https://press.princeton.edu/books/paperback/9780691082424/surprises-in-theoretical-physics

"Each chapter focuses on a specific area:

General Quantum Mechanics

Quantum Theory of Atoms

Statistical Mechanics

Condensed Matter

Transport Problems

Many-Body Problems

Nuclear Physics

Relativity"

So: confine some electrons to one-dimensional chains, cool the result down, and they organize themselves into rippling waves whose collective motions can outrun sound!

But as we cool blue bronze still further, the density waves lock into step with the underlying atomic lattice, and all these effects go away.

(6/n, n = 6)

Once the charge density wave has formed, it has two natural ways to wiggle:

• its amplitude (how strongly bunched the ripple is) can pulse;
• its phase (where exactly the peaks of the ripple sit along the chain) can shift.

Pulsing the amplitude costs real energy. But sliding the phase - pushing the entire density wave bodily along the chains - costs essentially nothing. Why? Because nothing in the crystal cares where the peaks of the ripple happen to land. This is a strange gift of the fact that the charge density waves have a wavelength that's an irrational multiple of the spacing between atom pairs. No position is preferred over any other, so the wave can slip freely!

The resulting slow, almost-free sliding excitation is called a phason, and very gentle, long-stretching versions of it cost vanishingly little energy to excite. There's a deep principle at work here, known as Goldstone's theorem, that says whenever a system spontaneously settles into one of infinitely many equivalent configurations, it must come with a corresponding gentle shimmer mode that explores the alternatives.

So the Peierls instability in blue bronze doesn't just give a static charge density wave: we get phasons too!

(6/n)

Goldstone boson - Wikipedia

(en.wikipedia.org)

Blue bronze is K₀.₃MoO₃ — a deep-blue, metallic-looking crystal in the family of alkali molybdenum bronzes first prepared by Wold, Kunnmann, Arnott and Ferretti in 1964. As far as we can tell, these substances don't exist in nature. The name "bronze" is jargon inherited from the sodium tungsten bronzes Wöhler made back in 1825. It refers to the brassy luster of these compounds, not to the copper-tin alloy.

Blue bronze is built from parallel chains of molybdenum and oxygen atoms threading through the crystal, with potassium ions tucked between the layers. The electrons free to carry current are essentially trapped on these chains, like cars on a one-lane highways with no exits. This makes blue bronze a textbook example of a quasi-one-dimensional metal.

This is the setup for a remarkable result Rudolf Peierls established in the 1930s: a one-dimensional metal cannot stay metallic in its ground state! Instead, its atoms spontaneously bunch themselves into pairs, raising a barrier that stops electrons from flowing freely and turning the metal into an insulator. This is the 'Peierls instability'.

Simultaneously, there is a slight periodic rippling in the density of conduction electrons: a 'charge density wave'. But the wavelength of this wave can be an irrational multiple of the spacing between atom pairs! And in blue bronze, this does happen. It's thus an example of an 'aperiodic crystal'.

(5/n)

(4/n)

Molybdenum bronze - Wikipedia

(en.wikipedia.org)

So we get 'phasons'!

(3/n)

Phason - Wikipedia

(en.wikipedia.org)

(2/n)