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Physical review. B, Condensed matter

Excluded-volume effects in magnetic polymers

Mustansir Barma

Published Dec 1, 1978• 3 citations
Publisher

Abstract

It is pointed out that the spin-spin correlation function in a magnetic-polymer model is proportional to the generating function of the probability function describing the spatial distribution of monomers. This observation is used to generalize Thorpe's study of a freely jointed magnetic polymer, to take the excluded-volume effect into account using a scaling theory. The inverse magnetic correlation length measured along the polymer plays the role of a scaling field which can, therefore, be controlled experimentally. However, strong disorder in the magnetic coupling can prevent this field from reaching its critical value, zero.

DOI
10.1103/physrevb.18.6398

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Journal of Physics A Mathematical and General•Jul 7, 1996•0 citations
It is pointed out that the spin-spin correlation function in a magnetic-polymer model is proportional to the generating function of the probability function describing the spatial distribution of monomers. This observation is used to generalize Thorpe's study of a freely jointed magnetic polymer, to take the excluded-volume effect into account using a scaling theory. The inverse magnetic correlation length measured along the polymer plays the role of a scaling field which can, therefore, be controlled experimentally. However, strong disorder in the magnetic coupling can prevent this field from reaching its critical value, zero.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.
For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.
We study the general behaviour of the correlation length for the two-point correlation function of the local fields in an Ising chain with binary distributed fields. At zero field it is shown that is the same as the zero-field correlation length for the spin - spin correlation function. For the field-dominated behaviour of we find an exponent for the power-law divergence which is smaller than the exponent for the spin - spin correlation length. The entire behaviour of the correlation length can be described by a single crossover scaling function involving the new critical exponent.