Two-dimensional Classical Model Research Articles

Tensor network based finite-size scaling for two-dimensional Ising model

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

UNIVERSITY AND LIFELONG EDUCATION IN THE EUROPEAN EDUCATIONAL SPACE: THE INTERDEPENDENCE OF GOALS AND STRATEGIES

The article analyzes the interdependence of goals and strategies of university education and lifelong learning in the European educational space. The aim of the article is to substantiate the interaction of university education and lifelong learning and to determine their role in the modern European social space. The research is carried out on the basis of methods of theoretical generalization, comparative analy-sis; methods of grouping and systematization. It was emphasized that the rethinking of social tasks, the need to adapt to the expectations of society, the growth of the demand for future skills motivate university education at the beginning of the 21st century to prepare a successful professional who is focused on the continu-ous updating and replenishment of knowledge, skills and abilities. It is revealed that in modern conditions in university and lifelong education the emphasis is on the formation of such competencies as an optimistic view of life situations, that is, the ability to adapt and interact effectively in the growing demands of volatile and uncertain social environments, the ability to analytically process large amounts of data, detection of algo-rithms, prediction of dynamics, ability to model optimal production processes and environments for systematically achieving desired results in each individual situation, ability to establish direct and deep interpersonal connections, ability to recognize causes and consequences of key trends, notice tendencies on the periphery of social systems. It is concluded that today more and more the question arises about the need to expand the scope of universities and their exit beyond the two-dimensional classical model, which includes educational activities and scientific research and the social purpose of universities, and within their framework and education throughout their lives, the direction and formation of socially important competencies.

Quantification of anticipation of excitement with a three-axial model of emotion with EEG

Objective. Multiple facets of human emotion underlie diverse and sparse neural mechanisms. Among the many existing models of emotion, the two-dimensional circumplex model of emotion is an important theory. The use of the circumplex model allows us to model variable aspects of emotion; however, such momentary expressions of one’s internal mental state still lacks a notion of the third dimension of time. Here, we report an exploratory attempt to build a three-axis model of human emotion to model our sense of anticipatory excitement, ‘Waku-Waku’ (in Japanese), in which people predictively code upcoming emotional events. Approach. Electroencephalography (EEG) data were recorded from 28 young adult participants while they mentalized upcoming emotional pictures. Three auditory tones were used as indicative cues, predicting the likelihood of the valence of an upcoming picture: positive, negative, or unknown. While seeing an image, the participants judged its emotional valence during the task and subsequently rated their subjective experiences on valence, arousal, expectation, and Waku-Waku immediately after the experiment. The collected EEG data were then analyzed to identify contributory neural signatures for each of the three axes. Main results. A three-axis model was built to quantify Waku-Waku. As expected, this model revealed the considerable contribution of the third dimension over the classical two-dimensional model. Distinctive EEG components were identified. Furthermore, a novel brain-emotion interface was proposed and validated within the scope of limitations. Significance. The proposed notion may shed new light on the theories of emotion and support multiplex dimensions of emotion. With the introduction of the cognitive domain for a brain-computer interface, we propose a novel brain-emotion interface. Limitations of the study and potential applications of this interface are discussed.

Analogue simulation of two-body quantum dynamics with classical setup

One-dimensional Bose-Hubbard models provide a variety of intriguing predictions including the formation of repulsively bound two-boson pairs, their collapse and revival and the realization of the two-boson topological states. While probing such systems in quantum regime is quite a challenging task, there is another appealing option to explore the properties of repulsively bound bosons based on mapping of the one-dimensional interacting system onto the two-dimensional classical model described by the standard coupled-mode equations. Adopting the latter approach, we develop the procedure to probe the energy of bound boson pairs as well as to perform the tomography of the bound pair quantum state in realistic dissipative systems.

Tensor renormalization group with randomized singular value decomposition.

An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.

