Exponent Nu Research Articles

Magnetic phase transition in nickel with uniaxial compressive and tensile stress

Small-angle neutron scattering experiments have been performed on single crystals of nickel using the diffractometer D11 at the Institute Laue-Langevin, Grenoble. The correlation length zeta of the critical spin fluctuations and the static susceptibility xi have been measured for reduced temperatures 10-6<or= tau =(T-Tc)/Tc<or=10-2 with Tc approximately=630K. Uniaxial compressive and tensile stress could be applied to the sample as external and variable anisotropy parameters. The results show the expected decrease of the values of the critical exponents gamma and nu along the induced easy magnetic directions as well as the finite size of the fluctuations for Tc from T along the hard directions, below the cross-over temperature. An analysis of the scattered intensities in the two-dimensional multidetector plane gives information on the influence of crystalline and stress anisotropy on the critical fluctuations at small reduced temperatures above Tc.

Universality of neighbour-avoiding and self-avoiding walks on the square lattice

It is shown that, within position-space renormalisation group (PSRG) theory, neighbour-avoiding walks (NAWs) and self-avoiding walks (SAWs) on the square lattice obey the same 'end-to-end distance' critical exponent nu . This universality is shown by applying a recent finite lattice renormalisation transformation to ordinary and oriented NAW and SAW problems and then using an exact equivalence between SAWs on an oriented lattice and NAWs on its covering lattice. The universality class includes several oriented walk problems.

Cluster-quench simulation of the two-dimensional random Ising model

A simulation technique is introduced which generated states stable against spin reversal of all spin clusters of n or fewer spins. The distribution of energy barriers for reversing m-spin clusters (m>n) and the distribution of energy gained are computed. The results are interpreted using a renormalisation-group picture with n1/2 as the length scale. The data scale with correlation length exponent nu =2.6 and an average energy barrier increasing approximately as 1-n-1/2 nu . These results provide a detailed test of the phenomenological scaling theory of spin glasses.

Lower critical dimension of Ising spin glasses: a numerical study

A transfer matrix method is used to study the variation with length scale L of the distribution PL(Jeff) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod Jeff mod ) varies as L-1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L-1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.

Series study of the one-dimensional 'true' self-avoiding walk

The 'true' self avoiding walk problem is formulated using a grand canonical approach, and exact enumeration methods are used to calculate the average end-to-end distance for one-dimensional 'true' self-avoiding walks with up to 21 steps. The results are in agreement with a universality picture obtained both from Monte Carlo simulations and from scaling and crossover arguments. The extrapolated value of the end-to-end distance exponent nu is nu =0.67+or-0.04.

Topological properties of percolation clusters

Percolation properties, including the total cluster mass S, the shell mass B, and the linear geometrical size R, are studied as a function of the topological 'chemical distance' parameter L. All critical exponents are shown to be related to an apparently new exponent nu , defined by R approximately Lv. Critical exponents are calculated exactly for percolation clusters on the Cayley tree (a model for 6D percolation), for which S approximately L2, B approximately L, and R2 approximately L. For diffusion on such clusters one finds that L3 approximately t. Numerical estimates of the exponents are obtained for other dimensions. A conjecture which relates nu to beta and nu is discussed.

Bootstrap percolation: a renormalisation group approach

In bootstrap percolation, sites are occupied at random with probability p, but each site is considered active only if at least m of its neighbours are also active. Within an approximation position-space renormalisation group framework on a square lattice the authors obtain the behaviour of the critical concentration pc and of the critical exponents nu and beta for m=0 (ordinary percolation), 1,2 and 3. They find that the bootstrap percolation problem can be cast into different universality classes, characterised by the values of m.

Real-space renormalisation group for directed systems

A new real-space renormalisation group method is developed to treat systems with directionally dependent critical behaviour. Here two effective lengths S/sub ///(b) and Sperpendicular to (b) are defined for a given cell of linear size b. The renormalised lattice is constructed from these effective lengths and is deformed from the original lattice. It is the anisotropic rescaling of these effective lengths which give the anisotropic exponents nu /sub /// and nu perpendicular to . This method gives the exact result for the case of fully directed SAWs on a square lattice in the b to infinity limit. For directed bond and directed site lattice animals, the results obtained here are also in good agreement with the other known values.

Scaling theory of dead-end distribution in percolation clusters

On the basis of scaling assumptions, the critical behaviour of the distribution (number) of dead (dangling) ends in the infinite cluster of percolating systems is investigated. Critical exponents describing the dead-end distribution are introduced and scaling relations between them are derived. The exponents are also expressed in terms of other exponents nu , beta and beta B for percolation and their explicit values are evaluated from known estimates for nu , beta and beta B.

Anomalous acoustic behaviour and backbone structure of percolation clusters

A real-space renormalisation group method is used to investigate acoustic properties of percolation lattices in the vicinity of the percolation threshold. Scaling form an expressions for critical exponents x, y describing the sound velocity are derived. The existence of an anomalous dispersion relation is elucidated. Dimensionalities dT and dL which represent the geometrical structure of cluster backbones are introduced. The exponents x, y and the dimensionalities dT, dL are expressed in terms of other exponents nu , beta , t, s and beta B for percolation and their explicit values are evaluated from known estimates for nu , beta , t, s and beta B. A geometrical picture of the backbone structure is obtained in terms of the dimensionalities.

