What is the integer quantum Hall effect?

And why is it so amazing?

Surprises [about physics in two dimensions] occurred in the 1980s when it became possible to study Landau levels [the quantised energy levels of electrons in a magnetic field] in Flatland. This happens when the electrons are completely constrained to move in only two dimensions. The surface within which the electrons move needs to be extremely flat and free from defects and impurities. Advances in semiconductor technology in the 1970s led to two realisations of this Flatland. Both were developed for technological reasons: the desire to have transistors in which the electrons and holes can move extremely fast. One class of device is silicon MOSFETs (Metal Oxide Semiconductor Field Effect Transistors). The second class is heterostructures, where layers of ultrapure semiconductors such as gallium arsenide are grown on top of each other, one layer of atoms at a time. In both classes of device, a fixed density of electrons (or holes) can be injected at the surface. These charge carriers can move freely in Flatland, acting like a fluid. Things get interesting when the number of charge carriers is small enough and the magnetic field is large enough that the number of charge carriers is comparable to the number of quanta of magnetic flux that pass through the system. Then, the quantum state of most of the charge carriers is one of the lowest Landau energy levels. 

To achieve this regime for the cleanest possible systems requires magnetic fields more than a hundred thousand times stronger than that of the Earth. Furthermore, the magnetic field must be spatially uniform in the region where the semiconductor system is located, stable over the time of the measurements, and the interior of the electromagnet producing the field must be large enough to contain a refrigerator that can cool the charge carriers in the system down to a few degrees above absolute zero. By 1980, all these conditions became possible. Klaus von Klitzing was able to perform measurements of the Hall resistance versus magnetic field in a special high magnetic field laboratory in Grenoble, France. The results were surprising and are shown schematically in Figure 35 below. There are four noteworthy features. 

 

Figure 35. The quantum Hall effect. The Hall resistance is shown as a function of the strength of the magnetic field and has a step-like structure. The integer n is related to the quantized energy that the charge carriers have.

First, there are distinct steps in the curve. At small magnetic fields the Hall resistance versus field is a straight line, as expected for the classical Hall effect. However, at larger fields there are plateaus in the curve.

Second, each of the plateaus is extremely flat. Von Klitzing found that the magnitude of the Hall voltage on each plateau did not vary to one part in ten million. As he varied the magnetic field, he noticed that the first seven digits on the voltmeter he was using did not change. He wondered if the voltmeter was broken and had become jammed. But it was working.

Third, the magnitude of the Hall resistance for all the plateaus has a simple relationship to fundamental physical constants. The quantum of resistance is defined as equal to h/2e^2 . When you calculate this quantity, the answer (25,812.827 ohms) is in the units of electrical resistance. The value of the Hall resistance is precisely equal to this value divided by an integer (n=1,2,3 …) which is related to the highest quantized energy (Landau level) that an electron can have at that magnetic field. That is why it is known as the integer quantum Hall effect.

Fourth, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes. [This independence is characteristic of the universality associated with emergent phenomena]. 

These four features are similar to those for the steps associated with the macroscopic quantum effects (magnetic flux in superconducting cylinders, circulation in a superfluid, Josephson effects) discussed in the previous chapter. Again, it is astonishing that a macroscopic measurement – of electrical resistance - of a macroscopic system can determine fundamental constants that are normally associated with properties of atomic systems. Just as the Josephson effect led to a new standard measure for voltage, the quantum Hall effect led to a new standard measure for electrical resistance.

Anyone familiar with building electronic circuits will have used resistors of varying values in ohms (Ω), e.g., 10 Ω or 25 kΩ. When these resistors are made, they are calibrated against some standard. For making integrated circuits with billions of transistors this standard needs to be extremely accurate. In 1990 the international standard for the ohm was changed to be that defined by the quantum Hall effect. Previously, the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. 

An extract from Topology Matters, Chapter 8, Condensed Matter Physics: A Very Short Introduction.

Superconducting order in organic charge transfer salts

A long-standing question for superconductivity in organic charge transfer salts concerns the symmetry of the superconducting order parameter. Is it unconventional (i.e. not s-wave) and if so are there nodes in the energy gap? Over the years there have been a wide range of claims, both theoretical and experimental.

Most recently a combined theory-STM experiment claimed the symmetry was d + s and that there were 8 nodes on the Fermi surface.

