Anisotropic Scaling Research Articles

Entanglement entropy of excited states in the quantum Lifshitz model

In this work we calculate the entanglement entropy of certain excited states of the quantum Lifshitz model (QLM). The QLM is a 2 + 1-dimensional bosonic quantum field theory with an anisotropic scaling symmetry between space and time that belongs to the universality class of the quantum dimer model and its generalizations. The states we consider are constructed by exciting the eigenmodes of the Laplace–Beltrami operator on the spatial manifold of the model. We perform a replica calculation and find that, whenever a simple assumption is satisfied, the bipartite entanglement entropy of any such excited state can be evaluated analytically. We show that the assumption is satisfied for all excited states on the rectangle and for almost all excited states on the sphere and provide explicit examples in both geometries. We find that the excited state entanglement entropy obeys an area law and is related to the entanglement entropy of the ground state by two universal constants. We observe a logarithmic dependence on the excitation number when all excitations are put onto the same eigenmode.

Quantum critical scaling and holographic bound for transport coefficients near Lifshitz points

The transport behavior of strongly anisotropic systems is significantly richer compared to isotropic ones. The most dramatic spatial anisotropy at a critical point occurs at a Lifshitz transition, found in systems with merging Dirac or Weyl point or near the superconductor-insulator quantum phase transition. Previous work found that in these systems a famous conjecture on the existence of a lower bound for the ratio of a shear viscosity to entropy is violated, and proposed a generalization of this bound for anisotropic systems near charge neutrality involving the electric conductivities. The present study uses scaling arguments and the gauge-gravity duality to confirm the previous analysis of universal bounds in anisotropic Dirac systems. We investigate the strongly-coupled phase of quantum Lifshitz systems in a gravitational Einstein-Maxwell-dilaton model with a linear massless scalar which breaks translations in the boundary dual field theory and sources the anisotropy. The holographic computation demonstrates that some elements of the viscosity tensor can be related to the ratio of the electric conductivities through a simple geometric ratio of elements of the bulk metric evaluated at the horizon, and thus obey a generalized bound, while others violate it. From the IR critical geometry, we express the charge diffusion constants in terms of the square butterfly velocities. The proportionality factor turns out to be direction-independent, linear in the inverse temperature, and related to the critical exponents which parametrize the anisotropic scaling of the dual field theory.

On the regularization of Lifshitz-type field theories

We consider Lifshitz-type scalar theories with explicit breaking of the Lorentz symmetry that, in addition, exhibit anisotropic scaling laws near the ultraviolet fixed point. Using the proper time regularization method on the spatial coordinates only, we derive the regularized form of the one-loop effective potential in such theories. We study the main features of the one-loop effective potential and, also, the RG flow of the scale-dependent potential both in the IR and UV regimes. The beta functions for the couplings are derived.

Anisotropic scaling lengths of colloidal monolayers near a water–air interface

Abstract

Uniaxial compressive strength estimation based on the primary wave velocity in coal: considering scale effect and anisotropy

ABSTRACT Coal mass strength estimation has attracted broad attentions for decades. Investigating the effect of scale and anisotropy on the Uniaxial Compressive Strength (UCS) estimation via P-wave velocity is crucial in upscaling the lab-obtained correlation to coal mass strength evaluation. Microstructures were considered as the major impact factors for the scale difference and anisotropy of mechanical properties in coal. Thus, the microstructure correlated influence of specimen sizes and anisotropy angles on the UCS estimation based on the primary wave (P-wave) velocity were investigated here. Specimens with four diameters (25 mm to 75 mm) and six anisotropy angles (angles between bedding plane orientation and coring direction, 0° to 90°) were processed. Roles microstructures played for the P-wave propagation were explored by the X-ray computed tomography scanning and P-wave velocity tests. The UCS estimation via P-wave velocity in coal were investigated by regression analysis, significances of scale and anisotropy effect on this correlation were quantitatively evaluated by an introduced one-way analysis of variance method. P-wave velocity for specimens with different diameters exhibits a U-shaped variation against anisotropy angles, it reduces at anisotropy angles 0° ~ 45°, while raises within that of 45° ~ 90°, the minimum, maximum, and second largest value were acquired, respectively, at the anisotropy angle 45°, 90°, and 0°. This was considered caused by the complementary variation of angles between directions of P-wave propagation and orientations of bedding plane and major cleats. P-wave velocity wakens in larger specimens, velocity reductions are significantly at the anisotropy angle of 45°, while less obviously at 0° and 90°. While the anisotropy of P-wave velocity strengthens as diameters gain. Specimen sizes and anisotropy angles were proved to affect the UCS estimation via P-wave velocity by the one-way analysis of variance. Exponential correlation was verified more appropriate in the UCS estimation via P-wave velocity in coal, specimens with diameter 38 mm and anisotropy angle 45° were proved optimal for the UCS estimating by P-wave velocity in coal, since the correlation between them is stronger than other diameters and anisotropy angles. This research facilitates us in quickly and accurately strength estimating in anisotropy coal, meanwhile, it also provides references in extending the lab-obtained mechanical properties of coal/rock specimens to coal/rock mass.

