Local Tensor Research Articles

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the $((5\pm\sqrt{5})/2)$-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.

Exact Mobility Edges in 1D Mosaic Lattices Inlaid with Slowly Varying Potentials

AbstractA family of 1D mosaic models inlaid with a slowly varying potential is proposed. Combining the asymptotic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs), and pseudo‐MEs (PMEs) in their energy spectra are solved semi‐analytically, where ME separates extended states from weakly localized ones and PME separates weakly localized states from strongly localized ones. The nature of eigenstates in extended, critical, weakly localized, pseudo‐critical, and strongly localized is diagnosed by the local density of states, the Lyapunov exponent, the localization tensor, and fractal dimension. Numerical calculation results are in excellent quantitative agreement with theoretical predictions.

Machine Learning for the Flow Prediction of Fluids with Memory Effects on the Stress

We have recently developed a machine learning method capable of inferring the constitutive equation for the stress of a fluid with memory (e.g., polymeric fluid) from microscopic simulations. For this, we used a Gaussian Process regression scheme to learn the constitutive relation, given as the time-derivative of the stress, as a function of the local stress and velocity strain tensors. Crucially, no assumptions are made regarding the functional form of this relation. We have applied our method to the Hookean dumbbell model, for which the exact analytical constitutive relation (Maxwell equation) is known, in order to validate the approach. Our results are in excellent agreement with the analytical solution, showing that we are able to capture the history dependence of the flow, as well as the elastic effects in the fluid. Compared to full multi-scale simulations, in which the micro and macro degrees of freedom are directly coupled, our method provides a similar degree of accuracy, at a small fraction of the cost. In addition, the method can be easily generalized to more complex and realistic polymer models. (As part of this article, we include a Japanese summary of our main results, published in [7], CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).)

Nonreciprocal localization of light in magnetoactive cholesteric liquid crystals in an external magnetic field

The optical properties of cholesteric liquid crystals (CLCs) exhibiting magneto-optical activity in the presence of an external static magnetic field have been investigated. The boundary-value problem of light transmission through a finite layer of a magnetoactive CLC at normal incidence of light is considered in the limiting case of the absence of local linear birefringence in the medium. That is a CLC with parameters Re∆=0,Im∆≠0, where ∆=ε1−ε2/2, andε1,ε2 are the principal values of the local dielectric tensor, is considered. In this case, in the absence of an external magnetic field, all linearly polarized incident waves undergo practically the same reflection in the photonic band gap (PBG). An external magnetic field removes this degeneracy. For the first time, the features of the localization of light in a layer of a magnetoactive CLC are investigated, and it is shown that an external magnetic field leads to a change in the localized light energy in the CLC layer and nonreciprocal accumulation of light energy takes place.

Robust Multi-Frame Super-Resolution Based on Adaptive Half-Quadratic Function and Local Structure Tensor Weighted BTV.

It is difficult to improve image resolution in hardware due to the limitations of technology and too high costs, but most application fields need high resolution images, so super-resolution technology has been produced. This paper mainly uses information redundancy to realize multi-frame super-resolution. In recent years, many researchers have proposed a variety of multi-frame super-resolution methods, but it is very difficult to preserve the image edge and texture details and remove the influence of noise effectively in practical applications. In this paper, a minimum variance method is proposed to select the low resolution images with appropriate quality quickly for super-resolution. The half-quadratic function is used as the loss function to minimize the observation error between the estimated high resolution image and low-resolution images. The function parameter is determined adaptively according to observation errors of each low-resolution image. The combination of a local structure tensor and Bilateral Total Variation (BTV) as image prior knowledge preserves the details of the image and suppresses the noise simultaneously. The experimental results on synthetic data and real data show that our proposed method can better preserve the details of the image than the existing methods.

