceptible
to fluctuations, and we can imagine that the Z_2 symmetry
breaks first without O(3) symmetry breaking. This state,
with Z_2 symmetry breaking but no O(3) symmetry breaking, is
to be identified with electronic nematicity above the SDW transition.

In this picture, the nematicity comes from the spin physics, and other
things follow because they are coupled to the spin physics.

To put some flesh on this, in the last part of the lecture, Rafael moves on to consider the role of spatial fluctuations in \vec{m}_1 and \vec{m}_2
starting with F_mag derived earlier:
 
To do this, we decouple both quartic terms (g and u)  by two
separate Hubbard Stratanovich fields. The HS field dual
to the g term is the nematic order parameter. Integrating
out \vec{m}_1 and \vec{m}_2, one gets an
action functional for the two HS fields---one of these, \phi,
is the nematic order parameter, and the other \psi, gives
the magnitude of the tendency to magnetic order.
Next, we look for the saddle point of this action.
The saddle-point condition gives two coupled equations
for \psi and \phi at the saddle point.

Punchline: \phi is non-zero at the saddle-point *without* any magnetic
order developing---this happens because of
terms in the action coming from magnetic fluctuations (which
we integrated out to get the action for the HS fields). This is the transition to the nematic.  In this picture, this nematic order parameter (in the spin physics)
then couples to orbital degrees of freedom and induces
orbital order and everything else follows.
This is a nice way to rationalize, if not actually explain, the essential
features of what is seen experimentally.

And this concludes Rafael's very nice overview of the theory
for pnictide superconductors.
 
 


 

Friday, 10th August,

Hai Hu Wen (Nanjing University, China)
Materials and pairing mechanism in iron pnictides/chalcogenides:  
what we have learned and have still to learn.

Blogged by Leni Bascones and Piers Coleman















Hai Hu will start presenting the FeAs and FeSe based materials and structures. He talks about the discovery of superconductivity at 26 K in doped LaOFeAs by Hosono's group. He calls attention to  the large resisitivity of this material and the resistivity anomaly which was observed and that was later known to be an spin density wave transition. He notes that many of the parent materials had been synthesized some time before but were not studied in detailed.  Quickly after the discovery of superconductity above 50 K was found in related materials of the same family, so called 1111. Later on superconductivity was discovered in other families, like 122 or 111, what also contained FeAs layers, and in 11 famlily with simple structure based in FeSe layers. Nowdays there are seven related  families showing superconducting. Superconductivity is also found in Ni-compounds.

In most cases parent materials are non-superconducting. Superconductivity can be induced by electron or hole doping, by non-dopant substitution or by the application of pressure. Parent materials show an antiferromagnetic instability . It was early proposed that antiferromagnetism was due to Fermi nesting. Later on a different interpretation in terms of local moments was put forward. Magnetic order is called stripe or co-linear and show antiferromagnetic order along one direction and ferromagnetic along the other direction. Through the phase diagram, plotted as a function of pressure, the superconducting critical temperature shows a dome shape. A similar phase diagram is found when doping with electrons or holes. Antiferromagnetism disappears and superconductivity shows up.

Hai Hu compares the phase diagram of iron superconductors with that of cuprates. Cuprates are single band antiferromagnetic Mott insulators . with an exchange constant of order 150 meV while pnictides are multi-band metallic materials with an estimated second nearest neighbor exchange constant of order 50 meV.
He focuses on the transition towards antiferromagnetism and presents experimental results which show that at temperatures slightly larger than the Neel temperature there is also an structural transition. The gap in temperatures between the  structural and Neel temperatures depends on the family. He says that there are  two main proposals to explain the structural transitions  one based in spin nematicity and the other one in orbital ordering. Andrey Chubukov points that so far there are no microscopic calculations to support the  orbital ordering proposal. Structural details like As height or Fe-As.Fe angle seem to determine the optimal superconductivity. Theoretical input on why this could happen was given by Kuroki et al in 2009.

Hai-Hu wonders whether nematicity is important for superconductivy or not and whether it is just a consequence of magnetic or orbital fluctuations. He talks about transport measurements in detwinned samples by I. R. Fisher group which showed resistivity anisotropy with conductivity larger in the antiferromagnetic direction. Anisotropy was observed even above the structural transition and it has been discussed in terms of nematicity. Rafael Fernandes points out that because strain is applied to detwin the samples, above the structural transition one cannot talk about symmetry breaking. Other experiments showing anisotropic features where  published by Davis's and L. Green's group. Recent experiments in Matsuda's group suggest that the nematicity starts at temperatures much higher than the structural transition.

Hai-Hu suggests to try all kinds all of possibilities to make new materials. In particular he suggests to try systems with FeB and FeGe layers. So far only Tc ~5K have  been achived in these compoung but he hopes that Tc could become larger. To try new possibilities for materials lead to the observation of superconductivity in hole doped 1111 compounds, oxygen free 1111 materials and in more complex FeAs based materials. More recently superconductivity was observed in KFe2Se2 this compound is heavily electron doped. Some related compounds show Fe vacancies and are antiferromagnetic insulators. This family is called 245. Two different superconducting phases have been found in this family as a function of pressure. Q: Rafael Fernandes asks whether superconductivity in this materials is a bulk property . A:Hai Hu notes that the superconducting volume in not very large. Q: Andrey Chubukov asks whether there is phase separation A: yes it seems so.   There are current efforts to avoid Fe vacancies.

He concludes the talk saying that there is still some space to be explored for making new superconductors.

Q:  About Sr4V2O6Fe2As2 at high magnetic fields, whether a 2D vortex pancakes has been observed. A: Not directly.
Q: Why is iron special? A: Because of a possible balance between correlations and Drude weight.
Hai-Hu believes that many other superconductors without Cu or Fe will be found
Q: About the mechanism of superconductivity. A: Phonons seem not to mediate superconductivity, which seems related .to antiferromagnetic or orbital fluctuations.
Some other questions were asked but I could not hear too well the participants from my place.



Part II

Well, we're off to the races again!  After a short break, Hai Hu is begining his second lecture. He mentions that in his opinion, the iron based superconductors are intermediate coupling in character -
lying between the two extremes of spin fluctuation theory and the RVB theory of superconductivity.  He writes down the BCS formula, noting that over regions where the interaction is strongly repulsive, the gap likes to have different signs. Since the strongest interaction is betwen the electron and hole pockets, this lead to the proposal(by Igor Mazin and others), that the order parameter must have s+- symmetry, changing sign between the electron and hole pockets, but without a node crossing the Fermi surfaces. Proving that this is the nature of the pairing proves to be difficult, he says, and we don't have much direct evidence for the change of sign of the OP at this stage, though we have lots of evidence for a fully gapped Fermi surface.

Measurement by ARPES, shows that there are in fact two gaps, a feature of multiband superconductivity.  (I was not sure which part has the largest gap - the electrons or the holes? ) 

Using Hall probes, it becomes possible to measure the superfluid density.  There is an interesting temperature dependence, which can be fit to a two-component form, in which rho_a and rho_b both have BCS temperature dependence, with gaps

Deltaa= 1.6 meV,  Delta_b = 9.2 meV

rho = x rho_a + (1-x) rho_b

with a fit value of x =

Now the Hall constant is also consistent with multi-band physics. The strong temperature dependnece of the Hall constant is said to be a consequence of the multi-band character. This is contraversial, and some interpret this as a strong correlation effect, as a result of temperature dependent Z. I prefer to keep the interpretation simple, he says.

