Non-Hermitian dynamics of slowly-varying Hamiltonians

We develop a theoretical description of non-Hermitian time evolution that accounts for the break- down of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigen-energies as well as inter-band Berry connections calculated using basis sets from appropriately-chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of "sudden transitions", where the system state abruptly jumps from one eigenstate to another.Comment: 12 pages, 4 figure

Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions

This paper introduces and studies multivariate spacings. The spacings are developed using the order statistics derived from data depth. Specifically, the spacing between two consecutive order statistics is the region which bridges the two order statistics, in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics. These multivariate spacings can be viewed as a data-driven realization of the so-called ``statistically equivalent blocks.'' These spacings assume a form of center-outward layers of ``shells'' (``rings'' in the two-dimensional case), where the shapes of the shells follow closely the underlying probabilistic geometry. The properties and applications of these spacings are studied. In particular, the spacings are used to construct tolerance regions. The construction of tolerance regions is nonparametric and completely data driven, and the resulting tolerance region reflects the true geometry of the underlying distribution. This is different from most existing approaches which require that the shape of the tolerance region be specified in advance. The proposed tolerance regions are shown to meet the prescribed specifications, in terms of $\beta$-content and $\beta$-expectation. They are also asymptotically minimal under elliptical distributions. Finally, a simulation and comparison study on the proposed tolerance regions is presented.Comment: Published in at http://dx.doi.org/10.1214/07-AOS505 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Wide Band Equivalent Source Reconstruction Method Exploiting the Stoer-Bulirsch Algorithm with the Adaptive Frequency Sampling

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Realizing Hopf Insulators in Dipolar Spin Systems

The Hopf insulator represents a topological state of matter that exists outside the conventional ten-fold way classification of topological insulators. Its topology is protected by a linking number invariant, which arises from the unique topology of knots in three dimensions. We predict that three-dimensional arrays of driven, dipolar-interacting spins are a natural platform to experimentally realize the Hopf insulator. In particular, we demonstrate that certain terms within the dipolar interaction elegantly generate the requisite non-trivial topology, and that Floquet engineering can be used to optimize dipolar Hopf insulators with large gaps. Moreover, we show that the Hopf insulator's unconventional topology gives rise to a rich spectrum of edge mode behaviors, which can be directly probed in experiments. Finally, we present a detailed blueprint for realizing the Hopf insulator in lattice-trapped ultracold dipolar molecules; focusing on the example of ${}^{40}$K$^{87}$Rb, we provide quantitative evidence for near-term experimental feasibility.Comment: 6 + 7 pages, 3 figure

Symmetries and Lie algebra of the differential-difference Kadomstev-Petviashvili hierarchy

By introducing suitable non-isospectral flows we construct two sets of symmetries for the isospectral differential-difference Kadomstev-Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.Comment: 9 page