Confined vortex surface and irreversibility. 1. Properties of exact solution

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

Light diffraction by a nanograting with bimetallic metamaterial

Diffraction optical spectra for a periodic array of bimetallic metamaterial bars, overlying a dielectric substrate, are calculated and analysed within the Fourier formalism. The optical response of the metamaterial is here described by a local frequency-dependent permittivity tensor according to the effective medium approach (EMA). The zero- and first-order diffraction spectra, predicted by EMA, are compared with exact numerical calculations considering the real nanostructure of the bimetallic metamaterial, as well as realistic metals permittivities. In particular, results for a nanograting with Al–Ag layered metamaterial are presented. It is demonstrated that the diffraction spectra, for incident light polarized along the metamaterial bars, strongly depend on the orientation of the metal layers. For this reason, EMA can only provide a qualitative description of diffraction spectra. For a nanograting with Al–Ag-metamaterial layers, being perpendicular to the bar axis, the enhancement of the intensity of zero-order transmitted wave, accompanied by the strong suppression of the specular reflectivity and all the nonzero-order diffraction spectra, is predicted to be observed near a matching frequency.

Generalized Taylor dispersion for translationally invariant microfluidic systems

We consider Taylor dispersion for tracer particles in microfluidic planar channels with strong confinement. In this context, the channel walls modify the local diffusivity tensor and also interactions between the tracer particles and the walls become important. We provide a simple and general formula for the effective diffusion constant along the channel as well as the first non-trivial finite time correction for arbitrary flows along the channel, arbitrary interaction potentials with the walls, and arbitrary expressions for the diffusion tensor. The formulas are in particular amenable to a straightforward numerical implementation, rendering them extremely useful for comparison with experiments. We present a number of applications, notably for systems that have parabolically varying diffusivity profiles, to systems with attractive interactions with the walls as well as electro-osmotic flows between plates with differing surface charges within the Debye–Hückel approximation.

Emergent geometry through quantum entanglement in Matrix theories

In the setting of the Berenstein-Maldacena-Nastase Matrix theory, dual to light-cone M-theory in a PP-wave background, we compute the Von Neumann entanglement entropy between a probe giant graviton and a source. We demonstrate that this entanglement entropy is directly and generally related to the local tidal acceleration experienced by the probe. This establishes a new map between local spacetime geometry and quantum entanglement, suggesting a mechanism through which geometry emerges from Matrix quantum mechanics. We extend this setting to light-cone M-theory in flat space, or the Banks-Fischler-Shenker-Susskind Matrix model, and we conjecture a new general relation between a certain measure of entanglement in Matrix theories and local spacetime geometry. The relation involves a ‘c-tensor’ that measures the evolution of local transverse area and relates to the local energy-momentum tensor measured by a probe.

The importance of reference frame for pressure at the liquid–vapour interface

ABSTRACT The local pressure tensor is non-unique, a fact which has generated confusion and debate in the 70 years since the seminal work by Irving Kirkwood. This non-uniqueness is normally attributed to the interaction path between molecules, especially in the interfacial-science community. In this work, we reframe this discussion of non-uniqueness in terms of the location, or reference frame, used to measure the pressure. By using a general mathematical description of the liquid–vapour interface, we obtain a reference frame that moves with the interface through time, providing new insight into the pressure. We compare this instantaneous moving reference frame with the fixed Eulerian one. Through this process, we show the requirement that normal pressure balance at the moving surface is satisfied by surface fluxes; however, an additional corrective term based on surface curvature is required for the average pressure in a volume. We make the case that a focus on the path of integration is the cause of confusion in the literature. Using an explicit reference frame with a more general derivation of pressure clarifies some of the issues of uniqueness, providing a pressure tensor which is defined at any instant in time and valid away from thermodynamic equilibrium.

Identification of elastic properties of interphase and interface in graphene-polymer nanocomposites by atomistic simulations

This article tackles the problem of identification of elastic continuum model by atomistic simulations for graphene polymer nanocomposite. The Atomistic Local IdentificAtion of Stiffness method, so-called ALIAS method, is developed to estimate the local stiffness tensor at all points of polymer graphene laminate nanocomposite. Results suggest that the graphene can be modeled at continuum scale by a general imperfect interface with zero thickness. Moreover, the identification procedure reveals the existence of interphase on either side of the graphene with a thickness of 1 nm, which is one and a half times stiffer than the polymer bulk matrix. The identified continuum model is used to study the effective elastic properties of nanocomposites with sandwich microstructure. This study at continuum scale reveals a softening effect due the very low stiffness of slip along graphene plane. The softening due to the interfaces is preponderant in relation to the interphase stiffening. Finally, the continuum model also suggests that the wrinkling of graphene increases the stiffness of nanocomposites.