Now Hai Hu turns to NMR measurements. He notes the relationship

1/T_1T ~ Sum A(q)^2 chi''(q,omega)/omega at \omega

at omega = 0.  The upturn in 1/(T1 T) is consistent with antiferromagnetic spin fluctuations.

Now RPA spin fluctuation theory is predicted to have a sharp peak  below Tc - a resonant peak that is indeed observed in experiments.  This is consistent with s+-.   A second  measurement is using Fourier transformed STM measurements, carried out by Haneguri et al.  The magnetic field enhances sign-changing scattering, and this is what is observed in Fe(Se,Te).  Unfortunately so far, we have been unable to reproduce this scattering.

Q: what about sc loop experiments with half flux by Tsui et al?
HHW: Unfortunately, this also has not been reproduced and I am very sceptical.

Now he turns to the debate on gap structure.  There is a table -


ARPES - sees isotropic gap

SUPERFLUID density - most time shows powerlaw dependence, evidence of nodal gap.

NMR low T  1/ T1T shows power law, consistent with nodal gap or the s+- under impurity scattering

THERMODYNAMICS : also shows specific heat consistent with nodal gap


Now the spin fluctuation models the iron pnictides are found to give rise to gap anisotropy, but ARPES has not see it.

Q: Isn't this apples and oranges, doesn't a gap anisotropy exist in the 111
A: yes - (but blogger didn't understand the details of the answer...)

Linear dependence of the superfluid density (from penetration depth) is consistent with line-nodes in some materials, but with full gap on other materials.  Now many samples have lambda ~ (T)^n, with n =2 . Not consistent with ARPES data.

Thermal conductivity in Ba(Fe(1-x)Ni_xAs)_2, for current along the c-axis, there is evidence for a node in this direction. Now in Ba(FeAs1-xPx)_2, there is evidence from four fold oscllation in the thermal conductivity, that there may be nodal lines in two directions (see figure below).  Yet Zhang et al on the same material have data consistent with a different gap assignment.

Hai Hu also sees a four-fold oscillation of the C/T in a field in Fe-Se. (see Nature Comm, 1, 157 (2010). This has been phenomenologically fitted by Vorontosv and Vekhter, PRL 105, 187003 (2010) and Chubukov and Eremin (reference not gotten).

Continuing the discussion about gap anisotropy, in the sf theory, intrapocket scattering will enhance gap symmetry.  This remains a very contraversial issue and many groups are working hard on this topic.

Quantum Critical Point:  Much circumstantial evidence for a hidden quantum critical point.

(1) Entropy S(T) ~ T ln T
(2) Resistivity rho(T)= rho_0 + AT^n, where n is found to go down to n=1 at x = 0.4, but rises towards n=2 either side.
(3) Penetration depth seen to have a divergence.

How strong is the pairing: do we have a pairing group?  In the K-doped BaFe2As2, the jump in C/T is about   100mJ/mol/K^2.  This is consistent with delta C/C_n ~ 2, like strong coupling.  Canfield gorup has found C/Tc ~ T_c^2,  which has been interpreted by different groups in different ways.
(impurities + s+- one group, quantum criticality another group).

STM -

Hai Hu talks about STM in BaK Fe2As2.  His group has seen a dip-hump structure in the STM measurements. Giving an Eliashberg style interpretation, they can read-off

Delta = 7.5meV
Omega = 14meV
Delt + Omega = 21.5 meV

Hai Hu compares with the classic electron-phonon example, Rowell et al, and others used d^2I/dV^2 to read off the phonon spectrum. This is the classic example from lead.  HHW compares the two -
On the left hand side, dI/dV,  x=1 corresponds to omega = gap + resonant frequency.  Now shows results for the irons.

Now compares with Eliashberg simulation, puts in a strong mode at 14.5meV, for inter-pocket scattering. Comparing the blue simulation with the red expt, the two look qualitatively similar. 
Similar calculations for cuprates,

To convince you that the 14meV mode is not phonons, we have done this on more than one material - now on NaFeCoAs- still see a wiggle feature similar to the 122. Gap energy 5meV, mode energy 7-8 meV, consistent with the neutron resonance feature. Putting them all together,  neutron resonance energy versus STM wiggle data, find that

mode energy/ Tc ~ 4.5

Emerging Challenges

(1) Why does superconductivity survive with enormous impurity centers.  ON site doping induces stron impurity scattering, but superconductivity survives.  25% doping still gives superconductivity.
This is not consistent with an Abrikosov-Gorkov theory of splusminus.  A glaring inconsistency!

Remark:  Pressure effect and doping effect give comparable Tcs, even though one adds disorder, one does.

Q: What happens if you dope zinc onto the Fe site.
HHW:  No effect, but maybe on overdoped side.

Further data - Co doped Ba122 has an enormous gamma, but Tc is not effected.
Further data,  Cu and Mn doping on the Fe site,  exactly smae Tc reduction.

Rafael - one of the difficulties is that we don't know how to model the detailed impurity potentials.
HHW - still very hard to understand - one should still get large momentum transfer.

Further data - STM shows that Co doping does not affect the DOS very much. One would expect an ingap state forming from cobalt, but not seen. Small scattering potential or small momentum transfer scattering?   (The blogger is very sceptical about all of this - it all sounds like adding epicycles!)

(2) KyFe2-ySe2

arXiv1012.5164 

Absence of hole pocket - yet still pairing. The absence of the pocket is different to DFT, and it is not seen in ARPES. My group found that these samples are not homogenious, a dip in the field dependent magnetization. Signs of phase separation. Big debate in china

(1) I think the sample is inhomgenious. Second phase K2Fe4Se5
(2) Many disagree.

But I think now we know that the sample is in homogenious. 

Just recently STM data  - white area is 245, only the grey area is the superconducting phase - islands? Perhaps consistent with small sc fraction.

But with this sample, K0.8Fe1.6 Se2 - Keimers group sees a resonance - which they suggest corresponds to coupling between 0,pi and pi,0 electron pockets.  This experiment claims to reconcile the basic picture based on the AF SF mediated pairing.

Q:  Why is the sc volume so small still see this feature in the neutron scattering. ?

(3) Drude Weight. Slide from Basov. If you want high Tc, need strong correlation, but mobile carriers.

Concluding remarks.
s+- model - support from many experiments. Multiband widely observed.

(2) In most vases, the gap is nodeless. While gap anisotropy should ne an event with high probbablity, whcih so far disagrees with ARPES>

(3) The weak coupling picture with the Splus minus can account for some results, but it is challenged by some recent expts - the pair breaking effect induced by nm disorder is weak and the missing of hole pockets in KyFe2-ySe2.

(4) QCP, electronic nematicity, orbital physics have been observed in some systems. It needs more efforts to resolve how they relate to superconductivity.




Q: Is there any evidence for superconducting fluctuations.
HHW:  They are weak. No Nernst effect like in the cuprates.

Q:  Given you have multi-band, you can have diversity of band structure, and then you probably don't need a node to fit the data.
HHW:  At low T, the small gap will still dominate, but we don't go to low enough temperature.

