Research interests: My background is in mathematical physics, specifically, functional analysis with concentration on operator theory, spectral analysis and constructive quantum field formalism. For several years now, my research focused entirely on the physics of the condensed matter. Over the years, I learned how to combine rigorous mathematical analysis with computer simulations to seek original solutions to problems arising in condensed matter physics and to connect with experiments.
While I am pursuing a fairly broad research, my present focus is on topological insulators, where I am trying to understand the effect of disorder on the topological properties on these materials. I am studying the classic works in Non-Commutative Geometry, a branch of mathematics that seems to provide just the right tools for understanding and characterizing the topological phases. I have invested a fair amount of time to study the classic works of Atiyah in K-theory (and the modern notes in K-Theory by Koroubi), giving a special attention to the representation by Fredholm operators. It appears to me that the generalization of the Fredholm index, from an integer number to an element of the K(X) group, has not been yet explored in condensed matter. But in general, I believe there are many applications of K-Theory in condensed matter physics that awaits to be discovered. My recent contributions include an analysis of the robustness of the edge states in Chern and Quantum Spin-Hall insulators, a general framework for topological quantization of currents, a general methodology to define bulk topological invariants for topological insulators, a study on the phase diagram of topological insulators with strong disorder.
My current numerical applications (mostly done on my five MacPro workstations) deal with the physics of disordered lattice systems. In particular, computations of various topological invariants, detectection and visualization of the robust extended states in disordered topological insulators, definition and exploration of the non-commutative Fermi surface for disordered systems, numerical implementations of the C*-algebra approach to transport in aperiodic systems.
The soft matter is also a subject that interests me. Inspired by the protein lattice of microtubules, I and my collaborators have recently found the mechanical analoug of graphene, a harmonic dimer lattice which display Dirac points in the phonon spectrum. We were also been able to show that the Dirac points can be broken such that the phonon bands acquire finite Chern numbers, leading to topological edge phonon modes. Based on this observation, we put forward a novel mechanism for the dynamic instability of microtubules. We have also found filamentary structures that support topological phonon modes, structures that were inspired by the actin microfilaments.
I am also interested in the Riemann structure of the bands in periodic systems. A recent result lead to a novel and simple expression for the Green's functions in periodic insulators. This finding allowed me and Professor Roberto Car to develop a semi-analytic theory of tunneling transport. Numerical implementations of the formalism enabled theoretical investigations of experimental data that otherwise were out of reach.
I have a strong interest in the Algebraic Bethe Ansatz, Conformal Field Theories and Constructive Field Theory. In particular I have explored various Riemann structures in spin-chains with twisted boundaries with the same goal of deriving simple expressions for the Green's functions. Together with Professor Duncan Haldane, I have studied the non-abelian properties of the anyons in the Moore-Read state.
I have also completed several studies on the self-consistent models of the condensed matter, touching on issues like existence and uniqueness of the self-consistent solution, symmetry breaking and thermodynamic limit. I have wrote (from scratch) several atomistic codes based on Density Functional Theory or Hartree and Hartree-Fock approximation. With Professor Walter Kohn I have studied "Nearsightedness," the property of electron density of being insesitive to large perturbations.
I am also actively involved in developing quantum and semi-classical formalisms for optical response of atomic clusters and nano-particles. With Professor Peter Nordlander I have studied the plasmonic properties of complex metalo-dielectric nano-structures and developed what now is known as "Plasmon Hybridization Formalism."
Assistant Professor, Department of Physics
Stern College for Women of Yeshiva University
245 Lexington Avenue, New York, NY 10016
Office: Room 510
Tel: 212 340 7831