Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk—Monte Carlo Tests

Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map $z \to \sqrt z$ , then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.     

Anisotropic effects in surface acoustic wave propagation from a point source in a crystal

Strong anisotropic effects in the propagation of surface acoustic waves (SAWs) from a point-like source are studied experimentally and theoretically. Nanosecond SAW pulses are generated by focused laser pulses and detected with a cw probe laser beam at a large distance from the source compared to the SAW wavelength, which allows us to resolve fine intricate features in SAW wavefronts. In our theoretical model, we represent the laser excitation by a localized impulsive force acting on the sample surface and calculate the far-field surface response of an elastically anisotropic solid to such a force. The model simulates the measured SAW waveforms very well and accounts for all experimentally observed features. Using the data obtained for the (111) and (001) surfaces of GaAs, we describe a variety of effects encountered in the SAW propagation from a point source in crystals. The most interesting phenomenon is the existence of cuspidal structures in SAW wavefronts resulting in multiple SAW arrivals for certain ranges of the observation angle. Cuspidal edges correspond to the phonon focusing directions yielding sharp peaks in the SAW amplitude. A finite SAW wavelength results in internal diffraction whereby the SAW wavefront spreads beyond the group velocity cusps. Degeneration of a SAW into a transverse bulk wave is another strong effect influencing the anisotropy of the SAW amplitude and making whole sections of the SAW wavefront including some phonon focusing directions unobservable in the experiment. The propagation of a leaky SAW mode (pseudo-SAW) is affected by a specific additional effect i.e. anisotropic attenuation. We also demonstrate that many of the discussed features are reproduced in powder patterns, a simple technique developed by us earlier for visualization of SAW amplitude anisotropy.Received: 17 June 2003, Published online: 15 October 2003PACS: 43.35.+d Ultrasonics, quantum acoustics, and physical effects of sound - 68.35.Gy Mechanical properties; surface strains - 62.65.+k Acoustical properties of solidsA.M. Lomonosov: On leave from the General Physics Institute, 117942 Moscow, Russia     

Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk

The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with kappa = 8/3 leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents but also probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE(8/3).     

A Fast Algorithm for Simulating the Chordal Schramm–Loewner Evolution

The Schramm–Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N ^p ) with p<1. Simulations with κ=8/3 and κ=6 both give a value of p of approximately 0.4.     

Modulation of the electronic properties of GaAs quantum wells induced by surface acoustic waves

We investigated the modulation of the optical properties of GaAs-based structures by a surface acoustic wave (SAW) by microscopic measurements of reflectance and photoluminescence. We demonstrate that for photon energies away from electronic resonances, the modulation of the optical properties is associated with the strain field of the SAW (elastooptical mechanism). Close to the E₀ resonance, the electrooptical modulation due to the SAW piezoelectric fields becomes important and leads to a spatial modulation of the photoluminescence intensity.     

Self-avoiding walk model for proteins

We study a model for the backbone of proteins on a square lattice which consists of the path traced out by a self-avoiding walk (SAW) on the lattice and bridges not belonging to sites on the SAW but connecting nearest neighbor sites of the SAW. We calculated the fractal dimensiond _w for random walk on this model and found thatd _w2.6, in disagreement with a recent suggestion thatd _w should be 2.     

Computing the Loewner Driving Process of Random Curves in the Half Plane

We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N ^1.35) rather than the usual O(N ²), where N is the number of points on the curve.     

On Conformal Field Theory of SLE(κ,ρ)

SLE(κ ρ), a generalization of chordal Schramm-Löwner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = $\alpha_j = \frac{\rho_j}{2 \sqrt{\kappa}}SLE(κ ρ), a generalization of chordal Schramm-Löwner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = $\alpha_j = \frac{\rho_j}{2 \sqrt{\kappa}}SLE(κ ρ), a generalization of chordal Schramm-L?wner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = . The total charge vanishes and therefore the partition function has a simple product form. We also suggest a generalization of SLE(κ ρ)     

Bridge Decomposition of Restriction Measures

Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Itô’s excursion decomposition of a Brownian motion according to its zeros.     

Effect of melting on the excitation of surface acoustic wave pulses by UV nanosecond laser pulses in silicon

The influence of melting on the excitation of Surface Acoustic Wave (SAW) pulses in silicon is studied both theoretically and experimentally. The developed theory of Rayleigh-type SAW laser-induced thermoelastic excitation in a structure composed of a liquid layer on a solid substrate predicts that the SAW is predominantly generated in the solid phase due to the absence of shear rigidity in a liquid. The characteristic changes in the SAW pulse shape as well as the saturation and even the decrease of the SAW pulse amplitude observed above the melting threshold are explained theoretically to be a result of the decrease of the heat flux into the solid phase as well as due to the decrease of the volume of the solid phase caused by melting. Although the heat flux into the solid phase is decreased both as a consequence of the reflectivity increase and the additional energy losses (latent heat of melting) at the phase transition, it is demonstrated that the influence of reflectivity changes on the SAW pulse is negligible in comparison with the effect of melt-front motion. For laser pulses of 7 ns duration at 355 nm, the threshold value of laser fluence for meltingF _m=0.23±0.04 J/cm² and for the ablationF _a=1.3±0.2 J/cm² were determined experimentally as the points of characteristic changes in the observed SAW pulses.     

