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Physical Review Letters

Droplet Theory of Low-Dimensional Ising Models

Ann Bruce, D. J. Wallace

Published Dec 14, 1981• 40 citations
Publisher

Abstract

A droplet theory of Ising systems is constructed, valid near space dimension $d=1$, and based on a model incorporating only the configurational energy of surface tension. The droplet distribution function is calculated explicitly, with use of renormalization-group methods to control the droplet-shape fluctuation effects essential to its scaling form. Universal quantities are calculated; those (e.g., $\ensuremath{\beta}$) which reflect the droplet concentration have an essential singularity in $\ensuremath{\epsilon}=d\ensuremath{-}1$.

DOI
10.1103/physrevlett.47.1743

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A droplet theory of Ising systems is constructed, valid near space dimension d=1d=1d=1, and based on a model incorporating only the configurational energy of surface tension. The droplet distribution function is calculated explicitly, with use of renormalization-group methods to control the droplet-shape fluctuation effects essential to its scaling form. Universal quantities are calculated; those (e.g., \ensuremathβ\ensuremath{\beta}\ensuremathβ) which reflect the droplet concentration have an essential singularity in \ensuremathϵ=d\ensuremath−1\ensuremath{\epsilon}=d\ensuremath{-}1\ensuremathϵ=d\ensuremath−1.
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.
The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi = 1 / 2 follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.
By an exact calculation on the planar Ising model we show that for all subcritical temperatures, the bubble model predicts the correct average distribution of matter contributing to the pair-spin correlation function in the two-particle sector. This considerably substantiates the bubble model which is generalized to higher dimensions with an external field h and then used to show that the free energy has an essential singularity in h of Andreev-Fisher form at all points along the phase boundary. We argue that such a singularity would also occur in the Baxter model.
The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T c , and becomes essentially independent of temperature below T c . Both in the vicinity of T c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.
The equation of state and the critical amplitudes are calculated for the Ising model in two and three dimensions on the basis of a modified form of the droplet model for critical phenomena proposed by Reatto. A droplet formula for the spin-spin correlation function is proposed. This formula has a homogeneous form but the scaling laws involving the dimensionality do not hold necessarily. The droplet model for the Heisenberg Hamiltonian is considered in detail by taking into account that the magnetization of a droplet can be in any direction.