ICTP2012: Innovations in Strongly Correlated Electronic Systems

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.