Kay Kirkpatrick

Lattice Models

The cubic nonlinear Schrodinger equation and its nonlocal versions emerge in forthcoming work with Enno Lenzmann and Gigliola Staffilani, where we consider a lattice with a quantum particle sitting at each site, interacting with the others. Such lattice systems are used to understand electron transport in biopolymers like organic semiconductors, molecular crystals, and DNA.

As a first approximation, we consider a discrete model of quantum particles at lattice points with two kinds of interactions: nearest-neighbor interactions appearing as a discrete Laplacian term, representing interactions between base pairs in DNA; and self-interactions appearing as a cubic nonlinear term, representing interactions within a base pair. The dynamics are governed by the discrete NLS on the lattice of mesh size h, and we prove that taking the mesh size of the lattice to zero (the continuum limit), gives macroscopic behavior described by the focusing cubic NLS:
      i ∂t φ = - Δ φ - |φ|2 φ .
As a better approximation, we want to account for long-range interactions (which need not be of fixed range, because DNA is constantly in flux). So we consider the same cubic self-interaction term as before, and inverse power-law long-range interactions for a parameter α.

Our main result is another continuum limit: for certain values of α, solutions of the discrete model converge weakly to a solution of the NLS with fractional-order Laplacian:

      i ∂t φ = (- Δ)α φ - |φ|2 φ .
We again follow the main outline of the first approximation with the NLS, but the new context is much more delicate. The main difficulties are that there is no canonical discretization of the fractional derivative and that the most physical one doesn't obviously play well with the fractional derivative. We construct a discrete fractional calculus and an interpolation of the discrete functions based on a special mollification. This framework is compatible with the fractional derivative, only now to exchange the order of interpolation and discrete differentiation, we have discovered an asymptotic fractional commutation relation. We also develop some ingenious harmonic analysis techniques to conduct the key steps of the proof in Fourier space.

We also prove conservation of energy and well-posedness for the fractional NLS. We expect scattering to linear solutions for the defocusing case because of a Morawetz-type inequality. By contrast, we expect solitons in the focusing case. We also hope to deepen the understanding of computation for these systems and address Grunwald-Letnikov-type discretizations for numerical efficiency.
(Figure: Biopolymers like DNA coil up in a way that allows electrons to jump long distances along the polymer. Courtesy of Azatoth and the Protein Data Bank.)

Other projects on lattice models of ferromagnets and spin glasses would clarify our understanding of glass and phenomena like neural networks. In current work with Elizabeth Meckes, we are studying the asymptotic distribution of appropriately normalized sums of spins in the spherical Ising model. We plan to prove this non-central-limit theorem by Stein's method of exchangeable pairs, in the spirit of other recent work.

I am also working on computational algorithms for quantum many-body systems with Jose Blanchet. We have analyzed a dynamic quantum Curie-Weiss-type model, and we plan to modify our techniques to handle other models approximating Bose-Einstein condensation. A dynamic quantum Ising model is the next step.
(Figure: Optical fibers, made of glass, courtesy of Sandia National Lab.)
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