Phase diagram and exotic spin-spin correlations of anisotropic Ising model on the Sierpiński gasket

The anisotropic antiferromagnetic Ising model on the fractal Sierpiński gasket is intensively studied, and a number of exotic properties are disclosed. The ground state phase diagram in the plane of magnetic field-interaction of the system is obtained. The thermodynamic properties of the three plateau phases are probed by exploring the temperature-dependence of magnetization, specific heat, susceptibility and spin-spin correlations. No phase transitions are observed in this model. In the absence of a magnetic field, the unusual temperature dependence of the spin correlation length is obtained with 0 ≤ J_b/J_a< 1, and an interesting crossover behavior between different phases at J_b/J_a = 1 is unveiled, whose dynamics can be described by the J_b/J_a-dependence of the specific heat, susceptibility and spin correlation functions. The exotic spin-spin correlation patterns that share the same special rotational symmetry as that of the Sierpiński gasket are obtained in both the 1 / 3 plateau disordered phase and the 5/9 plateau partially ordered ferrimagnetic phase. Moreover, a quantum scheme is formulated to study the thermodynamics of the fractal Sierpiński gasket with Heisenberg interactions. We find that the unusual temperature dependence of the correlation length remains intact in a small quantum fluctuation.     

The anisotropic quantum antiferromagnet on the Sierpiński gasket: Ground state and thermodynamics

We investigate an antiferromagnetic s = 1/2 quantum spin system with anisotropic spin exchange on a fractal lattice, the Sierpiski gasket. We introduce a novel approximative numerical method, the configuration selective diagonalization (CSD) and apply this method to a the Sierpiski gasket with N = 42. Using this and other methods we calculate ground state energies, spin gap, spin-spin correlations and specific heat data and conclude that the s = 1/2 quantum antiferromagnet on the Sierpinski gasket shows a disordered magnetic ground state with on the Sierpinski gasket shows a disordered magnetic ground state with a very short correlation length of and an, albeit very small, spin gap. This conclusion holds for Heisenberg as well a for XY exchange.Received: 18 February 2004, Published online: 20 April 2004PACS: 75.10.-b General theory and models of magnetic ordering - 05.45.Df Fractals - 75.40.Mg Numerical simulation studies     

Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice

A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ ₁=Q,l=1,2,..., on the Sierpiski-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM _k-spin correlation functions, whereM _k=3 ^k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM _k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM _k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.     

The absence of phase transition for the classical XY-model on Sierpiński pyramid with fractal dimension D=2

For the spin models with continuous symmetry on regular lattices and finite range of interactions, the lower critical dimension is d?=?2. In two dimensions the classical XY-model displays Berezinskii–Kosterlitz–Thouless (BKT) transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids (SPs) whose fractal dimension is D = log?4/log?2?=?2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of a long-range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions, we draw a conclusion that in the thermodynamic limit there is no BKT transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of SP.     

Philosophical Implications of Kadanoff’s Work on the Renormalization Group

This paper investigates the consequences for our understanding of physical theories as a result of the development of the renormalization group. Kadanoff’s assessment of these consequences is discussed. What he called the “extended singularity theorem” (that phase transitons only can occur in infinite systems) poses serious difficulties for philosophical interpretation of theories. Several responses are discussed. The resolution demands a philosophical rethinking of the role of mathematics in physical theorizing.     

Response to Kowalczyński on tachyons

I answer and briefly comment upon a paper on tachyons by J. K. Kowalczyski. Suitable answers are already contained in the recent literature about extended relativity (ER), apparently unknown to that author. My answer is threefold. (1) About causality: No paradoxes can be sensibly discussed without studying in detail the tachyon-exchange dynamics; but, once one knows tachyon mechanics, the solution of the paradox is straightforward. As an example, I exploit and solve the Tolman-Regge paradox. (2) About superluminal frames and transformations: I agree that (as I have noted elsewhere) in four dimensions such language is unfortunate; it was borrowed from two dimensions, where it is completely justified. Formulations in terms of a new language can be found in my recent papers on ER. (3) The statement that the pseudo-Euclidean space-time is a particular Riemannian manifold is wrong. It ispseudo-Riemannian, or Lorentzian. When dealing with tachyons the difference between pseudo-Riemannian and Riemannian is essential.     

A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.     

Some exact solutions of the Plebański and Pellegrini tetrade gravitational equations

Tetrades are found which satisfy the gravitational equations found recently by Plebaski and Pellegrini in the case of Peres and Bondi plane gravitational waves and the energy density and energy-current density in these waves are determined.     

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Nonlocal symmetries of Plebański’s second heavenly equation

We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering. Also, we find two new infinite hierarchies of commuting nonlocal symmetries in this covering and describe the structure of the Lie algebra of the obtained nonlocal symmetries.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Analysis of the influence of node location on transducer performance

This paper,taking the distance between the piezoelectricceramic center and the displacement node location of the transducer asa parameter,investigates the relation of characteristic parameters oftransducer,such as force factor,equivalent resistance,potential maximumelectroacoustical efficiency,and loading performance,to the displacementnode location.Some design considerations about the selection of nodelocation are noted.     

Analysis of the Zeeman effect on D_α spectra on the EAST tokamak

Based on the passive spectroscopy,the D_α atomic emission spectra in the boundary region of the plasma have been measured by a high resolution optical spectroscopic multichannel analysis(OSMA) system in EAST tokamak.The Zeeman splitting of the D_α spectral lines has been observed.A fitting procedure by using a nonlinear least squares method was applied to fit and analyze all polarization π and ±σ components of the D_α atomic spectra to acquire the information of the local plasma.The spectral line shape was investigated according to emission spectra from different regions(e.g.,low-field side and high-field side) along the viewing chords.Each polarization component was fitted and classified into three energy categories(the cold,warm,and hot components) based on different atomic production processes,in consistent with the transition energy distribution by calculating the gradient of the D_α spectral profile.The emission position,magnetic field intensity,and flow velocity of a deuterium atom were also discussed in the context.     

Application of Corner Transfer Matrix Renormalization Group Method to the Correlation Function of a Two－Dimensional Ising Model ＊

The correlation function of a two-dimensional Ising model is calculated by the corner transfer matrix renormal-ization group method. We obtain the critical exponent η= 0.2496 with few computer resources.     

Functional Renormalization Group Flows on Friedman–Lemaître–Robertson–Walker backgrounds

We revisit the construction of the gravitational functional renormalization group equation tailored to the Arnowitt–Deser–Misner formulation emphasizing its connection to the covariant formulation. The results obtained from projecting the renormalization group flow onto the Einstein–Hilbert action are reviewed in detail and we provide a novel example illustrating how the formalism may be connected to the causal dynamical triangulations approach to quantum gravity.     

Renormalization group treatment of the quantum nonlinear σ-model

The quantum nonlinear -model in (d+1)-dimensional space-time is investigated by a renormalization group approach. The beta-functions for the couplingg and the temperaturet are given. The renormalisation group equations of theN-point functions are derived for finite coupling and finite temperature. It is known that the model shows a phase transition at zero temperature at some critical couplingg _c . The behaviour near this critical point is investigated. The crossover exponent , describing the crossover between different regimes near the critical point is calculated, verifying a conjecture by Chakravarty, Halperin and Nelson, who have argued that ind dimensions should have the same value as the critical exponent of the correlation length in a (d+1)-dimensional classical system. A subtraction scheme appropriate to calculate the renormalisation factors and from these the beta-functions at finite temperature and finite coupling constant will be introduced. Using this method the beta-functions will be calculated to order two loops. The exponents obtained this way are in good agreement with the values found on other ways.     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999