Two-phase coupling for MMC-LES of spray combustion

A sparse-Lagrangian particle implementation of the multiple mapping conditioning / large-eddy simulation (MMC-LES) model for two-phase flows is developed with the aid of carrier-phase direct numerical simulation (CP-DNS) of a droplet-laden temporally evolving reacting double shear layer. In sparse-Lagrangian MMC for spray combustion, the liquid fuel droplets are represented by a first set of Lagrangian particles, while the reacting gas phase is described by a second set of (stochastic) particles. The two phases exchange heat and mass and a one-to-one coupling technique of liquid and gas particles with particle pair selection conditional on mixture fraction and/or temperature is introduced. The simulations show that unconditional mean and rms of mixture fraction are accurately reproduced by MMC and are largely independent of the particle coupling. Temperature, however, is significantly underpredicted when using the Eulerian mixture fraction for particle selection as is conventional for particle-particle selection in MMC. This is due to a lack of correlation between temperature and mixture fraction caused by excessive cooling of selected fluid elements due to evaporation and subsequent local flame extinction and reignition. Conditioning on temperature leads to mild improvements but does not prevent a decorrelation of the temperatures “seen” by the droplets. Alternatively, a minimization based on stochastic particle temperature is suggested. This leads to good agreement between our MMC-LES model and the CP-DNS. The comparison between the CP-DNS data and MMC-LES also confirms that the anisotropic mixing time scale derived for gaseous flows ensures an accurate degree of micro-mixing also for two-phase flows and provides very good predictions of conditional fluctuations of reacting scalars around the conditional means.

Scaling transition and edge effects for negatively dependent linear random fields on [formula omitted

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for a class of negatively dependent linear random fields X on Z2 with moving-average coefficients a(t,s) decaying as |t|−q1 and |s|−q2 in the horizontal and vertical directions, q1−1+q2−1<1 and satisfying ∑(t,s)∈Z2a(t,s)=0. The scaling limits are taken over rectangles whose sides increase as λ and λγ when λ→∞, for any γ>0. The scaling transition occurs at γ0X>0 if the scaling limits of X are different and do not depend on γ for γ>γ0X and γ<γ0X. We prove that the scaling transition in this model is closely related to the presence or absence of the edge effects. The paper extends the results in Pilipauskaitė and Surgailis (2017) on the scaling transition for a related class of random fields with long-range dependence.

Cross-correlation of the astrophysical gravitational-wave background with galaxy clustering

We investigate the correlation between the distribution of galaxies and the predicted gravitational-wave background of astrophysical origin. We show that the large angular scale anisotropies of this background are dominated by nearby non-linear structure, which depends on the notoriously hard to model galaxy power spectrum at small scales. In contrast, we report that the cross-correlation of this signal with galaxy catalogues depends only on linear scales and can be used to constrain the average contribution to the gravitational-wave background as a function of time. Using mock data based on a simplified model, we explore the effects of galaxy bias, angular resolution and the matter abundance on these constraints. Our results suggest that, when combined with galaxy surveys, the gravitational-wave background can be a powerful probe for both gravitational-wave merger physics and cosmology.

Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices

We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature $T_c$ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for suffciently large values of the scaling variable, and in consequence is not an analytical function. At $T > T_c$ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.

Towards a Unitary, Renormalizable, and Ultraviolet-Complete Quantum Theory of Gravity

For any fundamental quantum field theory, unitarity, renormalizability, and relativistic invariance are considered to be essential properties. Unitarity is inevitably connected to the probabilistic interpretation of the quantum theory, while renormalizability guarantees its completeness. Relativistic invariance, in turn, is a symmetry which derives from the structure of spacetime. So far, the perturbative attempt to formulate a fundamental local quantum field theory of gravity based on the metric field seems to be in conflict with at least one of these properties. In quantum Ho\v{r}ava gravity, a quantum Lifshitz field theory of gravity characterized by an anisotropic scaling between space and time, unitarity and renormalizability can be retained while Lorentz invariance is sacrificed at high energies and must emerge only as approximate symmetry at low energies. I review various approaches to perturbative quantum gravity with a particular focus on recent progress in the quantization of Ho\v{r}ava gravity, supporting its theoretical status as a unitary, renormalizable and ultraviolet-complete quantum theory of gravity.