Andy Schofield (University of Birmingham, UK)

Quantum Criticality - A tutorial


Blogged by Michael Norman and Piers Coleman

Andy is going to do a blackboard talk so let's see how good my eyesight is.  The focus will be on ZrZn2, a weak ferromagnet, with the emphasis of how current-current interactions affect metals, and whether the standard Hertz-Millis theory (time dependent Ginzburg-Landau) works or not.

To motivate all of this, let us consider a few cases first.  The "strange metal" behavior of cuprates exhibits linear T resistivity over a wide range of temperatures, but the nature of the "ordered" (pseudogap) phase is controversial.  CePd2Si2 is an antiferromaget, which is suppressed to zero under pressure.  Near the resulting quantum critical point, one sees unconventional superconductivity.  Or consider a first order transition (like liquid-gas).  At the end of the first order line, one has a second order critical end point, which in principle could be tuned to zero by varying some quantity like field, pressure, or what have you.

Q: What do you mean by critical fluctuations?
AS:  Near the ordering phase line, small variations of the order parameter field lead to large responses.

In ZrZn2, the electrical resistivity varies as T^5/3.  The thermal resistivity instead varies as T.  This is seen in nickel doped palladium as well, where Ni doping drives paramagnetic Pd into a ferromagnetic state.  The phase line varies as (x-xc)^3/4, where xc is the concentration of Ni that first induces ferromagnetism.   In Ni doped Pd, the specific head coefficient C/T varies as the log of T.







Now to the tutorial.  In Landau theory, there is a one to one mapping between non-interacting electrons and fermionic quasiparticles.  One can characterize the system by a distribution function, n(E), which varies with momentum, k.  The difference from the Fermi function is determined by interactions.  This can be determined by a scattering rate, which can be calculated using Fermi's golden rule, taking into account the presence of a filled Fermi sea.  One finds that the scattering rate depends on omega^2, where omega is the energy loss.  Including temperature, omega^2 changes to omega^2 + (pi*T)^2.

This is a phase space argument.  Now, let's assume the scattering potential depends on momentum and energy.  Exploiting energy and momentum conservation, one sees that the transferred energy, omega, is restricted to being (T=0) between zero and the energy E of the state, whereas the transferred momentum is restricted to being between omega/vF and 2kF, where vF is the Fermi velocity and kF is the Fermi momentum.  The result for the scattering rate (easily derived by power counting) is

1/tau ~ Integral(0 to E) omega domega Integral (omega/vF to 2kF) q^(D-3) dq |V(q,omega)|^2 where V is the scattering potential and D the dimensionality.  If V is constant, in D=3, this gives E^2 as before, whereas in D=2, one gets E^2log(E) instead.

If V has structure, then one gets something more interesting.  Coulomb scattering looks promising, since V ~ 1/q^2, but screening converts this to 1/(q^2 + qTF^2) where qTF is the Thomas-Fermi wavevector, and the infrared divergence is cutoff.

On the other hand, magnetic scattering is not screened, so this is more promising.  Assuming an Amperean potential between currents, j.  Using Maxwell equations and Ohm's law, the current-current scattering potential varies as 1/(q^2 - omega^2/c^2 + i*omega*sigma) where sigma is the conductivity.  The last term in the denominator is the so-called skin effect.  In the clean limit, sigma varies as 1/q.  So V^2 goes as 1/q^4 down to q of order omega^(1/3).  The result is that 1/tau (D=3) now goes like E instead of E^2.  Since 1/tau is a measure of the imaginary part of the self-energy, by Kramers-Kronig, one can show that the real part of the self-energy goes as E*log(E).  Therefore, the specific heat coefficient goes as log(T).

But the problem is that the ratio of the Amperean potential to the Coulomb one goes as vF^2/c^2, which is of order 10^(-6).  So, although one does indeed find a breakdown of Fermi liquid theory, this only shows up at extremely low temperatures.

Q:  This looks like a classical treatment?
AS:  Yes and no.  One has a Fermi surface, and then treats things in a semi-classical approximation.

Part II.  Andy begins with a summary of the first lecture.  The blackboard is empty and we are ready for another wonderful blackboard talk. Andy is going to use the results from the last lecture to gain insight into the magnet quantum critical point.  Andy says - if I think about an electric current and imagine how it is affected by small angle scattering.  A quasiparticle receives a tiny knock, and is now diverted through a small angle - this contributes to the decay rate of the quasiparticle, but it does not affect the transport current very much. (Bloggers aside: We saw this yesterday in Lara Benfatto's talk - this is the effect of the vertex corrections. ) The change in the current is

Delta j = k_F (1-cos (theta)) ~ k_F * (q/k_F)^2

which is reduced by the factor q^2s, so that transport rate is now

current decay rate = Integral omega domega Integral  d^D-1 q dq/ q^2  * |V(q,omega)|^2

                                * additional q^2 term

The q^D-1 cancels with q^2 in D=3, the |V(q,omega)|^2 ~ 1/q^4 at q^2  > omega/q, so the lower limit becomes omega^1/3, so when we do it, we have dq/q^2 from omega^1/3 , which gives 1/omega^1/3 from the q-integral, which gives in the end, with the frequency integral omega^5/3 -> T^5/3. (Blogger's aside - this result is consistent with simple dimensional analysis! AS answers that it is good to see how this works in detail. )

Q: wouldn't you have to do this self-consistently, because the resistivity appears in the damping rate?
AS: Potentially - but it comes back to whether the effective mass should be calculated self-consistently.  But the answer seems to be, it doesn't affect anything.

AS asks - can we use these results for U(1) gauge theories, where the coupling constant is much bigger.  AS reminds us about the method that was used by Yong Baek in his talk on spin liquids -

  fermion  = spinon creation * holon                         (slave boson approach)

Now this introduces a local gauge invariance, and to make this work once the spinons are delocalized, you need a fictitious electromagnetism.

Q: Is this really fictitious - after all it leads to a collective " artificial light ",
AS: I wanted to make the point that it is not conventional Electromagnetism.  It is observable, because it will lead to  a T^2/3 specific heat.  Andy points out that the energy will go like
T^2/3. (Blogger didn't quite catch it all).


Now we discuss Matter close to quantum criticality.    The action is written down

S = phi[ r0 + q^2 + |omega|/Gamma(q)]phi + u phi^4

Why do I have a term proportional to omega?  Answer - because this is allowing for the physics of damping?

quasiparticle interacts with medium that is almost magnetic, and it sends out modes with a propagator that is the inverse of the quadratic coefficient of S

1 / [r0 + q^2 + |omega|/Gamma(q)]


If you were in an insulator, the damping would be replaced by omega^2.  But here we have a metallic environment, the magnon interacts with the Fermi sea which makes particle-hole pairs.

Q: can this be derived?
AS: Yes, but we now know there are some problems with the derivation (non-analytic terms in frequency for example).

So in a system like ZrZn_2 close to criticality, the propagator of the critical magnetization is essentially identical with the current-current fluctuations we talked about this morning, but with a much stronger coupling.  And we see a T^5/3 resistivity as expected. Note that the thermal resistivity,  which is just governed by the quasiparticle relaxation rate, is still linear in T, as measured by Smith et al in the data shown below.

