The legacy of 40 years of cuprate superconductivity

In February 1986, Bednorz and Müller made a stunning discovery: superconductivity at a temperature of 35 K in a doped copper oxide (cuprate). Arguably, this discovery changed condensed matter physics. In April 1986, they submitted their results to Z. Phys. B. Only nineteen months later, they were awarded the Nobel Prize in Physics, the shortest time ever between a discovery and the award. A nice and short review of the history is here.

One measure of my estimate of the influence of this discovery is that it received about 5 pages of coverage in my Condensed Matter Physics: A Very Short Introduction. (See Chapter 5, Adventures in Flatland).

How things have developed over the past forty years, for better and worse, may be representative of how science advances: discovery by serendipity, hype about applications, unexpected secondary benefits, foundational questions, new concepts, unification, and incremental advances.

Hype about technological applications

On March 20, 1987, The New York Times had a front-page article, DISCOVERIES BRING A 'WOODSTOCK' FOR PHYSICS, by James Gleick. This followed the 1987 APS March meeting. It began 

"Physicists from three continents converged on the New York Hilton for a hastily scheduled special conference on a string of discoveries that seem certain to produce a rapid cascade of commercial applications in electricity, magnetism and electronics.There are many things we know and understand that we did not when they were first discovered."

This has largely been unfulfilled. There are a few niche applications, but cuprates are not used in electricity distribution or even in the superconducting magnets in hospital MRI machines, which are probably the main commercial application of superconductors. One of the significant obstacles is that it is hard to make wires from these materials, as they are ceramics. This is an example of the common gap between research laboratory science and commercially viable technology.

After 40 years, do we have a successful theory?

It depends on who you ask. But I would say there is a lot we do understand.

We have a phenomenological theory for all the macroscopic phenomena associated with the superconducting state: Ginzburg-Landau theory!

Properties of the superconducting state are well-described by a BCS wavefunction with a d-wave order parameter and the associated Bogoliubov quasiparticles. [This is somewhat puzzling, as in the metallic state quasi-particles are not well defined].

Although not everyone agrees, I think it is fair to say that the essential physics is in a one-band Hubbard model, and the key physics is:

strong electronic correlations,

a doped antiferromagnetic Mott insulator,

d-wave pairing that is "mediated"/caused from some mixture/variant of antiferromagnetic spin fluctuations or RVB spin singlets,.....

We certainly don't understand the cuprates at the same level as elemental superconductors. But we do understand the essential physics.

What is harder to describe and understand are the states adjacent to the superconducting state in the phase diagram: the pseudogap state and the strange metal.

Strongly correlated electron materials became a large, vibrant and unified field

Before 1986, there were small, disconnected communities intermittently interested in transition metal oxides, rare earths, Kondo impurities, Mott metal-insulator transitions, organic superconductors, heavy fermions, and quantum antiferromagnets.

The discovery of the cuprates brought together these communities as they found common interests, challenges, questions, concepts, and techniques.

The discovery of superconductivity in strontium ruthenate, alkali fullerides, iron pnictides and chalcogenides, twisted bilayer graphene and more cuprates, organic charge-transfer salts, and heavy fermions has shown how rich these systems are. The challenge is to understand the similarities and differences between these chemically and structurally diverse systems. In many of them, superconductivity is proximate to a Mott insulating state.

The unity and excitement were probably stimulated and enhanced by the activities and ideas of high-profile theorists such as Anderson, Schrieffer, Scalapino, Pines, Rice, and Varma. On the other hand, their acrimonious disagreements probably did not help.

Secondary theoretical benefits

The things I list below were not new ideas when the cuprate discovery happened. However, interest in the cuprates led them to become major research themes and ideas.

Importance of phase diagrams, including as a function of interaction parameters in toy models

Highlighting the limitations of electronic structure methods based on Density Functional Theory with approximate Exchange-Correlation functionals (i.e., anything computational). In the presence of strong correlations, DFT methods have spectacular failures. For example, predicting a metallic state instead of the Mott insulator.

Low dimensionality leads to qualitatively different behaviour, including the possibility of new types of order and quasiparticles. This is most dramatic in one dimension, where one has Luttinger liquids and spin-charge separation.

Spin liquids. Landau was wrong. Spontaneous symmetry breaking does not always occur in antiferromagnets.

