Spin-glass dynamics and the barrier model: Extraction of the Parisi physical order parameter

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Abstract
The fact that the model of barrier heights monotonically increases with increasing metastable state Hamming distance (the difference between the spin orientations of any two states) on a hierarchical (ultrametric) tree can account for the ageing phenomenon in spin glasses. We make use of this model to extract spin-glass dynamics, including the instantaneous value of the zero-field-cooled magnetization M ZFC, and the time dependences of the thermoremanent magnetization M TRM(t w, t) or, equivalently, of the zero-field magnetization M ZFC(t w, t), where t is the measurement time after waiting a time t w, below the spin-glass transition temperature T g. The pure states introduced by Parisi are thought to originate from metastable states because finite barriers diverge as the temperature is lowered. These infinite barriers encompass metastable states separated by a self-similar distribution of barrier heights, independent of temperature. The model for extraction of spin-glass dynamics which we have used relies on the assumption that the states separated by barrier heights 0 ≤ Δ ≤ E z empty instantaneously into the new ground states upon a corresponding change in magnetic field energy E z associated with a change in magnetic field H. Previous magnetic field cycling experiments have shown that the exchange of occupation ‘respects’ the barrier heights; the new instantaneous ground-state occupation is bounded by the same set of barrier heights, 0 ≤ Δ ≤ E z, as the original state occupation. Subsequent diffusion within the initial manifold of states takes place over barriers with reduced magnitudes Δ - Ez , and to a sink arising from the change in Zeeman energy overcoming the barriers Δ ≤ Ez. This model can account for M TRM(t w, t) if the initial manifold is identified as the field-cooled state (with magnetization M FC), and the ground state upon H → 0 as the zero-magnetizatiotion state. We have solved the diffusion problem on the finite hierarchical tree in the presence of a sink at Δ ≤ Ez , enabling us to calculate M ZFC(t w, t). Our results are representative of experiment. We also show that we can account quantitatively for thermal cycling results by use of the measured temperature dependence of the barrier heights. We address the infinite network by relating M TRM(t w, t) to the Parisi physical order parameter P(D) given by P(D) = ω(t w)[d/dD(E z)][M TRM(t w, t)/M FC], where ω(t w) is the number of explored states during t w, and D(E z) is the Hamming distance appropriate to a barrier height Δ(D) = E z. We have measured M TRM(t w, t)/M FC over a wide range of temperatures and a very fine mesh of magnetic fields. Extracting P(D), we find agreement with the form of the mean-field results at different temperatures T. We relate the instantaneous (stationary) value of M ZFC as H → 0 to the strength of P(D) at D = 0: a δ function with area 1 — [xbar](T) in the Parisi mean-field solution. The temperature dependence of 1 — [xbar](T) generates the shape of M ZFC(t w, t → 0) against temperature. The mean-field solution leads to [M ZFC(t w, t → 0)/M FC]/T→0 = ½. The ratios at T → 0 found experimentally for a metallic spin glass Ag-6at.% Mn and an insulating spin glass CdCr1·7In0·3S4 approach 0·4 and 0·2 or less respectively. This implies that, as 1 — [xbar](T → 0) diminishes more from the mean-field result of ½, the more discrete the probability density for the exchange interactions.

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