Multiscale Characterizations of Surface Anisotropies.

Anisotropy can influence surface function and can be an indication of processing. These influences and indications include friction, wetting, and microwear. This article studies two methods for multiscale quantification and visualization of anisotropy. One uses multiscale curvature tensor analysis and shows anisotropy in horizontal coordinates i.e., topocentric. The other uses multiple bandpass filters (also known as sliding bandpass filters) applied prior to calculating anisotropy parameters, texture aspect ratios (Str) and texture directions (Std), showing anisotropy in horizontal directions only. Topographies were studied on two milled steel surfaces, one convex with an evident large scale, cylindrical form anisotropy, the other nominally flat with smaller scale anisotropies; a µEDMed surface, an example of an isotropic surface; and an additively manufactured surface with pillar-like features. Curvature tensors contain the two principal curvatures, i.e., maximum and minimum curvatures, which are orthogonal, and their directions, at each location. Principal directions are plotted for each calculated location on each surface, at each scale considered. Histograms in horizontal coordinates show altitude and azimuth angles of principal curvatures, elucidating dominant texture directions at each scale. Str and Std do not show vertical components, i.e., altitudes, of anisotropy. Changes of anisotropy with scale categorically failed to be detected by traditional characterization methods used conventionally. These multiscale methods show clearly in several representations that anisotropy changes with scale on actual surface measurements with markedly different anisotropies.

The breaking of geometric similarity

Scaled experimentation plays an important role in prototype and process development but is recognised to be severely constrained by the need for similitude, founded on the concepts of geometric, kinematic and kinetic similarity.This paper examines the possibility of breaking the requirement for geometric similarity by introducing the law of finite similitude for anisotropic scaling, which applies to continuum mechanics on anisotropically-scaled spaces. The law confirms that similarity solutions on skewed spaces always exist separately for quasistatic deformational and thermal-continuum problems. Thermomechanical and continuum dynamic problems however are shown to suffer from the inclusion of a (non-physical) non-orthogonal metric arising from the skewed coordinate system associated with anisotropic-space scaling. Even in this case however, an important subclass of problems is shown to be physically realisable involving a dominant component of velocity (displacement), where geometric similarity can again be broken yet retain good accuracy. The ability to skew artefacts yet achieve similitude is recognised to be a particularly important outcome as it allows for example one experimental model to be used to predict the behaviour of multiple skewed models.To showcase and highlight the significance of the new concept various scaled numerical models with anisotropic scaling with different geometrical scaling ratios in different directions is considered. The applicability of the theory is tested on a number of case studies providing strong supporting evidence for the validity and applicability of the new theory.

Thickness effect on pinning properties for GdBa2Cu3O7-x films deposited with RF sputtering

We studied the relationship between JC and H, direction and crystallization orientation of GBCO films grown by RF sputtering with different thickness which is 0.3 um, 0.5um and 0.9um respectively. By de-convoluting the random pinning from the correlated pinning contributions, we find that all the three films exhibit a lower effective anisotropy parameter γ = 3 rather than γ = 5 in YBCO films. The thinnest film is the most anisotropic. The anisotropic scaling analysis reveals an enhanced random pinning for thick film the whole range of angles studied in the angle correlation Jc curve. The Jc-thickness dependence analysis indicates that there is a certain threshold field Hd and a certain thickness d for magnetic decoupling.

Baryon-Interacting Dark Matter: heating dark matter and the emergence of galaxy scaling relations