Q: how do you derive this from the partition function?
AS: Well - we'd like to rewrite the Boltzmann partition function in terms of states that are not eigenstates? Fortunately for me that question was answered long-ago by Feynman, who wanted to use position eigenstates in zero temperature qm.  He showed that you could evalue the expectations of

<r | exp[-i H t] r'>

provided you split it up into tiny increments, and then sum over all "histories" {r(t)}.  I'm trying to do exactly the same thing here,

< phi | exp[- H beta] |phi> = Sum over the {phi_j} configurations of

               < phi | exp [ - H \delta \tau] |phi_1> x ..... x <phi_N-1| exp[-H \delta \tau |phi

I now have imaginary time, and I have to split it up into all trajectories of the magnetization in imaginary time.

Q:  I understand all of this, but why does it give rise to the precise form in S that you have shown us?
AS: The answer is that this is the first allowable term in the expansion - why do I have a q^2 - the first that is allowed - you might have though omega^2,  but because this is in a damped environment, you can have a |omega|.  Actually what we have here is the Lindhardt function - a polarization bubble which has precisely this form.

Q: But why do you have a 1/q in the damping omega/q.
AS: I kind of lied to you, because  ferromagnet is an eigenstate. But when we look at long wavelength modes, (draws a wave) . In the FM, the decay rates are suppressed as the wavelength gets longer and longer, and this gives rise to a damping rate Gamma(q) ~ q.  Balistic motion - the amount of time is proportional to the wavelength, which gives tau ~ lambda, 1/tau ~ q.  But if it was a dirty ferromagnet, the electrons would diffuse, so now tau ~ lambda ^2, 1/ tau ~ q^2. Final thing I should mention is an antiferromagnet.  Now it doesn't depend on the slight variations about the AFM wavelength, so damping rate is constant.

So for an antiferromagnet, the propagator becomes

1 / [ r0 + (Q-q)^2 + |omega|/Gamma_0]

where we expanding the Gamma around the afm wavevector.  So now I can change my scattering rate formula, which now becomes centered around Q.

 (Q-q) -> qtilde

The denominator has now gone away becomes big Q^2.  The q^2/k_F^2 disappears because the scattering is now at large Q, which relaxes the current easily.

Q: Why don't you change the lower limits of integration in momentum?
AS: The upper cutoff isn't really important. Its kind of order (k_F). The lower cutoff is now when p^2 ~ omega, so lower limit is indeed changed to omega^(1/2)


But why is all of this wrong?  

Well experimentally - in the FM - almost all of them become first order before we reach the QCP. (Bloggers aside, there is a new Yb system which does have a second order transition, but its more localized)

MnSi is a particularly fascinating case.  It also doesn't go quantum critical - weakly first order - and worst, the power law, which is not actually T^5/3, but T^3/2, appears over a wide range of the tuning parameter (presssure) - and nobody understands this.

Theoretical reappraisal. 

It turns out that there are additional terms in the LGW action. A more detailed analysis tells us that terms I've left out should affect the answer.  Finally - almost everything I've discussed now applies to itinerant systems, where the magnetism is delocalized, but a lot of the crucial experiments are on f-electron systems where the magnetism is very localized, and the itinerant approach I've described here doesn't apply.

A better job of dealing with the FM QCP, there are non-analytic terms, eg in 3D, q^3/2 terms, which leads to fourth order terms that go negative, explaining why the transitions tend to go first order.
The current picture of the FM QCP is as follows:


Now the electrons are coupling to the transverse modes. The early pioneers did not quite take into effect the higher order transverse modes - but if you can gap them out by applying a field, you do get interesting quantum criticality at a metamagnetic QCP.

Puzzles in Heavy Fermion QCP.

Examples include CeCu6-xAu_x

The reason we thought things were fine - that we can ignore the mode-mode coupling terms can be ignored (above the upper critical dimension) - yet there is evidence that the system obeys E/T scaling - a result of something called "naive scaling" -  which suggests that the underlying quantum critical point is somehow below its upper critical dimension.  There is other evidence that the critical fluctuations are affecting all the Fermi surface  - some kind of local quantum criticality.

Another example is YbRh2Si2, which has an incredibly low ordering temperature that is killed off by a tiny magnetic field. Seems to be an additional scale that is collapsing to zero T^* at the QCP. This is one of the great puzzles.

Ends with a cartoon about what might be going on. Conduction electron zooms past a spin and exchanges spin with it. The key question, is what is the fate of the localized spins. One side - the ordered state - they are localized - on the other hand, there is another possibility that the interaction between the electrons and the local moments could undergo a Kondo effect, whereby the electrons and local moments bind in delocalized heavy fermions.

An example of a delocalized heavy fermi liquid, Andy gives UPt3, where the volume of the FS includes the f-moments. The picture thats emerging, is that the QCP could be the merging of Kondo and magnetism - a direct competition between the formation of the heavy fermi liquid and the magnetism. Its not clear that the magnetism and the failure of the heavy fermi liquid need coincide - but they seem to in some systems.

I just want to mention the new interest from String Theory - but which you will hear about on Monday - what little I know - there is a famous conjecture that says that strongly interacting theories in one dimension have a duality with a gravity theory in one higher dimension.  The idea is that if you can solve for the motion of particles in a classical gravity theory in one higher dimension will give you the interacting physics of a QFT in the lower dimension.  We might all have to learn, or at least remind ourselves about general relativity....

Strongly interacting QFT in D dims  <-------> Classical general relativity in at least D+1 dimensions.

Q: Andrey - maybe you should mention the names of Hlubina and Rice
AS:  It turns out, that shown by Robert Hlublina and Rice,  that you shouldn't do the average over the Fermi surface of the scattering rate, but the scattering time. And when  you do this in the clean limit, you get the result that the Fermi liquid T^2 scattering rate dominates.  Achim Rosch has shown that in the dirty limit, one can account for non-Fermi liquid behavior at a dirty 2D afm quantum critical point.

Q: Are there any cases where the simple derivation works?
AS: yes is the answer.  My favourite is the quantum critical end point of Strontium Ruthenate.  Even though the PT to AFM is driven first order by applying a field, you can still at finite field where we get QCP.  Does it work there?  Well, there is evidence that still more interesting things happen around the quantum critical end-point.

Q: Can you give some simple insight why here you have to take more and more terms, why its failing here, as opposed to BCS theory, where things work out.
AS: BCS is a simple case where you don't have to include fluctuations.  Here the problem is that we normally make the assumption that theres a single low-lying mode - but when we deal with metallic systems, we've go other low lying degrees of freedom - could be various types - what we're not able to do is to handle consistently all these low lying modes together.  But as a corollary, if you do this in insulators, everything works beautifully.  There are cases where quantum criticality does work beautifully, but I was limiting myself to the case of metals, where there are problems.

AChubukov.  If I were to answer the question of self consistency. I would answer that formally you can do self-consistency - there is some self-consistency at least at zero level. (We now know there are more subtle problems at higher order - blogger aside - this is a reference to the recent work of Metilsky et al. )

Thursday, 9th August

Elena Bascones
(Instituto de Ciencia de Materiales de Madrid (ICMM), Spain )
Mott physics: from basic concepts to iron superconductors

Blogged by Michael Norman and Natasha Perkins (with a little help from the start by Piers Coleman)

Elena begins with an introduction to Mott Physics. Kinetic energy from hopping likes to delocalize electrons.  In the atomic limit, we have a Coulomb repulsion term U which raises the cost of charge fluctuations and suppresses double occupancy.  In a single band system with one electron per site (on average)  as we increase U, we expect a Mott transition to localized electrons, because double occupation is forbidden.

