Publications
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Can we trust the relationship between resonance poles and lifetimes?
(with Ira Herbst). Physics A: Mathematical and Theoretical. 49(195204), (2016).
(chosen for IOP Select) -
Dynamical bounds for quasiperiodic Schrodinger operators with rough potentials.
(with Svetlana Jitomirskaya). International Mathematics Research Notices. (2016). -
Level Spacing for Non-Monotone Anderson Models.
(with John Imbrie). Journal of Statistical Physics. 162(6),1451-1484.(2016). -
Continuity of the measure of the spectrum for
quasiperiodic Schrodinger operators with rough potentials.
(with Svetlana Jitomirskaya). Communications in Mathematical Physics. 325(2), 585-601 (2014). -
Measure of the spectrum of the almost Mathieu operator.
Oberwolfach Reports, 9, (2012). -
Phase transitions of barotropic flow coupled to a massive rotating sphere
- derivation of a fixed point equation by the Bragg method.
(with Chjan Lim). Physica A. 380. 43 - 60. (2007).
Preprints
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Localization for the Ising model in a transverse field with generic aperiodic disorder.
arXiv:1605.06514 [math-ph]. -
Resonant tunneling in a system with correlated pure point spectrum
(with Jeffrey Schenker). arXiv:1705.03039 [math-ph]. -
Ground States for Exponential Random Graphs
arXiv:1706.09084 [math.PR].
For my papers and preprints, see my page on the arXiv.
In preparation
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Dynamical localization for the Holstein model with random disorder.
(with Jeffrey Schenker). -
Stretched exponential decay in the quasiperiodic continuum percolation model.
Current research
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Critical disorder in the transverse field Ising model.
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Increased smoothness of the density of states in the Anderson model
Discussion of projects
Anderson Localization
A cornerstone of classical physics, the ergodic hypothesis postulates that an isolated system will eventually attain an equilibrium state where all microstates of equal energy are equivalently probable. In 1958 P.W. Anderson observed that, in a quantum system with sufficiently strong disorder this hypothesis fails to hold due to localized states which prevent the system from reaching equilibrium. Mathematically, however, this is a very difficult theory to establish in general.
Beginning with a proof of single body localization in many dimensions by Frolich and Spencer in 1983 many rigorous results on the localization phase of a class of single body random models have been established. This may be seen as a mathematical confirmation of Anderson's observation in the case of non-interacting particles.
The class of single body models which is best understood are those which satisfy two canonical properties. The first (monotonicity) is that the bare energies of the system move monotonically with respect to the random parameters. The second (the covering condition) is, roughly speaking, a parity of the random parameters and the bare energy levels of the system. In this case the density of eigenvalues and the statistics of the eigenvalues of the finite system are relatively well understood which is fundamental to understanding the complete system.
In a project with John Imbrie we consider a single particle random system where the bare energy levels respond nonmotinically to the random parameters. Although this would seem to be a minor complication, standard proofs completely break down in this case. Nevertheless, through a novel mutiscale method we obtain localization of states, estimates on the density of states and separation statistics of the eigenvalues.
In a second project in Anderson localization, with Jeffery Schenker, we consider a random system with severe failing of the covering condition. The context is a polaron, a system where a particle exists in a lattice which interacts non trivially with the particle. The state of the lattice is described by excitations at sites of the lattice. The levels of the system measure the total excitation of the lattice but they do not respond to the location of the excitations. Therefore there is degeneracy of the bare energies as the excitation ranges over the positions in the lattice. We show localization of the fractional moments of the Green's function which are stronger than the physically expected dynamical localization. [A preprint will appear shortly]
Quasistable states (Resonances)
The prevailing wisdom regarding isolated eigenvalues of a Schroedinger equation is that under small perturbations, the eigenvalues move continuously into other eigenvalues or may be recovered as resonances, that is, singularities of the Green's function, below the real axis.
Indeed this is the picture we find for the Hydrogen atom under a perturbation by a small constant electric field. In the bare Hydrogen atom, the essential spectrum is the positive real axis with isolated eigenvalues below zero, but under a perturbation of a constant electric field the eigenvalues vanish and the essential spectrum is the entire real axis. Nevertheless, the eigenvalues can be recovered as singularities of the Green's function as it is continued below the real axis. These singularities - the resonances - move continuously toward the eigenvalues as the field approaches zero. The electrons associated to the eigenvalues are eventually stripped from the atom by the electric field at a rate corresponding to half the distance of the resonance to the real line.
The bare Helium atom is more complicated. There is again a lower threshold of the essential spectrum with discrete eigenvalues to the left. However, there are self-ionizing auger states which are represented as resonances below the real axis. The fate of these resonances is unclear (and technically very challenging) under a constant electric field, but analogy with the eigenvalues in the Hydrogen case would suggest they move continuously under a constant electric field.
