Relating frustrated spin models and flat bands in tight-binding models

What kind of theory paper to I enjoy?
Here are some personal tastes
- "simple" enough I can understand it
- physical insight
- some analytical results
- some pretty pictures that illuminate

This week I read the following paper which I consider nicely meets these criteria.

Band touching from real-space topology in frustrated hopping models
Doron L. Bergman, Congjun Wu, and Leon Balents

The quantum spin antiferromagnetic Heisenberg model on the kagome lattice attracts a lot of attention because it may have a spin liquid ground state, for spin-1/2 and spin 1. This is arguably driven by the large spin frustration. A reflection of this frustration is that the classical model has a non-zero entropy at zero temperature due to a manifold of degenerate states. For this reason, the kagome lattice is sometimes said to be "maximally frustrated". This is in contrast to the triangular lattice for which their is a unique classical ground state and the spin-1/2 model exhibits long-range order.

The kagome lattice is also of interest because of the band structure for the tight-binding model has a flat band, i.e. it is dispersionless. This means that in the presence of interactions the electrons in this band may be strongly correlated and susceptible to instability to new states of matter.

The question arises as to whether there is any connection between these two properties of models on a particular "frustrated" lattice: flat bands and a manifold of degenerate classical ground states.

The purpose of this paper is to show that for a whole class of lattices, in two and three dimensions, that there is an close relationship between these properties.
It turns out that a key feature is that the flat bands touch a dispersive band at one point in k-space.

My interest was stimulated by the work of some of my UQ colleagues on a class of organometallic compounds that exhibit a kagomene lattice (that interpolates between kagome and honeycomb (graphene). The associated band structure (taken from this paper) is shown below.

The abstract states:

We demonstrate that this band touching is related to states which exhibit nontrivial topology in real-space. Specifically, these states have support [i.e. non-zero values] on one-dimensional loops which wind around the entire system 􏰀with periodic boundary conditions􏰁. A counting argument is given that determines, in each case, whether there is band touching or none, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations, which also split the degeneracy of the flat band.

I know illustrate this with the kagome lattice.

It has a three site basis (mu=1,2,3) and so there are three bands. If q is the Bloch wave vector, the Bloch states for the flat band can be written

One of these plaquette states is shown on the left below. 

A key point is that there is constructive interference between these plaquette states. Thus, one can take superpositions of them. On the right is the superposition of three neighbouring plaquette states.

A whole line of plaquette states can lead to visualising something with nontrivial topology.

The authors then show how similar physics occurs in other two- and three-dimensional lattice models. The one below is the dice lattice.

Finally, they show that the corresponding Hubbard model leads to a Heisenberg model in the classical limit does have macroscopic degeneracy.

I thank Ben Powell for bringing the paper to my attention.

What needs to be said about mental health issues in universities?

On friday I am giving the UQ Physics Department colloquium on mental health issues for scientists. The talk may be similar to one I gave a few years ago.

I will update my talk incorporating some recent reading and the articles below.

A recent Editorial in Nature declared
Time to talk about why so many postgrads have poor mental health 
An outpouring on Twitter highlights the acute pressures on young scientists.

[I thank Tanglaw Roman for bringing the editorial to my attention. I never look at luxury journals unless someone refers me to a specific article.]

The Editorial was in response to the Twitter response to an article in a baby Nature
Evidence for a mental health crisis in graduate education

Poisonous science: the dark side of the lab 
The bullying and subsequent suicide of a talented Ivy League scientist exposes ugly truths about the cruelty and dysfunction at the heart of academic science

Mindfulness won't fix bad management

It also conveniently shifts the burden of wellbeing from the employer causing stress to the employee trying to deal with it. Worse, it allows what you might call "well-washing": employers who cloak themselves in a veneer of caring for their workers while hurting them with bad management practices. 

Five tips to get a good nights sleep

However, I would like some feedback and suggestions from readers.

What do you think needs to be said?

Update. The colloquium was postponed to avoid a scheduling conflict and to make it accessible to a broader audience. Thus, there is still time to send in your suggestions.

Laughing at your life in science

I watched Ph.D Movie 2: Still in Grad School, which is based on the legendary Ph.D comics, written by Jorge Cham.



It is worth watching as it is quite funny. On the other hand, some of the caricatures are getting a little too close to reality....

While on the funny side of science, my wife and I have been enjoying watching Young Sheldon. I am a Big fan of The Big Bang Theory, but was not sure whether this new show would be as good. This was partly influenced by a moderately negative review in the New York Times (albeit based on one episode). I disagree as I think that both shows do have an interesting cast of characters.
On the other hand, Young Sheldon does not have as much science as TBBT, at least for the first six episodes that I have seen. Here Schrodinger's cat gets discussed.

Junior faculty position in Experimental Condensed Matter available at UQ

The physics department at UQ has just advertised for a new faculty member in experimental condensed matter. The advert is here.

There is also a junior faculty position available in astrophysics.

Aside. The picture is of Lake McKenzie (no relation) on Fraser Island, which I just visited on mid-semester break.

Physics, Politics, Pride, and Moral Failure

Which scientist had the greatest political influence of all time? the greatest influence on government policy?
Oppenheimer? Any suggestions?

Not Schrodinger, but arguably Frederick Lindemann.
Until last week I had barely heard of Lindemann. I knew of the Lindemann criterion for estimating the melting temperature of a solid. At Oxford, I had been in the Lindemann building (the front of the Clarendon lab) many times, but had not bothered to find out who Lindemann was.

Last week I listened to a fascinating podcast by Malcolm Gladwell, The Prime Minister and the Prof, that recounts Lindemann's long relationship with Winston Churchill. The podcast draws heavily on two sources (interestingly both written by physicists). The first source is three lectures that C.P. Snow [of two cultures fame] gave at Harvard in 1960, and published as Science and Government.
The second source is a book, Churchill's Secret War: The British Empire and the Ravaging of India during World War II by Madhusree Mukerjee.

Lindemann was a lifelong friend and advisor of Churchill and his main scientific advisor. It was largely his advice that led Churchill to implement policies that literally led to the deaths of millions.

One disaster, considered by Mukerjee, was the Bengal famine of 1943, which resulted because the British refused to send food supplies to their colony in India, even though they had bountiful supplies and shipping at the time.
The second disaster, considered by Snow, was "strategic bombing" of German civilians. Lindemann argued the bombing would break morale, even though the data he had actually supported the opposite view.

I highly recommend the podcast. It is stimulating and disturbing. I thank my daughter for bringing it to my attention. It was great listening in the car on a recent holiday together.

What other scientists can you think of who have exerted such great political influence, for better or for worse?

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.