Two of my UQ colleagues recently posted a nice preprint that comes to a different conclusion.
Microwave Conductivity Distinguishes Between Different d-wave States: Umklapp Scattering in Unconventional Superconductors 
D. C. Cavanagh, B. J. Powell

Microwave conductivity experiments can directly measure the quasiparticle scattering rate in the superconducting state. We show that this, combined with knowledge of the Fermi surface geometry, allows one to distinguish between closely related superconducting order parameters, e.g., dx2−y2 and dxy superconductivity. We benchmark this method on YBa2Cu3O7−δ and, unsurprisingly, confirm that this is a dx2−y2 superconductor. We then apply our method to κ-(BEDT-TTF)2Cu[N(CN)2]Br, which we discover is a dxy superconductor.

In 2005 Ben Powell  (and others) showed that the simplest RVB theory gives such an order parameter with nodes required by symmetry.
[Aside: in our paper, this is denoted d_x2-y2, but that is because of how the x-y axes are defined].

Experimental observation of the Hund's metal to bad metal crossover

A definitive experimental signature of the crossover from a Fermi liquid metal to a bad metal is the disappearance of a Drude peak in the optical conductivity. In single band systems this occurs in proximity to a Mott insulator and is particularly clearly seen in organic charge transfer salts and is nicely captured by Dynamical Mean-Field Theory (DMFT).

An important question concerning multi-band systems with Hund's rule coupling, such as iron-based superconductors, is whether there is a similar collapse of the Drude peak. This is clearly seen in one material in a recent paper

Observation of an emergent coherent state in the iron-based superconductor KFe2As2 
Run Yang, Zhiping Yin, Yilin Wang, Yaomin Dai, Hu Miao, Bing Xu, Xianggang Qiu, and Christopher C. Homes

Note how as the temperature increases from 15 K to 200 K that the Drude peak collapses. 

The authors give a detailed analysis of the shifts in spectral weight with varying temperature by fitting the optical conductivity (and reflectivity from which it is derived) at each temperature to a model consisting of three Drude peaks and two Lorentzian peaks. Note this involves twelve parameters and so one should always worry about the elephants trunk wiggling.

On the other hand, they do the fit without the third peak, which is of the greatest interest as it is the sharpest and most temperature dependent, and claim it cannot describe the data.

The authors also perform DFT+DMFT calculations of the one-electron spectral function (but not the optical conductivity) and find it does give a coherent-incoherent crossover consistent with the experiment. However, the variation in quasi-particle weight with temperature is relatively small.

Relating the Hall coefficient to thermodynamic quantities

Previously, I have posted about how in certain contexts one can relate non-equilibrium transport quantities to equilibrium thermodynamic quantities. This is particularly nice because for theorists it is usually a lot easier to calculate the latter than the former.
But, it should be stressed that all of these results are an approximation or only hold in certain limits.

Here are some examples.

The thermoelectric power can be related to the temperature derivative of the chemical potential through the Kelvin formula (illuminated by Peterson and Shastry).

A paper argues that the Weidemann-Franz ratio in a non-Fermi liquid can be related to the ratio of two different susceptibilities.

Work of Shastry showing that the high frequency limit of the Hall coefficient, Lorenz ratio and thermopower can be related to equilibrium correlation functions.

It has been suggested that the transverse thermoelectric conductivity (Nernst signal) due to superconducting fluctuations is closely related to the magnetisation.

Haerter, Pederson, and Shastry conjectured that for a doped Mott insulator on a triangular lattice that the Hall coefficient can be related to the temperature dependence of diamagnetic susceptibility.

Here I want to discuss some interesting results for the Hall coefficient R_H.
First,  remember that in a simple Fermi liquid (or a classical Drude model) with only one species of charge carrier of charge q and density n that

R_H=1/q n

Clearly this is a case where a rather complex transport quantity (which is actually a correlation function involving three currents) reduces to a simple thermodynamic quantity.

But, what about in strongly correlated systems?

There is a rarely cited paper from 1993

Sign of equilibrium Hall conductivity in strongly correlated systems 
 A. G. Rojo, Gabriel Kotliar, and G. S. Canright

It gives an argument of just a few lines that relates the Hall coefficient to the orbital magnetic susceptibility (Landau diamagnetism)

BTW. I think there is a typo in the very last equation. It should also contain a factor of the charge compressibility.

