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.

Science job openings in sunny Brisbane, Australia

Bribie Island, just north of Brisbane.

The University of Queensland has just advertised several jobs that may be of interest to readers of this blog, particularly those seeking to flee the USA.

There is a junior faculty position for a theorist working at the interface of condensed matter, quantum chemistry, and quantum computing.

There is also a postdoc to work on the theory of strongly correlated electron systems with my colleagues Ben Powell and Carla Verdi.

There is a postdoc in experimental condensed matter, to work on scanning probe methods, such as STM, with my colleague Peter Jacobson.

Glasshouse Mountains. Just north of Brisbane.

The interplay of ecological and evolutionary dynamics: immigration, extinction, and chaos (and DMFT?)

"Ecological and evolutionary dynamics are intrinsically entwined. On short timescales, ecological interactions determine the fate and impact of new mutants, while on longer timescales evolution shapes the entire community."

 Spatiotemporal ecological chaos enables gradual evolutionary diversification without niches or tradeoffs       Aditya Mahadevan, Michael T Pearce, and Daniel S Fisher

Understanding this interplay is "one of the biggest open problems in evolution and ecology."

New experimental techniques for measuring the properties of large microbial ecosystems have stimulated significant theoretical work, including from some with a background in theoretical condensed matter physics. For an excellent accessible introduction see:

Understanding chaos and diversity in complex ecosystems – insights from statistical physics

This is a nice 2.5-page article by Pankaj Mehta at the Journal Club for Condensed Matter. He clearly introduces an important problem in theoretical ecology and evolution and describes how some recent work has provided new insights using techniques adapted from Dynamical Mean-Field Theory, which was originally developed to describe strongly correlated electron systems. 

Here are just a few highlights of the article. It may be better to just read the actual article.

Fifty years ago, Robert May "argued that the more diverse an ecosystem is (roughly defined as the number of species present), the less stable it becomes." He derived this counter-intuitive result using a simple model and results from Random Matrix Theory. This is an example of an emergent property: qualitative difference occurs as a system of interacting parts becomes sufficiently large.

"One major deficiency of May’s argument is that it does not allow for the possibility that complex ecosystems can self organize through immigration and extinction. The simplest model that contains all these processes is the Generalized [to many species] Lotka-Volterra model (GLV)".

"Despite its simplicity, this equation holds many surprises, especially when the number of species is large".

Another case of how simple models can exhibit complex behaviour.

One special case is when the interactions are reciprocal – how species i affects species j is identical to how species j affects species I. "In the presence of non-reciprocity the system can exhibit complex dynamical behavior including chaos." Understanding this case was an open problem until the two papers reviewed by Mehta. For a detailed but pedagogical introduction see:

Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics, Wenping Cui, Robert Marsland III, Pankaj Mehta

This is relevant to understanding the origin of the fine grained diversity observed in sequencing experiments of microbial ecosystems.

Aside: de Pirey and Bunin "derive analytic expressions for the steady-state abundance distribution and an analogue of the fluctuation-dissipation theorem for chaotic dynamics relating static and dynamics correlation functions."

"Using a DMFT solution, they derive a number of remarkable predictions... in the chaotic system the species fall into two groups: species at high abundances and species at low abundances near the immigration floor. de Pirey and Bunin show that even in the chaotic regime, the number of high abundance species in the ecosystem will always be less than the May stability bound. This result is quite surprising since it suggests that ecosystems self-organize in such a way that the high abundance species still follow May’s diversity bound even when they are chaotic."

Probing the relationship between superexchange and superconductivity in cuprates

One of the most basic ideas in science is the controlled experiment. A single "independent" variable is changed while all others are held fixed. One then observes how the properties of the system change. Unfortunately, reality is more complicated and there are rarely any truly independent variables, particularly in materials science.

Since the discovery of cuprate superconductors one-quarter of a century ago there has been a constant struggle to tease out systematic trends that can provide insight into the underlying physics causing the superconductivity. This is a challenge because it is difficult to change only one variable. For example, a key property is how the superconductivity changes with the chemical composition of the material, particularly with regard to the doping level, i.e., the density of charge carriers. The problem is that with changes in doping, many other things change as well: the amount of disorder, the periodicity and strength of magnetic interactions, crystal structure, ... 