Non-Fermi liquids. Landau was wrong. Not all metals are Fermi liquids.

Quantum criticality. Although this is a robust concept for certain toy models, whether it is relevant to the cuprates remains contentious.

Systematic improvements in approximation schemes and numerical techniques - exact diagonalisation, DMRG, DMFT, quantum Monte Carlo,...

Emergence. Chemical complexity and strong interactions can lead to new states of matter.

Secondary experimental benefits

Better probes. The desire to characterise the cuprates helped drive significant improvements in the resolution of ARPES (Angle-Resolved PhotoEmission Spectroscopy), STM (Scanning Tunnelling Microscopy), and inelastic neutron scattering. These advances have born fruit in the study of a wide range of other materials, beyond the cuprates.

Growth of single crystals. The early days of the cuprates produced a lot of junk experimental results because of the poor quality of the samples produced by "shake and bake". However, the involvement of solid-state chemists has improved things. The techniques have also led to the production of single crystals for a wide range of strongly correlated materials.

Why is there so little research on cuprates today?

Today, there is little research directly on cuprates, both theoretically and experimentally. It is hard to get funding to work on them, even though there is a lot we don't understand really well.

This is because of the problem of fashion in science. The low-lying fruit has been picked. There is a continuous new stream of materials being discovered with exotic properties, the latest being twisted bilayer van der Waals compounds.

Single orbital + multiple sites = Rich physics

Since the discovery of the iron-based superconductors, it has become clear that the combination of multiple orbitals and strong correlations can lead to rich physics, beyond what one sees in single orbital routes.
An alternative route to rich physics is a single orbital model on a lattice with multiple sites in a unit cell.

Henry Nourse, Ben Powell, and I just posted a preprint
Multiple insulating phases due to the interplay of strong correlations and lattice geometry in a single-orbital Hubbard model

We find ten distinct ground states for the single-orbital Hubbard model on the decorated honeycomb lattice, which interpolates between the honeycomb and kagome lattices and is the simplest two-dimensional net. The rich phase diagram includes a real-space Mott insulator, dimer, and trimer Mott insulators, a spin-triplet Mott insulator, flat band ferromagnets, and Dirac metals. It is determined as a function of interaction strength, band filling, and hopping anisotropy, using rotationally invariant slave boson mean-field theory.

We welcome comments.

What was the greatest discovery of the past decade?

To my readers my best wishes for the New Year and the new decade!

It is worth reflecting on what has been achieved in condensed matter and chemical physics over the past decade. Which discovery or achievement would you rate as the most exciting, surprising, or significant?

To benchmark things, this is what I would say about previous decades, with regard to hard condensed matter, with a personal bias towards strongly correlated electron systems.

1970s: scaling and the renormalisation group

1980s: quantum Hall effects, cuprate superconductivity, heavy fermions, scanning tunneling microscopy (STM)

1990s: Dynamical Mean-Field Theory (DMFT), Kondo effect in quantum dots, superconducting qubits, (Angle-Resolved PhotoElectron Spectroscopy) ARPES advances, DMRG

2000s: iron-based superconductors, graphene, DMFT+DFT, topological insulators

2010s: ?

To be honest, I am worried that with each decade the discoveries are somewhat less exciting or significant. On the other hand, incremental advances, particularly steady ones over several decades should not be looked down on. An example is computational electronic structure methods and increases in the energy and momentum resolution associated with inelastic neutron scattering spectroscopy and ARPES.

What would you nominate for the past decade?

Imaging orbital-selective quasi-particles in a Hund's metal

Over the past two decades, a powerful new technique has been developed to determine quasi-particle properties in strongly correlated electron systems, based on STM (scanning tunneling microscope) measurements. Quasi-particle interference (QPI) has proved to be particularly useful for studying cuprates (e.g. in revealing the d-wave pairing) and now for iron-based superconductors. The basic physics is as follows. One measures the changes in the local tunneling density of states N(r,E), associated with a single impurity that scatters quasi-particles with a change in momentum q. Then the Fourier transform of this change is

The text above is taken from a nice paper
Imaging orbital-selective quasiparticles in the Hund’s metal state of FeSe 
A. Kostin, P.O. Sprau, A. Kreisel, Yi Xue Chong, A.E. Böhmer, P.C. Canfield, P.J. Hirschfeld, B.M. Andersen and J.C. Séamus Davis

They show theoretically that the intensity of the interference pattern is quite sensitive to the quasi-particle weights of the different d-orbital bands. The experiments are consistent with

The key figure is below. It shows shaded intensity plots of the change in DOS as a function of wavevector. The central column is experimental data with E increasing from -20 meV to +15 meV as one goes down the column. The left and right columns show theoretical values for the same quantities, calculated with all the quasi-particle weights Z=1 (left) and the Z values above (right).   

