Indubala I. Satija

Realistic time-reversal invariant topological insulators with neutral atoms.

We lay out an experiment to realize time-reversal invariant topological insulators in alkali atomic gases. We introduce an original method to synthesize a gauge field in the near field of an atom chip, which effectively mimics the effects of spin-orbit coupling and produces quantum spin-Hall states. We also propose a feasible scheme to engineer sharp boundaries where the hallmark edge states are localized. Our multiband system has a large parameter space exhibiting a variety of quantum phase transitions between topological and normal insulating phases. Because of their remarkable versatility, cold-atom systems are ideally suited to realize topological states of matter and drive the development of topological quantum computing.

Harper equation, the dissipative standard map and strange nonchaotic attractors: relationship between an eigenvalue problem and iterated maps

The almost periodic eigenvalue problem described by the Harper equation is connected to other classes of quasiperiodic behavior; the dissipative dynamics on critical invariant tori and quasiperiodically driven maps. Firstly, the strong coupling limit of the supercritical Harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems. Secondly, a simple transformation is used to relate the Harper equation to a quasiperiodically forced one-dimensional map. In this case, the localized eigenstates of the supercritical Harper equation correspond to strange but nonchaotic attractors of the driven map. Furthermore, the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map.

Self-similarity and localization.

The localized eigenstates of the Harper equation exhibit universal self-similar fluctuations once the exponentially decaying part of the wave function is factorized out. For a fixed quantum state, we show that the whole localized phase is characterized by a single strong coupling fixed point of the renormalization equations. This fixed point also describes the generalized Harper model with next nearest neighbor interaction below a certain threshold. Above the threshold, the fluctuations in the generalized Harper model are described by a strange invariant set of the renormalization equations.

A spectrum of universality classes in period doubling and period tripling

Abstract We have studied the functional dependence of the period-doubling and period-tripling universal bifurcation ratios α and δ on the order of the critical point z in the iterated map f;(χ) = 1 − a ¦χ¦ z , where z can be even, odd o r fractional. For the period-doubling sequence, α( z ) is seen to be a monotonically decreasing function of z , and δ( z ) a monotonically increasing function. For the period-tripling sequence, α( z ) is still monotonically decreasing; however, δ( z ) displays an interesting minimum at z = 2.

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.

Lifshitz-like transition and enhancement of correlations in a rotating bosonic ring lattice

We study the effects of rotation on one-dimensional ultra-c old bosons confined to a ring lattice. For commensurate systems, at a critical value of the rotation frequ ency, an infinitesimal interatomic interaction energy opens a gap in the excitation spectrum, fragments the ground state into a macroscopic superposition of two states with different circulation and generates a sudden change in the topology of the momentum distribution. These features are reminiscent of the topological changes in the F ermi surface that occurs in the Lifshitz transition in fermionic systems. The entangled nature of the ground state induces a strong enhancement of quantum correlations and decreases the threshold for the Mott insulator t ransition. In contrast to the commensurate case, the incommensurate lattice is rather insensitive to rotation. Our studies demonstrate the utility of noise correlations as a tool for identifying new physics in strongly correlated systems.

Renormalization approach to quasiperiodic quantum spin chains

A renormalization scheme which takes into account the natural frequency of the system is developed to study an anisotropic quantum XY spin chain in a quasiperiodic transverse field. The quasiparticle excitations of the model exhibit extended, localized as well as critical phase, with fractal characteristics, in a finite parameter interval. The scaling properties of the critical phase fall into four distinct universality classes. The isotropic limit of the model describes the extensively studied Harper equation. The renormalization approach provides a new method for determining energies and transition thresholds with extremely high precision.

A universal strange attractor underlying the quasiperiodic transition to chaos

Abstract We use a Monte Carlo approach to study the universal properties associated with the breakdown of two-torus attractors for arbitrary winding numbers. We demonstrate that the renormalization equations have a universal strange attractor, compute its critical exponents, and discuss its structure. The fractal dimension of this attractor is 1.8±0.1.

Metal-Insulator Transition Revisited for Cold Atoms in Non-Abelian Gauge Potentials

We discuss the possibility of realizing metal-insulator transitions with ultracold atoms in two-dimensional optical lattices in the presence of artificial gauge potentials. For Abelian gauges, such transitions occur when the magnetic flux penetrating the lattice plaquette is an irrational multiple of the magnetic flux quantum. Here we present the first study of these transitions for non-Abelian U(2) gauge fields. In contrast to the Abelian case, the spectrum and localization transition in the non-Abelian case is strongly influenced by atomic momenta. In addition to determining the localization boundary, the momentum fragments the spectrum. Other key characteristics of the non-Abelian case include the absence of localization for certain states and satellite fringes around the Bragg peaks in the momentum distribution and an interesting possibility that the transition can be tuned by the atomic momenta.

The re-entrant phase diagram of the generalized Harper equation

We study the phase diagram of the tight-binding model for an electron on an anisotropic square lattice with a four-dimensional parameter space defined by two nearest-neighbour and two next-nearest-neighbour couplings. Using a renormalization scheme, we show that the inequality of the two next-nearest-neighbour couplings destroys the fat critical regime found in the isotropic case above the bicritical line and replaces it with another re-entrant extended phase. The scaling properties of the model are those of the corresponding tight-binding models on the nearest-neighbour square and triangular lattices. The triangular universality class also describes the quantum Ising chain in a transverse field with the only exception being the conformally invariant state of the Ising model which has no analogue in the triangular-lattice case.