In a project with Ira Herbst, we consider a problem of pre-existing resonances under a perturbation and observe the results. The shape resonances is constructed simply from a trap of two positive delta functions in one dimension. The spectrum in this case is the positive real axis and the shape resonances correspond to the energies of the Dirichlet problem for the interval. As is typical of shape resonances, the lifetime of a particle initialized in one such quasistable state corresponds to half the distance of the resonance to the real line. In turning on a small constant electric field we find these resonances do not move continuously but are `erased and replaced' non-continuously by a different set of resonances roughly corresponding to a shape resonance of a delta function and the electr10ic field. On the other hand, the lifetimes of the quasistable states are approximated by the now erased resonances. That is, the system appears to obey behavior corresponding resonances which, properly speaking, do not exist.
Quasiperiodic Schroedinger Operators
The most famous example of a quasiperiodic Schroedinger operator is the almost Matthieu operator (AMO). The AMO originates in the study of the quantum version of the Hall effect, the integer quantum Hall effect, which was discovered experimentally in 1980 by von Klitzing. The operator is defined on the Hilbert space \( \ell^2(\mathbb{Z})\) for the integer lattice and has the secular equation \[\lambda V(2 \pi \alpha x + \theta ) \psi(x) +\Delta \psi(x) = E \psi(x) \] where \( V = 2\cos\) and \(\alpha,\theta \in \mathbb{R}\).
A plot of the spectra of the AMO at \(\lambda = 1\) over rational \(\alpha\) was produced by Douglas Hofstadter in 1976. Later, in 1980, Aubry and Andre conjectured that the Lebesgue measure of the spectrum of the AMO at irrational \(\alpha\) is given by \( 4 |1 - \lambda|\). The conjecture is supported - visually at least - by observing sequences of rational \( \alpha_n \) approaching an irrational \(\alpha\). The conjecture was finally proven in 2006 by Avila and Krikorian. For \(\lambda > 1 \) a critical part of the proof is the continuity of the Lebesgue measure of the spectrum as certain subsequences of rational \(\alpha_n\) approach irrational \(\alpha\). The continuity of the Lebesgue measure in fact was shown to hold up to analytic \(V\).
In a project with Svetlana Jitomirskaya, we extend these results beyond the analytic class of functions to periodic potentials \(V\) which are only required to be Holder - \(^1 /_2\) continuous.
Ground state of the transverse field Ising model
In studying thermal systems, a fundamental quality to determine is the phase portrait of the system. That is, as parameters or the temperature is varied, what phase of matter does the system exhibit? The point at which one phase transforms to another phase is known as the phase transition. If a phase transition is determined to hold for a translation invariant system, a natural question to ask is whether this transition is robust. That is, if `dirt' is added to the system does the phase transition persist?
The quantum Ising model has no phase transition as the parameters are varied at vanishing temperature (the ground state). On the other hand, the Ising model with a transverse field does exhibit a phase transition in the ground state.
The transverse field Ising model is defined by the Hamiltonian \[ H = \sum_{x,y: |x-y| = 1} J \sigma_x^{(3)} \sigma_y^{(3)} + \sum_x \delta \sigma_x^{(1)}. \] Here, \(\sigma^{(1)},\sigma^{(3)}\) are Pauli matrices and the sites \(x,y\) range over \(\mathbb{Z}^d\). The order parameter is \(\rho = J / \delta\), controls spontaneous magnetization. For \(\rho\) smaller than the critical value \(\rho_d\) there is no spontaneous magnetization, but for \(\rho\) above the critical value spontaneous magnetization increases monotonically with \(\rho\).
We ask whether the phase transition persists for disordered \(J, \delta\). That is, for \(\delta_x\) i.i.d. and \(J_{x,y} = \kappa \lambda_{x,y}\) for \(\lambda_{x,y}\) i.i.d. what moment conditions on the distributions allow for a phase transition as \(\kappa\) is varied? The strongest conditions demonstrating the existence of a phase transition were demonstrated by Klein 1991. On the other hand, conditions on the distribution ensuring the absence of a phase transition were demonstrated by Aizenman, Klein and Newman.
It is also interesting to consider other forms of disorder. Jitomirskaya and Klein found persistence of a phase transition for a model with constant \( J \) and quasiperiodically ordered \(\delta\). The quasiperiodic order here was defined by a function over the torus \(\mathbb{R}^n/\mathbb{Z}^d\) sampled by an irrational shift. I found this persistence extends to any aperiodically ordered transverse field, in fact, I found transverse fields with a phase transition are typical in the set of aperiodically ordered fields. Moreover, for certain `nice' irrational shifts in one dimension, I showed that the results of Jitomirskaya and Klein are optimal.