Showing papers in "Journal of Physics A in 1982"

On the computational complexity of Ising spin glass models

Abstract: In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied. In a finite two-dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice. In contrast to this fact, the same problems are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional case. NP-hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it.

Proof of uniqueness of the kerr-newman black hole solution

Abstract: The electrovacuum Ernst equations are formulated as a nonlinear sigma -model on the symmetric (Kahler) space SU(1,2)/S(U(1)*U(2)). It is shown, using this formulation, that a generalised Robinson-type identity for the electrovacuum Ernst equations may be derived. A special role played in the derivation of this identity by the hidden symmetry group SU(1,2) is established. A theorem is proven that the only possible exterior solution for a (pseudo-) stationary, rotating, electrovacuum black hole with non-degenerate event horizon is the Kerr-Newman solution with m2-a2-P2-Q2>0.

Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model

Abstract: It is shown that the power expansion of the Gibbs potential of the SK model up to second order in the exchange couplings leads to the TAP equation. This result remains valid for the general (including a ferromagnetic exchange) SK model. Theorems of power expansions and resolvent techniques are employed to solve the convergence problem. The convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations.

Large fluctuations for a nonlinear heat equation with noise

Abstract: Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which are stable. The solution of the complete forced equation is a stochastic process in space and time that has a unique stochastic equilibrium. The authors study this process in the limit of small noise, and obtain lower and upper bounds for the probability of large fluctuations. They then apply these estimates to calculate the transition probability between the stable configurations (tunnelling). This model problem can be interpreted as a rigorous version of some recent attempts to describe Euclidean quantum systems in terms of stochastic equilibrium states of a nonlinear stochastic differential equation in infinite dimensions. However, its significance goes beyond this situation and the authors' methods may be applicable to models in other areas of natural science.

Cluster structure near the percolation threshold

Abstract: Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.

Motions in a Bose condensate. IV. Axisymmetric solitary waves

Abstract: Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 PK~/C' and minimum energy about 0.145 p~~/c, where p is density, c is the speed of sound and K is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation K; as p+m its radius G-(~/TK)"* becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p +a; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be aE/ap, the derivative being taken along the family. At great distances, the disturbance in the condensate is pseudo-dipolar (dipolar in a stretched coordinate system); the strength of the pseudo-dipole moment is obtained numerically. Analogous calculations are presen- ted for the corresponding two-dimensional problem. Again, a continuous sequence of solitary waves is obtained, but the momentum per unit length p and energy per unit length E have no minima. For small forward velocities, the wave consists of two widely separated parallel, oppositely directed line vortices. As the forward velocity increases the wave loses its vorticity and becomes a rarefaction pulse of ever increasing spatial extent but ever decreasing amplitude. As its velocity approaches c, both p and E tend to zero, and Elp + c.

A transfer-matrix approach to random resistor networks

Abstract: Introduces a transfer-matrix formulation to compute the conductance of random resistor networks. The authors apply this method to strips of width up to 40, and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions, t=1.28+or-0.03.

N=1 supersymmetry algebras in d=2,3,4 mod 8

Abstract: The authors derive Poincare, de Sitter and conformal supersymmetry algebras, in all dimensions allowing Majorana spinors. They consider only minimal gradings (N=1), and show that these always exist. A brief discussion of fermionic central charges is given.

Analytical calculation of two leading exponents of the dilute Potts model

Abstract: A Potts model on a square lattice with two- and four-spin interaction and site and bond dilution is shown to be dual to itself. The model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model. By means of these mappings the dilute Potts model can be written as a Gaussian-like model with staggered and direct periodic fields. The leading and next-to-leading exponents of the Potts model are calculated, subject to the validity of certain assumptions.

Self-avoiding walks interacting with a surface

Abstract: Discusses two models of polymer adsorption. In one model a self-avoiding walk on a D-dimensional lattice interacts with a (D-1)-dimensional hyperplane; in the other model, this walk must also lie in or on one side of this hyperplane. Both models exhibit non-analytic behaviour corresponding to a phase transition, but these phase transitions do not occur at the same point. The authors give numerical estimates of the locations of these transitions.

The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics

Abstract: Deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second-order ordinary equations the structure of Euler-Lagrange equations. The approach is based on a generalisation of previous studies on linear systems. The main result concerns a set of necessary and sufficient conditions for the existence of a multiplier, which contains an infinite set of algebraic equations, the coefficients of which can be used to derive necessary conditions involving only the given right-hand sides of the differential equations. An outline is given of interesting points for future studies, and an example is presented for which all multipliers are explicitly constructed.

