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.