Restriction Properties of Annulus SLE

For κ∈(0,4], a family of annulus SLE(κ;Λ) processes were introduced in (Zhan in arXiv:) to prove the reversibility of whole-plane SLE(κ). In this paper we prove that those annulus SLE(κ;Λ) processes satisfy a restriction property, which is similar to that for chordal SLE(κ). Using this property, we construct n≥2 curves crossing an annulus such that, when any n−1 curves are given, the last curve is a chordal SLE(κ) trace.     

X-ray diffraction analysis of the surface acoustic wave propagation in langatate crystal

X-ray diffraction on a langatate crystal (La₃Ga_5.5Ta_0.5O₁₄, LGT) modulated by a Λ=12 μm Rayleigh surface acoustic wave (SAW) was studied in a double axis X-ray diffractometer scheme at the BESSY synchrotron radiation source. SAW propagation in the crystal causes sinusoidal modulation of the crystal lattice and the appearance of diffraction satellites on the rocking curves, with their number, angular positions, and intensities depending on the wavelength and amplitude of acoustic vibrations of the crystal lattice. Strong absorption of X-ray radiation in LGT enables the observation of the diffraction spectra extinction at certain SAW amplitudes. X-ray diffraction spectra analysis makes it possible to determine SAW amplitudes and wavelengths, to measure the power flow angles, and investigate the diffraction divergence in acoustic beam in LGT.     

Transforming Fixed-Length Self-avoiding Walks into Radial SLE<Subscript>8/3</Subscript>

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE_8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE_8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE_8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.     

Effects of Mn‐doping on surface acoustic wave properties of ZnO films

Surface acoustic wave (SAW) filters based on Mn‐doped ZnO films have been fabricated and effects of Mn‐doping on SAW properties are investigated. It is found that the electromechanical coupling coefficient (K²) of Zn_0.913Mn_0.087O films is 0.73 ± 0.02%, which is 73.8% larger than that of undoped ZnO films (0.42 ± 0.02%). Zn_0.913Mn_0.087O film filters also exhibit a lower absolute value of insertion loss (|IL|) of 16.1 dB and larger bandwidth (BW) of 5.9 MHz compared with that of undoped ZnO film filter. However, Zn_0.952Mn_0.048O film filters exhibit a smaller K² of 0.34 ± 0.02%, larger |IL| of 26.9 dB and smaller BW of 3.5 MHz. It is suggested that the SAW properties can be improved by appropriate Mn‐doping and Mn–ZnO/Si multilayer structure with large d₃₃ is promising for wide‐band and low‐loss SAW applications. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear acoustoelectronic interaction in thin superconducting lead films

The effect of high-power surface acoustic waves (SAW) on the superconducting state of Pb films has been studied in different transverse magnetic fields. It was established that a high-intensity sound wave affects the T _c of the film and the character of the superconducting transition, and that the observed changes are the larger, the higher is the SAW intensity. It was found that high-power SAW are capable of inducing vortex depinning in the film and reducing the critical current. Various mechanisms which are responsible for the nature of these effects are discussed. Fiz. Tverd. Tela (St. Petersburg) 39, 1753–1760 (October 1997)     

Identification of the Stress-Energy Tensor Through Conformal Restriction in SLE and Related Processes

We derive the Ward identities of Conformal Field Theory (CFT) within the framework of Schramm-Loewner Evolution (SLE) and some related processes. This result, inspired by the observation that particular events of SLE have the correct physical spin and scaling dimension, and proved through the conformal restriction property, leads to the identification of some probabilities with correlation functions involving the bulk stress-energy tensor. Being based on conformal restriction, the derivation holds for SLE only at the value κ = 8/3, which corresponds to the central charge c = 0 and the case when loops are suppressed in the corresponding O(n) model.     

声表面波MIM隧道发光二极管

我们在研究MIM隧道发光结(金属-绝缘层-金属)的过程中,首次以表面声波代替表面随机粗糙度的作用,将MIM结制作在声表面波场中,结果使结的发光效率、发光强度、稳定性、均匀性都有改善.本文介绍了结的基本结构,基本工艺,阐明了结的发光机理,讨论了结的Ⅰ-Ⅴ特性曲线和光谱曲线,根据结具有显著的负阻现象,我们设计出了一种新型的开关器件.     

Thermodynamics of the Casimir effect

A complete thermodynamic treatment of the Casimir effect is presented. Explicit expressions for the free and the internal energy, the entropy and the pressure are discussed. As an example we consider the Casimir effect with different temperatures between the plates (T) resp. outside of them (T'). For T'<T the pressure of heat radiation can eventually compensate the Casimir force and the total pressure can vanish. We consider both an isothermal and an adiabatic treatment of the interior region. The equilibrium point (vanishing pressure) turns out instable in the isothermal case. In the adiabatic situation we have both an instable and a stable equilibrium point, if T'/T is sufficiently small. Quantitative aspects are briefly discussed. Received 24 February 1999 and Received in final form 26 April 1999     

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We find=2.63820 ± 0.00004 ± 0.00030=1.352 ± 0.006 ± 0.025v=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.