The empirical scaling relations observed in disk galaxies remain challenging for models of galaxy formation. The most striking among these is the Mass Discrepancy-Acceleration Relation (MDAR), which encodes both a tight baryonic Tully-Fisher relation (BTFR) and the observed diversity of galaxy rotation curves through the central surface density relation (CSDR) . Building on our earlier work [1], we propose here that the MDAR is the result of interactions between baryons and `Baryon-Interacting Dark Matter' (BIDM), which heat up the dark matter. Following a bottom-up, hydrodynamical approach, we find that the MDAR follows if: i) the BIDM equation of state approximates that of an ideal gas; ii) the BIDM relaxation time is order the Jeans time; iii) the heating rate is inversely proportional to the BIDM density. Remarkably, under these assumptions the set of hydrodynamical equations together with Poisson's equation enjoy an anisotropic scaling symmetry. In the BIDM-dominated regime, this gives rise to an enhanced symmetry which fully captures the low-acceleration limit of the MDAR . We then show that, assuming a cored pseudo-isothermal profile at equilibrium, this set of equations gives rise to parameters reproducing the MDAR . Specifically, in the flat part of the rotation curve the asymptotic rotational velocity matches the parametric dependence of the BTFR . Moreover, in the central region of high-surface brightness galaxies, the profile reproduces the CSDR . Finally, by studying the time-dependent approach to equilibrium, we derive a global combination of the BTFR and CSDR, which matches the expectations in low surface-brightness galaxies. The form of the heating rate also makes model-independent predictions for various cosmological observables. We argue that our scenario satisfies existing observational constraints, and, intriguingly, offers a possible explanation to the EDGES anomaly.

Simulation and validation of residual deformations in additive manufacturing of metal parts

Selective laser melting (SLM) is gaining increasing relevance in industry. Residual deformations and internal stresses caused by the repeated layerwise melting of the metal powder and transient cooling of the solidified layers still presents a significant challenge to the profitability and quality of the process. Excessive distortions or cracking may lead to expensive rejects. In practice, critical additively manufactured parts are either iteratively pre-compensated or redesigned based on production experience. To satisfy the need for improved understanding of this complex manufacturing process, CAE software providers have recently developed solutions to simulate the SLM process. This study focuses on the evaluation of two solutions by ANSYS, i.e. ANSYS Additive Print and ANSYS Additive Suite.ANSYS Additive Print (AAP), a user-oriented software, and ANSYS Additive Suite (AAS), a software requiring advanced experience with Finite Element Methods (FEM), are investigated and validated with regard to residual deformations. For the evaluation of the two programs, calibration and validation geometries were printed by SLM in Ti–6Al–4V and residual deformations have been measured by 3D scanning. The results have been used for the calibration of isotropic and anisotropic strain scaling factors in AAP, and for sensitivity analyses on the effect of basic model parameters in AAS. The actual validation of the programs is performed on the basis of different sample geometries with varying wall thickness and deformation characteristic.While both simulation approaches, AAP and AAS, are capable of predicting the qualitative characteristics of the residual deformations sufficiently well, accurate quantitative results are difficult to obtain. AAP is more accessible and yields accurate results within the calibrated regime. Extrapolation to other geometries introduces uncertainties, however. AAS, on the other hand, features a sounder physical basis and therefore allows for a more robust extrapolation. Numerical efforts and modelling uncertainties as well as requirements for an extensive set of material parameters reduce its practicality, however. More appropriate calibration geometries, continuing extension of a more reliable material database, improved user guidelines and increased numerical efficiency are key in the future establishment of the process simulation approaches in the industrial practice.

Acoustic emission investigation on scale effect and anisotropy of jointed rock mass by the discrete element method

We investigate herein the scale effect and anisotropy of jointed rock mass (JRM) from the perspective of acoustic emission (AE) characteristics and estimate the size of the representative element volume (REV). First, we propose an AE calculation based on the discrete element method (DEM) and use the statistical results of joint planes to generate a discrete fracture network. Next, we calibrate the micro-parameters of the rock matrix and the joint plane based on physical experiment and the single plane of weakness theory, respectively. Finally, we use the proposed model to numerically simulate the AE characteristics of a JRM and estimate the size of the REV. The use of different model sizes and layer orientations shows that the AE event magnitude follows a power-law distribution. The frequency of AE events with different magnitudes follows a normal distribution, and a negative exponential relationship is found between the frequency of AE events and the number of cracks associated with each AE event. When the model size exceeds 10 × 10 m, the parameters of AE events (accumulated frequency, magnitude, frequency of AE events associated with only one crack, frequency of AE events associated with over ten cracks, and maximum crack number associated with each AE event) remain essentially constant. This means that the size of the REV is 10 × 10 m. The proposed model should also prove useful for further research into the fracture mechanism of JRM.

