We propose a new stochastic algorithm (generalized simulated annealing) for computationally finding the global minimum of a given (not necessarily convex) energy/cost function defined in a continuous D-dimensional space. This algorithm recovers, as particular cases, the so called classical ("Boltzmann machine") and fast ("Cauchy machine") simulated annealings, and can be quicker than both.

We study the critical properties of a two-dimensional Ising model with competing ferromagnetic exchange and dipolar interactions, which models an ultrathin magnetic film with high out-of-plane anisotropy in the monolayer limit. We present numerical evidence showing that two different scenarios appear in the model for different values of the exchange to dipolar intensities ratio, namely, a single first-order stripe-tetragonal phase transition or two phase transitions at different temperatures with an intermediate Ising nematic phase between the stripe and the tetragonal ones. Our results are very similar to those predicted by Abanov et al. ͓Phys. Rev. B 51, 1023 ͑1995͔͒, but suggest a much more complex critical behavior than predicted by those authors for both the stripe-nematic and the nematic-tetragonal phase transitions. We also show that the presence of diverging free energy barriers at the stripe-nematic transition makes possible to obtain by slow cooling a metastable supercooled nematic state down to temperatures well below the transition one.

Barci and Stariolo Reply: The focus of our work [1] was to identify conditions for the presence of an isotropicnematic phase transition in the context of a generic system with isotropic competing interactions. By taking into account nontrivial angular momentum contributions from the interaction, we found a second order isotropic-nematic phase transition at mean field level, which becomes a Kosterlitz-Thouless one [2] when fluctuations are taken into account.In his Comment [3], Levin criticizes our results by showing that the low temperature fluctuations of a stripe phase in 2d diverge linearly in the thermodynamic limit. His analysis is restricted to the stripe phase and, contrary to what is suggested in the Comment, does not apply to the central result of our Letter which is the existence of an isotropic-nematic phase transition. In fact, as clearly anticipated by us in the Letter [1], the corresponding analysis of the fluctuations of the nematic order parameter displays a logarithmic divergence leading to a low temperature phase with quasi-long-range order.In our model, despite the involved calculations, it is straightforward to understand this fact. Introducing the nematic order parameterQ ij n inj ÿ 1 2 i;j [wherê n i cos; sin is the director field] through a Hubbard-Stratonovich transformation, it is possible to decouple the quartic terms. Integrating out the field, we obtain the following long wavelength effective free energy for the nematic order parameter: FQ a 2 =2Tr Q 2 a 4 =4Tr Q 4 =4Tr QDQ . . . , where the symmetric derivative tensor D ij r i r j and a 2 , a 4 , and are temperature dependent coefficients given in terms of the parameters of the original model. At mean field, the last term is zero, and we find ÿa 2 =a 4 p for a 2 < 0, going continuously to 0 for a 2 > 0. Note that any global rotation of the order parameter costs no energy. Therefore, parametrizing the order parameter by a modulus and an angle, the long wavelength angle fluctuations x dominate the low energy physics. Computing the free energy at lowest order in the derivatives of the angle fluctuations, we find F 2 R d 2 xjrj 2 , where F is the excess of free energy relative to the saddle point value. Therefore, the free energy of fluctuations corresponds to that of the XY model. The only difference with the usual vector orientational order is that the system should have the symmetry ! modifying the vorticity of the topological defects. Thus, one finds for the angle fluctuations hxx 0 i lnk 0 x ÿ x 0 , which in turn lead to an algebraic decay of the order parameter correlations. In an extended paper we will show the explicit dependence of the Frank constant KT 2 with the parameters of our

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We show results from Monte Carlo simulations of a two dimensional Heisenberg model for ultrathin films with perpendicular anisotropy. A complete phase diagram is obtained as a function of anisotropy and temperature, spanning a wide range of behavior. We discuss our results in relation with experimental findings in different ultrathin films. We observe and characterize a line of Spin Reorientation Transitions . This transition from out of plane stripe order to in plane ferromagnetic order presents a paramagnetic gap in between in a finite region in parameter space, as reported in experiments. For large anisotropies direct transitions from a low temperature stripe phase to a paramagnetic or tetragonal phase with dominant perpendicular magnetization is observed, also in agreement with experiments.We also show the phase diagram for a system without exchange, i.e. with pure dipolar and anisotropy interactions. It shows a similar behavior to the ferromagnetic case with antiferromagnetic instead of stripe phases at low temperatures. A Spin Reorientation Transition is also found in this case.