There is a beautiful experimental paper that recently overcomes these problems. 

On the electron pairing mechanism of copper-oxide high temperature superconductivity

Shane M. O’Mahony, Wangping Ren, Weijiong Chen,  Yi Xue Chong, Xiaolong Liu, H. Eisaki, S. Uchida, M. H. Hamidian, and J. C. Séamus Davis 

In a very clever way they can do all their measurements on a single material of fixed chemical composition, and yet vary a key parameter, the size of the energy difference between the relevant oxygen and copper electronic states, Epsilon.

In the material under study,  Bi2Sr2CaCu2O8+x, there are CuO5 units, as pictured below. In the crystal there is a modulation of delta, the distance at which the fifth oxygen sits above the CuO4 squares that form the square lattices that comprise the layers responsible for the superconductivity.

Due to electrostatics, the distance delta has an effect on the energy Epsilon. This in turn changes the size of the magnetic superexchange between neighbouring copper spins, as pictured below.

In the experiment, a STM is used to measure how Epsilon varies as delta varies (see the red dots in the Figure below). We then expect this to vary the superexchange.

An electron-pair (Josephson) STM is used to measure the magnitude of the superfluid density (electron-pair density) and how it changes with delta (see the blue dots in the figure below).

These two sets of measurement are combined in the second figure below. 

The yellow band in the figure above is the range of values expected from theory, including the recent paper.

Oxygen hole content, charge-transfer gap, covalency, and cuprate superconductivity

Nicolas Kowalski, Sidhartha Shankar Dash, Patrick Sémon, David Sénéchal, and André-Marie Tremblay

The theory is based on DMFT calculations for a three-band Hubbard model, following earlier work including by Weber, Haule, Kotliar, and independently by Maier.

Quanta magazine has a popular report on the experiment. The headline, "High-Temperature Superconductivity Understood at Last", overstates the significance of the experiment.

There are still issues of correlation versus causality. I would also like to see what other theories predict for the relationship between Epsilon and the pairing density. Nevertheless, it is a beautiful experiment and marks a significant advance.

Vertex corrections do matter

For an experimentalist one of the "easiest" quantities to measure for a metal is the electrical resistivity. Yet, for a many-body theorist working on models for strongly correlated electron systems this is one of the most difficult quantities to calculate, without making strong and debatable assumptions. One of the key questions is whether vertex corrections do matter. Ten years ago I summarised some of the issues.

This issue is nicely addressed in this nice paper from 2019.

Conductivity in the Square Lattice Hubbard Model at High Temperatures: Importance of Vertex Corrections

J. Vučičević, J. Kokalj, R. Žitko, N. Wentzell, D. Tanasković, and J. Mravlje

Besides the general issue of understanding the importance of vertex corrections, the paper is partly motivated by recent experiments on ultracold atoms that were compared to the results of calculations for a Hubbard model, using the finite-temperature Lanczos method (which essentially gives exact results on small finite lattices (e.g. 4 x 4)) and cluster Dynamical Mean-Field Theory (DMFT) (which does not include vertex corrections and has some momentum dependence in the self energy).

Before looking at the results I should point out the parameter values for the calculations. They are done for a Hubbard model on a square lattice. The half-bandwidth D=4t where t is the hopping parameter and U=10t. For the graphs below the doping p=0.1 (comparable to optimal doping in the cuprates).

Most importantly, the lowest temperature for which reliable calculations can be performed is T=0.2D=0.8t. In the cuprates, t is about 0.3 eV and so this lowest temperature corresponds to about 3000 K!, i.e. well above the superconducting Tc and the range of resistivity measurements on real materials. Most solids melt at these high temperatures.

Nevertheless, the results are important for two reasons. 

First, the experiments on ultracold atoms are in this temperature regime. [Aside: again this shows how fermion cold atom experiments are a long long way from simulating cuprates, contrary to some hype a decade ago]. 

