What did Kondo do?

Chapter 2 of Alex Hewson's The Kondo problem to heavy fermions reviews what Kondo actually did to get his name on the problem. Here is a brief summary of the highlights from last weeks reading group.

He considered the experimental data on the temperature dependence of the resistivity of metals containing magnetic impurity atoms. It was particularly puzzling that there was a minimum. Generally, one expects scattering (and thus resistivity) to increase with increasing temperature.

First, Kondo recognised that the experimental data suggested that it was a single impurity problem, i.e, one could neglect interactions between the impurities.
Second, the effect seemed to scale with magnitude of the local magnetic moments.
This led him to consider the simplest possible model Hamiltonian the s-d model proposed by Zener in 1951, but now known as the Kondo model.

According to Boltzmann/Drude/Kubo at low temperatures the resistivity of a metal is proportional to the rate at which electrons with momentum k are elastically scattered into different states with momentum k'

Here T_kk' is the scattering T matrix.
Considering Feynman diagrams to second order in J, Kondo showed

One then substitutes this in the formula for the conductivity.

Integrating over energy leads to the famous logarithmic temperature dependence.

I am not really clear on what the essential physics is that leads to this logarithmic divergence, except something to do with spin flips in the particle-hole continuum above the Fermi energy.

The Kondo problem is that this leads to a logarithmic divergence at low temperatures. This suggests perturbation theory diverges. It also suggests an infinite scattering cross section which violates the unitarity limit. Somehow, this divergence must be cut off at lower temperatures by different physics.

The new physics turns out to be formation of spin singlets between the impurity spin and the conduction electron spins. These are known as Kondo singlets, although it was actually Yosida, Anderson, Nozieres, and Wilson who introduced/developed/showed this idea.

Killing comparisons

It is a natural human tendency to compare oneself to ones peers.
I suggest that this can be quite unhelpful for your mental health and for harmonious relationships.
A natural consequence of such comparisons may be discouragement or hubris depending on your personality.

Grad students and postdocs may compare hours worked, numbers of papers, number of interviews, numbers of invited conference talks, attention from their advisor....

Faculty may compare total funding, size of their latest grant, numbers of students, size of office, speed of promotion, h-index, lab space, ...
This can lead to bitterness and friction.

When I was younger I struggled due to making such comparisons. Mostly they led to unnecessary anxiety and discouragement. Furthermore, with hindsight my "metrics" turned out to be pretty irrelevant indicators of future success [i.e. survival] in science. I never considered luck, perseverance, flexibility, passion, communication and personal skills...

Now I am careful not to make comparisons. I don't think they help anyone.
I urge you not to make comparisons. Your mental health may be much the better for it.

First-order transition into the pseudogap state

There is a nice paper

Pseudogap temperature as a Widom line in doped Mott insulators

by Giovanni Sordi, Patrick Sémon, Kristjan Haule, and Andre-Marie Tremblay

The abstract ends with the important and articulate claim:

Broken symmetry states appear in the pseudogap and not the other way around.

The figure below shows the phase diagram that the authors calculated for the doped Hubbard model with cluster DMFT. The key point is that at low temperatures there is a first-order phase transition from the pseudogap to a correlated Fermi liquid. Furthermore, there is no symmetry breaking associated with this transition. In this respect the phase diagram is analogous to a liquid-vapour transition in a simple fluid and so the authors identify the metal-pseudogap crossover line with the Widom line for the former class of transitions.

This is an elegant new idea.

In the actual materials this first-order transition is masked by the presence of superconductivity.
Surely, this means that in high magnetic fields, which destroy the superconductivity, one should see this transition. In a single material (i.e. fixed doping) observing this may be a little tricky, requiring the first-order line to have a negative slope, and extremely high magnetic fields.

Another really nice and interesting result is connecting the pseudogap to fluctuating RVB type singlets. The figure below shows the temperature dependence of the probability of finding a singlet state on a single plaquette. [See earlier post one and two on how these RVB states appear in four-site Heisenberg models.]

Another question concerns what happens in the half-filled Hubbard model and the organic charge transfer salts. Figure 4 of an earlier PRL by the same authors gives a more general phase diagram (temperature vs. doping and U/t). I am not quite sure how to decode it and connect it to the organics and the bandwidth driven Mott transition that occurs at half-filling.

From RVB theory to parliament

When I was a grad student at Princeton, Phil Anderson had a number of students (Zhou Zou, Ted Hsu, Joe Wheatley, ...) who worked on RVB theory. They all eventually left physics for Wall Street. Phil used to joke that they were all making more money than him!

