The classical Lieb-Robinson bounds provide control over the speed of
propagation in quantum spin systems. In analogy to relativistic systems,
they establish a ``light cone'' $x \leq vt$ outside of which commutators
of initially localized observables are exponentially small. We consider an
XY spin chain in a quasiperiodic magnetic field and prove a new anomalous
Lieb-Robinson bound which features the modified light cone $x \leq
vt^\alpha$ for some $0<\alpha<1$. In fact, we can characterize $\alpha$
exactly as the upper transport exponent of a one-body Schr\"odinger
operator. This may be interpreted as a rigorous proof of anomalous quantum
many-body transport. Joint work with David Damanik, Milivoje Lukic and
William Yessen.

For discrete Schrödinger operators with potential given by a trigonometric polynomial of cosines (called generalized Harper's model), we use the complexified Lyapunov exponent to prove a criterion for subcritical energies in the spectrum and a criterion for supercritical energies. This work was done through the Caltech SURF program, with mentor Christoph Marx.

Abstract:  A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, known as Maclaurin’s inequalities, relating the 1/kthpowers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end (when k = n) we have the geometric mean, and on the right end (k = 1) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves f (n) steps away from either extreme. We also study the limiting behavior of such means for quadratic irrational α.

(Joint work with Francesco Cellarosi, Doug Hensley and Steven J. Miller)

Abstract: The Hall effect is the production of a voltage difference across a conductor, transverse to an electric current, in a presence of a magnetic field in the normal direction. At very low temperatures,  the (quantum) Hall conductance as a function of the strength of the magnetic field exhibited a staircase sequence of wide plateaus. The successive values of the Hall conductance turn out to be integer multiples of e^2/h, with remarkable precision (here e is the elementary charge and h is Planck's constant). This quantization can be understood in terms of topological invariant given by the Kubo-Streda formula. I will discuss the properties of the Kubo-Streda formula and its derivation in the adiabatic setting.