Delocalized electrons (U << t) ---> Mott Transition ---> Localized electrons (U >> t)

Likewise, starting from the insulating limit, we have an upper and lower Hubbard band, separated by an energy U, corresponding to electron removal (lower band) and electron addition (upper band). These bands will broaden as we turn on the hopping between sites, with a width W proportional to the hopping t. 

Q: Isn't it more subtle. As you move your hole, don't you scramble your antiferromagnetic order?
EB: I haven't yet talked about AFM order.  Mott insulators can exist without magnetic order - in principle, you don't need to have magnetic ordering (e.g. in frustrated or low dimensional systems).

Hopping lowers the kinetic energy, whereas double occupancy costs you an energy U.

Now, let's approach the problem from the metal.  There is a band with width W.  There are four states per site - empty, spin up, spin down, and double occupancy. The average interaction energy is thus U/4.  The kinetic energy (constant density of states) is -W/4.  Therefore, we expect a transition when U=W from metal to insulator.  This simple picture assumes that double occupancy has a step jump to zero at U=W.

A better approximation (Gutzwiller) is to have the double occupancy vary linearly with U.  The kinetic energy is raised, but the interaction energy is lowered, with the resulting total energy being lower, and the transition is moved to U=2W.  The quasiparticle residue steadily decreases in the metal, disappearing at U=2W (Brinkman-Rice transition).

The real situation lies in between, and can be captured by dynamical mean field theory (DMFT).  On the metal side, there are now three bands, lower Hubbard centered at -U/2, a metallic band centered at 0, and an upper Hubbard band centered at +U/2.

We now turn to magnetism at half filling.  An electron can virtually hop to a neighboring site if the spins are opposite, leading a a savings in energy of J = t2/U where t is the hopping.  This leads to a AF (Neel) state.

Now, what about multiple orbitals?  For d electrons, crystal field splitting leads to t2g and eg states.  This modifies the on-site energies.  The hopping now depends on the orbital index.  The interaction contains intra-orbital and inter-orbital pieces, as well as Hunds rule contributions that tend to align the spins of different orbitals.


The interaction contains terms proportional to U (intra-orbital), U' (inter-orbital, different spin) = U-2J, and U'-J (inter-orbital, same spin) where J is now the Hunds coupling.  If we take the limit that J goes to zero, the problem simplifies.  One can show that in the Gutzwiller approximation, the Mott transition occurs at NW, where N is the number of orbitals.  This is reflected in DMFT, where the bands widen with increasing N.
With N greater than 1, one can now have a Mott transition away from half filling unlike the N=1 case where the transition is confined to half filling.


Q: What fillings are allowed?
EB:  One must have an integer number of electrons per site for a Mott transition.


Now lets switch on the Hunds J.  The energy is lowered when the spins are aligned on a given site.  With increasing J, the Mott transition is moved to lower U at half filling.  The opposite occurs away from half filling for n (site occupancy) of 1.  This is due to two competing effects, J reducing the degeneracy, and J changing the interaction energy.  The net effect is that at half filling, J increases localization, whereas away from half filling for n=1, J promotes metallic behavior.  This is confirmed by DMFT simulations.

For cases where n is neither 1 or N, J typically promotes "bad metallic" behavior.  Take N=3.  According to DMFT, Mott behavior is promoted at n=3.  n=2 and 4 lead to low coherence temperatures ("bad" metal with "spin freezing").  More metallic behavior is found at n=1 and 5.

Q: How does Hunds coupling affect the Mott gap?
EB: For half filling, gap = U+(N-1)J.  Away from half filling, gap=U-3J.


Q:  What is the physical origin of J?
EB:  This comes from the dependence of the Coulomb interaction energy on spin and orbital indices.

Leni part II (Blogged by Natasha Perkins)

Non-equivalent bands.

Two  different transitions for two bands give a possibility for orbital selective Mott transition.

 First consider Hund=0.  Two bands are degenerate.  Because of degeneracy, large difference between bands is required for orbital selective MT (OSMT).

Now  Hund is non zero. Hund’s coupling decouples orbitals. With finite Hund’s coupling the metallic state does not benefit from the degeneracy. The  band with smaller bandwidth becomes insulating first. OSMT can be obtained also in case of 3 and 4 bands, and is also affected by Crystal field.  Leni shows two examples of OSMT in which it is clear how the quasiparticle (QP) weight varies with the ratio Hund/Coulomb.

Iron based SC.

During last years many Fe-based SC were discovered. These are multi-orbital systems. Leni shows the PD for Fe-based SC.

 Correlations in iron-based SC are probably weaker than in cuprates, but are still important. One can see from the experiment Lu et al (Nature 2008), that mass enhancement is  about 3. From Basov talk (optics), we also saw that correlations are weaker. Contrary to cuprates, parent Fe-compounds are not Mott insulators. Does this mean that they are not correlated?

We should include all 5 orbitals to explain Fe-based SC. These 5 orbitals are different, there are 6 electrons on 5 orbitals, so we are not at half-feeling.  We can think about pnictides as about doped Mott insulator.

Iron SC are Hund’s metals. Correlations are enhanced by Hund’s coupling. This is possible because of the multiorbital character, which plays an important role – we can have coexistence of localized and itinerant electrons.  Leni shows PD from which we can see that correlated metallic state appears due to Hund’s coupling. Different Fe  compound are either more or less correlated.

Leni compares results from 2- and 5-bands model.

Liebsch(2010) also shows that hole-doping increases correlations.  Doping with electrons decreases correlations.

Andrey: the  PD is at T=0. Then what the words FL mean? Leni: Yes, it is T=0 because Liebsch is doing exact diagonalization.  ( I did not get Leni’s answer).

BeFe2As2 – PD with FL and NFL. Crossover T is governed by Hund’s coupling. NFL is seen when we are moving towards half-filling. There the QP weight is much smaller.

Rafael: You  expect that 3d5 will be insulating? Leni: Yes. Rafael: but if you dope  3d5 with electrons it becomes metal as other pnictides.  Leni and Rafael  discuss resistivity in different compounds.

Orbital differentiation in iron SC – degree of correlation is orbital depend. ( plot of QP weight for different orbitals). XY orbital has lowest QP weight.

Do we expect an OSMT in pnictides? XY orbital is the most correlated. In the plot we can see that in some materials the orbital differentiation is significant and in then we can expect OSMT, for example, in FeSe. OSMT is induced by hole-doping in FeSe ( results of M. Capone).

Andrey: Is it known where the spectral weight goes? Leni: there is a spread of spectral weight. Andrey: there are ARPES experiments which are interpreted as OSMT.  I want to relate your talk to Andy’s  talk this morning. What is about the frequency dependence of Z-factor? Does it change? Leni: I do not know such calculations.

Summary: one can have weak correlation due to U but still be correlated due to Hund.

How correlated are electrons? Which is the nature of magnetism in pnictides?

Let us discuss it in a more general sense. Magnetism can come even from weak correlations, Fermi surface instabilities, renormalized FL behavior. And of course, from simply localized picture, for example, from J1-J2 model which has been proposed to explain the magnetism in Fe-compounds ( stripes).

