Lectures on renormalization and asymptotic safety

A short introduction is given on the functional renormalization group method, putting emphasis on its nonperturbative aspects. The method enables to find nontrivial fixed points in quantum field theoretic models which make them free from divergences and leads to the concept of asymptotic safety. It can be considered as a generalization of the asymptotic freedom which plays a key role in the perturbative renormalization. We summarize and give a short discussion of some important models, which are asymptotically safe such as the Gross–Neveu model, the nonlinear σ$\sigma$ model, the sine–Gordon model, and we consider the model of quantum Einstein gravity which seems to show asymptotic safety, too. We also give a detailed analysis of infrared behavior of such scalar models where a spontaneous symmetry breaking takes place. The deep infrared behavior of the broken phase cannot be treated within the framework of perturbative calculations. We demonstrate that there exists an infrared fixed point in the broken phase which creates a new scaling regime there, however its structure is hidden by the singularity of the renormalization group equations. The theory spaces of these models show several similar properties, namely the models have the same phase and fixed point structure. The quantum Einstein gravity also exhibits similarities when considering the global aspects of its theory space since the appearing two phases there show analogies with the symmetric and the broken phases of the scalar models. These results be nicely uncovered by the functional renormalization group method.     

Constraints on an asymptotic safety scenario for the Wess–Zumino model

Using the nonrenormalization theorem and Pohlmeyer's theorem, it is proven that there cannot be an asymptotic safety scenario for the Wess–Zumino model unless there exists a non-trivial fixed point with (i) a negative anomalous dimension (ii) a relevant direction belonging to the Kähler potential.     

On the behavior of the surface tension for spin systems in a correlated porous medium

We investigate the behavior of various spin-systems that are subject to the highly correlated and extremely diluted quenched disorder as provided by the fractal aerogel model. For these systems, it is (easily) established that, at all temperatures, the free energy is identical to that of the corresponding uniform system. The surface tension, however, behaves quite differently. Foremost, at any fixed temperature corresponding to the low temperature phase in the uniform system, there is a non-trivial curve in the aerogel phase plane dividing high-temperature behavior (zero surface tension) from low-temperature behavior (positive surface tension). The fractal aerogel has two distinctive phases in its own right: gel and sol. In the gel phase, the spin system has zero surface tension at all temperatures. In one region of the sol phase, the surface tension is shown to be equal to its value in the uniform system. Since part of this region borders on the gel phase, a certain portion of the sol/gel phase boundary is also the dividing line between high- and low-temperature behavior. Evidently, in this case, the surface tension is discontinuous at the phase boundary. on the other hand, there are well-defined length scales that diverge as the phase boundary is approached. This demonstrates an absence of scaling in these systems.     

Wave radiation in lattice fracture

The square and triangular lattices are considered, where the uniform crack growth is accompanied by the wave radiation. The radiation energy and structure are studied. The energy radiated to the bulk of the lattice is found in a direct way. The radiation structure is described based on the crack problem solution and by means of the analysis of two-dimensional dispersion relations for the intact lattice. The mode III problem for square lattice is discussed in detail, whereas, in the case of the plane problem for the triangular lattice, the only those results are derived which follow from the two-dimensional dispersion relations. It is shown that there exists a finite crack-speed-dependent region of wavenumbers corresponding to the waves radiated to the bulk of the lattice. In the case of the triangular-cell lattice, in addition, one or several lattice Rayleigh waves are radiated. For the square lattice a complete solution for the wave field is presented with the crack-speed-dependent far-field asymptote. The latter is characterized by the wave amplitude asymptotically decreasing as the distance from the crack front in power −1/3. The asymptotically significant crack-speed-dependent direction of the radiation is determined. Such asymptotic results are also valid for the triangular lattice.     

Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.     

Vortex deformation and breaking in superconductors: a microscopic description

Vortex breaking has traditionally been studied for non-uniform critical current densities, although it may also appear due to non-uniform pinning force distributions. In this article we study the case of a high-pinning/low-pinning/high-pinning layered structure. We have developed an elastic model for describing the deformation of a vortex in these systems in the presence of a uniform transport current density J for any arbitrary orientation of the transport current and the magnetic field. If J is above a certain critical value, J_c , the vortex breaks and a finite effective resistance appears. Our model can be applied to some experimental configurations where vortex breaking naturally exists. This is the case for YBa₂Cu₃O_7?δ (YBCO) low-angle grain boundaries and films on vicinal substrates, where the breaking is experienced by Abrikosov–Josephson vortices (AJV) and Josephson string vortices (SV), respectively. With our model, we have experimentally extracted some intrinsic parameters of the AJV and SV, such as the line tension ? _l and compared it to existing predictions based on the vortex structure.     

Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction–diffusion systems

In bistable systems, the long-term behavior of solutions depends on the location of the initial conditions. In a deterministic setting, where the initial condition is kept fixed in one particular basin of attraction, repeated numerical simulations will always lead to the same long-term behavior. The other possible asymptotic solution type will never be observed. This clear distinction does not hold anymore if the system is forced by random fluctuations. In this case, both asymptotic solutions can be attained, and the relative frequency of different long-term behaviors observed in many repeated simulation runs will follow a certain probability distribution. We present a simple reaction–diffusion model of spatial predator–prey interaction, where depending on the initial spatial distribution of the two populations either spatially homogeneous or spatiotemporal irregular oscillations may be observed. We show by repeated stochastic simulations that, when starting in the basin of attraction of the spatiotemporal irregular solution, in the randomly forced system the probability to observe spatially homogeneous oscillations instead of spatiotemporally irregular oscillations follows a non-trivial bimodal distribution.     

非均匀采样的功率谱反演大气湍流相位屏的快速模拟

对大气湍流功率谱非均匀采样可以有效改善传统功率谱反演法低频采样严重不足的缺陷, 实现高精度的大气湍流相位屏的模拟. 但采用的直接求和运算计算复杂度高, 相位屏的模拟速度极慢. 将非均匀快速傅里叶变换(NUFFT)引入到大气湍流相位屏的模拟, 可以实现相位屏的快速模拟. 从随机过程的谱分解出发, 将大气湍流相位随机过程表示为有限谐波分量叠加和的均方极限. 通过一个高斯核函数的卷积, 将非均匀分布的谐波复振幅映射到均匀网格空间, 进而利用快速傅里叶变换, 降低计算复杂度, 加快大气湍流相位屏的模拟速度. 以大气湍流的Kolmogorov 谱为例, 利用NUFFT仿真得到大气湍流相位屏, 并对相位屏的模拟精度、模拟速度和误差进行统计分析. 结果表明, NUFFT的引入可以实现快速、高精度的大气湍流相位屏的模拟.     

Bound states and scattering resonances induced by spatially modulated interactions

We study the two-body problem with a spatially modulated interaction potential using a two-channel model, in which the interchannel coupling is provided by an optical standing wave and its strength modulates periodically in space. As the modulation amplitudes increase, there will appear a sequence of bound states. Part of them will cause a divergence of the effective scattering length, defined through the phase shift in the asymptotic behavior of scattering states. We also discuss how the local scattering length, defined through short-range behavior of scattering states, modulates spatially in different regimes. These results provide a theoretical guideline for a new control technique in the cold atom toolbox, in particular, for alkaline-earth(-like) atoms where the inelastic loss is small.     

Evolution of paramagnetic quasiparticle excitations emerged in the high-field superconducting phase of CeCoIn5

We present (115)In NMR measurements in a novel thermodynamic phase of CeCoIn(5) in a high magnetic field, where exotic superconductivity exists with the incommensurate spin-density wave order. We show that the NMR spectra in this phase provide direct evidence for the emergence of the spatially distributed normal quasiparticle regions. The quantitative analysis for the field evolution of the paramagnetic magnetization and newly emerged low-energy quasiparticle density of states is consistent with the nodal plane formation, which is characterized by an order parameter in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The NMR spectra also suggest that the spatially uniform spin-density wave is induced in the FFLO phase.     

功率谱反演大气湍流随机相位屏采样方法的研究

从产生大气湍流随机相位屏的功率谱反演法原理出发,分析了均匀采样造成的随机相位屏大量低频信息泄漏的不足,提出利用非均匀采样方法对功率谱进行滤波产生随机相位屏.建立了非均匀采样模型,并分析比较了两种采样方法覆盖的采样频率范围和单边采样频率区域的湍流功率,理论证明了非均匀采样功率谱反演产生大气湍流随机相位屏的可行性和有效性.针对大气湍流的Kolmogorov谱,分别仿真计算了两种采样模型下功率谱反演生成的大气湍流随机相位屏.仿真结果表明,在不增加采样点和计算量条件下,非均匀采样方法生成的大气湍流随机相位屏具有丰富的低频和高频信息,有效改善了传统功率谱反演大气湍流随机相位屏时低频信息严重不足的缺陷.     

