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Journal of Physics C Solid State Physics

Probability density functions for collective coordinates in Ising-like systems

Ann Bruce

Published Sep 10, 1981• 116 citations
Publisher

Abstract

This paper describes a study of the behaviour of 'block' coordinates, measuring the spatial average, over a block of side L, of the local scalar ordering coordinates in a system undergoing a continuous phase transition. A general renormalisation group argument suggests that, in the limit in which both L and xi (the correlation length) are large compared with the lattice spacing, the block coordinate probability density function tends to a universal form, P infinity dependent on the ratio L/ xi . The function P infinity is calculated explicitly for dimension d=1. It is controlled by the statistical mechanics of kinks; intra-cluster ripples (phonons in the structural phase transition context) contribute identifiable corrections to the large L, xi limit. In the critical limit, L/ xi to 0, P infinity tends to a fixed point form P* with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* m with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* for d=3 and d=2; the latter result is substantiated by a calculation based on exactly established properties of the planar Ising model. In d=3P infinity is singly-peaked; in d=2 it has a strongly double-peaked character reminiscent of the d=1 system at low temperatures. The results are used to illuminate the nature of short-range order at criticality and the character of the collective excitation spectrum in the critical region, with particular reference to scattering experiments near structural phase transitions.

DOI
10.1088/0022-3719/14/25/012

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•Jan 1, 2008•1 citation
This paper describes a study of the behaviour of 'block' coordinates, measuring the spatial average, over a block of side L, of the local scalar ordering coordinates in a system undergoing a continuous phase transition. A general renormalisation group argument suggests that, in the limit in which both L and xi (the correlation length) are large compared with the lattice spacing, the block coordinate probability density function tends to a universal form, P infinity dependent on the ratio L/ xi . The function P infinity is calculated explicitly for dimension d=1. It is controlled by the statistical mechanics of kinks; intra-cluster ripples (phonons in the structural phase transition context) contribute identifiable corrections to the large L, xi limit. In the critical limit, L/ xi to 0, P infinity tends to a fixed point form P* with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* m with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* for d=3 and d=2; the latter result is substantiated by a calculation based on exactly established properties of the planar Ising model. In d=3P infinity is singly-peaked; in d=2 it has a strongly double-peaked character reminiscent of the d=1 system at low temperatures. The results are used to illuminate the nature of short-range order at criticality and the character of the collective excitation spectrum in the critical region, with particular reference to scattering experiments near structural phase transitions.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law $\ensuremath{\sim}{x}^{\ensuremath{-}a}$ for large separations xxx. We extend the Harris criterion for this case, finding that for a<da<da<d (where ddd is the spatial dimension) the disorder is irrelevant if $a\ensuremath{\nu}\ensuremath{-}2>0$, while if a>da>da>d we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An mmm-vector system of this type is studied with the use of a renormalization-group expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=4\ensuremath{-}a$. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent $\ensuremath{\nu}=\frac{2}{a}$. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.
PART I: DISORDER The effect of quenched impurities on systems which undergo first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random field Ising model is introduced which provides a simple physical explanation of the absence of any latent heat in two dimensions, and suggests that in higher dimensions such systems should exhibit a tricritical point with a correlation length exponent related to the exponents of the random field model by \nu = \nu_RF/(2-\alpha_RF-\beta_RF). A phase diagram unifying pure, percolative and non-trivial random behaviour is proposed. In two dimensions we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a magnetic exponent \beta/\nu which varies continuously with q, and a weakly varying correlation length exponent \nu ~ 1. For q>4 the first-order transitions of the pure model are softened due to the impurities, and the resulting universality class is different from that of the pure Ising model. We find strong evidence for multiscaling of the correlation functions, as expected for such random systems. PART II: FRUSTRATION Exact results for conformational statistics of compact polymers are derived from the two-flavour fully packed loop model on the square lattice. This loop model exhibits a two-dimensional manifold of critical fixed points each one characterised by an infinite set of geometrical scaling dimensions. We calculate these dimensions exactly by mapping the loop model to an interface model whose scaling limit is described by a Liouville field theory. The formulae for the central charge and the first few scaling dimensions are compared to numerical transfer matrix results and excellent agreement is found. Compact polymers are identified with a particular point in the phase diagram of the loop model, and the non-mean field value of the conformational exponent \gamma = 117/112 is calculated for the first time. We also present the very precise numerical estimate \kappa = 1.472801(10) for the connective constant. Interacting compact polymers are described by a line of fixed points along which \gamma and \kappa vary continuously. Our identification of compact polymers with a critical model has bearings on the cooperativity of protein folding thermodynamics. Namely, in the large chain limit, the absence of an energy gap separating the denatured states from the native (compact) ones implies that homopolymer collapse is a one-state process. We comment on the prospects of modeling specific sequence (amino acid) information through the imposition of quenched disorder.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
We discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two-parameter scaling form G(k)=kc(-2)g(kxi,k/kc), where k is the wave vector and xi is the correlation length, and the interaction-dependent nonuniversal momentum scale kc remains finite at the critical fixed point. The correlation function describes the entire critical regime and captures the classical to critical crossover. One-parameter scaling is recovered only in the limit k/kc-->0. We present an approximate calculation of g(x,y) for the Ising universality class using the functional renormalization group.
Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities. However, a class of lattice statistics models show that rather more complex behaviour can be obtained, under conditions which are suffciently general that they might also occur in less idealised models. A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called phase. The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions) occur over a finite range of temperatures. The state of the massless phase is characterised by a topological ordering with analogues of spin-waves being the dominant perturbations. The transition from the ordered to massless is analysed using new exact series expansions obtained by the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared to other studies in the small number of cases where such series expansions exist for comparable models, including the 6-state clock model. One of the properties that makes the massless phase dificult to study is the weak nature of the associated Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent cross-over to power law behaviour. This suggests that behaviour analogous to the phases will be diffcult to characterise as one moves beyond idealised lattice systems.