I am going to discuss first metallic AFM state (details in Rafael‘s talk). Columnar state with ( pi,0) ordering. Local-moment description – Heisenberg J1-J2 model. In this case the exchange appears from the second order perturbation theory, as in Mott state. J2 is rather large in pnictides due to hopping through arsenic ion. It has been proposed that J2>J1/2. There columnar order is stabilized.

 We computed J1 and J2. We showed that at small Hund,  the state is not columnar, but instead Neel state with (pi,pi). However, at large Hund, we can get columnar order with (pi,0). In fact, the orbital states in (pi,pi) and (pi,0) phases are not the same. There is of course a crystal field sensitivity of the orbital state.

Also, what one can get from localized picture is electron-hole doping asymmetry. We see that hole and electrons are going to different orbitals. Electron doping has a tendency to FM compared with n=6 parent compound ( LaOCoAs).

Andrey: in J1j2- the way how you select (pi,0) is due to quantum fluctuations. Is it also present in your analysis? Leni: no, these are higher order terms. It is probably quite complicated, one need to have bi-quadratic  exchange. We do Hartree-Fock.

Leni discusses Hartree-Fock PD, in which one can see (pi,pi)-( pi,0) transition with increasing Hund’s coupling.

YBK: Do you include orbital fluctuations?  Leni: yes, this is included. YBK: I have impression that you consider different orbital configuration and then compute exchange. Leni: Yes. YBK: But then you consider rigid orbital configurations, without fluctuations? Leni: This is the leading term.

What is the nature of (pi,0) state?

We focus on this state and try to understand what’s going on in this state. Leni shows the PD in parameter’s space of J/U  vs U. They have found three different regions: Itinerant phase with strong orbital differentiation, Non-magnetic insulating and Magnetic insulating phases. In itinerant phase not all orbitals are itinerant:  while xy and yz orbitals are gapped and insulating, zx, 3z^2-r^2 and x^2-y^2 are itinerant orbitals but with correlation features. It is interesting to see how one can go from one phase to another by doping.

Summary: multiorbital physics is important. Orbital differentiation is important and we should play attention to them.

Rafael: I have small issue about LDA+DMFT. There is a double counting of correlations in LDA and DMFT.

Leni: I din’t do this, but I know that people try to take this into account. We are working with tight-binding.  But the point is that different methods give similar results.

Gabovich: what is your opinion are about magnetic correlations?  Do they support SC?

Leni: I did not work on this issue, but I think in these systems fluctuations would support it.

Wednesday 8th August.

Lara Benfatto
(University of Rome, La Sapienza, ISC CNR, Italy)
Optical properties of correlated electron systems: basic theoretical aspects and optical sum rule

Blogged by Rafael Fernandes

 Lecture I

 Lara started with the basic definitions, defining the current operator from the minimal coupling to the gauge field (vector potential). In the lattice, it is convenient to use the Peierls ansatz to modify the creation operators by adding the appropriate extra phase. The paramagnetic and diamagnetic parts of the current assume simple forms in momentum space, given in terms of derivatives of the band dispersion. Using linear response theory, Lara showed how one can calculate the tensor, frequency-dependent conductivity in terms of the current-current correlation function: this is nothing but Kubo formula.

 The analysis shows that the real part of the optical conductivity (i.e. the frequency dependent conductivity) has a delta function at omega=0, with a prefactor. Does this delta contribution really exists? The answer is readily obtained by considering charge conservation and gauge invariance, which requires the vanishing of the delta-function prefactor. Furthermore, by using Kramers-Kronig relations, it is possible to derive the famous sum rule. Thus, the optical sum rule is a direct consequence of charge conservation.

 Lara mentions that, in calculating the optical response, one has to be careful and ensure that the approach satisfies these constraints imposed by charge conservation and gauge invariance. Vertex corrections play an important role in enforcing these conservation laws when computing the current-current correlation function. The simplest way to calculate the current-current correlation function is the bare bubble, which contains two Greens functions and the appropriate current vertices. However, the bubble is not a conserving approximation. The latter is enforced by the vertex corrections, which are given by integral equations. Answering a question from the audience, Lara explains that the vertex correction - i.e. the dressed current - depends on the model under study. To illustrate these issues, she will tell us about two paradigmatic examples.

 The first example is the case of impurity scattering. The evaluation of the bare bubble in this case is straightforward, and is given in terms of the spectral function. In the presence of impurities, the spectral function has a Lorentzian form, with a finite lifetime due to impurity scattering. As a result, the optical conductivity is also a Lorentzian function of frequency: Lara just obtained the famous Drude formula for the optical conductivity. However, the scattering rate does not have the same form that comes from Boltzmann theory, which takes into account the changes in the momentum of the quasi-particle after a scattering event. The difference is due to the absence of vertex corrections... Questions are asked to clarify technical details. Now Lara is solving the integral equation satisfied by the vertex (dressed current) in the case of impurity scattering. The solution can be cast as a redefinition of the scattering rate, without changing the Lorentzian shape of the optical conductivity (question of the blogger). Questions trigger a discussion on the equivalence between Kubo formula (with vertex corrections) and Boltzmann equations.

  Lara now moves to the second example: superconductivity. She defines the superfluid density (Ds), given by the static limit of the transverse correlation function. In the BCS approximation, Ds can be calculated in a straightforward way from the bare bubble: it coincides with the diamagnetic tensor at low temperatures (i.e. all electrons are superconducting) and vanishes at Tc. Why it works even without vertex corrections? How can the latter be included? The answer comes from the phase fluctuations. Lara shows us the action of the phase variable, where Ds is the stiffness. Integrating out the phase fluctuations gives a contribution only to the longitudinal part of the correlation function, which does not change the bare bubble result for the transverse part, responsible for Ds. This, however, is only true for clean systems: in dirty systems, the phase fluctuations also couple to the transverse part of the gauge field, changing the BCS result (bare bubble). Lara also tells that as long as the interaction is momentum independent, the vertex corrections vanish. This is the case of Eliashberg theory and DMFT calculations.

 The next topic is the sum rule. In real experiments, there is always a cutoff in the integrated optical conductivity, and one has to consider only the bands near the Fermi level, and their corresponding masses, which can in principle be calculated via first-principles (DFT). Comparing experiment with DFT, one can then estimate the impact of correlations. In the non-interacting limit, the sum rule should decrease with temperature as T^2, but this change is expected to be small. However, in the cuprates, these variations are much larger. In the pnictides, on the other hand, the sum rule increases with temperature. Lara points out that the use of the cutoff in the sum rule always introduces a temperature dependence.

Lecture II

 Lara started the second lecture reviewing the sum rule properties she explained in the last talk. She reminds us that correlations will change not only the absolute value of the sum rule, but also its temperature dependence (in a regular metal, it should weakly decrease with increasing temperature). To explore the effects of such correlations, she presents some results of DMFT calculations for the one-band Hubbard model. There are two Hubbard sub-bands and a quasi-particle peak at zero frequency. Transitions between the Hubbard bands and between them and the quasi-particle peak are manifested in the different frequency regions of the spectral weight. She shows that the intraband integrated spectral weight scales with the strength of the quasi-particle peak, whereas the coefficient of the temperature-dependent term scales with the inverse of the latter.

