22,794 research outputs found

    Non-Universal Critical Behaviour of Two-Dimensional Ising Systems

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    Two conditions are derived for Ising models to show non-universal critical behaviour, namely conditions concerning 1) logarithmic singularity of the specific heat and 2) degeneracy of the ground state. These conditions are satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising models with short- or long-range interactions and even Ising systems without the translational or the rotational invariance.Comment: 17 page

    Over-constrained Weierstrass iteration and the nearest consistent system

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    We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least kk common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give explicitly several iteration functions to compute the optimum. The arithmetic complexity of the iterations is detailed

    On the absence of chiral fermions in interacting lattice theories

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    We consider interacting theories with a compact internal symmetry group on a regular lattice. We show that the spectrum is necessarily vector-like provided the following conditions are satisfied: (a)~weak form of locality, (b)~relativistic continuum limit without massless bosons, and (c)~pole-free effective vertex functions for conserved currents. The proof exploits the zero frequency inverse retarded propagator of an appropriate set of interpolating fields as an effective quadratic hamiltonian, to which the Nielsen-Ninomiya theorem is applied. The main results of this paper have been reported in WIS-93/56-JUNE-PH, hep-lat/9306023.Comment: WIS-93/57-JULY-PH, LaTeX, 24 page

    Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence

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    Using the fusion rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous non-perturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We shall call this generalization of non-perturbative locality to multiple applications of the nonlinear interactions operator "multilocality". The resulting cross-terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to non-perturbative locality of generalized structure functions on all orders, and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.Comment: 25 pages, 24 figures, resubmitted to Physical Review

    Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

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    We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound the distance between an arbitrary parameter E>0E > 0 and the spectrum {Ej}\{E_j \} in terms of the boundary L2L^2-norm of a normalized trial solution uu of the Helmholtz equation (Δ+E)u=0(\Delta + E)u = 0. We also bound the L2L^2-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all EE greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber E\sqrt{E}. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure

    Spinor gravity and diffeomorphism invariance on the lattice

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    The key ingredient for lattice regularized quantum gravity is diffeomorphism symmetry. We formulate a lattice functional integral for quantum gravity in terms of fermions. This allows for a diffeomorphism invariant functional measure and avoids problems of boundedness of the action. We discuss the concept of lattice diffeomorphism invariance. This is realized if the action does not depend on the positioning of abstract lattice points on a continuous manifold. Our formulation of lattice spinor gravity also realizes local Lorentz symmetry. Furthermore, the Lorentz transformations are generalized such that the functional integral describes simultaneously euclidean and Minkowski signature. The difference between space and time arises as a dynamical effect due to the expectation value of a collective metric field. The quantum effective action for the metric is diffeomorphism invariant. Realistic gravity can be obtained if this effective action admits a derivative expansion for long wavelengths.Comment: 13 pages, proceedings 6th Aegean Summer School, Naxos 201
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