Well conditioned mathematical modeling for homogenization of thermo-electro-mechanical behaviors of piezoelectric composites

In the present paper, a mathematical modeling of effective properties of fully coupled thermo-electro-mechanical heterogeneous materials is elaborated. Constitutive equations combined with equilibrium equations lead to a coupled partial differential system. A methodological procedure is elaborated based on thermo-electro-elastic Green’s functions, integral equations, and a micro-macro approach. Localization tensors are derived using the Mori–Tanaka mean-field assumptions and the fully coupled effective behavior is obtained through averaging techniques. Due to the large dispersion between elastic, dielectric, and piezoelectric coefficients ill-conditioned localization tensors are resulted. Block matrix decomposition is elaborated to overcome this drawback and the so-regularized localization tensor’s inverses are used to get a well conditioned problem. Numerical computations are done in the general case of ellipsoidal inclusions and anisotropic behavior. Effective thermal conductivity and heat capacity coefficients are obtained with and without the heat strain effect for various volume fractions and types of inclusions. The global properties of thermo-electro-elastic reinforced composites are presented with respect to shape and volume fractions of inclusions.

On mean field homogenization schemes for short fiber reinforced composites: Unified formulation, application and benchmark

This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class.

A novel fusing semantic- and appearance-based descriptors for visual loop closure detection

Loop-closure detection plays an important role in visual simultaneous localisation and mapping;it is an independent part of the visual odometer and can effectively reduce its accumulated error, in addition to helping with loop-closure detection for relocalisation. With the development of deep learning methods in recent years, the training models of convolutional neural networks for major data sets have been improved for loop-closure detection. Presently, some high-level engineering problems still rely on auxiliary equipment, such as panoramic cameras and radar lasers, which greatly increase the expensive extra cost; however, owing to the extreme appearance and viewpoint changes involved in such problems, loop-closure detection that relies on two-dimensional images is not applicable. Based on the two nearest neighbour vector of locally aggregated descriptors (TNNVLAD) method, a novel feature descriptor called two nearest neighbour local sensor tensor(TNNLoST) is proposed herein by combining the semantic features of high-level neural networks with dense descriptors. This approach introduces a semantic concept similar to human cognition for the surrounding environment, thus enabling better understanding of the environment. The proposed method was applied to publicly available benchmark datasets to show its performance.

Smoothed particle radiation hydrodynamics: two-moment method with local Eddington tensor closure

ABSTRACTWe present a new smoothed particle hydrodynamics-radiative transfer method (sph-m1rt) that is coupled dynamically with sph. We implement it in the (task-based parallel) swift galaxy simulation code but it can be straightforwardly implemented in other sph codes. Our moment-based method simultaneously solves the radiation energy and flux equations in sph, making it adaptive in space and time. We modify the m1 closure relation to stabilize radiation fronts in the optically thin limit. We also introduce anisotropic artificial viscosity and high-order artificial diffusion schemes, which allow the code to handle radiation transport accurately in both the optically thin and optically thick regimes. Non-equilibrium thermochemistry is solved using a semi-implicit sub-cycling technique. The computational cost of our method is independent of the number of sources and can be lowered further by using the reduced speed-of-light approximation. We demonstrate the robustness of our method by applying it to a set of standard tests from the cosmological radiative transfer comparison project of Iliev et al. The sph-m1rt scheme is well-suited for modelling situations in which numerous sources emit ionizing radiation, such as cosmological simulations of galaxy formation or simulations of the interstellar medium.

Approximate optimization of the MaxCut problem with a local spin algorithm

Local tensor methods are a class of optimization algorithms that was introduced in [Hastings,arXiv:1905.07047v2][1] as a classical analogue of the quantum approximate optimization algorithm (QAOA). These algorithms treat the cost function as a Hamiltonian on spin degrees of freedom and simulate the relaxation of the system to a low energy configuration using local update rules on the spins. Whereas the emphasis in [1] was on theoretical worst-case analysis, we here investigate performance in practice through benchmarking experiments on instances of the MAXCUT problem.Through heuristic arguments we propose formulas for choosing the hyperparameters of the algorithm which are found to be in good agreement with the optimal choices determined from experiment. We observe that the local tensor method is closely related to gradient descent on a relaxation of maxcut to continuous variables, but consistently outperforms gradient descent in all instances tested. We find time to solution achieved by the local tensor method is highly uncorrelated with that achieved by a widely used commercial optimization package; on some MAXCUT instances the local tensor method beats the commercial solver in time to solution by up to two orders of magnitude and vice-versa. Finally, we argue that the local tensor method closely follows discretized, imaginary-time dynamics of the system under the problem Hamiltonian.

