arXiv subjects

Indubala Satija

Publications and source records attributed to Indubala Satija.

5 recordsLinked to original sources

arXiv2025

The Hofstadter Butterfly: Bridging Condensed Matter, Topology, and Number Theory

Celebrating its golden jubilee, the Hofstadter butterfly fractal emerges as a remarkable fusion of art and science. This iconic X shaped fractal captivates physicists, mathematicians, and enthusiasts alike by elegantly illustrating the energy spectrum of electrons within a two dimensional crystal lattice influenced by a magnetic field. Enriched with integers of topological origin that serve as quanta of Hall conductivity, this quantum fractal and its variations have become paradigm models for topological insulators, novel states of matter in 21st century physics. This paper delves into the theoretical framework underlying butterfly fractality through the lenses of geometry and number theory. Within this poetic mathematics, we witness a rare form of quantum magic: Natures use of abstract fractals in crafting the butterfly graph itself. In its simplest form, the butterfly graph tessellates a two dimensional plane with trapezoids and triangles, where the quanta of Hall conductivity are embedded in the integer sloped diagonals of the trapezoids. The theoretical framework is succinctly expressed through unimodular matrices with integer coefficients, bringing to life abstract constructs such as the Farey tree, the Apollonian gaskets, and the Pythagorean triplet tree.

cond-mat.mes-hall↗

arXiv2021

Geometry, Number Theory and the Butterfly Spectrum of Two-Dimensional Bloch Electrons

We take a deeper dive into the geometry and the number theory that underlay the butterfly graphs of the Harper and the generalized Harper models of Bloch electrons in a magnetic field. Root of the number theoretical characteristics of the fractal spectrum is traced to a close relationship between the Farey tree -- the hierarchical tree that generates all rationals and the Wannier diagram -- a graph that labels all the gaps of the butterfly graph. The resulting Farey-Wannier hierarchical lattice of trapezoids provides geometrical representation of the nested pattern of butterflies in the butterfly graph. Some features of the energy spectrum such as absence of some of the Wannier trajectories in the butterfly graph fall outside the number theoretical framework, can be stated as a simple rule of "minimal violation of mirror symmetry". In a generalized Harper model, Farey-Wannier representation prevails as the lattice regroups to form some hexagonal unit cells creating new {\it species} of butterflies

nlin.CD↗

arXiv2018

Pythagorean Triplets, Integral Apollonians and The Hofstadter Butterfly

Hierarchical sets such as the Pythagorean triplets ($\cal{PT}$) and the integral Apollonian gaskets ($\cal{IAG}$) are iconic mathematical sets made up of integers that resonate with a wide spectrum of inquisitive minds. Here we show that these abstract objects are related with a quantum fractal made up of integers, known as the {\it Hofstadter Butterfly}. The "butterfly fractal" describes a {\it physical system} of electrons in a crystal in a magnetic field, representing exotic states of matter known as {\it integer quantum Hall} states. Integers of the butterfly are the quanta of Hall conductivity that appear in a highly convoluted form in the integers of the $\cal{PT}$ and the $\cal{IAG}$. Scaling properties of these integers, as we zoom into the self-similar butterfly fractal are given by a class of quadratic irrationals that lace the butterfly in a highly intricate and orderly pattern, some describing a {\it mathematical kaleidoscope}. The number theoretical aspects are all concealed in Lorentz transformations along the light cone in abstract Minkowski space where subset of these are related to the celebrated {\it Pell's equation}.

nlin.CD↗

arXiv2014

Topology and Self-Similarity of the Hofstadter Butterfly

We revisit the problem of self-similar properties of the Hofstadter butterfly spectrum, focusing on spectral as well as topological characteristics. In our studies involving any value of magnetic flux and arbitrary flux interval, we single out the most dominant hierarchy in the spectrum, which is found to be associated with an irrational number $ζ=2+\sqrt{3}$ where nested set of butterflies describe a kaleidoscope. Characterizing an intrinsic frustration at smallest energy scale, this hidden quasicrystal encodes hierarchical set of topological quantum numbers associated with Hall conductivity and their scaling properties. This topological hierarchy maps to an {\it integral Apollonian gasket} near-$D_3$ symmetry, revealing a hidden symmetry of the butterfly as the energy and the magnetic flux intervals shrink to zero. With a periodic drive that induces phase transitions in the system, the fine structure of the butterfly is shown to be amplified making states with large topological invariants accessible experimentally.

cond-mat.dis-nn↗

arXiv2003

Anholonomy and Geometrical Localization in Dynamical Systems

We characterize the geometrical and topological aspects of a dynamical system by associating a geometric phase with a phase space trajectory. Using the example of a nonlinear driven damped oscillator, we show that this phase is resilient to fluctuations, responds to all bifurcations in the system, and also finds new geometric transitions. Enriching the phase space description is a novel phenomenon of ``geometrical localization'' which manifests itself as a significant deviation from planar dynamics over a short time interval.

nlin.CD↗