ICTP2012: Innovations in Strongly Correlated Electronic Systems

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.