Calculation Of Critical Exponents Research Articles

Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method.

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ,andδ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.

On the neural network flow of spin configurations

We study the so-called neural network flow of spin configurations in the 2-d Ising ferromagnet. This flow is generated by successive reconstructions of spin configurations, obtained by an artificial neural network like a restricted Boltzmann machine or an autoencoder. It was reported recently that this flow may have a fixed point at the critical temperature of the system, and even allow the computation of critical exponents. Here we focus on the flow produced by a fully-connected autoencoder, and we refute the claim that this flow converges to the critical point of the system by directly measuring physical observables, and showing that the flow strongly depends on the network hyperparameters. We explore the network metric, the reconstruction error, and we relate it to the so called intrinsic dimension of data, to shed light on the origin and properties of the flow.

Statistical Properties of the Quantum Internet.

Steady technological advances are paving the way for the implementation of the quantum internet, a network of locations interconnected by quantum channels. Here we propose a model to simulate a quantum internet based on optical fibers and employ network-theory techniques to characterize the statistical properties of the photonic networks it generates. Our model predicts a continuous phase transition between a disconnected and a highly connected phase and that the typical photonic networks do not present the small world property. We compute the critical exponents characterizing the phase transition, provide quantitative estimates for the minimum density of nodes needed to have a fully connected network and for the average distance between nodes. Our results thus provide quantitative benchmarks for the development of a quantum internet.

Holographic entanglement entropy in the QCD phase diagram with a critical point

We calculate the holographic entanglement entropy for the holographic QCD phase diagram considered in [Knaute, Yaresko, K\"ampfer (2017), arXiv:1702.06731] and explore the resulting qualitative behavior over the temperature-chemical potential plane. In agreement with the thermodynamic result, the phase diagram exhibits the same critical point as the onset of a first-order phase transition curve. We compare the phase diagram of the entanglement entropy to that of the thermodynamic entropy density and find a striking agreement in the vicinity of the critical point. Thus, the holographic entanglement entropy qualifies to characterize different phase structures. The scaling behavior near the critical point is analyzed through the calculation of critical exponents.

Free energy of 3D Ising-like system near the phase transition point`

An urgent problem in describing critical phenomena lies in the development of a generalizedtheory which allows one to obtain (except the calculation of critical exponents and some other uni-versal characteristics) explicit expressions for physical quantities like in the case of classic theories[1,2]. The scaling theory turns out to be the most developed with respect to this problem. It isbased on the similarity hypothesis proposed in [3{6] and on the scheme of constructing e ectiveKadano spin blocks [5]. We consider the system based on the Ising model on a simple cubic latticehaving a lattice constant c. The initial lattice is split into blocks with linear sizes s c, where sis an arbitrary number (s > 1). Then, instead of N initial sites with period c we get N

Calculation of critical exponents by self-similar factor approximants

The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric phi^4 theory and of the Ising glass. It is demonstrated that this method, being much simpler than other known techniques of series summation in calculating the critical exponents, at the same time, yields the results that are in very good agreement with those of other rather complicated numerical methods. The principal advantage of the method of self-similar factor approximants is the combination of its extraordinary simplicity and high accuracy.

The application of the Gibbs equation for stability limit to the description of phase transitions of the second kind. Calculation of critical exponents

Phase transitions of the second kind occur at the limit of stability (spinodal) common to both phases, and this defines the continuous pattern of these transitions. The equation of the limit of stability, formulated by Gibbs, turns out to be the basic input equation of the theory of phase transitions of the second kind. The characteristic property of spinodal states lies, as is demonstrated in this paper, in the quadratic dependence of second variation of internal energy on the temperature difference. Proceeding from this regularity, the method of calculation of critical exponents is developed.

Monte Carlo results for three-dimensional self-avoiding walks

We discuss possible sources of systematic errors in the computation of critical exponents by renormalization-group methods, extrapolations from exact enumerations and Monte Carlo simulations. A careful Monte Carlo determination of the susceptibility exponent γ for three-dimensional self-avoiding walks has been used to test the claimed accuracy of the various methods.