I am a bit puzzled by the derivation, because it appears to be completely rigorous and general. The derivation of the Kelvin formula for thermopower, also has this deceptive general validity. It turns out that "devil in the details" turns out to be that the two limits of sending frequency and wave vector to zero do not commute.

A related paper shows that with a certain limiting procedure the Hall response (at zero temperature) is related to the derivative of the Drude weight with respect to the density.

Reactive Hall Response
X. Zotos, F. Naef, M. Long, and P. Prelovšek

This is valuable because it gives a simple explanation of why in a doped Mott insulator the Hall coefficient can change sign as the doping changes.

The mysterious origin of resistivity in Fermi liquids

It is hard to believe that we really don't understand the basic issues that I am going to discuss.

Resitivity occurs in a metal because scattering causes decay of charge currents. This means that the total momentum of the electrons in the presence of an electric field decays.
However, in a Fermi liquid metal with strong electron-electron interactions the main scattering of electrons is due to electron-electron scattering. But, in such collisions the total momentum of the two electrons is the same before and after the collision.
One can calculate the life time of the quasi-particles and it is inversely proportional to the temperature squared. The quasi-particle scattering rate ~ T^2.
Suppose one makes the relaxation time approximation in the Boltzmann equation or equivalently, neglects vertex corrections in the corresponding current-current correlation function associated with the Kubo formula for the conductivity. Then the resistivity is proportional to the quasi-particle scattering rate and one has resistivity ~ T^2. We say the transport lifetime is the same as the quasi-particle lifetime.
However, these are approximations, and strictly speaking there is no decay of the total electron momentum (or current) by electron-electron scattering and so the resistivity should be zero!
One way to save the situation is when there is Umklapp scattering. However, this requires a special relation between the shape of the Fermi surface and the Brillouin zone, as illustrated below.

These issues and puzzles are highlighted in a beautiful paper

Scalable T^2 resistivity in a small single-component Fermi surface 
Xiao Lin, Benoît Fauqué, Kamran Behnia

By chemical doping they tune the charge density and Fermi energy by several orders of magnitude, with the size of the Fermi surface increasing from some very small fraction of the Brilloiun zone.
In all cases the resistivity equals A T^2, characteristic of electron-electron scattering.
The figure below shows how the proportionality factor A scales with the density.
They also find A scales with the inverse of the effective mass squared as one expects from the Kadowaki-Woods ratio.

Yet for small densities (and Fermi surfaces) it is just not clear how one can have electron-electron scattering since Umklapp scattering is not relevant.

This major puzzle awaits an explanation.

I thank David Cavanagh, Jure Kokalj, Jernez Mravlje, and Peter Prevlosek for stimulating discussions about this topic.

Note added. The theoretical issues are nicely reviewed in
Resistivity of non-Galilean-invariant Fermi- and non-Fermi liquids 
 H. K. Pal, V. I. Yudson, D. L. Maslov

Signatures of strong electron correlations in the Hall coefficient of organic charge transfer salts

Superconducting organic charge transfer salts exhibit many signatures of strong electron correlations: Mott insulator, bad metal, renormalised Fermi liquid, ...

Several times recently I have been asked about the Hall coefficient. There really is little experimental data. More is needed. But, here is a sample of the data for the metallic phase.
Generally, increasing pressure reduces correlations and moves away from the Mott insulator. Almost all of these materials are at half filling and at high pressures there is well defined Fermi surface, clearly seen in angle dependent magnetoresistance and quantum oscillation experiments.

The figure below is taken from this paper. At low temperatures the Hall coefficient is weakly temperature dependent and has a value consistent with the charge carrier density, i.e., what one expects in a Fermi liquid. However, about 30 K, which is roughly the coherence temperature, corresponding to the crossover to a bad metal, R_H decreases significantly, and appears to change sign.

The next data is from this paper and shows measurements on two different samples of the same material.

Note how in the two samples for a pressure of 4 kbar the temperature dependence and magnitude is not the same. This should be a point of concern about the reliability of the measurements.
But, broadly one sees again a significant temperature dependence, particularly on the scale of the coherence temperature.