 She now turns to the effects of electron-boson coupling to the optical conductivity, using Eliashberg equations. The bosons may be phonons, spin fluctuations, etc. Since there is no momentum dependence in the interaction, vertex corrections are not necessary (in the sense of a well-defined conserving approximation) and one can restrict the calculation to the bare bubble. Technical questions about this last point are asked, and addressed by Lara. The result at low energies is the recovery of the Drude formula, with renormalized mass and scattering rate; at higher energies, one obtains the extended Drude formula, with frequency dependent mass and scattering rate. She shows explicitly results for the particular case of an Einstein mode, for both the optical conductivity and the sum rule. The sum rule is more or less satisfied with the addition of coherent and incoherent parts of the spectrum. Lara explains that the extended Drude formula, based on the Eliashberg results, can be used to analyze experimental data - she particularly focus on pnictides.

 A point is made to the existence of two effects: a high-energy (~1ev) correlation effect, associated with a significant renormalization of the bandwidth due to strong correlations, affecting the sum rule. There is also a low-energy (~0.1eV) effect due to coupling to bosonic modes, which changes the Drude formula and the low-frequency behavior of the optical conductivity.

 Lara is now discussing the iron pnictides, introducing the main properties of the materials. She highlights the proximity of superconductivity to a magnetically ordered phase, and the fact that the pnictides are multi-band systems. Due to the compensated nature of these multi-bands metals, the bands are almost empty, and one expects very little temperature dependence of the sum rule. However, this is not the experimental observation: the sum rule increases with increasing temperature, unlike other correlated metals. While the reduction of the absolute value of the sum rule can be rationalized as an effective mass renormalization by correlations, the increasing T behavior of the sum rule is harder to understand.

 A point is made that due to the multi-band nature of the pnictides, Fermi surfaces shrink due to the coupling to bosonic modes (but the total charge is conserved, as expected). Lara shows experimental data (ARPES on pnictides) displaying a systematic shrinking of the Fermi surface as function of temperature and doping. Such a shrinking has important effects to the sum rule. It can be captured by an Eliashberg calculation considering inter-band coupling due to spin fluctuations. Besides a reduction of the Fermi surfaces, the calculation also reveals a redistribution of spectral weight, showing that this effect is not a mere rigid band shift. Lara mentions that while the Fermi surface areas are changed, the total carrier concentration (integrated over all energies) is practically unchanged.

 Lara explains that this inter-band interaction also results in a transfer of spectral weight, leading to the occupation of otherwise unoccupied bands. Presenting results for the optical conductivity, she points out that the Drude formula is renormalized by the spin fluctuations, but the incoherent part is extended to much larger energies. Thus, the cutoff in the sum rule may not capture the correct asymptotic limit of the high-frequency part, resulting in an apparent (but not actual) decrease of the sum rule. By considering that the bosonic mode weakens as temperature is raised (in agreement with neutron scattering experiments), Lara's calculations are able to reproduce the anomalous increase of the sum rule as function of temperature. This effect is then actually a consequence of the redistribution of spectral weight to very high energies and the introduction of the cutoff. Thus, this increase is basically reflecting the temperature dependence of the coupling to the bosonic modes (spin fluctuations).

 She points out that the coupling constant (between electrons and the collective modes) extracted from the extended Drude model analysis is not consistent with estimates from other probes. The flatness of the Drude peak might instead be not only related to the coupling to the collective mode, but also to inter-band transitions not considered in the model.

 During the questions part, it is pointed out by Girsh Blumberg (Rutgers U.) that changes in the temperature should also affect the band structure by changing the relative positions of Fe and As and the crystalline field splitting. This also contributes to changes in the Fermi surface areas, besides the interaction effects discussed in the talk. Lara points out that the anomalous temperature dependence of the sum rule, however, might be difficult to describe with this mechanism only.

Wednesday, 8th August

Dimitri Basov (University of California, San Diego)

An infrared probe of electronic correlations and many body effects in solids: a case study of high-Tc pnictides and graphene 

Blogged by Piers Coleman

Good morning folks. Andre Marie is introducing Dmitri Basov. He remembers Dimitri from his Toronto days, when, with Tom Timusk,  he discovered the optical signatures of the pseudogap in the cuprates.  Andre Marie mentioned how much new insight into high temperature superconductivity have come from Dimitri's work. 

Dimitri mentions how many of his collaborations began here at the ICTP.   He has recently written a review on optics in RMP, cited below

D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, K. Haule “Electrodynamics of correlated electron materials” Rev. Mod. Phys. 85(2), 471 (2011).

Dimitri begins with a review of frequencies, emphasizing the breadth of energies that govern cuprates and graphene, from 10 to 30,000cm^-1. (1cm^-1 = 0.124 meV = 1.4K). Dimitri reminded us that j = sigma E, that the dielectric constant is related to the optical conductivity

epsilon = 1 + (4 pi i) sigma/omega

Sigma obeys the remarkable sum rule

This sum rule, he said, derives from the fact sigma is a response function (blogger: its really a statement of impulse - this integral is the instantaneous current that responds to the impulse from a pulse of electric field - at short times, the acceleration is given by Newtons law).

Dimitri discusses the Drude model - which he says plays the same roll to electrodynamics as Shakespeare does to literature!  Drude's model came out just three years after the discovery of the electron. Here is a summary of the main points:

DB shows the optical conductivity of a weakly interacting metal.  The area under the Drude peak is just the electron density (see above). The relaxation rate tau(\omega) ~ omega^2 is frequency dependent.  In an electron phonon system, the scattering rate is determined by the phonon spectrum, alpha^2 F(omega), and one can actually invert the expression. C-60 is an example of an electron-phonon superconductivity with a relatively high Tc. But what about strongly interacting systems - here the Drude weight is dramatically reduced below the band-structure value. The frequency dependence is substantially different to a simple Lorentzian - the coherent part drops away more rapidly, there is also a large incoherent background.

Question - what is the connection of sigma(omega) to the reflection of IR light? 
DB - what we actually measure is the reflectance R(omega). From this we can extract both the real and imaginary part of the conductivity using Kramers-Kronig. If you have a transparent system, you can directly pull out the real and imaginary parts of sigma. 

Now he turns to the optical conductivity as a probe of correlations in cuprate and iron-based superconductors. These pose many interesting questions:

(1) Exotic superconductors - are they all alike - at least within one family?  (cf Tolstoy  "Happy Families are all alike")
(2)  Unusual normal state properties - are they a pre-requisite for Hi Tc?
(See D. N. Basov and A. V. Chubukov, Nature Phys 7, 272, (2011) ). 

Dmitri contrasts the phase diagram of the two systems.  Dimitri points out that both have a pseudo-gap.  Doping has a different meaning in the two types of system. The Co and P doped Ba(Fe2As2) show a nice Drude peak, with a spectral weight that does not change a lot with doping.  (See below, rhs - note that the log scale exaggerates the difference between doped and undoped).
By contrast, the doped cuprates show a strong growth of the Drude peak with doping - because you are doping an insulator.  He notes that the scattering rates must be driven by interactions.

Q: How does one know where the interband processes begin? 
DB: I know that most of the low energy weight must come from intraband weight (in the iron-based). This is the only way to get consistency with other probes. 