Prismatic mesh generation based on anisotropic volume harmonic field

In this paper, we present an effective prismatic mesh generation method for viscous flow simulations. To address the prismatic mesh collisions in recessed cavities or slit areas, we exploit 3D tensors controlled anisotropic volume harmonic field to generate prismatic meshes. Specially, a well-fitting tetrahedral mesh is first constructed to serve as the discrete computation domain of volume harmonic fields. Then, 3D tensors are exploited to control the volume harmonic field that better fits the shape geometry. From the topological perspective, the generation of the prismatic mesh can be treated as a topology-preserved morphing of the viscous wall. Therefore, iso-surfaces in the volume harmonic field should be homeomorphic to the viscous wall while fitting its shapes. Finally, a full prismatic mesh can be induced by estimating the forward directions and visible regions in the volume harmonic field. Moreover, to be compatible with different simulation situations, the thickness of prismatic meshes should be variable. Our approach provides local adjustable thickness of prismatic meshes, which can be achieved by controlling local 3D tensors. The proposed anisotropic volume harmonic field based prismatic meshes are efficient and robust, and a full prismatic mesh can be guaranteed without low quality collisions. Various experiments have shown that our proposed prismatic meshes have obvious advantages in terms of efficiency and effectiveness.

Fast computation of scattering by isolated defects in periodic dielectric media

Scattering by an isolated defect embedded in a dielectric medium of two-dimensional periodicity is of interest in many sub-fields of electrodynamics. Present approaches to compute this scattering rely on either the Born approximation and its quasi-analytic extensions or ab initio computation that requires large domain sizes to reduce the effects of boundary conditions. The Born approximation and its extensions are limited in scope, while the ab initio approach suffers from its high numerical cost. In this paper, I introduce a hybrid scheme in which an effective local electric susceptibility tensor of a defect is estimated by solving an inverse problem efficiently. The estimated tensor is embedded into an S-matrix formula based on the reciprocity theorem. With this embedding, the computation of the S-matrix of the defect requires field solutions only in the unit cell of the background. In practice, this scheme reduces the computational cost by almost two orders of magnitude, while sacrificing little in accuracy. The scheme demonstrates that statistical estimation can capture sufficient information from cheap calculations to compute quantities in the far field. I outline the fundamental theory and algorithms to carry out the computations in high dielectric contrast materials, including metals. I demonstrate the capabilities of this approach with examples from optical inspection of nano-electronic circuitry where the Born approximation fails and the existing methods for its extension are also inapplicable.

Role of Long-Range Electrostatic Interactions and Local Topology of the Hydrogen Bond Network in the Wettability of Fully and Partially Wetted Single and Multilayer Graphene

We report the wetting behavior of graphitic surface–water interfaces through the calculation of the local stress tensor based on the Irving–Kirkwood–Noll theory using detailed atomistic molecular d...

Can we define a unique microscopic pressure in inhomogeneous fluids?

The estimation of a microscopic pressure tensor in an adsorbed thin film on a planar surface remains a challenge in both experiment and theory. While the normal pressure is well-defined for a planar surface, the tangential pressure at a point is not uniquely defined at the nanoscale. We report a new method that allows us to calculate the local pressure tensor and its spatial integral using an arbitrary contour definition of the "virial-route" local pressure tensor. We show that by integrating the local tangential pressure over a small region of space, roughly the range of the intermolecular forces, it is possible to define a coarse-grained tangential pressure that appears to be unique and free from ambiguities in the definition of the local pressure tensor. We support our argument by presenting the results for more than ten types of contour definitions of the local pressure tensor. By defining the coarse-grained tangential pressure, we can also find the effective thickness of the adsorbed layer and, in the case of a porous material, the statistical pore width. The coarse-grained in-layer and in-pore tangential pressures are determined for Lennard-Jones argon adsorbed in realistic carbon slit pores, providing a better understanding of the pressure enhancement for strongly wetting systems.