Periodic Minimizers in 1D Local Mean Field Theory

There are not many physical systems where it is possible to demonstate rigorously that energy minimizers are periodic. Using reflection positivity techniques we prove, for a class of mesoscopic free-energies representing 1D systems with competing interactions, that all minimizers are either periodic, with zero average, or of constant sign. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.     

Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems

The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.     

Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions

We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to a sharp interface energy with screened Coulomb interaction. Near the onset the minimizers of the sharp interface energy consist of nearly identical circular droplets of small size separated by large distances. In the limit the droplets become uniformly distributed throughout the domain. The precise asymptotic limits of the bifurcation threshold, the minimal energy, the droplet radii, and the droplet density are obtained.     

Effect of Local Magnetic Field in G Measurement with Time-of-Swing Method

The effect of the local time- varying magnetic field in our G measurement with the time-of-swing method is studied by magnifying the magnetic field to cause a perceptible change in the pendulum＇s period. The experimental result shows that the coefficients of the change in the period to the magnetic field are 37（1） and 12（1） ms/gauss in the two horizontal directions respectively, which means that the systematic uncertainty due to the local magnetic field is less than 0.4ppm in our G measurement.     

On the Third Critical Field in Ginzburg-Landau Theory

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, , describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables.The two authors are supported by the European Research Network ‘Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems’ with contract number HPRN-CT-2002-00277, and the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). Part of this work was carried out while S.F. visited CIMAT, Mexico.     

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain TT _c , H0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang–Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose extended analyticity; roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated extended dispersion relation.     

Froth-like Minimizers of a Non-Local Free Energy Functional with Competing Interactions

We investigate the ground and low energy states of a one dimensional non-local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasi-optimal profiles are “froth-like”: if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasi-optimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective “sharp interface” one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.     

运动磁场中的力与能量问题探讨

运用电动力学的规律,分析运动磁场中的电磁感应问题,探讨自由电子所受的洛伦兹力与涡旋电场力在提供非静电力时的作用,总结了运动磁场中功能关系的特点.     

Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group ₃(S²)=. We prove in the full space situation that there exists an infinite subset of such that for any m, the Faddeev energy, E, has a minimizer among the topological class Q=m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for =. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality EC|Q|^3/4, where C>0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.     

Multi-Vortex Non-radial Solutions to the Magnetic Ginzburg-Landau Equations

We show that there exists multi-vortex, non-radial, finite energy solutions to the magnetic Ginzburg-Landau equations on all of ${\mathbb{R}^2}$ . We use Lyapunov-Schmidt reduction to construct solutions which are invariant under rotations by ${\frac{2 \pi}{k}}$ (but not by rotations in O(2) in general) and reflections in the x? axis for some k ≥ 7.     

Magnetic Resonance of Impurity Spin in Slow Fluctuating Local Field

A basic model of magnetic resonance is considered. The model takes into account external static and orthogonal to it rotating magnetic fields together with fluctuating (local) field directed along the static field. The local field is considered as smooth normal stochastic process. New solutions for longitudinal relaxation are obtained in the region of adiabatic slow fluctuations and nonadiabatic losses are estimated.     

Spin Filter Effect in Organic Polymers in the Presence of Local Magnetic Field

Using a nonadiabatic evolution method, we investigate the spin filter effect in organic polymers in the presence of a local magnetic field. Through a spin-dependent magnetic field, polarons （charge carrier） with different spins will feel repulsive or attractive force determined by their spins. Our simulations show that in a single-site magnetic field （affecting electrons at a single site）, for example V150 = 0.35 eV, or V150 = 0.45 eV, a spin-up polaron accelerated to saturated velocity by an electric field can pass through the field while the spin-down polaron is trapped. When the local field extends over several sites （Vn～ exp[-（n - nc）2/nw^2]）, similar behaviour is also found. Simultaneously we find that it is more likely to realize the spin filter effect in a comparatively large field since the polaron which feels attractive force is easily trapped by a local magnetic field.     

微扰法计算中等强磁场中氢原子的能级

本文在考虑氢原子轨道运动磁矩与磁场之间、自旋磁矩与磁场之间和感生磁矩与外磁场之间的相互作用的基础上,根据角动量和球谐函数的性质,应用简并态微扰方法研究了在中等强磁场中氢原子的能级,给出了计算中等强磁场中氢原子的一级近似能级的方法,具体计算了 T范围内氢原子 的各能级的数值,结果与有关文献给出的理论计算值是相近的,表明本文所给出的方法是简单的、计算结果是正确的。     

磁场中多原子晶体中极化子的激发能量

研究磁场中多原子晶体中极化子的激发态的性质。采用线性组合算符和幺正变换方法。分别导出强、弱耦合情形下磁极化子的激发能量和平均声子数。结果表明，磁极化子的激发能量和平均声子效不仅包括不同支LO声子与电子耦合的贡献，而且还存在不同支LO声子间相互作用所贡献的附加项。     

反向导引场自由电子激光器的三维非线性模拟

 对Conde–Bekefi反向导引场自由电子激光（FEL）放大器实验进行了三维非线性分析。当引入一类似于回旋自谐振脉塞收缩角(pinch angle)参数的电子束入射角参数后，其模拟计算结果与实验相符。这预示了反向导引场FEL中束波互作用的一种新机理。     

Microcanonical Analysis of the Random Energy Model in a Random Magnetic Field

We study the spin glass system consisting of a Random Energy Model coupled with a random magnetic field. This system was investigated by de Oliveira Filho et al. (Phys Rev E 74:031117, 2006) who computed the free energy. In this paper, we recover their result rigorously using elementary large deviations arguments and a conditional second moment method. Our analysis extends at the level of fluctuations of the ground states. In particular, we prove that the joint distribution of the minimal energies has the law of a Poisson process with exponential density after a recentering, which is random as opposed to the standard REM. One consequence is that the Gibbs measure of the model exhibits a one-step replica symmetry breaking as argued by de Oliveira Filho et al. using the replica method.     

Modeling Spectral Properties in Stationary Processes of Varying Dimensions with Applications to Brain Local Field Potential Signals

In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, a frequency specific spectral ratio (FS-ratio) statistic is proposed and its asymptotic properties are derived. The FS-ratio is blind to the dimension of the stationary process and captures the proportion of spectral power in various frequency bands. Here we develop a technique to automatically identify frequency bands that carry significant spectral power. We apply our method to track changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. At every epoch (a distinct time segment from the duration of the experiment), the nonstationary LFP signal is decomposed into stationary and nonstationary latent sources and the complexity is analyzed through these latent stationary sources and their dimensions that can change across epochs. The analysis indicates that spectral information in the Beta frequency band (12–30 Hertz) demonstrated the greatest change in structure and complexity due to the stroke.     

NMR in Magnetic Field of the Earth: Pre-Polarization of Nuclei with Alternating Magnetic Field

The polarization of nuclei in the low static magnetic field \(B_0\) with an alternating magnetic field \(B^{*} (B^{*} \gg B_0)\) at a very low frequency \(f_m\) (but \(f_m\gg 1\) / \({T_1}\) , where \(T_1\) is the spin-lattice relaxation time) has been investigated. The question of the optimization of the energy consumption during the pre-polarization is also considered. The possibilities of the method are illustrated by the observation of nuclear magnetic resonance signals from a few liquids.     

Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions

We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that interactions up to ? ⁶-tensorial type are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning properties of melonic graphs, the second order expansion of melonic two point subgraphs needed for wave-function renormalization.