Wednesday, August 9, 2017

Subtle paths to effective Hamiltonians in complex materials

Many of the most interesting materials involve significant chemical and structural complexity. Indeed, it is not unusual for a unit cell for a crystal to contain the order of one hundred atoms.
Yet, for a given class of materials, one would like to find an effective Hamiltonian involving as few degrees of freedom and parameters as possible.

Following Kino and Fukuyama, twenty years ago I argued that the simplest possible effective Hamiltonian for a large class of superconducting organic charge transfer salts was a one-band Hubbard model on an anisotropic triangular lattice at half filling.
It seemed natural to then argue that the relevant model for the spin degrees of freedom in the Mott insulating phase is the corresponding frustrated Heisenberg model with spatial anisotropy determined by the anisotropy in the tight-binding model.

However, it turns out this is not the case.
There are some subtle quantum interference effects that I overlooked in the "derivation"  of these effective models, leading to a different spatial anisotropy.
This is shown by some of my colleagues in a nice recent paper, accepted for PRL.

Dynamical reduction of the dimensionality of exchange interactions and the "spin-liquid" phase of κ-(BEDT-TTF)2X 
B. J. Powell, E. P. Kenny, J. Merino


This raises questions about what the relevant effective Hamiltonian is for the metallic, superconducting, and (possibly) ferroelectric phases.

The paper's significance goes beyond organic charge transfer salts to the general problem of finding effective Hamiltonians in complex materials.

Similar interference effects have been found to arise in quite a different class of materials.

Heisenberg and Dzyaloshinskii-Moriya interactions controlled by molecular packing in trinuclear organometallic clusters 
B. J. Powell, J. Merino, A. L. Khosla, and A. C. Jacko

This also reminds me of subtleties (and debates) that occur in the Zhang-Rice "derivation" of the t-J model from a three-band Hubbard model.

Saturday, August 5, 2017

Who was the greatest theoretical chemist of the 19th century?

Dimitri Mendeleev, who proposed the periodic table of the elements, purely from phenomenology and without quantum mechanics!
He even successfully predicted the existence of new elements and their properties.

A friend who is a high school teacher [but not a scientist] asked me about how he should teach the periodic table to chemistry students. It is something that students often memorise, especially in rote-learning cultures, but have little idea about what it means and represents. It makes logical sense, even without quantum mechanics. This video nicely captures both how brilliant Mendeleev was and the logic behind the table.



A key idea is how each column contains elements with similar chemical and physical properties and that as one goes down the column there are systematic trends.
It is good for students to see this with their own eyes.
This video from the Royal Society of Chemistry shows in spectacular fashion how the alkali metals are all highly reactive and that as one goes down the column the reactivity increases.



The next amazing part of the story is how once quantum theory came along it all started to make sense!

Tuesday, August 1, 2017

The role of the Platonic ideal in solid state physics

In the book Who Got Einstein's Office?, about the Institute for Advanced Study at Princeton, the author Ed Regis, mocks it as the "One True Platonic Heaven" because he claims its members are Platonic idealists, who are interested in pure theory, and disdain such "impurities" as computers and applied mathematics.


This stimulated me to think about the limited but useful role of pure mathematics, Platonic idealism, and aesthetics in solid state theory. People seem particularly excited when topology and/or geometry plays a role.

The first example I could think of is the notion of a perfect crystal.

Then comes Bloch's theorem, which surely is the central idea of introductory solid state physics.

Beautiful examples where advanced pure maths plays are role are
Chern-Simons theory of edge states in the Quantum Hall Effect
and topological terms in the action for quantum spin chains, as elucidated by Haldane.

As I have said before I think topological insulators is a beautiful, fascinating, and important topic. However, I am concerned by the disproportionately large number of people working on the topic and the associated hype. I wonder if some of the appeal and infatuation is driven by Platonic idealism.

For a classic example of how Platonism leads to imperfect theory is Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596).


Good theory finds a balance between beauty and the necessity of dirty details.

Can you think of other examples where Platonic idealism plays a positive role in condensed matter theory?