Second, we are desperate for reliable results, and so it is worth knowing something about the possible importance of vertex corrections, even at very high temperatures. [Aside: my first guess would have been that they are not very important since I would have thought that correlations would be short-range and hand waving from Ward's identity would suggest that it follows the vertex corrections are small. This is wrong.]

In the figure above the top panel is the charge compressibility versus temperature. This is a thermodynamic quantity and the results show that most of the methods give similar results suggesting that the corresponding vertex corrections are small, at least above 0.1D.

The lower panel shows the temperature dependence of the resistivity and suggests that vertex corrections do lead to quantitative, but not qualitative differences. I guess the resistivity is in units of the quantum of resistance. Each rectangle has a vertical dimension of 5 units and so the resistivity is in excess of the Mott-Ioffe-Regel limit, i.e. the system is a bad metal. 

The figure above shows the frequency dependence of the optical conductivity for T=0.5D. There is a Drude peak at zero frequency and the broad peak near omega=2.5D=U corresponds to transitions between the lower and upper Hubbard band. DMFT is qualitatively correct but does differ from FTLM, showing the importance of vertex corrections.

The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

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?

2019 Nobel Predictions

It is that time of year again. I have not made predictions for a few years.

For physics this year I predict
Experiments for testing Bell inequalities and elucidating the role of entanglement in quantum physics
Alan Aspect, John Clauser, and Anton Zeilinger
They received the Wolf Prize in 2010, a common precursor to the Nobel.

My personal preference for the next Nobel for CMP would be centred around Kondo physics, since that is such a paradigm for many-body physics, maybe even comparable to BCS.

Kondo effect and heavy fermions
Jun Kondo, Frank Steglich, David Goldhaber-Gordon

Arguably the latter two might be replaced with others who worked on heavy fermions and/or Kondo in quantum dots.
Steglich discovered heavy fermion superconductivity.
Goldhaber-Gordon realised tuneable Kondo and Anderson models in quantum dots (single-electron transistors).

Unlike many, I still remain to be convinced that topological insulators is worthy of a Nobel.

For chemistry, my knowledge is more limited. However, I would go for yet another condensed matter physicist to win the chemistry prize: John Goodenough, inventor of the lithium battery.
He also made seminal contributions to magnetism, random access memories, and strongly correlated electron materials.

What do you think?

Postscripts (October 10).

I got confused about the day of the physics prize and I think when I posted my ``prediction'' the prize may have already been announced.

A few years ago I read Goodenough's fascinating autobiography. It was actually in that book that I learned about U. Chicago requiring PhD students to publish a single author paper. This observation featured in my much commented on recent post about PhD theses.

I also have a prediction for the Peace Prize. First, I hope it is not Greta Thunberg, as much as I admire her and agree with the importance of her cause. I worry whether it may ruin her life.
My wife suggested the Prime Minister of Ethiopia, Abiy Ahmed and the President of Eritrea, Isaias Afwerki. I find it truly amazing what Ahmed has achieved.
Another great choice would be some of the leaders of Armenia, which has seen significant increases in human rights, political freedoms, and freedom the press. It was selected as The Economist's country of the year in 2018.

Postscript (October 30).
I was really happy about the economics prize. Six years ago, I read Poor Economics, by Banerjee and Duflo, with my son (an economics student), and blogged about it. Below a respond to a commenter who was critical of this prize.

Bad metallic behaviour in ultracold atoms

There is a nice paper
Bad metallic transport in a cold atom Fermi-Hubbard system 
Peter T. Brown, Debayan Mitra, Elmer Guardado-Sanchez, Reza Nourafkan, Alexis Reymbaut, Simon Bergeron, A.-M. S. Tremblay, Jure Kokalj, David A. Huse, Peter Schaus, and Waseem S. Bakr

The paper represents a significant experimental advance in using ultracold atoms to investigate questions directly relevant to strongly correlated electron systems. In this case, the system Hamiltonian can be tuned to be a Hubbard model on a square lattice, such that the model parameters, U and t, and the doping, n are known.
One limitation is that current experiments can only be performed down to the lowest temperature of T/t =0.3. [For comparison, for cuprates this is of the order of 1000 K!].
Using imaging techniques the authors are able to directly extract the density (charge) diffusion constant D and the density susceptibility, chi, shown below. The experimental data are red dots. The blue curve is the result of calculations based on the Finite Temperature Lanczos Method (FTLM). Green dots the results of Dynamical Mean-Field theory. Gamma is the density relaxation rate.