   

The large variation between the Z values for different orbitals shows how the effect of the correlations are orbital selective.

The same Z values were used by the same cast of characters in a study of the superconducting state that showed orbital selectivity played a key role in the Cooper pairing, including the significant variation of the energy gaps over the different Fermi surfaces. The quantitative agreement between experiment and the associated theory is quite impressive.

I thank Alejandro Mezio for bringing the papers to my attention.

Superconductivity in a Hund's metal

The BCS theory of superconductivity is one of the towering intellectual achievements of the twentieth century. There are many ingredients to the theory and many significant results. One key step is to consider an effective interaction that is responsible for the Cooper pairing. A key result is that many properties are universal in that one can rescale temperatures and energies by the energy gap (at zero temperature), Delta(0) or the transition temperature Tc. In the limit of weak-coupling there is a universal ratio
2 Delta(0)/kTc = 3.5
Most elemental superconductors are consistent with this value. Some such as Hg and Pb have larger values, but these can actually be calculated when strong coupling effects are taken into account, via the Eliashberg equations.

Unconventional superconductors (cuprate, organic, heavy fermion, iron based) have resisted a simple unifying theory and universal trends, comparable to the stellar success of BCS theory. For example, the gap/Tc ratio is all over the place. However, there has been some progress for the iron-based superconductors. Recent ARPES results (summarised in the figure at the bottom below) have shown a universal ratio, of about 7.2 for a wide range of materials.

A fascinating feature of these iron-based materials is the nature of the metallic state that undergoes the superconducting instability. I have written several blog posts about the Hund's metal. One important feature is that there is relatively low coherence temperature below which a Fermi liquid metal forms, and there is a correspondingly low energy scale Omega0 associated with spin fluctuations, which become very slow. This arises from the rich Kondo physics associated with the multi-orbital character of the system. Furthermore, the spin fluctuation spectrum has a power law dependence above Omega0.

The above ideas come together in an interesting preprint
On the Superconductivity of Hund's Metals 
Tsung-Han Lee, Andrey Chubukov, Hu Miao, Gabriel Kotliar

They consider a single band superconductor described by the strong-coupling Eliashberg equations where the frequency dependence of the (effective) electron-electron attraction is given by

where the exponent gamma is treated as a variable. The Eliashberg equations are solved (for a single band) and give the following relationship between the gap ratio and the exponent gamma.

The value of gamma=1.2 is that associated with the relevant Kondo problem above the coherence temperature. The gap ratio corresponds to the black dashed line in the graph below.

One thing should be stressed here is that one is observing a transition from an incoherent metal into a superconductor, unlike in the BCS situation where the transition is from a coherent Fermi liquid.
I thank Alejandro Mezio for bringing the paper to my attention.

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.

Questions about slave-particle mean-field theories of Hund's metals

One of most interesting new ideas about quantum matter from the last decade is that of a Hund's metal. This is a strongly correlated metal that can occurs in a multi-orbital material (model) as a result of the Hund's rule (exchange interaction) J that favours parallel spins in different orbitals.
Above some relatively low temperature (i.e. compared to the bare energy scales such as non-interacting band-widths, J, and Hubbard U) the metal becomes a bad metal, associated with incoherent excitations.
An important question concerns the extent to which slave mean-field theories can capture the stability of the Hund's metal, and its properties including the emergence of a bad metal above some coherence temperature, T*.

In a single-band Hubbard model, the strongly correlated metallic phase that occurs in proximity to a Mott insulator is associated with a small quasi-particle weight and suppression of double occupancy, reflecting suppressed charge fluctuations. This is captured by slave-boson mean-field theory, including the small coherence temperature.

