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arXiv:cond-mat/0612476 · Physical Review Letters

Comment on “Random-Field Spin Models beyond 1 Loop: A Mechanism for Decreasing the Lower Critical Dimension”

Yoshinori Sakamoto, Hisamitsu Mukaida, Chigak Itoi

Published Jun 29, 2007• 4 citations
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Abstract

A comment on the Letter by Le Doussal and Wiese, Phys. Rev. Lett. 96, 197202 (2006).
DOI
10.1103/physrevlett.98.269703

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Kühn and Mazzeo Reply:

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We reply to a comment of Van Enter, Kuelske and Maes (cond-mat/0005176) on our letter "Critical Behavior of the Randomly Spin-Diluted 2-d Ising Model - A Grand Ensemble Approach", Phys. Rev. Lett. {\bf 73}, 2268-2271 (1994).

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In a recent letter [PRL 80 (1998) 3539] Fisher, Le Doussal and Monthus report new predictions for the persistence properties of Sinai's model, which they obtain by using an approximate real space renormalization group scheme. In this comment we show that these predictions can be derived in a rigorous and more transparent way by reducing the quantities of interest to particular statistical properties of a simple random walk in a homogeneous environment. We also point out that they are valid in a much more general context: our results are neither restricted to the random force model but are valid for any asymmetric hopping model, nor to the vicinity of the critical point.
The functional renormalization group for the random-field and random-anisotropy O(N) sigma models is studied to 2 loop. The ferromagnetic-disordered (F-D) transition fixed point is found to next order in d = 4 + epsilon for N > N(c) (N(c) = 2.834 740 8 for random field, N(c) = 9.441 21 for random anisotropy). For N < N(c) the lower critical dimension d = d(lc) plunges below d(lc) = 4: we find two fixed points, one describing the quasiordered phase, the other is novel and describes the F-D transition. d(lc) can be obtained in an (N(c)-N) expansion. The theory is also analyzed at large N and a glassy regime is found.