Both the experiment and the ability to make such a detailed comparison with concrete theoretical calculations is a significant and exciting achievement.

The dashed curve in the upper panel is the value of the diffusion constant associated with the Mott-Ioffe-Regel limit below which one expects bad metal behaviour.
Aside: one should always keep in mind that for the MIR limit, different authors use different criteria, leading to different factors of pi, sqrt(pi), ...

Using the Nernst relation, sigma = chi * D,  the data above gives the conductivity (sigma) and resistivity (rho), shown below.

The blue and green curves correspond to the predictions, of FTLM and DMFT, respectively. The dashed grey line is the Mott-Ioffe-Regel limit.

One comment I have concerns an additional comparison that the authors could make. Based on heuristic arguments and results from AdS-CFT, Hartnoll conjectured a lower bound for the diffusion constant, hv_F^2/T.
Previously, Nandan Pakhira and I showed that this bound was significantly violated in the bad metallic regime, as described by DMFT.

There is also a commentary on the paper by Ehud Altman at the Journal Club of Condensed Matter.
I thank Matt Davis for bringing the paper to my attention.

Emergent temperature scales and spin-orbital separation in the Hund's metal

An important and fascinating issue in many-body physics is the emergence of new energy scales, particularly scales that are orders of magnitude smaller than the energy scales in the underlying Hamiltonian. One example is the coherence temperature associated with the crossover from a Fermi liquid (with coherent quasi-particles) to a bad metal.

Recently, I posted about the crossover from a Hund's metal to a bad metal, seen in the collapse of the Drude peak in the optical conductivity, and the issue of capturing this slave-particle theories. One commenter mentioned the relevance of the paper below and another asked about the claim that the Kondo effect is associated with the collapse.

I agree that Kondo physics is associated with the crossover. Although, far from obvious this is also the case in the single-band Hubbard model. The Kondo effect was first studied with isolated magnetic impurities in metals and can be described by a single-impurity Anderson model (SIAM). Although there are no magnetic impurities in the Hubbard model, it turns out that when studied at the level of Dynamical-Mean-Field Theory (DMFT), the model is described by a self-consistent SIAM and close to the Mott metal-insulator transition Kondo physics does emerge. Specifically, the Kondo temperature for the self-consistent SIAM corresponds to the temperature at which there is a crossover from local unscreened local magnetic moments (associated with the almost-localised electrons near the Mott phase; the bad metal) to a Fermi liquid where the "magnetic moments" are screened.

What happens in a two-band Hubbard-Kanamori model with Hund's rule coupling?

The physics is richer because there is now the possibility screening of spin and/or orbital degrees of freedom, and of a orbital-selective Mott phase (or bad metal). 

This is nicely investigated in the following paper.

Dynamical Mean-Field Theory Plus Numerical Renormalization-Group Study of Spin-Orbital Separation in a Three-Band Hund Metal
K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum

For me, the figure below is the most interesting and illuminating. It shows how due to the Hund's rule coupling, two distinct energy scales (differing by about two orders of magnitude) emerge and associated with screening the spin and orbital degrees of freedom, respectively.

This is Kondo physics, but there are no magnetic impurties.

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.

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.

How are DMFT, DMET, and slave bosons related?

Several years ago I posted about Density Matrix Embedding Theory (DMET), proposed as a (computationally cheaper) alternative to Dynamical Mean-Field Theory (DMFT).

An important question is what is the relationship (if any) between the two methods?
Given that the two methods are formulated in quite different ways it was not clear to me at all whether these question could be answer in any sort of definitive way.

There is a very nice paper which does answers this question in a precise way, with the bonus of also giving the relationship of both methods with rotationally invariant slave bosons (RISB).

