arXiv subjects

Indubala I Satija

Publications and source records attributed to Indubala I Satija.

At least 19 recordsLinked to original sources

arXiv2024

Building the Butterfly Fractal: The Eightfold Way

The hierarchical structure of the butterfly fractal -- the Hofstader butterfly, is found to be described by an octonary tree. In this framework of building the butterfly graph, every iteration generates sextuplets of butterflies, each with a tail that is made up of an infinity of butterflies. Identifying {\it butterfly with a tale} as the building block, the tree is constructed with eight generators represented by unimodular matrices with integer coefficients. This Diophantine description provides one to one mapping with the butterfly fractal, encoding the magnetic flux interval and the topological quantum numbers of every butterfly. The butterfly tree is a generalization of the ternary tree describing the set of primitive Pythagorean triplets.

math.GM↗

arXiv2021

Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets

We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be encoded in transformations that forms the modular group $SL(2,Z)$. The asymptotic scalings of curvatures of the circles are given by a special set of quadratic irrationals with continued fraction $[n+1: \overline{1,n}]$ - that is a set of irrationals with period-2 continued fraction consisting of $1$ and another integer $n$. Belonging to the class $n=2$, there exists a nested set of self-similar kaleidoscopic patterns that exhibit three-fold symmetry. Furthermore, the even $n$ hierarchy is found to mimic the recursive structure of the tree that generates all Pythagorean triplets

math.GM↗

arXiv2020

What do Bloch Electrons in a Magnetic Field have to do with Apollonian packing of Circles ?

Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the "Hofstadter butterfly". In this Apollonian-Butterfly-Connection, dubbed as $\mathcal{ABC}$, the integer curvatures of the circles contain in a convoluted form, the topological quantum numbers of the butterfly graph -- the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. The $\mathcal{ABC}$ unfolds as the conformal maps describing butterfly recursions are related to the conformal maps describing nesting of circles in the Apollonian packing. Mapping of butterflies to Apollonian at all scales where Farey tree hierarchy plays the central role, reveals how geometry and number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.

nlin.CD↗

arXiv2017

Topology Induced Oscillations in Majorana Fermions in a Quasiperiodic Superconducting Chain

Spatial profile of the Majorana fermion wave function in a one-dimensional $p$-wave superconductors ($\cal{PWS}$) with quasi periodic disorder is shown to exhibit spatial oscillations. These oscillations damp out in the interior of the chain and are characterized by a period that has topological origin and is equal to the Chern number determining the Hall conductivity near half-filling of a two-dimensional electron gas in a crystal. This mapping unfolds in view of a correspondence between the critical point for the topological transition in $\cal{PWS}$ and the {\it strong coupling fixed point} of the Harper's equation. Oscillatory character of these modes persist in a generalized model related to an extended Harper system where the electrons also tunnel to the diagonals of a square lattice. However, beyond a bicritical point, the Majorana oscillations occur with a random period, characterized by an invariant fractal set.

cond-mat.dis-nn↗

arXiv2016

A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

nlin.PS↗

arXiv2014

Hidden Quasicrystal in Hofstadter Butterfly

Topological description of hierarchical sets of spectral gaps of Hofstadter butterfly is found to be encoded in a quasicrystal where magnetic flux plays the role of a phase factor that shifts the origin of the quasiperiodic order. Revealing an intrinsic frustration at smallest energy scale, described by $ζ=2-\sqrt{3}$, this irrational number characterizes the universal butterfly and is related to two quantum numbers that includes the Chern number of quantum Hall states. 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↗

arXiv2014

Solitons in a hard-core bosonic system: Gross-Pitaevskii type and beyond

A unified formulation that obtains solitary waves for various background densities in the Bose-Einstein condensate of a system of hard-core bosons with nearest neighbor attractive interactions is presented. In general, two species of solitons appear: A nonpersistent (NP) type that fully delocalizes at its maximum speed, and a persistent (P) type that survives even at its maximum speed, and transforms into a periodic train of solitons above this speed. When the background condensate density is nonzero, both species coexist, the soliton is associated with a constant intrinsic frequency, and its maximum speed is the speed of sound. In contrast, when the background condensate density is zero, the system has neither a fixed frequency, nor a speed of sound. Here, the maximum soliton speed depends on the frequency, which can be tuned to lead to a cross-over between the NP-type and the P-type at a certain critical frequency, determined by the energy parameters of the system. We provide a single functional form for the soliton profile, from which diverse characteristics for various background densities can be obtained. Using the mapping to spin systems enables us to characterize the corresponding class of magnetic solitons in Heisenberg spin chains with different types of anisotropy, in a unified fashion.

