# Three-dimensional Ising model with nearest- and next-nearest-neighbor interactions

**ABSTRACT** The phase diagram of the Ising model in the presence of nearest- and next-nearest-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by employing an effective-field theory with finite clusters consisting of a pair of spins. A functional form is also proposed for the free energy, similar to the Landau expansion, in order to obtain the phase diagram of the model. The transition from the ferromagnetic (or antiferromagnetic) phase to the disordered paramagnetic phase is of second order. On the other hand, a first-order transition is obtained from the lamellar phase to the paramagnetic phase, as well as from the lamellar phase to the ferromagnetic (or antiferromagnetic) phase, with the presence of a critical end point. An expected singular behavior of the first-order line at the critical end point is also obtained.

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**ABSTRACT:**The classical n-vector model on the face-centered cubic lattice with ferromagnetic and antiferromagnetic interactions is studied by using the framework of an effective field theory approach in cluster with two spins. For n=1 (Ising), n=2 (XY-planar rotator), n=3 (Heisenberg), and n→∞ (spherical) models the ferromagnetic system undergoes a second-order phase transition while the antiferromagnetic system presents a first-order behavior. The first-order character for n=3, which is inconclusive in the literature, seems actually to be the case for the isotropic Heisenberg antiferromagnet, and the same is expected to hold for the less studied XY and spherical models. In addition, the transition temperatures are quantitatively comparable to the exact one (when available) and to those from Monte Carlo simulations and series expansion.Physical review. B, Condensed matter 01/2008; 77(2). DOI:10.1103/PhysRevB.77.024419 · 3.66 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the bilayer pre-transition exhibited by some lipids at temperatures below their main phase transition, and which is generally associated to the formation of periodic ripples in the membrane. Experimentally we focus on the anionic lipid dipalmytoylphosphatidylglycerol (DPPG) at different ionic strengths, and on the neutral lipid dipalmytoylphosphatidylcholine (DPPC). From the analysis of differential scanning calorimetry traces of the two lipids we find that both pre- and main transitions are part of the same melting process. Electron spin resonance of spin labels and excitation generalized polarization of Laurdan reveal the coexistence of gel and fluid domains at temperatures between the pre- and main transitions of both lipids, reinforcing the first finding. Also, the melting process of DPPG at low ionic strength is found to be less cooperative than that of DPPC. From the theoretical side, we introduce a statistical model in which a next-nearest-neighbor competing interaction is added to the usual two-state model. For the first time, modulated phases (ordered and disordered lipids periodically aligned) emerge between the gel and fluid phases as a natural consequence of the competition between lipid-lipid interactions.Biochimica et Biophysica Acta 02/2009; 1788(5):954-63. DOI:10.1016/j.bbamem.2009.01.007 · 4.66 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study simple lattice systems to demonstrate the influence of interpenetrating bond networks on phase behavior. We promote interpenetration by using a Hamiltonian with a weakly repulsive interaction with nearest neighbors and an attractive interaction with second-nearest neighbors. In this way, bond networks will form between second-nearest neighbors, allowing for two (locally) distinct networks to form. We obtain the phase behavior from analytic solution in the mean-field approximation and exact solution on the Bethe lattice. We compare these results with exact numerical results for the phase behavior from grand canonical Monte Carlo simulations on square, cubic, and tetrahedral lattices. All results show that these simple systems exhibit rich phase diagrams with two fluid-fluid critical points and three thermodynamically distinct phases. We also consider including third-nearest-neighbor interactions, which give rise to a phase diagram with four critical points and five thermodynamically distinct phases. Thus the interpenetration mechanism provides a simple route to generate multiple liquid phases in single-component systems, such as hypothesized in water and observed in several model and experimental systems. Additionally, interpenetration of many such networks appears plausible in a recently considered material made from nanoparticles functionalized by single-strands of DNA.Physical Review E 05/2009; 79(4 Pt 1):041502. DOI:10.1103/PhysRevE.79.041502 · 2.33 Impact Factor