Extended {BRS} Invariance and Osp(4/2) Supersymmetry

Abstract: A superfield action is proposed within an OSp(4/2) framework whose component form reproduces the covariant xi -gauge Yang-Mills action, but with modified ghost-compensating terms. (The case xi =0 reduces to the usual Landau gauge.) 'Supertranslations' give rise to extended BRS transformations, and lead to constraints amongst the renormalisation constants. In addition, the system admits 'super-Lorentz' transformations, which mix vector and ghost fields. For other field representations, the ghost structure suggested by the space-time supersymmetry OSp(4/G) is also exhibited. This simplifies the results for counting ghosts and their own ghosts.

Roughening transitions and the zero-temperature triangular Ising antiferromagnet

Abstract: The zero-temperature triangular Ising antiferromagnet is mapped onto a solid-on-solid (SOS) model. The system undergoes a roughening transition characterised by a critical exponent alpha =1/2, by the absence of excitations in the smooth phase, and by domain wall excitations (stripes) in the rough phase. At infinite SOS temperature the height-height correlation function is explicitly calculated with the aid of known four-point Ising correlations. The authors point out that a certain six-vertex model with a comparable SOS interpretation has an identical critical temperature, critical exponent and critical amplitude. This is in support of existing ideas on university in systems with striped phases.

Problems with metric-teleparallel theories of gravitation

Abstract: A general Lagrangian formulation of the metric-affine, metric compatible theories of gravitation is given. The applicability of the metric-teleparallel geometry to gravitation is considered. It is pointed out that a teleparallel theory with the Lagrangian usually accepted in the literature leads to a non-predictable behaviour of torsion. A new choice of the Lagrangian is proposed.

Collapse transition and crossover scaling for self-avoiding walks on the diamond lattice

Abstract: A Monte Carlo study of self-avoiding walks on the diamond lattice is presented. This model incorporates some of the steric effects and short-range stiffness of real alkanes, and for a nearest-neighbour attractive interaction- epsilon is found to have a collapse transition at kB theta / epsilon =2.25+or-0.05. The behaviour of the chains in the vicinity of this theta -temperature is analysed with the help of recent 'crossover scaling' theories. It is shown that for finite chain length N there is a rather broad theta -region where the chains' behaviour is quasi-ideal. The width of this region wtheta behaves as omega 0 varies as N-12/, consistent with the blob picture. The peak of the specific heat occurs at the boundary between the theta -region and the region of collapsed chains. The authors also give rough estimates for the scaling functions describing the crossovers of the end-to-end distance and structure factor of the chains.

On the density of states of Schrodinger operators with a random potential

Abstract: Using very recent results on ergodic theorems for superadditive processes on Rd, the authors prove the existence of the density of states for a wide class of random Schrodinger operators. In particular, new non-asymptotic estimates on the density of states are obtained and examples are discussed.

Morphology of ground states of two-dimensional frustration model

Abstract: The problem of generating ground states of a quenched random Ising spin system with variable concentration of mixed-neighbour exchange couplings (Jij()0) on a planar lattice (frustration model) is mapped into the problem of the Chinese postman which has been solved by a polynomial algorithm known as Edmond's algorithm. This algorithm is transposed and applied to the frustration problem. Not only is one particular ground state generated, but a post-optimal algorithm is established which gives the map of the rigid bonds and solidary spins (bonds in the same state for all ground states). This study of the rigidity on a square lattice reveals three distinct regimes by varying x, the concentration of negative bonds: a low-concentration regime where the ground states are rigid and ferromagnetic; an intermediate regime 0.1

Diffusion on percolation clusters at criticality

Abstract: The concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters. The authors find from Monte Carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three-dimensional space is D=3.3+or-0.1 where the size of the cluster is restricted to be larger than the span of the walk, and is D'=3.9+or-0.1 for a walk on clusters not subject to this restriction. For two-dimensional space they find D approximately=D' approximately=2.7+or-P0.1. The exponent D (and D') is related to the scaling of the average length R of N steps via RD varies as N. The fracton dimensionality which is related to the density of states was found to be 1.26+or-0.1. These results are in good agreement with the predictions of Alexander and Orbach (1982).