Applications of directional wavelets, Universal Multifractals and anisotropic scaling in ensemble nowcasting; a review of methods with case‐studies

AbstractThis article begins with a review of the scale decomposition and stochastic noise generation aspects of the ensemble nowcasting process, as used in the Short Term Ensemble Prediction System (STEPS). Alternative methods are then suggested, with case‐study examples, which could lead to enhanced performance and may also be applicable in other fields. The standard FFT‐band‐pass‐filtering scale decomposition, essential to this process, is shown to be practically the same as a redundant Continuous Wavelet Transform (CWT). This immediately makes accessible a number of more advanced methods from the wavelet literature. Particular attention is given to directional wavelet transforms and their use for analysing anisotropic scaling behaviour, common in weather images. With a view to numerically efficient operational usage, the focus is then shifted to Discrete Wavelet Transforms (DWTs). While the standard DWT has a number of disadvantages, including limited orientational selectivity in two dimensions, the relatively recent Dual Tree Complex WT (DTCWT) overcomes most of these. Examples are given that show how its six‐fold directionality can be exploited, to create synthetic features that replace the unpredictable small‐scale parts of a (weather‐radar‐based) extrapolation nowcast with more realistic spatially varying anisotropy. Further, the Bounded Log‐normal Cascade noise model, as used in STEPS, is reconsidered and a proposal to use Universal Multifractal realizations is presented. For the noise parameters, some operationally viable estimation methods are considered, such as the Double Trace Moment technique and a structure‐function‐based approach, which could work on readily available DTCWT coefficients.

Isotropic Scaling Features Measured Locally in the Solar Wind Turbulence with Stationary Background Field

Abstract The scaling anisotropy is crucial to interpret the nonlinear interactions in solar wind turbulence. Previous observations provide diverse results and the structure function analyses are also reported to be an approach to investigate the scaling anisotropy based on a local magnetic field. However, the determination of the sampling angle with respect to the local background magnetic field requires that the observed time series for the average are time stationary. Whether or not this required time stationarity is compatible with the measurements has not been investigated. Here we utilize the second-order structure function method to study the scaling anisotropy with a time-stationary background field. We analyze 88 fast solar wind intervals each with time durations ≥ 2 days measured by WIND spacecraft in the period 2005–2018. We calculate the local magnetic field as the average of the time series whose time stationarity is fulfilled by our criterion ϕ &lt; 10° (ϕ is the angle between the two averaged magnetic fields after cutting into two halves). We find for the first time the isotropic scaling feature of the magnetic-trace structure functions with scaling indices and , respectively, with the local magnetic field parallel and perpendicular to the solar wind velocity directions. The scaling for the velocity-trace structure functions is also isotropic and the indices are and . We also find that with increasing ϕ threshold to 90°, the scaling index of the magnetic-trace structure function in the parallel direction decreases to −0.81, while the rms of the instantaneous angle between magnetic field and solar wind velocity increases up to 45° at the timescale 150 s, indicating a mix of perpendicular measurements into parallel ones at large scales.

Galactic magnetic field bias on inferences from UHECR data

A consequence of Liouville's theorem indicates that the recently observed large scale anisotropy in the arrival direction of Ultra-High-Energy Cosmic Rays (UHECRs) cannot be produced from an isotropic extragalactic flux, thus some anisotropy already needs to be present outside our Galaxy. But in this case, the observed spectrum and composition of UHECRs differ from the one outside of the Milky Way, due to the suppression or the amplification of the UHECR flux from certain directions by the Galactic magnetic field. In this work, we investigate this effect for the case of a dipole and a quadrupole anisotropy, respectively, for the widely-used JF12 and PT11 magnetic field models. We investigate bounds on the maximal amplitude of the observed anisotropy and the maximal charge number of UHECRs. Furthermore, the flux modification is discussed in the light of the recent Auger results about the dipolar anisotropy and the mass composition. We find that this effect yields a modification of the observed flux of up to ∼ 25% for the investigated magnetic field model and the observed dipole, in particular for a heavy chemical composition of UHECRs as suggested by Auger data interpreted assuming the `Sibyll2.3c' or the `EPOS-LHC' model of hadronic interactions in air showers.

2D Galilean field theories with anisotropic scaling

In this work, we study two-dimensional Galilean field theories with global translations and anisotropic scaling symmetries. We show that such theories have enhanced local symmetries, generated by the infinite dimensional spin-l Galilean algebra with possible central extensions, under the assumption that the dilation operator is diagonalizable and has a discrete and non-negative spectrum. We study the Newton-Cartan geometry with anisotropic scaling, on which the field theories could be defined in a covariant way. With the well-defined Newton-Cartan geometry we establish the state-operator correspondence in anisotropic GCFT, determine the two-point functions of primary operators, and discuss the modular properties of the torus partition function which allows us to derive Cardy-like formulae.