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J. Villain and J.F. Fernandez, Z. Phys. B 54, 139 (1984) and Ref. 1 gave the same estimates for the pinning and elastic energies as we do here, but implied that these estimates remain valid for L>(Formula presented), and concluded from that argument that the system was unstable towards the proliferation of free vortices at length scale L≫:(Formula presented). However, since these estimates are really obtained from perturbing about the unpinned state, their range of validity is restricted to L⩽(Formula presented), as was argued in Ref. 12.
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There exist other cases where two divergent length scales govern the physics of the problem. Examples are the zero-temperature quantum transition in the transverse-field Ising spin glass [M. Guo, R.N. Bhatt and D.A. Huse, Phys. Rev. Lett. 72, 4137 (1994)] and the phase transition(s) in the d=2 uniformly frustrated (UF) XY models [P. Olsson, Phys. Rev. Lett. 75, 2758 (1995)]. In the former, the static (Formula presented) and dynamic (Formula presented) correlation lengths diverge with different critical exponents at a unique critical transverse field. There, a finite-size scaling characterizing the transition could be done by varying the aspect ratio of the simulation cell. In the UF XY model, there exists the possibility of two very close phase transitions preventing a unique (independent of system size) crossing of Binder ratios of the staggered magnetization (Formula presented) (our results for the size and field dependence of various moments of the static uniform magnetization in the random-field XY model are qualitatively very similar to those for (Formula presented) in the UF XY model). In the UF XY model, some progress may have been made because of the theoretical analytical input used in analyzing the scaling behavior of various extracted thermodynamic quantities. In our case, we do not dispose of a well-established theoretical framework to allow us to perform a one-parameter scaling in terms of a "renormalized" coupling. (We thank E.S. Sørensen for suggesting that we clarify this issue.)
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