Dynamical mean-field theory, density-matrix embedding theory, and rotationallyinvariant slave bosons: A unified perspective 
Thomas Ayral, Tsung-Han Lee, and Gabriel Kotliar

The main results are summarised in the Figure below. A result that is useful and insightful is that DMET corresponds to RISB with the quasi-particle weight set to unity (Z=1). There is then no band narrowing associated with the correlations. Given this is a key aspect of strongly correlated electron systems, I think this is a significant shortcoming of DMET. I am also a bit confused about how DMET can then capture a Mott transition.

Observation of renormalised quasi-particle excitations

A central concept of quantum-many body theory is that of coherent quasi-particles. Their key property is a well-defined relationship between energy and momentum (dispersion relation). Prior to the rise of ARPES (Angle-Resolved Photo-Emission Spectroscopy) over the past three decades, the existence of electronic quasi-particles was only inferred indirectly.

A very nice paper just appeared which shows a new way of measuring quasi-particle excitations in a
strongly correlated electron system. Furthermore, the experimental results are compared quantitatively to state-of-the-art theory, showing several subtle many-body effects.

Coherent band excitations in CePd3: A comparison of neutron scattering and ab initio theory 
Eugene A. Goremychkin, Hyowon Park, Raymond Osborn, Stephan Rosenkranz, John-Paul Castellan, Victor R. Fanelli, Andrew D. Christianson, Matthew B. Stone, Eric D. Bauer, Kenneth J. McClellan, Darrin D. Byler, Jon M. Lawrence

The mixed valence compound studied is of particular interest because with increasing temperature it exhibits a crossover from a Fermi liquid with coherent quasi-particle excitations to incoherent excitations, an example of a bad metal.

The figure below shows a colour intensity plot of the dynamical magnetic susceptibility

at a fixed energy omega, and a function of the wavevector Q. The top three panels are from the calculations of DFT+DMFT (Density Functional Theory + Dynamical Mean-Field Theory).

The bottom three panels are the corresponding results from inelastic neutron scattering.
A and B [D and E] are both at omega=35 meV and in two different momentum planes. C [F] is at omega=55 meV.
The crucial signal of coherence (i.e. dispersive quasi-particles) is that the shift of the maxima between the G and R points at 35 meV to the M and X points at 55 meV.

It should be stressed that these dispersing excitations are not due to single (charged) quasi-particles, but rather spin excitations which are particle-hole excitations.

The figure below shows how the dispersion [coherence] disappears as the temperature is increased from 6 K (top) to 300 K (bottom). The solid lines are theoretical curves.

The figure below shows that the irreducible vertex corrections associated with the particle-hole are crucial to the quantitative agreement of theory and experiment. The top (bottom) panel in the figure below shows the calculation at low (high) temperatures. The black (blue) curves are with (without) vertex corrections. The red curves are a rescaling of the blue curves by a numerical factor.

The correction has two effects: First, it smooths out some of the fine structure in the energy dependence of the spectra while broadly preserving both the Q variation and the overall energy scale; and second, it produces a strong enhancement of the intensity that is both energy and temperature dependent, for example, by a factor of ~6.5 at w = 60 meV at 100 K. This shows that the Q dependence of the scattering is predomi- nantly determined by the one-electron joint density of states, as expected for band transitions, whereas the overall intensity is amplified by the strong electron correlations. 

This landmark study is only possible due to recent parallel advances in theory, computation, and experiment. 
On the theory side, it is not just DMFT but also including particle-hole interactions in DMFT.
On computation, it is new DMFT algorithms and increasing computer speed. 
On the experimental side, it is pulsed neutron sources, and improvements in the sensitivity and spatial and energy resolution of neutron detectors.

Postdoc available for strongly correlated electron systems

Ben Powell and I have just advertised for a new postdoc position to work with us at the University of Queensland on strongly correlated electron systems.

The full ad is here and people should apply before January 28 through that link.

The power and limitations of ARPES

The past two decades have seen impressive advances in Angle-Resolved PhotoEmission Spectroscopy (ARPES). This technique has played a particularly important role in elucidating the properties of the cuprates and topological insulators. ARPES allows measurement of the one-electron spectral function, A(k,E) something that can be calculated from quantum many-body theory. Recent advances have included the development of laser-based ARPES, which makes synchrotron time unnecessary.