In contrast, to a "Mott metal", a Hund's metal is associated with suppression of singlet spin fluctuations on different orbitals, without suppression of charge fluctuations and is seen in a Z_2 slave-spin mean-field theory at zero temperature.

Specific questions are whether slave mean-field theories at finite temperature can capture the following?

• The coherence temperature, T*.
• A suppression of spin singlet fluctuations at T increases towards T*.
• An orbital-selective bad metal may occur in proximity to an orbital selective Mott transition. This is where at least one band (orbital) is a Fermi liquid and another is a bad metal. This would mean that there are two different coherence temperatures. 
• The emergence of a single low-energy scale, common in both bands, as is seen in DMFT.
• The spin-freezing temperature.

Finally, how does the stability of the Hund's metal change with the number of orbitals?
Figures in this post suggest that the Hund's physics is more pronounced with increasing the number of orbitals. However, that may be because the critical U (and thus proximity to the Mott insulator) changes with the number of orbitals and all the curves are for the same U.

What do you call a mixture of a bad metal and a good metal?

It is fun to come up with clever names for new physical phenomena: quark, big bang, Janus, slepton,  chromodynamics, inflation, squashon, ...
There is an amusing article by David Mermin about how he managed to get boojum  accepted as a scientific term.
Can you think of others?

What is a good synonym for something that has both good and bad qualities?
A curate's egg?

I was wondering about this because of thinking about a metal that is a mixture of a good metal and a bad metal. This is relevant close to an orbital-selective Mott transition. There it may be possible to have multiple Fermi liquids (associated with multiple bands) at low temperatures with different coherence temperatures. For example, this does occur in strontium ruthenate.  As a result, when the temperature is increased one can enter a state in which one of the bands has coherent quasi-particles (and a well-defined Fermi surface) and another does not, i.e. it is a bad metal.

A relevant paper is
Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in AxFe2-ySe2 (A 1⁄4 K, Rb) Superconductors 
M. Yi, D. H. Lu, R. Yu, S. C. Riggs, J.-H. Chu, B. Lv, Z. K. Liu, M. Lu, Y.-T. Cui, M. Hashimoto, S.-K. Mo, Z. Hussain, C. W. Chu, I. R. Fisher, Q. Si, and Z.-X. Shen

They present ARPES data, including that below, that shows how the spectral intensity changes as the temperature increases. The blue and red curves are identified with different d-orbital bands.

Being cautious, I am a bit wary about how clearly the data do support the conclusions. Nevertheless, ...
The authors also present a slave-spin theory calculation for a five-band Hubbard-Kanamori model that is consistent with the experimental data.

I thank Alejandro Mezio for helpful discussions about this topic.

Low energy scales near the orbital-selective Mott transition

One of the most fundamental and profound concepts in quantum many-body theory is the emergence of low energy scales that are much smaller than the energy scales in the "bare" Hamiltonian.
For example, in a metallic phase near the Mott transition in a single band system, there is the energy scale associated with a Fermi liquid. Studies using Dynamical Mean-Field Theory (DMFT) have shown how this scale is associated with ``kinks'' in the quasi-particle dispersion relation and is related to the energy scale for spin fluctuations.

The problem of the Mott transition in multi-band systems (degenerate orbitals) is fascinating and of renewed interest since the discovery of iron-based superconductors. A basic question concerns how the Mott transition is qualitatively different from in single band systems. More specifically, how does a Hund's rule coupling change things?

One new concept is that of an orbital-selective Mott transition. This is where one or more of the bands remains metallic but others become Mott insulators. This concept was originally introduced to explain the intriguing properties of Ca_xSr_2-xRuO4 with x ~ 0.5: it is metallic but has localised spin-1/2 magnetic moments.
[For a critical discussion see the  nice review Strong correlations from Hund's coupling by Antoine Georges, Luca de' Medici, and Jernej Mravlje.]

One might expect that near this transition there are separate low energy scales associated with each of the bands and that these scales are quite different for the bands that become insulator.
However, this is not the case.

There is a nice paper
Emergence of a Common Energy Scale Close to the Orbital-Selective Mott Transition 
Markus Greger, Marcus Kollar, and Dieter Vollhardt

They use DMFT to study a two-band Hubbard model with different bandwidths. They calculate the one-electron spectral functions, the electronic self energy, and the dynamical spin susceptibilities.

