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Physical review. B, Condensed matter

Local magnetic field distributions. II. Further results

T. C. Choy, David Sherrington, M. F. Thomsen, Michael F. Thorpe

Published Jun 1, 1985• 7 citations
Publisher

Abstract

The concept and character of the local magnetic field distribution P(h), developed in an earlier paper, is extended to higher space and spin dimensions and to several special but spatially periodic systems, disorder being the subject of a future paper. For Ising spins on standard lattices, P(h) is shown to have a shape which is dimension dependent but relatively coordination independent. While in two dimensions (2D) P(h) has a dip at h=0 at ${T}_{c}$, in 3D it is extremely flat near the origin, in 4D it is pseudo-Gaussian, while in all dimensions it is qualitatively similar to a discrete form of the corresponding universal block-spin probability function. Bethe lattices, studied for all spin dimensions, are shown to be quite different. Results are also presented for P(h) and the ground-state degeneracies of some one-dimensional frustrated systems exhibiting high-degeneracy disorder points.

DOI
10.1103/physrevb.31.7355

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Physical Review•Dec 10, 1969•125 citations
The concept and character of the local magnetic field distribution P(h), developed in an earlier paper, is extended to higher space and spin dimensions and to several special but spatially periodic systems, disorder being the subject of a future paper. For Ising spins on standard lattices, P(h) is shown to have a shape which is dimension dependent but relatively coordination independent. While in two dimensions (2D) P(h) has a dip at h=0 at Tc{T}_{c}Tc​, in 3D it is extremely flat near the origin, in 4D it is pseudo-Gaussian, while in all dimensions it is qualitatively similar to a discrete form of the corresponding universal block-spin probability function. Bethe lattices, studied for all spin dimensions, are shown to be quite different. Results are also presented for P(h) and the ground-state degeneracies of some one-dimensional frustrated systems exhibiting high-degeneracy disorder points.
The structure of the three-dimensional (3D) random field Ising magnet is studied by ground-state calculations. We investigate the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at $[\ensuremath{\Delta}/J{]}_{c}\ensuremath{\simeq}2.27,$ where $\ensuremath{\Delta}$ is the standard deviation of the Gaussian random fields (J=1).(J=1).(J=1). With an external field H there is a disorder-strength-dependent critical field $\ifmmode\pm\else\textpm\fi{}{H}_{c}(\ensuremath{\Delta})$ for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. ${H}_{c}\ensuremath{\sim}(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{p}{)}^{\ensuremath{\delta}},$ where ${\ensuremath{\Delta}}_{p}=2.43\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\delta}=1.31\ifmmode\pm\else\textpm\fi{}0.03,$ implying a critical line for ${\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}.$ When, with zero external field, $\ensuremath{\Delta}$ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=1.00$ to ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=0.$ This is located at $\ensuremath{\Delta}=2.32\ifmmode\pm\else\textpm\fi{}0.01,$ i.e., above ${\ensuremath{\Delta}}_{c}.$ The spanning cluster has the fractal dimension of standard percolation, Df=2.53{D}_{f}=2.53Df​=2.53 at ${H=H}_{c}(\ensuremath{\Delta}).$ We provide evidence that this is asymptotically true even at H=0H=0H=0 for ${\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}$ beyond a crossover scale that diverges as ${\ensuremath{\Delta}}_{c}$ is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.
Abstract We revisit the Haake–Lewenstein–Wilkens approach to Edwards–Anderson (EA) model of Ising spin glass (SG) (Haake et al 1985 Phys. Rev. Lett. 55 2606). This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, q EA . We use saddle point/steepest descent (SPSD) method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that q EA > 0 at 0 < T < T c in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts SG order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: (i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; (ii) there is (no) SG transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak.
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.
We continue our investigation of an Ising model with immobile random impurities by studying the spin-spin correlation functions. These correlations are not probability-1 objects and have a probability distribution. When the random bonds have the particular distribution function studied in the first paper of this series, we demonstrate that the average value and the second moment of the temperature derivatives of these correlations are infinitely differentiable but fail to be analytic at Tc{T}_{c}Tc​, the temperature at which the observable specific heat fails to be analytic. When T<TcT<{T}_{c}T<Tc​, we consider ${S}_{\ensuremath{\infty}}(l)=limit of\text{}〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{l,m}〉\text{as}m\ensuremath{\rightarrow}\ensuremath{\infty}$. This limit is not independent of lll. In the special case that the random bonds are symmetrically distributed about the lllth row, the geometric mean of ${S}_{\ensuremath{\infty}}(l)$ is computed and shown to vanish exponentially rapidly when $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. We contrast this with a lower bound that shows that the spontaneous magnetization can vanish no more rapidly than ${T}_{c}\ensuremath{-}T$, and present a description of how the local magnetization ${S}_{\ensuremath{\infty}}{(l)}^{\frac{1}{2}}$ behaves as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.