Finally, the data below is from a recent PRL, and is for a material that is argued to be away from half filling (doped with 0.11 holes per lattice site (dimer)).

At high pressures there are a large number of charge carriers and weak temperature dependence, consistent with a Fermi liquid with a "large" Fermi surface.
However, at low pressure (i.e. when the metal is more correlated) the Hall coefficient becomes large and temperature dependent.

I thank Jure Kokalj, Jernez Mravlje, Peter Prelovsek, and Andre-Marie Tremblay for stimulating discussions about the data.

I welcome any comments.
Later I will post about the theoretical issues.

Quantum of thermal conductance

Here are a couple of things I find surprising about the electronic transport properties of materials.

1. One cannot simply have materials, particularly metals, that have any value imaginable for a transport coefficient. For example, one cannot make the conductance or the thermopower as large as one wishes by designing some fantastic material.

2. Quantum mechanics determines what these fundamental limits are. Furthermore, the limiting values of transport coefficients are often set in terms of fundamental constants [Planck's constant, Boltzmann's constant, charge on an electron].

The fact that this is profound is indicated by the fact that this was not appreciated until about 25 years ago. A nice clean example is the case of a quantum point contact with N channels. The conductance must be N times the quantum of conductance, 2e^2/h. This result was proposed by Rolf Landauer in 1957 but many people did not believe it until the first experimental confirmation in 1988.

The thermal conductance through a point contact should also be quantised. The quantum of thermal conductance is

Asides:
1. note that the Wiedemann-Franz ratio is satisfied.
2. this sets the scale for the thermal conductivity of a bad metal.

A paper in Science this week reports the experimental observation of this quantisation.

Physical manifestation of the Berry connection

Although I have written several papers about it I still struggle to understand the Berry phase and how it is or may be manifested in solids.
Recent reading, summarised below, has helped.

There is a nice short review Geometry and the anomalous Hall effect in ferromagnets
N. P. Ong and Wei-Li Lee

As late as 1999 Sundaram and Niu wrote down the semi-classical equations of motion for Bloch states in the presence of a Berry curvature, script F below.
(1) and (2) below. n.b. how there is a certain symmetry between x and k.

The last equation gives the "magnetic monopoles" associated with the Berry connection/. Aside: the Berry connection Omega_c is the analogue of the magnetic field. It is related to the curvature F tilde by (F tilde)_ab= epsilon_abc Omega_c.

The Berry connection is related to the Berry phase in the same sense that a magnetic field is associated with an Aharonov-Bohm phase.

The above text is taken from a beautiful paper Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property by Duncan Haldane.

The symmetry arguments above show why the anomalous Hall effect only occurs in the presence of time-reversal symmetry breaking, e.g. in a ferromagnet.

It is interesting that Robert Karplus (brother of Martin) and Luttinger wrote down what is now called the Berry connection as long ago as 1954! (30 years before Berry!)
They called it the anomalous velocity.
The connection with Berry and topology was only made in 2002 by Jungwirth, Niu, and MacDonald.
An extensive review of the anomalous Hall effect, both theory and experiment, is here.

Scaling plots near the Mott transition

Earlier this year Jure Kokalj brought to my attention an interesting PRL
Quantum Critical Transport near the Mott Transition
by H. Terletska, J. Vučičević, Darko Tanasković, and Vlad Dobrosavljević.

My interest in this paper increased this week when Vlad emailed me to tell me about a recent talk at KITP by Kazushi Kanoda. The right side of the slide below [click on it to see it larger] shows a scaling analysis of the temperature and pressure dependence of the resistivity of the organic charge transfer salt kappa-(ET)2(CN)3 near the pressure driven Mott transition. This scaling analysis is based on the theory in the PRL.

The left side shows the Dynamical Mean-Field Theory (DMFT) results [for a Hubbard model at half filling] in the PRL. The top shows the scaling of the resistivity curves and the bottom the T vs. U phase diagram where the yellow region is the "quantum critical" region above the Mott transition.

It is striking that the experimental curves involve scaling over about 6 orders of magnitude of the resistivity.

There are several things that are rather strange/exotic/interesting about the theory. 

First, there is a "duality" between the resistivity in the metallic and insulating sides of the transition.

Second, the critical resistivity is an order of magnitude larger than the Mott-Ioffe-Regel limit.