The left panel below shows the spectral functions for the two bands. Note how for one the quasi-particle peak width is much smaller than the other.
The right panel (top) shows the energy dependence of the real part of the self-energy for the two bands. Surprisingly, the kink occurs at the same energy.
Furthermore, the bottom of the right panel shows that this peak corresponds to the peak in the dynamical spin susceptibility for both bands.

The figure below shows that "If the Hund’s rule coupling is sufficiently strong, one common energy scale emerges which characterizes both the location of kinks in the self-energy and extrema of the diagonal spin susceptibilities."

The authors then give a physical explanation of this energy scale from a two-impurity Kondo model.

Strong correlations and thermal expansion in iron based superconductors

There is a nice preprint
Strong Correlations, Strong Coupling and s-wave Superconductivity in Hole-doped BaFe2As2 Single Crystals 
F. Hardy, A. E. Böhmer, L. de' Medici, M. Capone, G. Giovannetti, R. Eder, L. Wang, M. He, T. Wolf, P. Schweiss, R. Heid, A. Herbig, P. Adelmann, R. A. Fisher, C. Meingast

The figures below summarise some of the key physics. The top is the phase diagram.

The bottom shows the specific heat coefficient gamma as a function of alkali metal content (Cs to Rb to K, and then fractional K content (doping x).

Note that

a. The black curve shows values calculated from density functional theory (DFT) based calculations. The blue points are experimental data, which are as much as an order of magnitude larger, reflecting strong correlations.

b. As one goes K to Rb to Cs the correlations are enhanced, somehow reflecting the "negative pressure" associated with the increasing ion size.

c. The experimental trend is captured nicely by calculations using slave spins (SS) to treat the relevant multi-band Hubbard model with Hund's rule coupling and band structure from DFT.

The thermal expansion alpha is particularly interesting because it is dominated by electronic effects (unlike in most metals) and shows a coherent-incoherent crossover from a Fermi liquid (where alpha/T is constant) to a bad metal at a temperature T*.
As one goes K to Rb to Cs alpha/T is enhanced reflecting the increased correlations.

One reason I am particularly interested in the manifestation of strong correlations in the thermal expansion because this also occurs in organic charge transfer salts, as discussed at length in a recent paper I published with Jure Kokalj. But, we did struggle to obtain a detailed quantitative description of the experiments, partly because of the crystallographic complexity.
It would be nice to see if a DMFT + LDA treatment of the relevant model for these iron compounds could describe the data above.

I thank Christoph Meingast for bringing this work to my attention and helpful discussions about it.

Linear magnetoresistance in Dirac semi-metals turns out to be boring

An enduring theme on this blog is that one should always consider "boring" explanations for "surprising" experimental results before invoking the exotica beloved and promoted by luxury journals. An example was the extremely large magnetoresistance materials.

In most metals the magnetoresistance [change in electrical resistance with external magnetic field B] depends quadratically on the B.
The past few years there have been a plethora of papers about linear magnetoresistance in topological insulators, iron pnictide superconductors, and Dirac semi-metals. I wrote a post which discusses the issue and also links to an earlier post that considers different theoretical explanations.
Many of these papers, particularly those in the baby Natures, want to link the linear magnetoresistance to the Dirac cone and possibly the Berry geometric phase associated with it.

However, there are some critical and constructive papers. For example,
Magnetotransport of proton-irradiated BaFe2As2 and BaFe1.985Co0.015As2 single crystals 
D. A. Moseley, K. A. Yates, N. Peng, D. Mandrus, A. S. Sefat, W. R. Branford, and L. F. Cohen

By using proton-beam irradiation to change the defect scattering density, we find that the dependence of the magnitude of the linear magnetoresistance on scattering quite clearly contravenes this prediction [of Abrikosov's quantum model].

