 Award ID(s):
 1820734
 Publication Date:
 NSFPAR ID:
 10344815
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2021
 Issue:
 9
 ISSN:
 10298479
 Sponsoring Org:
 National Science Foundation
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A bstract We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator J over a null plane. Local operators with weight n ≥ 2 under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight n ≥ 2 can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of J over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into halfspaces.

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[2,2]$ and $H_0$ does not have eigenvalues embedded into $(2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } nV(n)=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+n}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, andmore »

Abstract
Site description. This data package consists of data obtained from sampling surface soil (the 07.6 cm depth profile) in black mangrove (Avicennia germinans) dominated forest and black needlerush (Juncus roemerianus) saltmarsh along the Gulf of Mexico coastline in peninsular westcentral Florida, USA. This location has a subtropical climate with mean daily temperatures ranging from 15.4 °C in January to 27.8 °C in August, and annual precipitation of 1336 mm. Precipitation falls as rain primarily between June and September. Tides are semidiurnal, with 0.57 m median amplitudes during the year preceding sampling (U.S. NOAA National Ocean Service, Clearwater Beach, Florida, station 8726724). Sealevel rise is 4.0 ± 0.6 mm per year (19732020 trend, mean ± 95 % confidence interval, NOAA NOS Clearwater Beach station). The A. germinans mangrove zone is either adjacent to water or fringed on the seaward side by a narrow band of red mangrove (Rhizophora mangle). A nearmonoculture of J. roemerianus is often adjacent to and immediately landward of the A. germinans zone. The transition from the mangrove to the J. roemerianus zone is variable in our study area. An abrupt edge between closedcanopy mangrove and J. roemerianus monoculture may extend for up to several hundred meters 
We study the onedimensional discrete Schr\"odinger operator with the skewshift potential $2\lambda \cos\2π(j^2\omega+jy+x)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\lambda>0$. In this paper, we introduce a novel perturbative approach for studying the zeroenergy Lyapunov exponent $L(\lambda)$ at small $\lambda$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(\lambda)$ is fully consistent with $L(\lambda)$ being positive and satisfying the usual FigotinPastur type asymptotics $L(\lambda)\sim C\lambda^2$ as $\lambda\to 0$. The analogous quantity behaves completely differently in the AlmostMathieu model, whose zeroenergy Lyapunov exponent vanishes for $\lambda<1$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.

Metalmediated crosscoupling reactions offer organic chemists a wide array of stereo and chemicallyselective reactions with broad applications in fine chemical and pharmaceutical synthesis.1 Current batchbased synthesis methods are beginning to be replaced with flow chemistry strategies to take advantage of the improved consistency and process control methods offered by continuous flow systems.2,3 Most crosscoupling chemistries still encounter several issues in flow using homogeneous catalysis, including expensive catalyst recovery and air sensitivity due to the chemical nature of the catalyst ligands.1 To mitigate some of these issues, a ligandfree heterogeneous catalysis reaction was developed using palladium (Pd) loaded into a polymeric network of a silicone elastomer, poly(hydromethylsiloxane) (PHMS), that is not air sensitive and can be used with mild reaction solvents (ethanol and water).4 In this work we present a novel method of producing soft catalytic microparticles using a multiphase flowfocusing microreactor and demonstrate their application for continuous SuzukiMiyaura crosscoupling reactions. The catalytic microparticles are produced in a coaxial glass capillarybased 3D flowfocusing microreactor. The microreactor consists of two precursors, a crosslinking catalyst in toluene and a mixture of the PHMS polymer and a divinyl crosslinker. The dispersed phase containing the polymer, crosslinker, and crosslinking catalyst is continuously mixed and thenmore »