Renormalisation group recursions by mean-field approximations

Abstract: Within a renormalisation group strategy, different 'rescaled' mean-field approximations for the magnetisation are combined. Good qualitative estimates are obtained for the critical properties of classical and quantum spin systems. A particularly simple realisation of the method yields recursions for arbitrary dimensionality d. The critical couplings are accurate for all d, and have the correct asymptotic behaviour for d approaching infinity.

Hard hexagons: interfacial tension and correlation length

Abstract: Functional equations are derived for the eigenvalues of the row-to-row transfer matrix of the generalised hard hexagon model. These equations are exact for a lattice with N columns and are solved in the limit N to infinity . The partition function per site is rederived without the previous analyticity assumptions. The authors are also able to calculate the interfacial tension and the correlation length; the associated critical exponents are mu = nu = nu '=5/6 in agreement with the scaling relations.

On the correlation between sizes and shapes of cells in epithelial mosaics

Abstract: It is shown that Lewis's empirical, linear relationship between the average area of a cell and the number of its sides in two-dimensional mosaics corresponds to maximal arbitrariness in the cellular distribution. An expression for the distribution is given in the general case.

A class of new exact solutions in general relativity

Abstract: A class of new exact solutions is obtained for spherically symmetric and static configurations by considering a simple relation enu varies as (1+x)n. For each integral value of n the field equations can be solved exactly and one gets a new exact solution. For physical relevance of the solutions, the pressure and the density should be finite and positive and the density, P/ rho and dP/d rho should decrease as one goes outwards from the centre to the surface of the structure. Most of the exact solutions known at present are irregular in this respect. The new exact solutions for n=3, 4 and 5 are regular in this respect for a certain range of values of u(=mass/radius). The cases corresponding to n=1 and 2 are already available in the literature, being obtained by other methods. For regular solutions with dP/d rho

Critical kinetics near gelation

Abstract: The authors consider j-mers Ai reacting irreversibly according to the scheme Aj+Ak to Aj+k. The kinetic equations for the concentration of Aj are examined, and particularly their behaviour near gelation. Only the case Rjk=jalpha kalpha (0

Semiclassically weak reflections above analytic and non-analytic potential barriers

Abstract: The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross)-1)); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross)N. If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n) gives r approximately exp(-h(cross)-n(n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10-1000.

On the Sz=0 excited states of an anisotropic Heisenberg chain

Abstract: The Sz=0 excited states of the anisotropic antiferromagnetic Heisenberg Hamiltonian H= Sigma j=1N(SjxSj+1x+SjySj+1y+ rho SjzSj+1z) are studied when 0

Off-shell electromagnetic duality invariance

Abstract: Reviews the theorem that the Maxwell action, and not just the field equation, is invariant under duality transformations.

Closed-vortex-type solitons with Hopf index

Abstract: The structure of three dimensional solitons with non-trivial Hopf index is investigated for the S2 nonlinear sigma -model. It is shown that the corresponding regular solutions are of closed-vortex type. The author prove the existence of regular vortex-like solutions which are used for the approximation of solitons with large values of Hopf index.

Enumeration of directed site animals on two-dimensional lattices

Abstract: Studies the problem of directed site animals on the square, triangular and hexagonal lattices. Closed form expressions are proposed for A(s), the number of animals of size s, on the square and triangular lattices. These expressions have been checked for s

Infrared properties of forced magnetohydrodynamic turbulence

Abstract: The dynamical renormalisation group (RG) is implemented to study the large-scale properties of incompressible conducting fluid stirred by random forces and currents. In contrast with Navier-Stokes turbulence, invariance properties and dimensional constraints do not always prescribe the renormalisation of the couplings. In dimensions d>dc approximately=2.8, the system displays two non-trivial regimes: a kinetic regime where the renormalisation of the transport coefficients is due to the kinetic small scales, and a magnetic regime where it is due to the magnetic small scales. The results for the magnetic regime are not identical with predictions from the direct interaction approximation; this is due to vertex renormalisation of the Lorentz force. In dimensions 2

Exact solutions of the Schrodinger equation (-d/dx2 + x2 + λx2/(1 + gx2))ψ(x) = Eψ(x)

Abstract: The authors prove the existence of a class of exact eigenvalues and eigenfunctions of the Schrodinger equation for the potential x2 + λx2/(1 + gx2) when certain algebraic relations between λ and g hold. Some of the properties of these solutions are discussed. It is shown that in a certain sense they may be regarded as Sturmians for the Schrodinger equation with the potential x2 - λ/(g + g2x2).