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arXiv:cond-mat/9903410 · Nuclear Physics B

Mean-field expansion for spin models with medium-range interactions

Andrea Pelissetto C, Paolo Rossi, Ettore Vicari

Published Aug 1, 1999• 18 citations
arXivHTMLPDF
DOI
10.1016/s0550-3213(99)00311-9

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We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, which corresponds to the limit N-->0 of an N-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed by using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions for the critical crossover functions, finding good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal crossover behavior of our data for any finite range.
The dynamical mean field theory of spin glasses is investigated in systems with non-zero mean of the infinitely-ranged random exchange interactions under uniform but alternating external magnetic fields. A perturbational expansion with respect to the interaction uσi4 introduced in a soft-spin version of the model is analyzed by means of the diagrammatic method. Restricting ourselves to its lowest order (single-loop) approximation, we examine relaxational dynamics of the model spin glass and clarify nature of its phase transition in the presence of an ac external field. The results provide a new clue to ac nonlinear measurements on spin glasses.
We introduce a systematic method for expanding general spin-glass Hamiltonians in terms of Mattis interactions, providing a novel perspective for understanding the fundamental differences between short-range Edwards-Anderson (EA) and mean-field Sherrington-Kirkpatrick (SK) spin glasses. By iteratively extracting patterns from the coupling matrix, we expand the original spin-glass system into a Hopfield-like model (a series of Mattis interactions) plus a residual system. Our analysis reveals profound distinctions between EA and SK models: while EA models in two and three dimensions break into isolated subconnected sections after expansion, the SK model exhibits remarkable self-similar behavior, with the residual system preserving the mean-field structure and Gaussian statistics throughout the expansion process. This self-similarity manifests in exponential decay of residual matrix norms and expansion coefficients, reflecting the inherent mean-field nature of the SK model. Furthermore, we demonstrate that pattern expansion can identify ultra-low energy excitations in EA models, revealing excitations with energies that decrease rapidly with expansion step. Through connected component analysis, we quantify the size-energy relationship of these independent excitation clusters, opening new avenues for understanding the low-energy landscape of spin glasses and providing insights into the nature of metastable states.