APS March Meeting 2017
Volume 62, Number 4
Monday–Friday, March 13–17, 2017;
New Orleans, Louisiana
Session X29: The Butterfly Plot Turns 40
8:00 AM–11:00 AM,
Friday, March 17, 2017
Room: 292
Sponsoring
Unit:
GSNP
Chair: Greg Huber, University of Southern California, Santa Barbara
Abstract ID: BAPS.2017.MAR.X29.4
Abstract: X29.00004 : Pure and Poetic: Butterfly in the Quantum World
9:48 AM–10:24 AM
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Abstract
Author:
Indubala Satija
(George Mason University)
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
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2017.MAR.X29.4