Critical properties of Ising models with long-range interactions on Sierpiński-gasket lattices in higher Euclidean dimensions

Ising systems with two-spin long-range interactions are studied on Sierpiński-gasket lattices generalized to Euclidean dimensions d = 3, 4, 5, 6. For each lattice, the long-range interactions are assigned to edges that are part of the appropriate self-similar structure. In consequence, these systems are infinitely ramified, and exhibit critical behavior at finite temperatures. An analysis of critical properties of the systems, in particular the calculation of critical exponents, is carried out by using an exact renormalization-group method. It is shown that there exists some similarity between critical properties of the long-range interacting fractal models under study here and critical properties of short-range interacting Ising models defined on abstract regular lattices interpolated to noninteger dimensionalities.

Infinite symmetry group in D-dimensional conformal quantum field theory

The diffeomorphism group of the D-dimensional space as the symmetry group for conformal quantum field theory models is proposed. The problem of calculation of critical exponents turns out to be reduced to the study of special diffeomorphism group representations. The general properties of these representations on the space of symmetric traceless tensors are investigated.

Exact evolution equation for the effective potential

We derive a new exact evolution equation for the scale dependence of an effective action. The corresponding equation for the effective potential permits a useful truncation. This allows one to deal with the infrared problems of theories with massless modes in less than four dimensions which are relevant for the high temperature phase transition in particle physics or the computation of critical exponents in statistical mechanics.

Complex excitations in the thermodynamic Bethe-anzatz approach.

We relate the complex solutions of the thermodynamic Bethe-ansatz equations to the excited states in factorizable massive models. This approach permits one to recover exactly the ultraviolet behavior of these excitations. In order to illustrate our method we consider several unitary and nonunitary integrable models. We also comment on the calculation of critical exponents in massless integrable models.

A self-avoiding walk exponent bound on the thermal Ising exponent on some hierarchical lattices

Calculation of critical exponents on a simple class of hierarchical lattice reveals that lambda s>or= lambda t, where lambda s is the self-avoiding walk fixed point eigenvalue and lambda t the Ising thermal eigenvalue. High-dimensional limits of some families of hierarchies obey lambda t to lambda s as D to infinity ; this convergence replaces the Euclidean concept of upper critical dimension on these lattices. However, families of hierarchies for which D to infinity but with constant connectivity do not show this convergence.

Renormalization-scheme ambiguity and perturbation theory near a fixed point

We consider the perturbative calculation of critical exponents in massless, renormalizable theories having a nontrivial fixed point. In conventional perturbation theory, all results depend on the arbitrary renormalization scheme used. We show how to resolve this problem, following the "principle of minimal sensitivity" approach. At least three orders of perturbation theory are required for quantitative results. We give scheme-independent criteria for determining the presence or absence of a fixed point in $n\mathrm{th}$ order, and discuss the conditions under which perturbative results might be reliable. As illustrations we discuss QED with many flavors, and ${({\ensuremath{\varphi}}^{4})}_{4}$ theory. In neither case do we find a fixed point, in contrast to naive perturbative expectations.

Monte Carlo Calculation of Renormalized Coupling Parameters

A practical method is presented for calculating many-parameter renormalization-group trajectories for both spin systems and lattice gauge theories from Monte Carlo computer simulations. It provides a direct check on the assumption of short-range renormalized interactions, and preliminary work indicates improvement of the convergence in the calculation of critical exponents.

Monte Carlo renormalization-group studies of critical phenomena

Monte Carlo renormalization-group (MCRG) methods are described and illustrated for models of current interest. The calculation of critical exponents and the determination of critical temperatures are discussed, as well as the application to multicritical points.

Calculation of critical exponents in two dimensions from quantum field theory in one dimension

We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.

Renormalization-group calculation of critical exponents for three-dimensional Ising-like systems

Renormalization-group calculation of critical exponents for three-dimensional Ising-like systems

On the calculation of critical exponents by renormalisation group techniques

The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u0c, the Feynman graph coupling constant, is discussed.