There is a nice paper that gives a rather mundane explanation for the experiments.
Linear magnetoresistance in metals: Guiding center diffusion in a smooth random potential
Justin C. W. Song, Gil Refael, and Patrick A. Lee

We predict that guiding center (GC) diffusion yields a linear and nonsaturating (transverse) magnetoresistance in 3D metals. Our theory is semiclassical and applies in the regime where the transport time is much greater than the cyclotron period and for weak disorder potentials which are slowly varying on a length scale much greater than the cyclotron radius. Under these conditions, orbits with small momenta along magnetic field B are squeezed and dominate the transverse conductivity. When disorder potentials are stronger than the Debye frequency, linear magnetoresistance is predicted to survive up to room temperature and beyond. We argue that magnetoresistance from GC diffusion explains the recently observed giant linear magnetoresistance in 3D Dirac materials.

In their calculations the Berry phase plays no role.

The large electronic entropy of bad metals

A common feature of bad metals is that at relatively low temperatures [of the order of the coherence temperature which is much less than the non-interacting Fermi temperature] they have an entropy per electron of the order of Boltzmann's constant, k_B. This is more characteristic of a classical than a quantum system. For localised non-interacting spins the entropy is ln(2) k_B. In contrast, in a Fermi liquid such as an elemental metal, the electronic entropy is of order k_B T/T_F where T is the temperature and T_F is the Fermi temperature (10,000s K in an elemental metal).

I don't think this bad metal property of the large electronic entropy is emphasised enough, although it was highlighted here.

I illustrate this below with two sets of experimental data. The first set is measurements on YBCO, with x related to the doping, small x corresponding to the under doped regime and x=1 approximately optimal doping.

In the metallic state, the entropy increases approximately linearly with temperature and has a value of the order of 0.6 k_B at T=300 K.

Aside: previously I discussed how the entropy is maximal at optimal doping and this is reflected in a change in sign of the thermopower.

The second is from the same research group (20 years later) concerning a family of iron pnictides
Electronic specific heat of Ba1−xKxFe2As2 from 2 to 380 K
G Storey, J W Loram, J R Cooper, Z Bukowski, and J Karpinski

Note that here the entropy is divided by the temperature, so in a simple Fermi liquid the graph would be flat. For reference, the value of S/T [and the specific heat] calculated from DFT-based band structure calculations is about 10 mJ/mol/K^2. Hence, we see how strong correlations enhance the entropy. Furthermore, S ~ ln(2) R ~ 50 mJ/mol/K for temperatures of the order of 100 K for x > 0.4.

Rich Kondo physics in iron-based superconductors.

One of the most fascinating, challenging, and frustrating aspects of the iron-based superconductors is the presence of many competing energy scales, particularly associated with Hund's rule coupling.
There are debates about just how strong the correlations are, how large the spin moments are, and how relevant Mott physics is.

Furthermore, Hund's rule seems to lead to an unusual metallic state that is both difficult to characterise and describe theoretically. It has some signatures of a bad metal, as emphasised by (amongst others) Haule, Kotliar, Si, and Abrahams, but this characterisation is disputed by Johnston in his review [see Section 3.8.2].

Given the chemical and structural diversity of this large class of materials, we should be cautious about claiming the same physics is dominant in all the materials.

There is a nice paper that illustrates the richness of this system.
Local Quantum Criticality of an Iron-Pnictide Tetrahedron 
T. Tzen Ong and Piers Coleman

It shows how the Hund's rule coupling and interplay of orbital and spin degrees of freedom can suppress formation of the Fermi liquid state that would occur in a one band system.

One of our key observations, is that in addition to their spin physics, the iron-based tetrahedra develop an orbital degree of freedom associated with the degenerate eg orbitals. For conventional transition metal ions, the Hund’s coupling JH locks the unpaired electrons together into a high-spin configuration, exponentially suppressing the spin-Kondo temperature to low temperatures according to an effect discovered by Schrieffer [15] and recently noted by others [16]. Here we show that unlike their spin counterparts, orbital fluctuations are not subject to the “Schrieffer effect”, giving rise to a unique situation in which the orbital degrees of freedom behave as fluctuating quantum mechanical variables that result in an incoherent “non-Fermi liquid” ground state. While departures from perfect tetragonality will reestablish the Fermi liquid, a large temperature range of incoherent metal behavior is expected to remain.

