SIMPLEXITY OF GEOMETRICALLY FRUSTRATED ISING ANTIFERROMAGNETS

Taras Yavors'kii

Applied Mathematics Research Centre, Coventry University

The notion of independent variables is central in mathematics. Sequences of independent random variables made up the setting for the first non-trivial mathematical results of the probability theory. The central limit theorem is perhaps the best known such result. [1]
Yet, the beauty, richness and complexity of the physical world has its origin in interactions and correlations between degrees of freedom. For instance, geometrically frustrated spin systems at temperatures, lower than the scale of the leading spin interactions, can form massively degenerate correlated dynamical states, called spin liquids or collective paramagnets. Non-trivial properties of these states are believed behind the many exotic, quantum and classical, phenomena observed in geometrically frustrated condensed matter systems [2], including observed disobedience to the third law of thermodynamics.
Since several years, graphics processing units (GPUs) have been increasingly used in physics research as a powerful and value general purpose computational alternative to regular computers [3]. In this talk I use GPUs to study the classical spin liquid states of the geometrically frustrated systems. Specifically, I run GPU-aided Monte Carlo simulations [4] on the nearest-neighbor antiferromagnetic Ising spin models on the two-dimensional kagome and the three-dimensional pyrochlore lattice.
I show that, down to the degenerate ground state manifolds, the spin correlations in such models show features consistent with the picture of independent random variables, usually describing spin models at high temperatures. Statistical physics properties of the models, such as pair correlation function, are shown to be well described by the variational single-particle mean-field theory [5] (MFT) ansatz at all T ≥ 0, provided the MFT temperature scale Θ, where Θc < Θ < ∞, is mapped onto the physical temperature scale 0 ≤ T < ∞ by considering Θ as a suitable function of T. The models are thus completely "transparent" to the paramagnetic MFT treatment deep below the MFT critical temperature Θc > 0, making MFT a simple and powerful tool for the study of perturbations at low T [6].
[1] A.N. Kolmogorov. Foundations of the theory of probability. Chelsea Publishing Company, 1956.
[2] Introduction to frustrated magnetism: materials, experiments, theory. Eds. C. Lacroix, P. Mendels, F. Mila. Springer, 2011.
[3] Computer simulations on graphics processing units. Eds. M. Weigel, A. Arnold, P. Virnau. Eur. Phys. J. Special Topics, 210 (2012).
[4] The GPU code used here has many common elements with the code described. In: T. Yavors'kii, M. Weigel. Optimized GPU simulations of continuous-spin glass models. p. 159 of Ref. [3].
[5] P.M. Chaikin, T.C. Lubensky. Principles of condensed matter physics. Cambridge University Press, Cambridge, UK, 1995.
[6] T. Yavorskii, M. Enjalran, M.J.P. Gingras. Spin Hamiltonian, competing small energy scales, and incommensurate long-range order in the highly frustrated Gd₃Ga₅O₁₂ garnet antiferromagnet. Phys. Rev. Lett. 97 (2006) 267203.
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