Session X29: The Butterfly Plot Turns 40

Bumping into the Butterfly, When I Was But a Bud

I will recount the main events that led me to discover the so-called "Hofstadter butterfly" when I was a physics student, over 40 years ago. A key moment in the tale was when, after years of futile struggle, I finally abandoned particle physics, and chose, with much trepidation, to try solid-state physics instead, a field of which I knew nothing at all. I was instinctively drawn to a long-standing classic unsolved problem in the field \textemdash\ What is the nature of the energy spectrum of Bloch electrons in a magnetic field? \textemdash\ when Professor Gregory Wannier told me that it involved a weird distinction between "rational" and "irrational" magnetic fields, which neither he nor anyone else understood. This mystery allured me, as I was sure that the rational/irrational distinction cannot possibly play a role in physical phenomena. I tried manipulating equations for a long time but was unable to make any headway, and so, as a last resort, I wound up using brute-force calculation instead. I programmed a small desktop computer to give me numbers that I then plotted by hand on paper, and one fine day, to my shock, my eyes suddenly recognized a remarkable type of pattern that I had discovered twelve years earlier, when I was an undergraduate math major exploring number theory. All at once, I realized that the theoretical energy spectrum I'd plotted by hand consisted of infinitely many copies of itself, nested infinitely deeply, and it looked a little like a butterfly, whence its name. This unanticipated discovery eventually led to many new insights into the behavior of quantum systems featuring two competing periodicities. I will briefly describe some of the consequences I found back then of the infinitely nested spectrum, and in particular how the baffling rational/irrational distinction melted away, once the butterfly's nature had been deeply understood.   

The Hofstadter Butterfly and some physical consequences.

Opening its beautiful wings for the first time four decades ago, the Hofstadter Butterfly emerged as a self-similar pattern of bands and gaps displaying the allowed energies for two dimensional crystalline electrons in a perpendicular magnetic field.$^{\mathrm{1\thinspace }}$Within the Harper model, as the external field parameter is varied well defined gaps traverse the spectrum, some closing at a Dirac point where two approaching bands touch. Such band edges degeneracy is lifted in more realistic models.$^{\mathrm{2}}$ Gaps have a unique label that determines the Hall conductivity of a noninteracting electron system, as observed in recent experiments.$^{\mathrm{3}}$ When the 2D electron assembly is allowed to interact in the absence of an underlying periodic potential, the mean field approximation predicts a liquid at integer filling fractions and electron density fluctuations otherwise, which if periodic may be represented again by a Harper equation. The intriguing odd denominator rule observed in experiment in the fractional quantum Hall regime is then a natural prediction of the model.$^{\mathrm{4}}$ \begin{enumerate} \item D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976) \item F. Claro, Phys. Status Solidi (b) 104, K31 (1981) \item C. R. Dean et al, Nature 497, 598 (2013) \item F. Claro, Phys. Rev. B 35, 7980 (1987) \end{enumerate}   

Bethe-Ansatz Solution of the Hofstdter Problem

I review the solution of the Hofsdater problem by means of the Bethe Ansatz. The solution has been obtained quite some time ago together with Anton Zabrodin and further developed together with Alexander Abanov and Joop Talstra. Solution is possible due to realization of the group of magnetic translations by means of the cyclic representation of the quantum group $U_q(SL_2)$. It equates the Hofstadter problem to the Heisenberg magnetic chain on just few sites but with a large spin.   

Pure and Poetic: Butterfly in the Quantum World

Story of the Hofstadter butterfly is a magical occurrence in a quantum flatland of two-dimensional crystals in a magnetic field. In this drama, the magnetic flux plays the role of Planck constant, linking the variables $x$ and $p$ in the butterfly Hamiltonian $ H = cos x + cos p$ as $[x, p] = i \hbar$. It is a story of reunion of Descartes and Pythagoras and tale of this quantum fractal is related to Integral Apollonian gaskets. Integers rule the butterfly landscape as quantum numbers of Hall conductivity while irrational numbers emerge as the asymptotic magnification of these topological integers in the kaleidoscopic images of the butterfly $^{1}$. \par Simple variations of the above Hamiltonian generates a wide spectrum of physical phenomenon. For example, the Hamiltonian $H = \cos x + \lambda \cos p$ with the parameter $\lambda \ne 1$ in its zero energy solution {\it hides} the critical point of a topological transition in a superconducting chain and thus barely misses the Majorana fermions$^{2}$. Another example is the Hamiltonian obtained by including terms like $\cos( x \pm p)$ which for flux {\it half} exhibits Dirac semi-metallic states in addition to all integer quantum Hall states corresponding to all possible solutions of the Diophantine equation for this value of the magnetic flux. In this analytically tractable model where the parameter $\lambda$ varies periodically with time, the topological states are described by edge modes whose dispersion is given by a pure cosine function$^{3}$. \par Finally, nature has composed beautiful variations of the Hofstadter butterfly not only in systems such as Penrose and Kagame lattices and also in the relativistic colorful world of quarks and antiquarks$^{4}$. \par\noindent (1) "Butterfly in the Quantum World", Indubala I Satija with contributions by Douglas Hofstadter, IOP Concise, Morgan and Claypool publication, 2016.\\ (2) I. Satija and G. Naumis, Phys Rev B ${\bf 88}$, 054204 (2013)\\ (3) I. Satija and E. Zhao, arXiv:1609.02807\\ (4) G. Endr\"{o}di, $32$nd Int. Symp. on Lattice Field Theory $2014$, New York: Columbia University   

Ultracold atoms in strong synthetic magnetic fields

The Harper-Hofstadter Hamiltonian describes charged particles in the lowest band of a lattice at high magnetic fields. This Hamiltonian can be realized with ultracold atoms using laser assisted tunneling which imprints the same phase into the wavefunction of neutral atoms as a magnetic field dose for electrons. I will describe our observation of a bosonic superfluid in a magnetic field with half a flux quantum per lattice unit cell. Subsequently, we have used laser assisted tunneling to realize synthetic spin orbit coupling and to observe a supersolid. A supersolid is superfluid and breaks translational symmetry, i.e. it has shape.