One of the effects of the projection into the high-spin manifold is an S-fold reduction of the spin-Kondo coupling, JS=J/4S but strikingly, the strength of the orbital Kondo interaction is unaffected. This means that the Hund’s interaction will exponentially suppress the spin-Kondo effect down to a lower scale TspinK/D∼(TorbK/D)2S, as in conventional transition metal ions, while leaving the orbital Kondo effect unaffected. This schism between the orbital and spin-Kondo effect drastically affects the physics,

Slow spin dynamics in the bad Hund's metal

I have the following picture of a bad metal. It is half way between a Fermi liquid and a Mott insulator. This means that although there is no energy gap, the electrons are almost localised. The imaginary part of their self energy is comparable to the electron bandwidth. Since they are almost localised they have slowly fluctuating local moments. Hence, the dynamical spin correlation function, chi_s(omega) should be narrow, on what energy scale I am not sure. Somehow I expect a qualitative change in chi_s(omega) as the temperature increases above the coherence temperature, as the system crosses over from the Fermi liquid to the bad metal.
But, I am not sure whether this picture is correct because as far as I am aware there are very few calculations of chi_s(omega), and particularly not its temperature dependence.

There are a few Dynamical Mean-Field Theory [DMFT] calculations at zero temperature, i.e., in the Fermi liquid regime, such as described here. 
In the Mott insulating phase there is a delta function peak, associated with non-interacting local moments, as described here.

There are a few calculations in imaginary time, but little discussion of exactly what this means in real time. I struggle to make the connection.
The figure below shows a DMFT calculation of the imaginary time spin correlation function for the triangular lattice, by Jaime Merino, and reported in this PRB.

Here, I want to highlight two DMFT calculations for multi-band Hubbard models including Hund's rule coupling.

Spin freezing transition and non-Fermi-liquid self-energy in a three-orbital model
Philipp Werner, Emmanuel Gull, Matthias Troyer, Andy Millis

Dichotomy between Large Local and Small Ordered Magnetic Moments in Iron-Based Superconductors
P. Hansmann, R. Arita, A. Toschi, S. Sakai, G. Sangiovanni, and Karsten Held

The first paper reports the phase diagram below, where n is the average number of electrons per lattice site. The solid vertical lines represent a Mott insulating phase.

The boundary determined between the Fermi liquid phase and the "frozen moment" phase is determined from the Figure below. The top set of curves are the spin-spin correlation function.

The important distinction is that in the Fermi liquid phase chi_s(tau beta=1/2) will go to zero linearly in temperature, whereas in the frozen moment regime it tends to a non-zero constant.

Similar results are obtained in the second paper, on a slightly different model, but they don't use the "frozen moment" language, and emphasise more the importance of the Hund's rule coupling.

Here, I have a basic question about the above results. The lowest temperature used in the Quantum Monte Carlo is T = t/100 [impressive], and so I wonder if this is above the Fermi liquid coherence temperature and if one could go to low enough temperatures one would recover a Fermi liquid.

It is known that the coherence temperature is often much smaller for two-particle properties, and Hund's rule can also dramatically lower it. Both points are discussed here.

Or is my question [bad metal vs. frozen moments] just a pedantic distinction? The important point is that, practically speaking, over a broad temperature range, the spins are effectively relaxing very slowly.

Anyway, I think there is some rich physics associated with spin dynamics in bad metals and hope it will be explored more in the next few years. I welcome discussion and particularly calculations!

Are ultracold atomic gases strongly correlated systems?

I recently heard a talk by someone working on cold atoms who kept saying again and again that these were strongly correlated systems. I may have missed it but the justification was never clear. This got me wondering, what criteria would I use as a signature of strong correlations?

Here is my tentative answer motivated by strongly correlated electron materials.

A key signature of strong correlations is a significant redistribution of spectral weight [i.e., the many-body eigenvalue spectrum] compared to the corresponding non-interacting electron problem.

Common phenomena associated with this redistribution are

• the emergence of new low-energy scales [e.g. Kondo temperature]
• large renormalisation of quasi-particle energies [heavy fermions]
• separation of the energy scales for spin and charge excitations
• incoherent spectral features [Hubbard "bands"] 
• breakdown of quasi-particle approximations [bad metals]

This redistribution is usually poorly [never?] described by weak coupling theories, static mean-field theories or the RPA [Random Phase Approximation].

