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# High-temperature series for the RPn-1 lattice spin model (generalized Maier-Saupe model of nematic liquid crystals) in two space dimensions and with general spin dimensionality n.

Physical review. B, Condensed matter (Impact Factor: 3.66). 12/1992; 46(17):11141-11144. DOI: 10.1103/PhysRevB.46.11141

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**ABSTRACT:**High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions d=1, 2, and 3, are expressed in terms of the two variables K1=J1/kT and K2=J2/kT, where J1 and J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the d=3 uniaxial XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter R=J2/J1. Universality with respect to R on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination γ=1.3177(5) and ν=0.6726(8) of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional XY universality class, can be achieved. For the exponents at the multicritical (m,d,N)=(1,3,2) Lifshitz point the estimates γl=1.535(25), ν⊥=0.805(15), and ν∥=0.40(3) are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.Physical review. B, Condensed matter 08/2008; 78(5). · 3.66 Impact Factor -
##### Article: Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

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**ABSTRACT:**We compute the phase diagram in the N→∞ limit for lattice RPN−1, CPN−1 and QPN−1σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.Nuclear Physics B 05/2001; 601(3):425–502. · 3.95 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We have considered a classical spin system, consisting of 3-component unit vectors, associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally invariant pair potentials, isotropic in spin space, and of the long-range form where ∊ is a positive constant setting energy and temperature scales (i.e. T* = kBT/∊). Extending previous rigorous results, one can prove the existence of an ordering transition at finite temperature when 0 < σ < 1, and its absence when σ ≥ 1. We have studied the border case σ = 1, by means of computer simulation. Similarly to the magnetic counterparts of the present model, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ = 0.475 ± 0.005.International Journal of Modern Physics B 01/2012; 09(25). · 0.46 Impact Factor

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