Third, connections are suggested with the "holographic duality" picture of in papers such as this one by Subir Sachdev.

Aside: This same organic material is also of considerable interest because the Mott insulating phase seems to exhibit a spin liquid ground state and metallic state becomes superconducting at low temperatures. (See this review).

Deconstructing the Hall effect in quasi-one-dimensional metals

Understanding the Hall effect in strongly correlated electron materials is a challenge. In simple metals, the Hall coefficient R_H is, to a good approximation, equal to the inverse of the charge carrier density, and so is weakly dependent on temperature.
Furthermore, it has the same sign as the charge carriers.

In contrast, in strongly correlated metals, R_H can vary significantly with temperature, including changing sign, and its magnitude and signh can be significantly different from the charge carrier density estimated from band structure calculations or alternative experimental measurements such as the Drude weight in the optical conductivity.
I have written several previous posts about this issue. A post on the cuprates illustrates the problem.

There is a really nice paper Hall effect in quasi-one-dimensional metals in the presence of anisotropic scattering by Nicholas Wakeham and Nigel Hussey.
They show that due to the highly anisotropic band structure in a quasi-one-dimensional Fermi liquid metal, a variation of the scattering rate of just a few per cent over the Fermi surface, can significantly modify the Hall coefficient.
The figure below compares experimental data for the cuprate PrBa2Cu4O8 to a simple model with a temperature dependent anisotropy of less than 4 per cent.
The measured Hall coefficient is an order of magnitude smaller than the band structure value and changes sign twice as a function temperature.

This relatively simple explanation of complex behaviour should caution against exotic and non-Fermi liquid interpretations of Hall effect data (e.g., this PRL).

Signatures of "band-like" transport in organic electronic materials

I used to regularly write posts about charge transport in organic electronic materials. Some of these generated lively discussion in the comments section.

This morning I read an interesting paper Band-Like Electron Transport in Organic Transistors and Implication of the Molecular Structure for Performance Optimization
by Nikolas Minder, Shimpei Ono, Zhihua Chen, Antonio Facchetti, Alberto Morpurgo

They correctly distinguish "band-like" transport where a charge carrier is delocalised over just a few molecules from true band transport where it is delocalised over a large number of molecules [or unit cells in a crystalline semiconductor such as silicon].

They claim that a signature of band-like transport is the common observation of a mobility that decreases with increasing temperature and a Hall effect signal. I agree with the former but am confused about the latter. I thought for incoherent polaron transport one could still have a Hall effect, as discussed in a classic paper by Friedman and Holstein. 

The authors overlook the fact that a signature of band-like transport is that the mobility should be larger than e a^2/hbar ~ 1 cm^2/Vsec. Ignorance of this old and important result seems to be common in the field.

Previously, I pointed out that comparing the relative magnitudes of the energy gaps respectively associated with mobility, optical conductivity, and thermopower is a nice way to distinguish coherent from incoherent transport.

Overdoped cuprates are an anisotropic marginal Fermi liquid II

Jure Kokalj, Nigel Hussey, and I have just completed a paper, Transport properties of the metallic state of overdoped cuprate superconductors from an anisotropic marginal Fermi liquid model.

We show how a relatively simple model self-energy [considered earlier in this PRL] gives a nice quantitative description of a wide range of experimental results on Tl2201 including intra-layer resistivity, frequency-dependent conductivity, and the Hall resistance. No new parameters are introduced beyond those needed to describe angle-dependent magnetoresistance experiments from Nigel's group.

One thing I found striking was just how sensitive the Hall conductivity is to anisotropies in the Fermi surface and the scattering rate [a point emphasized by Ong with his beautiful geometric interpretation].

We also show that our model self-energy successfully describes both the resistivity (with a significant linear in temperature T dependence) and the Hall angle ( ~T^2) without invoking exotic new theories.

A key outstanding challenge is to connect our model self-energy [which is valid in the overdoped region] to possible forms for the underdoped region where the pseudogap occurs.

We welcome comments.

Violating a text book rule

Solid state physics text books tell us that Matthiessen's rule is obeyed by simple metals: the temperature dependent resistivity is the sum of a temperature independent term due to elastic scattering off impurities and an impurity independent term which is temperature dependent due to inelastic scattering. I teach this to undergrads, but have struggled to actually find experimental data to show them.