I attempt to illustrate this idea with two figures below. The first color shaded plot shows the one-particle spectral density calculated from LDA+DMFT [Local Density Approximation for DFT (Density Functional Theory) + Dynamical Mean-Field Theory] for the parent compound of the iron pnictide superconductors.
The dashed lines are the band structure calculated from pure LDA [i.e. not including the strong correlation effects captured by DMFT].
The Figure is taken from a PRL by Haule, Shim, and Kotliar.

The figure below shows the spectral density measured by ARPES [Angle Resolved PhotoEmission Spectroscopy] for the iron pnictide LaOFeP. The solid red lines are the band structure calculated from LDA.

This is Figure 6 in a recent review from Z.X. Shen's group.

In the absence of strong correlations all of the spectral weight would lie on top of the band structure.
The important point is that it does not.

In the cuprates these effects are even more dramatic.

So what about cold atomic gases?

For fermionic systems I have not seen much discussion of redistribution of spectral weight.
Often a quasi-particle picture and mean-field techniques are used in theoretical calculations.

Chris Vale's group has done a beautiful series of experiments measuring dynamical spin and density correlation functions for a strongly interacting system with a BEC-BCS crossover. The figure below is taken from this PRL. One does see differences between the density [D] and spin [S] correlation functions and there is a redistribution of spectral weight. But, to me at least, it does not appear as dramatic as in strongly correlated electron materials.

For bosons near the Mott transition there has been some discussion of the spectral weight redistribution [see this PRA and references therein].

I think the condensate fraction in the first cold atom BEC's is close to unity. In contrast, in superfluid helium 4 the fraction is about 10 per cent. The vanishing of the condensate fraction as one approaches the Mott insulator has been observed, but seems to be captured by a mean-field theory.
For reference, in cuprate superconductors the superfluid density can be much less than the charge density.

So, my questions are:

Is large redistribution of spectral weight the best signature of strong correlations?

In what cold atom systems does one see the largest redistribution?

Seeking definitive signatures of quantum criticality

Generally I am skeptical about quantum criticality as an important organising principle for strongly correlated electron materials (see for example, this earlier post, Are elemental metals quantum critical?). The most significant evidence is probably in heavy fermion compounds. However, particularly in the cuprates, I see quantum criticality as one of several competing "hand waiving" explainations of unusual properties. It is important to keep coming back to the idea that a true quantum critical point (QCP) will be associated with some diverging correlation length for some type of "order". Furthermore, this should lead to power laws in physical quantities over several orders in magnitude.

There is an interesting paper
Transport near a quantum critical point in BaFe2(As1−xPx)2
James G. Analytis,  H-H. Kuo, Ross D. McDonald,  Mark Wartenbe,  P. M. C. Rourke, N. E. Hussey,  and I. R. Fisher

The relevant background and context is this review article which contains the figure below

A few comments on the paper.

1. At room temperature the resistivity becomes of the order of the Mott-Ioffe-Regel limit, characteristic of a bad metal.

2. The low temperatures the metal is always a Fermi liquid [at least has a resistivity with a quadratic temperature dependence] even at dopings very close to the putative QCP. This is seen in the next figure.

3. The graph below shows the temperature dependence of the slope of the resistivity. It is striking that at high temperatures it has the approximately the same value, independent of doping. This is consistent with an earlier claim of such universality, [that even extended to elemental metals!]

4. The data above only shows a linear in T resistivity [which is often associated with quantum criticality], over less than one decade of temperature.

5. The figure below shows the doping dependence of the A coefficient of the Fermi liquid resistivity and the effective mass. Note the range of values of A is at most one order of magnitude. The effective mass only changes by about a factor of three.
In quantum critical theory both should diverge at the critical point [dashed line]. However, in the actual experimental data A does not diverge but decreases for the smallest x=0.31. This can be seen with the naked eye in the data above.

6. The colour shaded plot below shows the exponent n of the resistivity temperature dependence. The dashed line shows the Fermi liquid temperature going to zero at the putative QCP. However, as noted above even at the critical doping a Fermi liquid is always seen at low temperatures.

7. What about alternative theories?
One candidate must be Selective Mottness, as described in this preprint.
It does predict a significant enhancement doping dependence of the effective mass and the Fermi liquid coherence temperature. I am not sure how the P doping translates to band filling. Hopefully the proponents can comment.
Unlike quantum criticality Selective Mottness is actually based on calculations from a microscopic model with realistic interactions.