垂直圆管内向上稀疏层流泡状流相分布的实验研究

采用三维照相法对垂直圆管内稀疏层流泡状流充分发展段的相分布进行了实验研究。得到了８个流动工况下均匀尺寸气泡形成的泡状流的空泡率分布以及６个流动工况下非均匀尺寸气泡形成的泡状流的总体和大、小气泡组各自的空泡率分布．实验结果表明当气泡组的平均直径小于约３．５ｍｍ时,其空泡率分布在管壁附近出现尖峰;当气泡组的平均直径大于约 ３．５ ｍｍ时,其空泡率分布的尖峰移向管中心;气泡尺寸对泡状流的相分布有重要影响．     

Orientation effects accompanying the propagation of ultrarelativistic electrons through crystals

The results obtained by experimentally investigating the dynamics of the propagation of 1.2-GeV electrons through a thin silicon single crystal are discussed. The orientation dependences of electron scattering into a small solid angle, which are measured at various scattering angles, under conditions where the effects of crystallographic axes and planes manifest themselves are interpreted. It is shown that there are such electron-scattering directions for which the orientation “independence” of the scattering intensity with respect to a specific crystallographic plane is observed and that there also exists a crystal-axis orientation for which the region of a uniform angular distribution of the intensity is observed.     

Photo-Detached Electron Flux Distribution in a Gradient Electric Field

This paper investigates the flux distributions of the electron photo-detached from H-ion localized in a gradient electric field. In contrast with the photodetachment in the uniform electric field [Phys. Rev. A 40(1989) 4983],where only two electron trajectories interfere at each given point on a detector, for the photodetachment in a gradient electric field, the electrons waves can travel along multiple paths from the negative ion to a given point on the detector plane, which makes the electron flux distributions on the detector plane become much complex. Using the semi-classical theory, we put forward a formula for calculating the electron flux. Our calculation results suggest that the electron flux distributions on a given detector plane is not only related to the propagation time of the detached electron, but also related to the detached electron's energy. With the increase of the detached electron's energy, the oscillating region in the electron flux distributions becomes enlarged and the oscillating structure in the flux distributions becomes much more complicated. This study will guide future experiment research on the photodetachment microscopy of the negative ions in the presence of non-uniform external fields.     

Asymptotic scaling laws for imploding thin fluid shells

Scaling laws governing implosions of thin shells in converging flows are established by analyzing the implosion trajectories in the (A,M) parametric plane, where A is the in-flight aspect ratio, and M is the implosion Mach number. Three asymptotic branches, corresponding to three implosion phases, are identified for each trajectory in the limit of A,M >1. It is shown that there exists a critical value gamma(cr) = 1+2/nu (nu = 1,2 for, respectively, cylindrical and spherical flows) of the adiabatic index gamma, which separates two qualitatively different patterns of the density buildup in the last phase of implosion. The scaling of the stagnation density rho(s) and pressure P(s) with the peak value M(0) of the Mach number is obtained.     

Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells

We investigate the polarization structure of coherent light, produced by a convergent light beam transmitted through nematic liquid crystal (NLC) cells with different director configurations. Employing solutions to the transmission problem for the case when plane wave propagates through an anisotropic layer, we analyze the arrangement of the topological elements, such as polarization singularities (C points with circular polarization and L lines with linear polarization), saddle points and extrema of polarization azimuth. We observe transformations of the topological structure under the variation of the incident light ellipticity and represent it by corresponding trajectories of topological elements in three-dimensional space. For the cells with uniform and non-uniform director configuration we describe the processes of creation/annihilation of C point pairs, which can be controlled precisely in the case of the cell with non-uniform director. Our experimental measurements for the homeotropically oriented NLC cells are in good agreement with the theoretical predictions.     

Taming perturbative divergences in asymptotically safe gravity

We use functional renormalization group methods to study gravity minimally coupled to a free scalar field. This setup provides the prototype of a gravitational theory which is perturbatively non-renormalizable at one-loop level, but may possess a non-trivial renormalization group fixed point controlling its UV behavior. We show that such a fixed point indeed exists within the truncations considered, lending strong support to the conjectured asymptotic safety of the theory. In particular, we demonstrate that the counterterms responsible for its perturbative non-renormalizability have no qualitative effect on this feature.