cond-mat.quant-gas↗

arXiv2013

Supermetallic and Trapped States in Periodically Kicked Lattices

A periodically driven lattice with two commensurate spatial periodicities is found to exhibit super metallic states characterized by enhancements in wave packet spreading and entropy. These resonances occur at critical values of parameters where multi-band dispersion curves reduce to a universal function that is topologically a circle and the effective quantum dynamics describes free propagation. Sandwiching every resonant state are a pair of anti-resonant {\it trapped states} distinguished by dips in entropy where the transport, as seen in the spreading rate, is only somewhat inhibited. Existing in gapless phases fo the spectrum, a sequence of these peaks and dips are interspersed by gapped phases assocated with flat band states where both the wave packet spreading as well as the entropy exhibit local minima.

cond-mat.dis-nn↗

arXiv2013

Chern and Majorana Modes of Quasiperiodic Systems

New types of self-similar states are found in quasiperiodic systems characterized by topological invariants-- the Chern numbers. We show that the topology introduces a competing length in the self-similar band edge states transforming peaks into doublets of size equal to the Chern number. This length intertwines with the quasiperiodicity and introduces an intrinsic scale, producing Chern-beats and nested regions where the fractal structure becomes smooth. Cherns also influence the zero-energy mode, that for quasiperiodic systems which exhibit exponential localization, is related to the ghost of the Majorana; the delocalized state at the onset to topological transition. The Chern and the Majorana, two distinct types of topological edge modes, exist in quasiperiodic superconducting wires.

quant-ph↗

arXiv2013

Soliton dynamics of an atomic spinor condensate on a Ring Lattice

We study the dynamics of macroscopically-coherent matter waves of an ultra-cold atomic spin-one or spinor condensate on a ring lattice of six sites and demonstrate a novel type of spatio-temporal internal Josephson effect. Using a discrete solitary mode of uncoupled spin components as an initial condition, the time evolution of this many-body system is found to be characterized by two dominant frequencies leading to quasiperiodic dynamics at various sites. The dynamics of spatially-averaged and spin-averaged degrees of freedom, however, is periodic enabling an unique identification of the two frequencies. By increasing the spin-dependent atom-atom interaction strength we observe a resonance state, where the ratio of the two frequencies is a characteristic integer multiple and the spin-and-spatial degrees of freedom oscillate in "unison". Crucially, this resonant state is found to signal the onset to chaotic dynamics characterized by a broad band spectrum. In a ferromagnetic spinor condensate with attractive spin-dependent interactions, the resonance is accompanied by a transition from oscillatory- to rotational-type dynamics as the time evolution of the relative phase of the matter wave of the individual spin projections changes from bounded to unbounded.

cond-mat.quant-gas↗

arXiv2012

Topological Insulators with Ultracold Atoms

Ultracold atom research presents many avenues to study problems at the forefront of physics. Due to their unprecedented controllability, these systems are ideally suited to explore new exotic states of matter, which is one of the key driving elements of the condensed matter research. One such topic of considerable importance is topological insulators, materials that are insulating in the interior but conduct along the edges. Quantum Hall and its close cousin Quantum Spin Hall states belong to the family of these exotic states and are the subject of this chapter.

cond-mat.quant-gas↗

arXiv2010

Other incarnations of the Gross-Pitaevskii dark soliton

We show that the dark soliton of the Gross-Pitaevskii equation (GPE) that describes the Bose-Einstein condensate (BEC) density of a system of weakly repulsive bosons, also describes that of a system of strongly repulsive hard core bosons at half filling. As a consequence of this, the GPE soliton gets related to the magnetic soliton in an easy-plane ferromagnet, where it describes the square of the in-plane magnetization of the system. These relationships are shown to be useful in understanding various characteristics of solitons in these distinct many-body systems.

quant-ph↗

arXiv2010

The Gross-Pitaevskii Soliton: Relating Weakly and Strongly Repulsive Bosonic condensates and the magnetic soliton