Yesterday I came across a 1944 paper by Fairbank which contained the figure below for copper with tin impurities:

It looks pretty convincing. However, the text points out that the temperature dependence actually varies significantly with the impurity concentration, in violation of Matthiesens rule!

Does anyone know of better data?

Any simple explanations for these deviations?

Deconstructing the metal-insulator transition in 2DEGs

There is an interesting preprint, Wigner-Mott scaling of transport near the two-dimensional metal-insulator transition by M. M. Radonjic, D. Tanaskovic, V. Dobrosavljevic, K. Haule, and G. Kotliar.

They argue that the density dependent metal-insulator transition seen in Silicon MOSFETs and other two dimensional electron gases (2DEGs) in semiconductor heterostructures is not driven by disorder (which has been claimed for many years) but rather by electronic correlations. Furthermore, the relevant experimental data can be described by a Dynamical Mean-Field Theory (DMFT) treatment of the Wigner-Mott transition in an extended Hubbard model on a lattice.

This means that the non-monotonic temperature dependence of the resistivity is associated with the crossover from a Fermi liquid at low temperatures to a bad metal at higher temperatures. I think thermopower measurements may be the most effective way to test this claim (see an earlier  post).

A very strange metal

The linear chain compound Li0.9Mo6O17 exhibits a subtle competition between superconductivity, a "bad" metal, and a strange "insulating" phase. Recently large deviations from the Weidemann-Franz law were reported by Nigel Hussey's group.

The graph below shows the temperature dependence of the electrical resistance for current parallel to the chain direction. It has a "metallic" temperature dependence above about 30 K, and an "insulating" temperature dependence between the superconducting transition temperature around 1 K and 30 K. This is rather unusual and puzzling since one normally sees a direct transition from a metallic phase to a superconducting phase. Although there are other cases such as reported in this PRB [see Fig. 2 inset] for an organic charge transfer salt where a superconducting state occurs close to a charge ordered insulator [see also the Table in this PRL].

The data is taken from a Europhys. Lett. by Chen et al. which also reports a rather strange angular dependent magnetoresistance.

Deconstructing vertex corrections

Ultimately much of quantum many-body theory concerns calculating correlation functions which are measurable. For example, the conductivity can written as a current-current correlation function [Kubo formula]. The simplest approximation neglects vertex corrections and just calculates the "bubble" diagram consisting of the product of Green's functions.

What are vertex corrections? When do they matter? What sort of robust or general results are available about them?

Many people, including myself, often just ignore them. I fear this is partly motivated by difficulty rather good scientific criteria.

Below are a few things I am slowly learning, re-learning, and digesting.

Migdal showed that for the electron-phonon interaction the vertex corrections are small due to the smallness of the ratio of the electronic mass to the nuclear mass [alternatively the ratio of the speed of sound to the Fermi velocity].
But, Migdal's argument breaks down for an electron-magnon interaction.

Neglecting vertex corrections is equivalent to making the relaxation time approximation (RTA) when solving the Boltzmann equation. Then the quasi-particle lifetime equals the transport lifetime because one ignores dependence of the scattering rate on momentum transfer. Below is some helpful text from a review by Kontani:

 ....we have to take the Current Vertex Correction [CVC] into account correctly, which is totally dropped in the Relaxation Time Approximation [RTA]. In interacting electron systems, an excited electron induces other particle– hole excitations by collisions. The CVC represents the induced current due to these particle–hole excitations. The CVC is closely related to the momentum conservation law, which is mathematically described using the Ward identity [28–31]. In fact, Landau proved the existence of the CVC, which is called backflow in the phenomenological Fermi liquid theory, as a natural consequence of the conservation law [28]. The CVC can be significant in strongly correlated Fermi liquids owing to strong electron–electron scattering.

For specific types of interactions Ward identities allow one to relate the vertex function to derivatives of the self energy. Mahan's book (Section 8.1.3) discusses this in detail.

In the limit of infinite dimensions [in which Dynamical Mean-Field Theory (DMFT)] becomes exact, vertex corrections can be neglected.

In a recent PRB, Bergeron, Hankevych, Kyung, and Tremblay calculated the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. They found that at "high" temperatures (T > 0.2t) vertex corrections did not matter much, but were significant at lower temperatures near a quantum critical point.