ICTP2012: Innovations in Strongly Correlated Electronic Systems

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.