On the Dynamical Stability and Instability of Parker Problem

We investigate a perturbation problem for the three-dimensional compressible isentropic viscous magnetohydrodynamic system with zero resistivity in the presence of a modified gravitational force in a vertical strip domain in which the velocity of the fluid is non-slip on the boundary, and focus on the stabilizing effect of the (equilibrium) magnetic field through the non-slip boundary condition. We show that there is a discriminant $\Xi$, depending on the known physical parameters, for the stability/instability of the perturbation problem. More precisely, if $\Xi<0$, then the perturbation problem is unstable, i.e., the Parker instability occurs, while if $\Xi>0$ and the initial perturbation satisfies some relations, then there exists a global (perturbation) solution which decays algebraically to zero in time, i.e., the Parker instability does not happen. The stability results in this paper reveal the stabilizing effect of the magnetic field through the non-slip boundary condition and the importance of boundary conditions upon the Parker instability, and demonstrate that a sufficiently strong magnetic field can prevent the Parker instability from occurring. In addition, based on the instability results, we further rigorously verify the Parker instability under Schwarzschild's or Tserkovnikov's instability conditions in the sense of Hadamard for a horizontally periodic domain.Comment: 51 page

On Linear Instability and Stability of the Rayleigh-Taylor Problem in Magnetohydrodynamics

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh-Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving \emph{a priori} estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field.Comment: 33 page

On the Inhibition of Thermal Convection by a Magnetic Field under Zero Resistivity

We investigate the stability and instability of the magnetic Rayleigh--B\'enard problem with zero resistivity. An stability criterion is established, under which the magnetic B\'enard problem is stable. The proof mainly is based on a three-layers energy method and an idea of magnetic inhibition mechanism. The stable result first mathematically verifies Chandrasekhar's assertion in 1955 that the thermal instability can be inhibited by strong magnetic field in magnetohydrodynamic (MHD) fluid with zero resistivity (based on a linearized steady magnetic B\'enard equations). In addition, we also provide an instability criterion, under which the magnetic Rayleigh--B\'enard problem is unstable. The proof mainly is based on the bootstrap instability method by further developing new analysis technique. Our instability result presents that the thermal instability occurs for a small magnetic field.Comment: 4

Ground State Properties of Cold Bosonic Atoms At Large Scattering Lengths

In this letter, we study bosonic atoms at large scattering lengths using a variational method where the condensation amplitude is a variational parameter. We further examine momentum distribution functions, chemical potentials and speed of sound, and spatial density profiles of cold bosonic atoms in a trap in this limit. The later two properties turn out to bear similarities of those of Fermi gases. Estimates obtained here are applicable near Feshbach resonances, particularly when the fraction of atoms forming three-body structures is small and can be tested in future cold atom experiments.Comment: 4 pages, 3 figures, published versio

Probabilistic Topic and Syntax Modeling with Part-of-Speech LDA

This article presents a probabilistic generative model for text based on semantic topics and syntactic classes called Part-of-Speech LDA (POSLDA). POSLDA simultaneously uncovers short-range syntactic patterns (syntax) and long-range semantic patterns (topics) that exist in document collections. This results in word distributions that are specific to both topics (sports, education, ...) and parts-of-speech (nouns, verbs, ...). For example, multinomial distributions over words are uncovered that can be understood as "nouns about weather" or "verbs about law". We describe the model and an approximate inference algorithm and then demonstrate the quality of the learned topics both qualitatively and quantitatively. Then, we discuss an NLP application where the output of POSLDA can lead to strong improvements in quality: unsupervised part-of-speech tagging. We describe algorithms for this task that make use of POSLDA-learned distributions that result in improved performance beyond the state of the art.Comment: Currently under review for the journal Computational Linguistic

Non-Hermitian skin effect and chiral damping in open quantum systems

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary condition but exponential under open boundary condition. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wavefront. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.Comment: 9 pages, 4 figures, including supplemental materia

Non-Hermitian Chern bands

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.Comment: 11 pages, 12 figures, including supplemental materia

Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the "fluid region", as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the "fluid region".Comment: 19 page

Nonlinear Rayleigh-Taylor Instability for Nonhomogeneous Incompressible Viscous Magnetohydrodynamic Flows

We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.Comment: 46 pages. arXiv admin note: substantial text overlap with arXiv:1205.227

Majorana corner modes in a high-temperature platform

We introduce two-dimensional topological insulators in proximity to high-temperature cuprate or iron-based superconductors as high-temperature platforms of Majorana Kramers pairs of zero modes. The proximity-induced pairing at the helical edge state of the topological insulator serves as a Dirac mass, whose sign changes at the sample corner because of the pairing symmetry of high-$T_c$ superconductors. This sign changing naturally creates at each corner a pair of Majorana zero modes protected by time-reversal symmetry. Conceptually, this is a topologically trivial superconductor-based approach for Majorana zero modes. We provide quantitative criteria and suggest candidate materials for this proposal.Comment: 15 pages, 9 figures, including supplemental material. Published versio