Classical and refined shell models for the analysis of nano-reinforced structures

The use of nanocomposites could be an interesting innovation in aeronautics and space structures such as multilayered plates and shells. The static analysis of layered shells embedding nanocomposites is here accomplished by means of classical and refined two-dimensional models. The shells analyzed are simply supported with a sinusoidal pressure applied at the top. These boundary and loading conditions allow the governing equations to be solved in a closed form. In nanocomposites a small amount of nanoscale reinforcements is embedded which have effects on the macroscale properties. The reinforcements considered are carbon nanotubes (CNTs) and clays, the former ones are nanometer-diameter cylinders, the second fillers are nanometer-thin platelets. Several types of nanocomposites are used in the static analysis of the shells proposed, their elastic properties have been obtained from a critical literature review. Classical two-dimensional models such as Classical Lamination Theory (CLT) and First order Shear Deformation Theory (FSDT), and a refined mixed model based on Carrera Unified Formulation (CUF) are used to investigate four shell configurations: a single layered cylindrical shell panel with CNT reinforcements in elastomeric or thermoplastic polymers, a single layered shell with CNT reinforcements in a polymeric matrix embedding carbon fibers, a sandwich shell with external skins in aluminium alloy and an internal core in silicon foam filled with CNTs and a single layered shell with clay reinforcements in a polymeric matrix. The aims of the paper are to demonstrate that the use of classical shell theories is inadequate to investigate the static response of nanocomposite shells, to quantify the beneficial effect of the nanoreinforcements in terms of displacements and stresses, to show that the curvature of shells does not give further effects with respect to the plate case.

Shannon and entanglement entropies of one- and two-dimensional critical wave functions

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant ${S}_{0}$. In $c=1$ critical systems (Tomonaga-Luttinger liquids), we find that ${S}_{0}$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 $XXZ$ chain. In a massive (crystal) phase, ${S}_{0}$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a $c=1/2$ critical point.

Introduction to integrable many-body systems I

Introduction to integrable many-body systems IThis is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/

Resveratrol inhibits cell growth in a human cholangiocarcinoma cell line

Cholangiocarcinoma is a devastating tumour with a poor prognosis. An efficient therapy is unavailable in unoperable patients and new drugs are widely sought for and required. Resveratrol (RES) is a natural molecule with a reported anticancer effect, evaluated on different tumour cell lines. We tested the efficacy of RES on a cholangiocarcinoma cell line for the first time. We used the human SK-ChA-1 cell line, cultured in the classical two-dimensional model and in the three-dimensional spheroids. After RES exposure morphology, cell viability (colony-forming assay), lactate dehydrogenase (LDH), alkaline phosphatase (ALP) and cancer antigen (CA) 19-9 medium releases, cellular transglutaminase activity, karyotype and cell cycle were evaluated. Resveratrol inhibited cell growth in both the cell culture systems used (from -15 to -80% vs untreated controls) and induced a 40-fold increase of LDH and ALP activities in the culture medium. Also, transglutaminase (TG) activity increased in the cell lysates, together with a cell cycle perturbation characterised by an accumulation in the G(1)/S phase. Karyotype and CA 19-9 expression were not influenced by the treatment. The observed cytotoxic effect of RES on the human cholangiocarcinoma SK-ChA-1 cell line cultured two- and three-dimensionally suggests to further analyse its chemotherapic/chemopreventive possibilities for this kind of cancer.

Elastic properties of two-dimensional soft discs of various diameters at zero temperature

Elastic properties of two classes of two-dimensional models, consisting of soft particles of various sizes have been determined numerically and compared: (1) polydisperse discs (with Gaussian probability distribution of diameters) and (2) binary discs (with probability density distribution of diameters composed of two Dirac delta functions of equal amplitudes). Simple analytical formulae approximating the numerical results are found. It is shown that the elastic properties of both systems are very close to each other if the first two moments (i.e. the mean value and the standard deviation) of the probability density distribution in the systems are equal.

The dilute AL models and the Φ1,2 perturbation of unitary minimal CFTs

Motivated by recent studies by Dorey, Pocklington and Tateo for unitary minimal models perturbedby ϕ1,2, we examine the thermodynamics of one-dimensional quantum systems,whose counterparts in the two-dimensional classical model are the diluteAL models in regime 2. The functional relations for arbitrary values ofL are established. Guided by numerical evidence, we obtain a set of coupled integralequations from the established relations, which yields the evaluation of the free energy atarbitrary temperature. In the scaling limit, the integral equations coincide with thethermodynamic Bethe ansatz equations (TBA) proposed by Dorey, Pocklington and Tateo,thereby supporting their results. The new fermionic representations of the Virasorocharacters are remarked upon briefly.

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z 2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a Z 2 deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

Normal mode spectra of two-dimensional classical atoms confined by a Coulomb potential.