We show that the dark soliton of the Gross-Pitaevskii equation (GPE) that describes the Bose-Einstein condensate (BEC) density of a system of weakly repulsive bosons, also describes that of a system of strongly repulsive hard core bosons at half filling. This connection establishes a relationship between the GPE soliton and the magnetic soliton of an easy-plane ferromagnet, where the BEC density relates to the square of the in-plane magnetization of the system. This mapping between well known solitons in two distinct physical systems provides an intuitive understanding of various characteristics of the solitons.

cond-mat.quant-gas↗

arXiv2010

Exploring complex phenomena using ultracold atoms in bichromatic lattices

With an underlying common theme of competing length scales, we study the many-body Schrödinger equation in a quasiperiodic potential and discuss its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We propose a possible visualization of such connection in experimentally accessible many-body observables. Those observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson and band insulator phases. In addition, they exhibit fingerprints of non-linear phenomena such as Arnold tongues, bifurcations and devil's staircases. Our numerical treatment is complemented with a perturbative analysis which provides insight on the underlying physics. The perturbation theory approach is particularly useful in illuminating the distinction between the Anderson insulator and the band insulator phases in terms of paired sets of dimerized states.

cond-mat.quant-gas↗

arXiv2008

Symmetry-Breaking and Symmetry-Restoring Dynamics of a Mixture of Bose-Einstein Condensates in a Double Well

We study the coherent nonlinear tunneling dynamics of a binary mixture of Bose-Einstein condensates in a double-well potential. We demonstrate the existence of a new type of mode associated with the "swapping" of the two species in the two wells of the potential. In contrast to the symmetry breaking macroscopic quantum self-trapping (MQST) solutions, the swapping modes correspond to the tunneling dynamics that preserves the symmetry of the double well potential. As a consequence of two distinct types of broken symmetry MQST phases where the two species localize in the different potential welils or coexist in the same well, the corresponding symmetry restoring swapping modes result in dynamics where the the two species either avoid or chase each other. In view of the possibility to control the interaction between the species, the binary mixture offers a very robust system to observe these novel effects as well as the phenomena of Josephson oscillations and pi-modes

quant-ph↗

arXiv2007

Optical phase shifts and diabolic topology in Mobius-type strips

We compute the optical phase shifts between the left and the right-circularly polarized light after it traverses non-planar cyclic paths described by the boundary curves of closed twisted strips. The evolution of the electric field along the curved path of a light ray is described by the Fermi-Walker transport law which is mapped to a Schrödinger equation. The effective quantum Hamiltonian of the system has eigenvalues equal to $0, \pm κ$, where $κ$ is the local curvature of the path. The inflexion points of the twisted strips correspond to the vanishing of the curvature and manifest themselves as the diabolic crossings of the quantum Hamiltonian. For the Möbius loops, the critical width where the diabolic geometry resides also corresponds to the characteristic width where the optical phase shift is minimal. In our detailed study of various twisted strips, this intriguing property singles out the M"{o}bius geometry.

cond-mat.other↗

arXiv2005

Noise Correlations of Hard-core Bosons: Quantum Coherence and Symmetry Breaking

Noise correlations, such as those observable in the time of flight images of a released cloud, are calculated for hard-core bosonic (HCB) atoms. We find that the standard mapping of HCB systems onto spin-1/2 XY models fails in application to computation of noise correlations due to the contribution of multiply occupied virtual states in HCB systems. Such states do not exist in spin models. An interesting manifestation of such states is the breaking of particle-hole symmetry in HCB. We use noise correlations to explore quantum coherence of strongly correlated bosons in the fermionized regime with and without external parabolic confinement. Our analysis points to distinctive new experimental signatures of the Mott phase.

cond-mat.other↗

arXiv2005

Noise Correlations and Quantum Coherence in Hard-core Bosons in One-dimensional Lattices

Noise correlations, such as those observable in the time of flight images of a released cloud, are calculated for hard-core bosonic (HCB) atoms. We find that the standard mapping of HCB systems onto spin-1/2 XY models fails in application to computation of noise correlations. This is due to the contribution of multiply occupied virtual states to noise correlations in bosonic systems. Such states do not exist in spin models. We use these correlations to explore quantum coherence of the ground states and re-address the relationship between the peaks present in noise correlation and the Mott phase. Our analysis points to distinctive new experimental signatures of the Mott phase. The importance of these correlations is illustrated in an example of a quasiperiodic potential that exhibits a localization transition. In this case, in contrast to the momentum distribution, the noise correlations reveal the presence of quasiperiodic order in the localized phase.

quant-ph↗