I just learned that Ted Hsu is now a member of parliament in Canada!
I saw this in an article in Physics Today that raises concerns about changes in funding direction for physics in Canada.
There is also an interview with him on the Physics Today site.

Thermodynamics of a transition between a bad metal and a Mott insulator

Jure Kokalj and I just finished a paper
Thermodynamics of a bad metal-Mott insulator transition in the presence of frustration

We study the temperature dependence of a range of thermodynamic properties (charge susceptibility, specific heat, entropy and spin susceptibility) of the Hubbard model on the anisotropic triangular lattice at half filling by means of the numerical finite-temperature Lanczos method. This Hubbard model describes several important families of superconducting organic charge transfer salts.

The results include

• Clear signatures of a metal-Mott insulator transition in the charge susceptibility.
• The metal-insulator transition can be driven either by increasing interactions or by reducing frustration.
• The metallic phase is characterized by a small charge susceptibility, large entropy, low coherence temperature, large renormalized quasiparticle mass, and large spin susceptibility.
• The coherence temperature corresponds to destruction of quasi-particles and crossover from a Fermi liquid to a bad metal. Our estimate of the temperature is comparable to what is observed in the organics. 
• The local magnetic moment in the metallic phase is large and comparable to the local moment in the insulating phase. This is characteristic of a bad metal.
• Frustration increases the density of low-lying spin excitations in the Mott insulating phase and decreases longer range spin correlations.

Are the iron pnictide superconductors strongly correlated?

Yes. According to a nice review by Yu, Si, Goswami, and Abrahams.

A key piece of the evidence for strong correlations is recent inelastic neutron scattering experiments reported in this Nature Physics paper.
They show that even in the superconducting materials [which are doped from parent Antiferromagnetic compounds] there are sizeable fluctuating magnetic moments. The figure below shows the dynamical spin susceptibility versus energy. For both superconducting and antiferromagnetic materials the susceptibility is essentially the same for energies above 100 meV.

The area under the curve is equal to the square of the fluctuating local moment. The magnitude is a few Bohr magnetons, as one would expect in a doped Mott insulator.

Furthermore, a weak coupling RPA treatment [which is invoked to explain the superconductivity] cannot capture the magnitude of these spin fluctuations. In contrast, a DMFT treatment from Park, Haule, and Kotliar is consistent with the experimental data, highlighting the role of strong correlations and Hund's rule coupling.

Caveats about thermopower interpretation

The thermoelectric power of a metal is rather complex. Even Ashcroft and Mermin suggested that it was difficult to interpret and relate to the theoretical calculations.
Earlier posts have considered some of the subtleties, particularly in strongly correlated electron systems.

To me a couple of recent experimental papers present beautiful data but are not cautious enough in their interpretation. They need to rule out alternative explanations [see below] before I will be convinced of the explanations that they propose.

Fermi-surface reconstruction by stripe order in cuprate superconductors
F. Laliberté, J. Chang, N. Doiron-Leyraud, E. Hassinger, R. Daou, M. Rondeau, B.J. Ramshaw, R. Liang,  D.A. Bonn, W.N. Hardy, S. Pyon, T. Takayama, H. Takagi, I. Sheikin, L. Malone, C. Proust, K. Behnia, and Louis Taillefer

They observe a sign change in the thermopower and associate this with a Fermi surface reconstruction, stripe formation, and a quantum phase transition.

Spin fluctuation driven enhancement of thermopower near the quantum critical point in Ba(Fe(1-x)Co(x))2As2

S. Arsenijević, H. Hodovanets, R. Gaál, L. Forró, S. L. Bud'ko, P. C. Canfield

In both papers, the fact that S/T for a specific doping has a logarithmic temperature dependence over about a decade in temperature is equated with quantum criticality.

Why am I not convinced?

1. A recent preprint shows the cuprate data can be explained using the semi-classical Boltzmann equation and a Functional Renormalisation Group treatment of the Hubbard model (see Figure 10). The sign change is associated with a the van Hove singularity and a Lifshitz transition.

2. Dynamical mean-field theory (DMFT) shows how the thermopower can change sign as a function of temperature due to strong correlations and the associated low coherence temperatures. (See e.g. Figure 4 in this preprint which I discussed in an earlier post).

I suspect that a DMFT treatment of the relevant multi-band Hubbard model with Hund's rule coupling [a la Park, Haule, Kotliar] will be able to explain the thermopower data for the pnictides.