The normal mode spectra of two-dimensional finite clusters of charged particles (-e) confined by a Coulomb potential resulting from a displaced positive charge Ze are obtained. This is a classical two-dimensional model system for atoms. We obtain the frequencies of the normal modes as a function of the confinement charge Z and the number of particles N. The analysis of the lowest normal mode frequency reveals a good agreement with the experimental results obtained in a system with screened interaction between charged particles. The dependence of the normal mode spectra as a function of a perpendicular magnetic field is also discussed and we found that the shear-like character of the modes is enhanced in the presence of the magnetic field. For large values of the magnetic fields the normal modes fall into two bands, a low frequency band with frequency approximately 1/omega(c) and a high frequency band with frequency approximately omega(c).

Incommensurate structures studied by a modified density-matrix renormalization-group method

A modified density matrix renormalization group (DMRG) method is introduced and applied to classical two-dimensional models: the anisotropic triangular nearest- neighbor Ising (ATNNI) model and the anisotropic triangular next-nearest-neighbor Ising (ANNNI) model. Phase diagrams of both models have complex structures and exhibit incommensurate phases. It was found that the incommensurate phase completely separates the disordered phase from one of the commensurate phases, i. e. the non-existence of the Lifshitz point in phase diagrams of both models was confirmed.

Numerical solution for various inverse problems

This study considers the classical two-dimensional model of jet flow using an ideal, incompressible, weightless fluid. Solving a direct problem consists in determining the pressure (or velocity) distribution on an obstacle in a flow. The aim of this paper is to describe and illustrate a numerical method for constructing wetted walls to obtain arbitrary velocity distributions, i.e. solving the inverse problem. This method is applied to airfoil design and to the construction of an infinite wall impinged by a jet. The specified distributions may entail constraints that require an analysis of the existence of the solution.

Dynamic topological solitons in a two-dimensional ferromagnet

It is shown that stable, skyrmion-type, dynamic solitons can be constructed for a wide class of two-dimensional models of anisotropic ferromagnets. These solitons are stabilized as a result of the conservation of various integrals of motion: the z projection of the total spin S z or the orbital angular momentum L z of the magnetization field. A class of two-parameter solitons with quite complicated (almost periodic) magnetization-field dynamics exists for a purely uniaxial model (in the sense of both spin and spatial rotations) with maximum symmetry. Stable solitons with periodic magnetization dynamics exist for ferromagnets with lower symmetry (only S z or L z or the total angular momentum J z =L z +S z is conserved).

Ideal free streamline flow over a curved obstacle

In the classical two-dimensional model the description of Helmholtz's (Kirchhoff) flow is a problem of complex analysis which can be solved analytically only for a few simple bodies or polygonal contours, using the Schwarz-Christoffel map. This paper presents a practical method for computing flows over arbitrary obstacles whose boundaries may be piecewise smooth curves, while the impinging flow may be an unbounded flow, a jet, or a semi-infinite stream, i.e. the ocean.

Quantum and classical two-dimensional analysis of rainbow structures in the Xe+CO2 rotational excitation at 0.2 eV collision energy and on a repulsive potential.

Rotational rainbows in the collision Xe+${\mathrm{CO}}_{2}$ are analyzed using quantum and classical two-dimensional models. We discuss the effect of the single- and the multiple-collision rotational rainbows on the differential cross sections, both for the fixed transitions and varying scattering angle and for the fixed scattering angle and varying transitions. A comparision is made with the existing centrifugal sudden calculations for this system.

A BILOCAL FIELD APPROACH TO THE LARGE N EXPANSION OF TWO-DIMENSIONAL (GAUGE) THEORIES

We consider a wide class of two-dimensional models as gauge theories, the Gross-Neveu model, O(N) and CPN−1-like models using a formalism based on the introduction of bilocal fields that permits us to perform easily the large N expansion of this set of models in a unified and general way. We mainly discuss the SU(N) gauge field theory minimally coupled to fermionic plus bosonic matter in the fundamental representation, and we obtain within the path integral approach exact equations for the particle spectrum, also in the presence of renormalizable polynomial potentials. Finally, we discuss the correspondence between this new approach and the one previously used in the context of the O(N) vector models.

Canonical analysis of a class of two-dimensional dilaton-gravity models

We exhibit the classical and quantum constraints that arise in a canonical hamiltonian formulation of a class of two dimensional models of gravitation. We present new classical solutions and discuss the closure of the classical and quantum constraint algebras.