Q: Can the changes in the optical conductivity be connected to electron electron interactions
DB: Yes - the spectral weight is much smaller than non interacting theories predict. Secondly the form of the conductivity is quite different that predicted by non-interacting theories. 

DB replots the data on a linear scale to contrast the cuprates and iron sc. The low frequency part is coherent, the high frequency part, with a large excess over and above the Drude, is the  "incoherent" 
part

Intraband = coherent + incoherent

Now we turn to SC dynamics.  DB notes that Michael Tinkham measured the energy gap before BCS theory (1956), with an onset of conductivity (absorption) at twice the energy gap.  In the early days, it was not known how to connect the absence of absorption at low energies, with the unchanged spectrum at high energies.  Tinkham showed there is a threshold.  Now what about the cuprates and iron sc?  In

Q: The shape of the curve - does it depend on the symmetry of the gap?
DB: Yes, the symmetry of the gap can impact the form of the optical conductivity.  d-wave sc have certain peculiarities - where there is a node - but the gross features are not that different from an s-wave sc.  The dominant effect is the suppression of absorption below the gap. 

Q: Optical conductivity averages all k, so what gap is measured?
DB: The strongest absorption measures the maximum gap.

However the situation is more complicated than in lead.  There is a lot of extra low energy absorption
associated with multi-band behavior.  

DB shows the optical spectrum of pseudo-gap materials, where there are signs of the pseudo-gap even in the normal state, above Tc.  This gap-like behavior is called the "pseudogap". One is gapping part of the Fermi surface, suppressing some scattering, which leads to a narrowing of the remaining Drude peak. 

Its particularly interesting to look at the frequency-dependent scattering rate. The scattering just above Tc has all the features of the fully developed sc. Amazingly, there is a similar, though slightly less marked pseudo-gap feature in the iron optical conductivity.  The pseudogap scale determines a crossover from weak low frequency scattering to strong high-frequency scattering rate. In the iron-sc, the pseudogap is related to the SDW.

Q: How is this related to transport?
DB: Transport is the dc limit.  But we get more information from the frequency dependence - it shows us how we are deviating from Drude.  These spectra highlight what is significant in the response. 

Q: Can one probe stripe order or phase coexistence using optical c?
DB: Yes, we tried to probe stripe order in La 214, and you can look at the anisotropy of the conductivity parallel and perpendicular to the stripes using polarized light. This has also been done in the pnictides, where recently Leo De Georgi is investigating anisotropies associated with nematicity. 

R: You associated the pgap in the pnictides with the SDW, whereas AM Tremblay associated it with Mott physics.  Could you contrast the two in the second lecture?
DB: Yes I will.


Lecture 2.  

DB returned to the issue of the origin of the pseudogap.  There is clear similarity in 1/tau (w) data.
In pnictides, pseudogap physics disappears beyond optimal doping. 
DB then turned to the discussion of the coupling to the bosonic spectrum.

Tutorial 3: the pairing glue.

Discussion of the Allen formula and how to extract \alpha^2 F(w) from the second derivative of 1/tau.
The discussion first focused on the cuprates, but then was extended to pnictides. 

Back to the pseudogap behavior of 1/tau in cuprates and pnictides. In pnictides, the structure associated with the pseudogap is quite different from the SC gap.  New aspect of the pseudogap story -- presence of SDW. The pseudogap scale in the pnictides == the same SDW scale as extracted from ARPES. 
Q. How \alpha^2 F was extracted from the data?
A. By using the theoretical formula with phenomenological pseudogap parameter and a coupling to "some" mode.

Tutorial 5 (Tutorial 4 was probably eliminated) -- condensate formation below T_c and from what frequencies the condensate is formed.

 DB discussed Homes Law - the scaling of rho_s with sigma_DC * T_c.  This can be understood from the sum rule - an area sigma_DC * 2 Delta ~ Sigma_DC * Tc condenses.  This indicates the presence of strong inelastic dissipation at Tc - dissipation that does not depend on disorder.

Q: Does the ratio of Delta/T_c remain constant.
DB: No - there is quite a bit of spread of this ratio, but within the log-log scaling, this doesn't matter.

Q: Wouldn't this also work for say lead.
DB: Yes it would, but the resistivity would be driven by dirt, not inelastic scattering.

Tutorial 6 Electronic Kinetic Energy and Correlations.

At the two extremes, we have Mott insulators - localized, Coulomb arrests metallic transport, at the other fully itinerant simple metals.  What we are dealing with is "correlated metals" that lie between these two extremes.  One way to characterize this is using Kinetic energy, in ref to the KE from band calculations.

Exptal KE = integral of sigma(omega)domega
Kband <---- band theory

The ratio of the two is a measure of correlation strength in the material.

Q: What cut-off do you use in obtaining these plots?
DB: We integrate up to the bottom of the inter-band transitions, trying to catch the coherent component of the sigma.

Remarkably, the iron-based superconductors lie around 0.4 on this scale - comparable with the cuprate cousins.  Ba 122 is around 0.3, LeFePO around 0.5. LasCO is around 0.2. MgB2 and doped C60 are at 0.8 and 1.0 respectively.  You can conclude that electron phonon does not renormalize the electron Kinetic energy.

KE is suppressed in all exotic superconductors. Iron, cuprate, ruthenate, heavy fermion and organics all experience this. This implies that there are changes in the spectrum at high energies - appears to be a condition for correlated superconductors. High Tc systems are not in general, super-correlated, because of course, strong e correlations kills the Drude peak, which also implies a reduced superfluid density and thus suppresses the superconductor.  This appears to be going on in the under-doped side of the cuprate phase diagram.

Q: Is there a sharp transition from correlated to conventional metals?
DB: Not as far as we know - but the data isn't sufficiently dense to completely answer this question.


Anisotropy. DB turns to the question of anisotropy in the optical conductivity. What happens to the conductivity in the in-plane direction? In the cuprates, the c-axis response often looks like an insulator. But in the iron-based, there is no qualitative difference between inplane and interlayer transport. In the cuprates, you can still pass current along the insulating c-axis direction.  Remarkably you can still see the plasma edge in the c-axis response - this derives from c-axis Josephson coupling. A moderate field H~ 8T eliminates the plasma edge, yet sc still survives. So its clear that interplane coupling does not matter for the superconductivity.

In the c-axis optical properties of the iron-based sc, one sees almost no difference between normal and sc state.  This has recently been resolved for122 samples. It turns out to be an artifact that is connected with "cleaning the surface".  Newer measurements show a marked drop in low frequency optical conductivity.

Infrared nano-scopy

  A new innovation in IR spectroscopy, actually a hybrid of STM and IR spectroscopy. The tip has a radius of 8-10nm, allowing a 3 or 4 order of magnitude increase in the resolution from direct optical methods. It also gives you access to much shorter wavelength than available with light.

It also gives you access to much shorter wavelength than available with light. Dimitri showed images of monolayer graphene using this method.












Addendum.  In the discussion on Thursday, Dimitri was asked to show the optical conductivity of graphene.  He talked about the work of Z. Q. Li, Nature Physics 2008 obtained with the above method. Its flat, undoped, but doped, as an interband threshold E_F, a Drude peak that is very sharp, and an incoherent part that is not yet understood. He said that higher mobility samples (best are now above 100,000) need to be measured. He finds also that sigma_1/sigma_2 has oscillations in the Friedel oscillations.