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Physical review. B, Condensed matter

Lower critical dimensionality of Heisenberg spin-glasses

Marek Cieplak, Jayanth R. Banavar

Published Jan 1, 1984• 29 citations
Publisher

Abstract

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at $T=0$ are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.

DOI
10.1103/physrevb.29.469

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Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at T=0T=0T=0 are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.
A new equilibrium characterization of the spin-glass-paramagnet transition is presented. It is based on the size dependence of the sensitivity of the free energy to the boundary conditions. Results of numerical studies of a three-dimensional Heisenberg spin-glass at T=0T=0T=0 are consistent with an algebraic decay of this sensitivity and suggest a zero-temperature phase transition.
Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at T=0T=0T=0 are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a $\ifmmode\pm\else\textpm\fi{}J$ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.
Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.
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This thesis addresses the surprising features of zero-temperature statics and dynamics of several spin glass models, including correlations between soft spins that arise spontaneously during avalanches, and the discovery of localized states that involve the presence of two-level systems. It also presents the only detailed historiographical research on the spin glass theory. Despite the extreme simplicity of their definition, spin glasses display a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis the author sheds light on some of these, focusing on both the search for phase transitions under perturbations of Hamiltonians and the zero-temperature properties and responses to external stimuli. After introducing spin glasses and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are presented: The first of these examines the de Almeida Thouless transition, and proposes a new finite-size scaling ansatz, which accelerates the convergence to the thermodynamic limit. The second reconstructs the phase diagram of the Heisenberg spin glass with random exchange anisotropy
Results of numerical studies on the equilibrium scaling stiffness energy for 2D nearest-neighbour Heisenberg systems at T=0 are presented. The authors first consider the percolation problem and estimate the critical exponent for the spin wave stiffness to be close to 1. They then consider systems in which -J-bonds appear with probability x and +J-bonds with probability (1-x). The scaling stiffness energy, a measure of sensitivity to changes in the boundary conditions, disappears exponentially with x at x c approximately=0.13 (an estimate for large systems). This value of concentration is identified as an upper boundary of the usual ferromagnetic phase. Nevertheless the magnetisation persists above x c and disappears at X c approximately=0.30 when the system enters into the spin glass phase. The region between x c and X c is identified as corresponding to the Gabay-Toulouse mixed phase in which the system looks like a spin glass but has some non-zero magnetisation. This interpretation receives further support from studies of the two- and four-spin correlation functions in local energy minima. The correlations decay exponentially, at least on small distances. The correlation lengths of the two-spin correlations start to drop at x c . The spin components parallel to the magnetisation are actually less correlated than the transverse ones.