A recent PRL shows the quality of data that can be achieved.

Orbital-Dependent Band Narrowing Revealed in an Extremely Correlated Hund’s Metal Emerging on the Topmost Layer of Sr2RuO4 
Takeshi Kondo, M. Ochi, M. Nakayama, H. Taniguchi, S. Akebi, K. Kuroda, M. Arita, S. Sakai, H. Namatame, M. Taniguchi, Y. Maeno, R. Arita, and S. Shin

The figure below shows a colour density plot of the intensity [related to A(k,E)] along a particular direction in the Brillouin zone.  The energy resolution is of the order of meV, something that would not have been dreamed of decades ago.
Note how the observed dispersion of the quasi-particles is much smaller than that calculated from DFT, showing how strongly correlated the system is.

The figure below shows how with increasing temperature a quasi-particle peak gradually disappears, showing the smooth crossover from a Fermi liquid to a bad metal, above some coherence temperature.

The main point of the paper is that the authors are able to probe just the topmost layer of the crystal and that the associated electronic structure is more correlated (the bands are narrower and the coherence temperature is lower) than the bulk.
Again it is impressive that one can make this distinction.

But this does highlight a limitation of ARPES, particularly in the past. It is largely a surface probe and so one has to worry about whether one is measuring surface properties that are different from the bulk. This paper shows that those differences can be significant.

The paper also contains DFT+DMFT calculations which are compared to the experimental results.

Desperately seeking Weyl semi-metals. 2.

Since my previous post about the search for a Weyl semimetal in pyrochlore iridates (such as R2Ir2O7, where R=rare earth) read two more interesting papers on the subject.

Metal-Insulator Transition and Topological Properties of Pyrochlore Iridates 
Hongbin Zhang, Kristjan Haule, and David Vanderbilt

Using a careful DMFT+DFT study they are able to reproduce experimental trends across the series, R=Y, Eu, Sm, Nd, Pr, Bi.

They show that when the self energy due to interactions is included that the band structure is topologically trivial, contrary to the 2010 proposal based on DFT+U.

They also find that the quasi-particle weight is quite small (about 0.1 for R=Sm, Nd and 0.2 for Pr). This goes some way towards explaining the fact that the infrared conductivity gives an extremely small Drude weight (about 0.05 electrons per unit cell), a puzzle I highlighted in my first post.

Field-induced quantum metal–insulator transition in the pyrochlore iridate Nd2Ir2O7 
Zhaoming Tian, Yoshimitsu Kohama, Takahiro Tomita, Hiroaki Ishizuka, Timothy H. Hsieh, Jun J. Ishikawa, Koichi Kindo, Leon Balents, and Satoru Nakatsuji

The authors make much of two things.

First, the relatively low magnetic field (about 10 Tesla) required to induce the transition from the magnetic insulator to the metallic phase. Specifically, the relevant Zeeman energy is much smaller that the charge gap in the insulating phase.
However, one might argue that the energy scale one should be comparing to is the thermal energy associated with the magnetic transition temperature.

Second. the novelty of this transition.
However, in 2001 a somewhat similar transition was observed in the organic charge transfer salt, lambda-(BETS)2FeCl4. It is even more dramatic because it undergoes a field-induced transition from a Mott insulator to a superconductor. The physics is also quite similar in that it can also be described by Hubbard-Kondo model, where local moments are coupled to interacting delocalised electrons.

A new picture of unconventional superconductivity

Two key ideas concerning unconventional superconductors are the following.

1. s-wave and p-wave pairing (in momentum space) are associated with spin singlet and spin triplet pairing, respectively. This can be shown with minimal assumptions (no spin-orbit coupling and spatial inversion symmetry).

2. If superconductivity is seen in proximity to an ordered phase (e.g. ferromagnetism or antiferromagnetism) with a quantum critical point (QCP) then the pairing can be "mediated" by low energy fluctuations (e.g. magnons) associated with the ordering.

