Dive into Hofstadter Butterfly

Joaquim Puig

Outline of the talk

• Introduce Hofstadter Butterfly. Features
• Physical Motivation.
• Mathematical Definition and Conjectures.

Note about sources of images

Links to the sources of all images can be found in the speaker notes of the slides. Press s to access the speaker notes. I have tried to use only CC images or sourced from the original scientific articles. For any comment, please contact me.

Hofstadter Butterfly

• It is a Diagram: a set of points in the space 2d. Black points represent inclusion.
• Resembles wings of buttefly.
• Symmetry wrt the two axis. Combination is point symmetry.

Magnification

• Self-similarity in many directions.
• The slopes ending at the top-right corner.
• The buttefly that repeat in a harmonic way.

Douglas Hofstadter

• Image credit.
• Original Article

Computing tools: Hewlett-Packard 9820A

• See the article for the history of the computation.
• Todays’s cost would be around $30,000USD.
• HP9820A

Gödel, Escher, Bach

• Won the Pulitzer Prize 1980. Contains a plot of the ‘Gplot’.
• Wikipedia entry for the book and image source

Can you spot the difference?

Vertical Coordinates!

• The original plot had a left key.
• You can check all the bands (bold horizontal intervals) occur at rational heights

Hofstadter Butterfly for rational values

Rational and Irrational Numbers

• Irrationals appear as limits of rational sequences.
• Location of irrational numbers: sqrt(2)-1, pi-3…

Hofstadter butterfly for irrationals (1) ?

\(\pi -3 \approx 0.14159265358... \in [0,1]\) (irrational)

Approximation	p/q	Bands
0.1	1/10	9
0.14	7/50	49
0.142	141/1000	999
0.1416	283/2000	1999
0.14159	14159/100000	999999

Does it have \(\infty\) many bands?

Hofstadter butterfly for irrationals (2) ?

• Consider the total length of bands for a given rational p/q.
• For p/q=0 or 1, the length is 8.
• For p/q=1/2 the length is \(2+2\sqrt{3} \approx 5.46...\) which is \(0.68..\) times the previous band.
• For p/q=1/3, this ratio is \(0.366...\).
• We can take the ruler and count. 😅
• Better with the help of a computer 🐶 !

Length of bands

Does it have \(\infty\) many bands BUT total length zero???

Does it even make sense for irrational values?

What is the definition of the Hostadter Butterfly?

Physical motivation

Original article

Sources:

• Harper, P. G. (1955). Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A, 68(10), 874.
• Azbel, M. Y. (1964). Energy spectrum of a conduction electron in a magnetic field. Soviet Physics Jetp-Ussr, 19(3), 634-645.

See the following article for more physical background:

• Avron, J. E., Osadchy, D., & Seiler, R. (2003). A topological look at the quantum Hall effect. Physics today, 56(8), 38-42.

Experimental confirmation

Source:

• Hofstadter butterfly quantum simulation by Roushan, P., Neill, C., Tangpanitanon, J., Bastidas, V. M., Megrant, A., Barends, R., … Martinis, J. (2017). Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science, 358(6367), 1175–1179. doi:10.1126/science.aao1401.

See also the experimental confirmations in 2013:

• Dean, C. R., Wang, L., Maher, P., Forsythe, C., Ghahari, F., Gao, Y., … & Kim, P. (2013). Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature, 497(7451), 598-602.
• Ponomarenko, L. A., Gorbachev, R. V., Yu, G. L., Elias, D. C., Jalil, R., Patel, A. A., … & Geim, A. K. (2013). Cloning of Dirac fermions in graphene superlattices. Nature, 497(7451), 594-597.
• Hunt, B., Sanchez-Yamagishi, J. D., Young, A. F., Yankowitz, M., LeRoy, B. J., Watanabe, K., … & Ashoori, R. C. (2013). Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science, 340(6139), 1427-1430.

Used the Peierls-Harper model

Mathematical Definition

Contains mathematical formulas… Can be skipped safely!

For a given \(\omega\) (Magnetic flux) and a phase \(\phi\) \[ (H^{}_{\omega,\phi} x)_n = x_{n+1}+x_{n-1} + 2\cos {2\pi (\omega n+\phi)} x_n, n \in \mathbb{Z}. \]

is called an Almost Mathieu Operator (AMO) on \(l^2(\mathbb{Z})\) the set of sequences \(\dots,x_{-1},x_0,x_1,\dots\) with \(\|x\|_2=\sqrt{\sum |x_n|^2}\) finite.

• Well-defined linear map: \(x \in l^2(\mathbb{Z}) \Rightarrow H^{}_{b,v} x \in l^2(\mathbb{Z})\)
• Bounded: \(\left\| H_{b,v} x \right\|_2 \le 4 \left\| x \right\|_2\)
• Self-adjoint (symmetric):\(\langle H_{\omega,\phi}x, y \rangle = \langle x, H_{\omega,\phi}y \rangle\)

:::

Experimental confirmation

Contains mathematical formulas… Can be skipped safely!

\[ H_{\omega,\phi}= \left( \begin{array}{cccc} \ddots & & 0 \\ & \begin{array}{ccc} 2\cos {2\pi (\omega +\phi)} & 1 & \\ 1 & 2\cos {2\pi (\phi)} & 1 \\ & 1 & 2\cos {2\pi (-\omega +\phi)} \end{array} & \\ 0 & & \ddots \end{array} \right). \]

Definition 1 (Hofstadter Butterfly)  

• Let \(\sigma_\omega(\phi)\) the spectrum of \(H_{\omega,\phi}\) (generalisation of eigenvalues).
• Let \(\sigma_\omega\) be the union of \(\sigma_\omega(\phi)\) for all \(\phi\). The Hofstadter butterfly is the set energies in \(\sigma_\omega\) (horizontal axis) for every \(\omega\) in \([0,1]\) (vertical axis).

Post-mathematical catchup

 

• The set \(\sigma_\omega\) exists for rational and irrational \(\omega\) alike.
• Hofstadter butterfly is a closed set included in the rectangle \([-4,4] \times [0,1]\).
• For rational \(\omega=p/q\), bands are given by the eigenvalues of two \(q \times q\) matrices: there can be at most \(q\) bands.

Some Mathematical Conjectures

Theorem 1 (Aubry-André Conjecture 1980-2006) The measure (length) of the \(\sigma_\omega\) is zero if \(\omega\) is irrational.

Theorem 2 (Ten Martini Problem 1981-2006) The set \(\sigma_\omega\) is a Cantor set if \(\omega\) is irrational:

• Perfect set: there are no isolated points.
• Nowhere-dense: contains no intervals.

How can a set of this type be?

Example: Cantor 1/3 set

 

• Every point is not isolated.
• Length of the excluded middles totals 1 \(\rightarrow\) length zero!

Total sum of excluded middles: \[\sum_{n=0}^\infty \frac{2^n}{3^{n+1}} = \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \cdots = \frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right) = 1\] and therefore the Cantor set has length zero!

Source of the image

Conclusions

• Hofstadter Butterfly has been a rich source of inspiration for physics and mathematics.
• Rational and Irrational numbers are graciously combined in a single diagram.
• Cantor sets are not an exotic mathematical object, but commonly found in physical models.
• Some of the questions that can be posed looking at the diagram can take many years to prove mathematically.