A hyperscaling relation in site-bond correlated percolation

A novel approach is used to construct a field theoretical model of site-bond percolation. The exponents nu and eta are calculated in an epsilon =6-d expansion for both Ising and percolation fields, and the scaling relation nu p=2/(d-2) is demonstrated to hold to order one loop for d>4. Using the equations of motion, the scaling relation nu p=2/(d-2+ eta 1) is shown to hold to all orders, extending below four dimensions. Although this scaling relation is shown to fail for d=1+ epsilon , it is argued that it is still valid for d>or=3.

A short note on directed self-avoiding walks

The critical behaviour of partially directed self-avoiding walks (SAWs) on a cubic lattice with only one axis directed is studied using exact enumerations. The critical exponents ( nu /sub ///=0.999+or-0.005, nu perpendicular to =0.501+or-0.006 and gamma =1.000+or-0.001) found here further support the universality hypothesis that various partially directed SAWs belong to the same universality class as fully directed SAWs and they are also dimension independent for d>or=2.

'True' self-avoiding walk on fractals

The authors investigate the 'true' self-avoiding walk on fractal spaces. The number SN of sites visited during an N-step walk and the root-mean-square displacement RN are calculated via extensive Monte Carlo simulations on the 2D Sierpinski gasket. For all positive values of the repulsion parameter g, they found the same exponents: s'=0.815+or-0.001 and nu '=0.510+or-0.005 respectively. Both RN and SN are shown to exhibit a simple scaling behaviour in N and g. The authors compare the results with a simple Flory-like prediction for the exponent nu '.

Statistics of the true self-avoiding walk in one dimension

Numerical results of a Monte Carlo study for the true self-avoiding walk in one dimension are presented. For any positive value of the strength parameter (0<g< infinity ), the root-mean-square displacement RN and the range SN of the walk are characterised by two universal exponents nu =2/3+or-0.003 and s=2/3+or-0.01 respectively. For negative g (self-attracting walk) both RN and SN exhibit a saturation effect: limN to infinity RN=Rinfinity (g) and limN to infinity SN=Sinfinity (g) with Rinfinity (g) approximately (-g)-1 and Sinfinity (g) approximately (-g)-1 at g<or approximately=0. A simple scaling analysis in N and g is proposed and found to be consistent with the Monte Carlo results.

Real space renormalisation theory of self-avoiding walks on a Manhattan lattice

The real space renormalisation group theory is applied to the self-avoiding walk (SAW) problem on a two-dimensional oriented lattice known as the Manhattan lattice. A finite lattice renormalisation transformation is used to calculate the connectivity constant mu and the size exponent nu . A more general SAW model is defined that contains the oriented and nonoriented lattice problems as special cases. A renormalisation group analysis of this model provides evidence that the SAW on the Manhattan lattice is the same universality class as the ordinary nonoriented SAW problem.

Generalised self-avoiding walk

A generalisation of the self-avoiding walk is introduced in which k or higher multiple points are forbidden (k=2 corresponds to the standard self-avoiding walk). The Flory theory gives the radius of gyration exponent nu k=(k+1)/((k-1)E+2) when E<or=Ec(k)=2k/(k-1). E is the Euclidean dimension of the problem and Ec(k) the upper critical dimension which is also obtained using the fractal set theory.

Numerical study of self-avoiding loops on d-dimensional hypercubic lattices

The loop gas in d=2, 3, 4 and 5 dimensions and with multiplicities m=0, 1, 1.5, 2, 3 and 4 is investigated by the Monte Carlo method. The critical temperatures and approximate values for the critical exponent nu corresponding to second-order phase transitions are obtained for d=2 and 3.

Critical exponent of a directed self-avoiding walk

The authors have investigated numerically and analytically the directed self-avoiding walk problem recently proposed and studied by Chakrabarti and Manna (1983). They find a different result for the exponent nu , namely nu =1.

X-ray scattering in the normal and incommensurate phases of Rb2ZnCl4

The static critical behaviour at the normal to incommensurate phase transition in Rb2ZnCl4 has been investigated by means of diffuse X-ray scattering. The measured values of the critical exponents gamma and nu are in good agreement with theoretical estimates for the d=3XY model. Measurements of the diffuse scattering in the incommensurate phase just below Tc are consistent with those expected if there are nearly gapless phase excitations. At lower temperatures an asymmetry develops in the predominantly acoustic mode scattering emanating from the primary satellite reflections which is attributed to coupling of acoustic and phason modes. The temperature dependence of the intensity of several higher-order satellites has been measured at temperatures just above the lock-in transition but is not understood.

Droplet theory in low dimensions: Ising systems in zero field

The authors develop a theory of the universal configurational physics underlying critical point phenomena in the Ising universality class. The theory is formally justifiable in d=1+ epsilon dimensions and may be regarded as the natural continuation of the kink-based theory of one dimension, which it incorporates as a limiting case. In d=1+ epsilon the configurational building block is the droplet. The typical droplet is not spherical and the many-droplet assembly is not dilute: the implied problems are handled with renormalisation group methods. It is found that droplet shape fluctuation effects control the correlation length exponent nu , while the nesting of droplets within droplets control the order parameter exponent beta . The exponents nu and beta thus effectively define, respectively, the fractal dimensions of the droplet surface and the droplet volume. The theory is used to determine and illuminate the critical behaviour of further quantities including the free energy, the susceptibility, the droplet number distribution and the distribution of the intra-droplet order.