3. Non-fermi liquid behaviour may be seen in the quantum critical region about the QCP.

However, an interesting paper shows that neither of the above is necessarily true.

Superconductivity from Emerging Magnetic Moments 
Shintaro Hoshino and Philipp Werner

They find spin triplet superconductivity with s-wave symmetry. This arises because there is more than one orbital per site and due to the Hund's rule coupling spin triplets can form on a single site.

They also find the pairing is strongest near the "spin freezing crossover" which is associated with the "Hund's metal", i.e. the bad metal arising from the Hund's rule interaction, and has certain "non-Fermi liquid" properties.

The results are summarised in the phase diagrams below, which has a striking similarity to various experimental phase diagrams that are usually interpreted in terms of 2. above.
However, all the theory is DMFT and so there are no long wavelength fluctuations.

Instability of the Fermi liquid near the Mott transition

In the metallic state of many strongly correlated electron materials, Fermi liquid properties are only observed at relatively low temperatures, at a scale (the coherence temperature T_coh) that can be orders of magnitude less than the Fermi temperature that is estimated from the relevant electronic band structure. Above T_coh one observes a "bad metal" and the absence of quasi-particles.

These features are nicely captured by Dynamical Mean-Field Theory (DMFT).

An interesting question is whether this low-temperature scale can be captured in simpler theories.

Alejandro Mezio and I just finished a paper

Low quasi-particle coherence temperature in the one band Hubbard model: a slave-boson approach

The phase diagram at half filling is shown below. Note how near the Mott insulator T_coh is orders of magnitude smaller than W/2, the scale of the Fermi temperature for U=0. It is also much smaller than this scale multiplied by Z, the band renormalisation due to interactions.

We welcome comments.

Emergence of the Hubbard bands near the Mott transition

Dynamical Mean-Field Theory (DMFT) has given many insights into the Mott metal-insulator transition in strongly correlated electron materials. In the metallic phase, DMFT nicely describes the interplay between the quasi-particles associated with Fermi liquid behaviour and the Hubbard bands that also exist in the insulating phase. DMFT gives a first-order phase transition and captures the emergence of bad metallic behaviour and the associated transfer of spectral weight.

On the down side DMFT is computationally expensive, particularly close to the Mott transition, as it requires solution of a self-consistent Anderson impurity problem. [If Quantum Monte Carlo is used one also has to do a tricky imaginary time continuation]. When married with atomistic electronic structure calculations (such as based on (Density Functional Theory) DFT-based approaches) DMFT becomes even more expensive. Sometimes I also feel DMFT can be a bit of a "black box."

Slave boson mean-field theory (SBMT) (and equivalently the Gutzwiller approximation (GA) to the Gutzwiller variational wave function) is computationally cheaper and also gives some insight. However, these approaches only describe the quasi-particles, completely miss the Hubbard bands and the associated physics, and give a second-order phase transition. This is sometimes known as the Brinkman-Rice picture.

There is a nice preprint that solves these problems.

Emergent Bloch Excitations in Mott Matter 
Nicola Lanatà, Tsung-Han Lee, Yongxin Yao, Vladimir Dobrosavljević

In addition to the physical orbitals they introduce "ghost orbitals" that are dispersionless (i.e. a flat band) and non-interacting. However, one starts with a Gutzwiller variational wave function that includes the "ghost orbitals". This enables capturing the charge fluctuations in the physical orbitals.
One sees that the Hubbard bands emerge naturally (as a bonus they are dispersive) provided one includes at least two ghost orbitals in the metallic phase and on in the Mott insulating phase.
There is a simple "conservation" of numbers of bands at play here.
The authors state that the Mott transition is a topological transition because of the change in the number of bands.

The metallic (insulating) phase is characterised by three (two) variational parameters.

The results compare well, both qualitatively and quantitatively, with DMFT.

The figure below shows plots of the spectral function A(E,k) for different values of the Hubbard U. The Mott transition occurs at U=2.9.
The color scale plots are DMFT results.
The green curves are from the ghost orbital approach with the size of the points proportional to the spectral weight of the pole in the one-electron Greens function.